{"id":290,"date":"2025-04-09T17:40:47","date_gmt":"2025-04-09T17:40:47","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/4-1-linear-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-19T20:49:41","modified_gmt":"2025-08-19T20:49:41","slug":"4-1-linear-functions-college-algebra-2e-openstax","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/4-1-linear-functions-college-algebra-2e-openstax\/","title":{"raw":"4.1 Linear Functions","rendered":"4.1 Linear Functions"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_f48a9644-4329-4387-9b38-4ac039f12570\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Represent a linear function.<\/li>\r\n \t<li>Determine whether a linear function is increasing, decreasing, or constant.<\/li>\r\n \t<li>Interpret slope as a rate of change.<\/li>\r\n \t<li>Write and interpret an equation for a linear function.<\/li>\r\n \t<li>Graph linear functions.<\/li>\r\n \t<li>Determine whether lines are parallel or perpendicular.<\/li>\r\n \t<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<ul id=\"list-00001\"><\/ul>\r\n<\/section><\/div>\r\n\r\n[caption id=\"attachment_647\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-647\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-300x169.jpg\" alt=\"\" width=\"300\" height=\"169\" \/> Figure 1. \"RTD Line Train 305 southbound at Broadway Station\" by FoamingInDenver is licensed under CC0 1.0[\/caption]\r\n\r\n<div>\r\n<div>\r\n\r\nSuppose the Denver RTD train travels at a speed of 79 mph for a period of time once it is 3 miles from the station.\u00a0 How can we analyze the train\u2019s distance from the station over a period of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train\u2019s distance from the station at a given point in time.\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id2374595\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Representing Linear Functions<\/h2>\r\n<p id=\"fs-id2673768\">The function describing the train\u2019s motion is a <span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">linear function<\/span>, which is defined as a function with a constant rate of change. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train\u2019s motion as a function using each method.<\/p>\r\n\r\n<section id=\"fs-id2052037\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Representing a Linear Function in Word Form<\/h3>\r\n<p id=\"fs-id1723576\">Let\u2019s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.<\/p>\r\n\r\n<ul id=\"fs-id2157898\">\r\n \t<li><em data-effect=\"italics\">The train\u2019s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.<\/em><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1502969\">The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. <span class=\"TextRun SCXW99595078 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW99595078 BCX2\">As<\/span><span class=\"NormalTextRun SCXW99595078 BCX2\"> the time(input) increases by 1 hour, the corresponding distance (output)<\/span> <span class=\"NormalTextRun SCXW99595078 BCX2\">increases by 79 miles.\u00a0 <\/span><span class=\"NormalTextRun SCXW99595078 BCX2\">The train began moving at this constant speed at a distance of 3 miles from the station.<\/span><\/span><\/p>\r\n\r\n<\/section><section id=\"fs-id2216588\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Representing a Linear Function in Function Notation<\/h3>\r\n<p id=\"fs-id1701341\">Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">slope-intercept form<\/span> of a line, where\u00a0[latex] x [\/latex] is the input value,\u00a0[latex] m [\/latex] is the rate of change, and\u00a0[latex] b [\/latex] is the initial value of the dependent variable.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll}\\text{Equation form} \\quad &amp; y = mx + b \\\\\\text{Function notation} \\quad &amp; f(x) = mx + b\\end{array} [\/latex]<\/p>\r\nIn the example of the Denver RTD, we might use the notation\u00a0[latex] D(t) [\/latex] where the total distance [latex] D [\/latex] is a function of the time\u00a0[latex] t. [\/latex] The rate [latex] m [\/latex] is 79 miles per hour. The initial value of the dependent variable\u00a0[latex] b [\/latex] is the original distance from the station, 3 miles.\u00a0 We can write a generalized equation to represent the motion of the train.\r\n<p style=\"text-align: center;\">[latex] D(t)=79t+3 [\/latex]<\/p>\r\n\r\n<\/section><\/section><\/div>\r\n<section id=\"fs-id1349346\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Representing a Linear Function in Tabular Form<\/h3>\r\n<p id=\"fs-id2040470\">A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in Figure 2. From the table, we can see that the distance changes by 79 miles for every 1 hour increase in time.<\/p>\r\n\r\n\r\n[caption id=\"attachment_1255\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1255\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2-300x184.png\" alt=\"\" width=\"300\" height=\"184\" \/> Figure 2. Tabular representation of the function D showing selected input and output values.[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Can the input in the previous example be any real number?<\/strong>\r\n\r\nA: <em data-effect=\"italics\">No. The input represents time so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1726229\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Representing a Linear Function in Graphical Form<\/h3>\r\n<p id=\"fs-id1630800\">Another way to represent linear functions is visually, using a graph. We can use the function relationship [latex] D(t)=83t+250, [\/latex] to draw a graph as represented in Figure 3. Notice the graph is a line<strong>.<\/strong> When we plot a linear function, the graph is always a line.<\/p>\r\n<p id=\"fs-id2059569\">The rate of change, which is constant, determines the slant, or <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">slope<\/span> of the line. The point at which the input value is zero is the vertical intercept, or [latex] y- [\/latex]intercept, of the line. We can see from the graph that the [latex] y- [\/latex]intercept in the train example we just saw is\u00a0[latex] (0, 250) [\/latex] and represents the distance of the train from the station when it began moving at a constant speed.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_649\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-649\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3-300x178.jpeg\" alt=\"\" width=\"300\" height=\"178\" \/> Figure 3. The graph of\u00a0[latex] D9t)=83t+250. [\/latex] Graphs of linear functions are lines because the rate of change is constant.[\/caption]\r\n<p id=\"fs-id1843158\">Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line [latex] f(x)=2x+1. [\/latex] Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Linear Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>linear <span id=\"term-00008\" data-type=\"term\">function<\/span><\/strong> is a function whose graph is a line. Linear functions can be written in the <strong><span id=\"term-00009\" data-type=\"term\">slope-intercept form<\/span> <\/strong>of a line\r\n<p style=\"text-align: center;\">[latex] f(x)=mx+b [\/latex]<\/p>\r\nwhere\u00a0[latex] b [\/latex] is the initial or starting value of the function (when input,\u00a0[latex] x=0[\/latex]) and\u00a0[latex] m [\/latex] is the constant rate of change, or slope of the function. The [latex] y- [\/latex]intercept is at [latex] (0, b). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Using a linear function to find the cost of ordering Tamales at the Mexican restaurant, Santiago\u2019s<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div>\r\n<div>\r\n\r\nThe cost,\u00a0[latex] C, [\/latex] in dollars of ordering Tamales depends on the number of tamales\u00a0[latex] t [\/latex] and this relationship may be modeled by the equation\u00a0[latex] C(t)=3t+6.99, [\/latex] where\u00a0[latex] \\$3 [\/latex] is the cost per tamale and [latex] \\$6.99 [\/latex] is the standard delivery charge.\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To restate the function in words, we need to describe each part of the equation. The cost as a function of the number of tamales equals 3 times the number of tamales ordered plus the delivery charge of $6.99.\r\n<h3>Analysis<\/h3>\r\nThe initial value, $6.99, is the standard delivery fee. The rate of change, or slope, is $3 per tamale. This tells us that the cost of the order increases $3 for each extra tamale ordered.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1630015\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining Whether a Linear Function Is Increasing, Decreasing, or Constant<\/h2>\r\n<p id=\"fs-id2209400\">The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">increasing function<\/span>, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in Figure 5<strong>(a)<\/strong>. For a <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">decreasing function<\/span>, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in Figure 5<strong>(b)<\/strong>. If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in Figure 5<strong>(c)<\/strong>.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_650\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-650\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-300x115.jpeg\" alt=\"\" width=\"300\" height=\"115\" \/> Figure 5[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Increasing and Decreasing Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe slope determines if the function is an <strong>increasing linear function<\/strong>, a <strong>decreasing linear function<\/strong>, or a constant function.\r\n<ul>\r\n \t<li>[latex] f(x)=mx+b [\/latex] is an increasing function if [latex] m&gt;0. [\/latex]<\/li>\r\n \t<li>[latex] f(x)=mx+b [\/latex] is a decreasing function if [latex] m&lt;0. [\/latex]<\/li>\r\n \t<li>[latex] f(x)=mx+b [\/latex] is a constant function if [latex] m=0. [\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Deciding Whether a Function Is Increasing, Decreasing, or Constant<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nStudies from the early 2010s indicated that teens sent about 60 texts a day, while more recent data indicates much higher messaging rates among all users, particularly considering the various apps with which people can communicate.<sup id=\"footnote-ref2\" data-type=\"footnote-number\"><a role=\"doc-noteref\" href=\"4-1-linear-functions#fs-id1759513\" data-type=\"footnote-link\">3<\/a><\/sup>. For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.\r\n\r\n(a) The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.\r\n\r\n(b) Lewis has a limit of 500 texts per month in their data plan. The input is the number of days, and output is the total number of texts remaining for the month.\r\n\r\n(c) Sofia has an unlimited number of texts in their data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Analyze each function.\r\n\r\n(a) The function can be represented as\u00a0[latex] f(x)=60x [\/latex] where\u00a0[latex] x [\/latex] is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.\r\n\r\n(b) The function can be represented as\u00a0[latex] f(x)=500-60x [\/latex] where\u00a0[latex] x [\/latex] is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after\u00a0[latex] x [\/latex] days\r\n\r\n(c) The cost function can be represented as\u00a0[latex] f(x)=50 [\/latex] because the number of days does not affect the total cost. The slope is 0 so the function is constant.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section><\/section><\/section><section id=\"fs-id2407166\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Interpreting Slope as a Rate of Change<\/h2>\r\n<p id=\"fs-id1716534\">In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input,\u00a0[latex] x_1 [\/latex] and\u00a0[latex] x_2, [\/latex] and two corresponding values for the output,\u00a0[latex] y_1 [\/latex] and\u00a0[latex] y_2 [\/latex] \u2014 which can be represented by a set of points,\u00a0[latex] (x_1, y_1) [\/latex] and\u00a0[latex] (x_2, y_2) [\/latex] \u2014 we can calculate the slope [latex] m. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{x_2-x_1} [\/latex]<\/p>\r\n<p id=\"fs-id1853016\">Note that in function notation we can obtain two corresponding values for the output\u00a0[latex] y_1 [\/latex] and\u00a0[latex] y_2 [\/latex] for the function\u00a0[latex] f, y_1=f(x_1) [\/latex] and\u00a0[latex] y_2=f(x_2), [\/latex] so we could equivalently write<\/p>\r\n<p style=\"text-align: center;\">[latex] m=\\frac{f(x_2)-f(x_1)}{x_2-x_1} [\/latex]<\/p>\r\n<p id=\"fs-id1503277\">Figure 6 indicates how the slope of the line between the points,\u00a0[latex] (x_1, y_1) [\/latex] and\u00a0[latex] (x_2, y_2), [\/latex] is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_651\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-651\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6-300x247.jpeg\" alt=\"\" width=\"300\" height=\"247\" \/> Figure 6. The slope of a function is calculated by the change in [latex] y [\/latex] divided by the change in [latex] x. [\/latex] It does not matter which coordinate is used as the [latex] (x_2, y_2) [\/latex] and which is the [latex] (x_1, y_1), [\/latex] as long as each calculation is started with the elements from the same coordinate pair.[\/caption]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Are the units for slope always<\/strong> [latex] \\frac{\\text{units for the output}}{\\text{units for the input}}? [\/latex]\r\n\r\nA: <em data-effect=\"italics\">Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Calculate Slope<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe slope, or rate of change, of a function\u00a0[latex] m [\/latex] can be calculated according to the following:\r\n<p style=\"text-align: center;\">[latex] m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{x_2-x_1} [\/latex]<\/p>\r\nwhere\u00a0[latex] x_1 [\/latex] and\u00a0[latex] x_2 [\/latex] are input values,\u00a0[latex] y_1 [\/latex] and\u00a0[latex] y_2 [\/latex] are output values.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given two points from a linear function, calculate and interpret the slope.<\/strong>\r\n<ol>\r\n \t<li>Determine the units for output and input values.<\/li>\r\n \t<li>Calculate the change of output values and change of input values.<\/li>\r\n \t<li>Interpret the slope as the change in output values per unit of the input value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Finding the Slope of a Linear Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf\u00a0[latex] f(x) [\/latex] is a linear function, and\u00a0[latex] (3, -2) [\/latex] and\u00a0[latex] (8, 1) [\/latex] are points on the line, find the slope. Is this function increasing or decreasing?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The coordinate pairs are\u00a0[latex] (3, -2) [\/latex] and\u00a0[latex] (8, 1). [\/latex] To find the rate of change, we divide the change in output by the change in input.\r\n<p style=\"text-align: center;\">[latex] m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-(-2)}{8-3}=\\frac{3}{5} [\/latex]<\/p>\r\nWe could also write the slope as\u00a0[latex] m=0.6. [\/latex] The function is increasing because [latex] m&gt;0. [\/latex]\r\n<h3>Analysis<\/h3>\r\nAs noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or [latex] y- [\/latex]coordinate, used corresponds with the first input value, or [latex] x- [\/latex]coordinate, used. Note that if we had reversed them, we would have obtained the same slope.\r\n<p style=\"text-align: center;\">[latex] m=\\frac{(-2)-(1)}{3-8}=\\frac{-3}{-5}=\\frac{3}{5} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf\u00a0[latex] f(x) [\/latex] is a linear function, and\u00a0[latex] (2, 3) [\/latex] and\u00a0[latex] (0, 4) [\/latex] are points on the line, find the slope. Is this function increasing or decreasing?\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Finding the Population Change from a Linear Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe population of Aurora increased from 338,774 in 2010 to 381,057 in 2020. Find the change of population per year if we assume the change was constant from 2010 to 2020.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The rate of change relates to the population to the change in time. The population increased by\u00a0[latex] 381,057-338,774=42,283 [\/latex] people over the 10-year interval.\u00a0 To find the rate of change, divide the change in the number of people by the number of years.\r\n<p style=\"text-align: center;\">[latex] \\frac{42,483 \\ \\ \\text{people}}{10 \\ \\ \\text{years}}\\approx\\frac{4,228}{\\text{year}} [\/latex]<\/p>\r\nSo, the population increased by approximately 4,228 people per year.\r\n<h3>Analysis<\/h3>\r\nBecause we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1700480\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Writing and Interpreting an Equation for a Linear Function<\/h2>\r\n<p id=\"fs-id1688492\">Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form and the point-slope form. Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function\u00a0[latex] f [\/latex] in Figure 7.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_653\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-653\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7-300x214.jpeg\" alt=\"\" width=\"300\" height=\"214\" \/> Figure 7[\/caption]\r\n<p id=\"fs-id1308976\">We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let\u2019s choose\u00a0[latex] (0, 7) [\/latex] and [latex] (4, 4). [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}m &amp;=&amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\&amp;=&amp; \\frac{4 - 7}{4 - 0} \\\\&amp;=&amp; -\\frac{3}{4}\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1574658\">Now we can substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}y - y_1 &amp;=&amp; m(x - x_1) \\\\\\quad y - 4 &amp;=&amp; -\\frac{3}{4}(x - 4)\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1591011\">If we want to rewrite the equation in the slope-intercept form, we would find<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}y - 4 &amp;=&amp; -\\frac{3}{4}(x - 4) \\\\y - 4 &amp;=&amp; -\\frac{3}{4}x + 3 \\\\y &amp;=&amp; -\\frac{3}{4}x + 7\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1804015\">If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the [latex] y- [\/latex]axis when the output value is 7. Therefore,\u00a0[latex] b=7. [\/latex] We now have the initial value\u00a0[latex] b [\/latex] and the slope\u00a0[latex] m [\/latex] so we can substitute\u00a0[latex] m [\/latex] and\u00a0[latex] b [\/latex] into the slope-intercept form of a line.\r\n<span id=\"fs-id1109555\" data-type=\"media\" data-alt=\"This image shows the equation f of x equals m times x plus b. It shows that m is the value negative three fourths and b is 7. It then shows the equation rewritten as f of x equals negative three fourths times x plus 7.\"><\/span><\/p>\r\n<img class=\"size-medium wp-image-654 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx-300x95.jpeg\" alt=\"\" width=\"300\" height=\"95\" \/>\r\n<p id=\"fs-id2065131\">So the function is\u00a0[latex] f(x)=-\\frac{3}{4}x+7, [\/latex] and the linear equation would be [latex] y=-\\frac{3}{4}x+7. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the graph of a linear function, write an equation to represent the function.<\/strong>\r\n<ol>\r\n \t<li>Identify two points on the line.<\/li>\r\n \t<li>Use the two points to calculate the slope.<\/li>\r\n \t<li>Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.<\/li>\r\n \t<li>Substitute the slope and y-intercept into the slope-intercept form of a line equation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Writing an Equation for a Linear Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite an equation for a linear function given a graph of\u00a0[latex] f [\/latex] shown in Figure 8.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_655\" align=\"aligncenter\" width=\"293\"]<img class=\"size-medium wp-image-655\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" \/> Figure 8[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Identify two points on the line, such as\u00a0[latex] (0, 2) [\/latex] and\u00a0[latex] (-2, -4). [\/latex] Use the points to calculate the slope.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}m &amp;=&amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\&amp;=&amp; \\frac{-4 - 2}{-2 - 0} \\\\&amp;=&amp; \\frac{-6}{-2} \\\\&amp;=&amp; 3\\end{array} [\/latex]<\/p>\r\nSubstitute the slope and the coordinates of one of the points into the point-slope form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}y - y_1 &amp;=&amp; m(x - x_1) \\\\y - (-4) &amp;=&amp; 3(x - (-2)) \\\\y + 4 &amp;=&amp; 3(x + 2)\\end{array} [\/latex]<\/p>\r\nWe can use algebra to rewrite the equation in the slope-intercept form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}y + 4 &amp;=&amp; 3(x + 2) \\\\y + 4 &amp;=&amp; 3x + 6 \\\\y &amp;=&amp; 3x + 2\\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nThis makes sense because we can see from Figure 9 that the line crosses the [latex] y- [\/latex]axis at the point\u00a0[latex] (0, 2), [\/latex] which is the [latex] y- [\/latex]intercept, so [latex] b=2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_656\" align=\"aligncenter\" width=\"293\"]<img class=\"size-medium wp-image-656\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" \/> Figure 9[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Writing an Equation for a Linear Cost Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSuppose Bailey starts a company in which they incur a fixed cost of $1,250 per month for the overhead, which includes their office rent. Their production costs are $37.50 per item. Write a linear function\u00a0[latex] C [\/latex] where\u00a0[latex] C(x) [\/latex] is the cost for\u00a0[latex] x [\/latex] items produced in a given month.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is $37.50. The variable cost, called the marginal cost, is represented by\u00a0[latex] 37.5 [\/latex] The cost Bailey incurs is the sum of these two costs, represented by [latex] C(x)=1250+37.5x. [\/latex]\r\n<h3>Analysis<\/h3>\r\nIf Bailey produces 100 items in a month, their monthly cost is found by substituting 100 for [latex] x. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}C(100) &amp;=&amp; 1250 + 37.5(100) \\\\&amp;=&amp; 5000\\end{array} [\/latex]<\/p>\r\nSo their monthly cost would be $5,000.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Writing an Equation for a Linear Function Given Two Points<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf\u00a0[latex] f [\/latex] is a linear function, with\u00a0[latex] f(3)=-2, [\/latex] and\u00a0[latex] f(8)=1, [\/latex] find an equation for the function in slope-intercept form.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can write the given points using the coordinates.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}f(3) &amp;=&amp; -2 \\rightarrow (3, -2) \\\\f(8) &amp;=&amp; 1 \\rightarrow (8, 1)\\end{array} [\/latex]<\/p>\r\nWe can then use the points to calculate the slope.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} m &amp;=&amp; \\frac{y_2-y_1}{x_2-x_1} \\\\ &amp;=&amp; \\frac{1-(-2)}{8-3} \\\\ &amp;=&amp; \\frac{3}{5} \\end{array} [\/latex]<\/p>\r\nSubstitute the slope and the coordinates of one of the points into the point-slope form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} y-y_1 &amp;=&amp; m(x-x_1) \\\\ y-(-2) &amp;=&amp; \\frac{3}{5}(x-3) \\end{array} [\/latex]<\/p>\r\nWe can use algebra to rewrite the equation in the slope-intercept form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} y+2 &amp;=&amp; \\frac{3}{5}(x-3) \\\\ y+2 &amp;=&amp; \\frac{3}{5}x-\\frac{9}{5} \\\\ y &amp;=&amp; \\frac{3}{5}x-\\frac{19}{5} \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf\u00a0[latex] f(x) [\/latex] is a linear function, with\u00a0[latex] f(2)=-11 [\/latex] and\u00a0[latex] f(4)=-25, [\/latex] write an equation for the function in slope-intercept form.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1577178\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Modeling Real-World Problems with Linear Functions<\/h2>\r\n<p id=\"fs-id1421845\">In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a linear function\u00a0[latex] f [\/latex] and the initial value and rate of change, evaluate [latex] f(c). [\/latex]<\/strong>\r\n<ol>\r\n \t<li>Determine the initial value and the rate of change (slope).<\/li>\r\n \t<li>Substitute the values into [latex] f(x)=mx+b. [\/latex]<\/li>\r\n \t<li>Evaluate the function at [latex] x=c. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Using a Linear Function to Determine the Number of Songs in a Music Collection<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEmma currently has 200 songs in her music collection. Every month, she adds 15 new songs. Write a formula for the number of songs,\u00a0[latex] N, [\/latex] in her collection as a function of time,\u00a0[latex] t, [\/latex] the number of months. How many songs will she own at the end of one year?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The initial value for this function is 200 because she currently owns 200 songs, so\u00a0[latex] N(0)=200, [\/latex] which means that [latex] b=200. [\/latex]\r\n\r\nThe number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that\u00a0[latex] m=15. [\/latex] We can substitute the initial value and the rate of change into the slope-intercept form of a line.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_658\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-658\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10-300x81.jpeg\" alt=\"\" width=\"300\" height=\"81\" \/> Figure 10[\/caption]\r\n\r\nWe can write this formula [latex] N(t)=15t+200. [\/latex]\r\n\r\nWith this formula, we can then predict how many songs Emma will have at the end of one year (12 months). In other words, we can evaluate the function at [latex] t=12. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} N(12) &amp;=&amp; 15(12)+200 \\\\ &amp;=&amp; 180+200 \\\\ &amp;=&amp; 380 \\end{array} [\/latex]<\/p>\r\nEmma will have 380 songs in 12 months.\r\n<h3>Analysis<\/h3>\r\nNotice that N is an increasing linear function. As the input (the number of months) increases, the output (number of songs) increases as well.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Using a Linear Function to Calculate Salary Based on Commission<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWorking as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya's weekly income\u00a0[latex] I, [\/latex] depends on the number of new policies,\u00a0[latex] n, [\/latex] he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for\u00a0[latex] I(n), [\/latex] and interpret the meaning of the components of the equation.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The given information gives us two input-output pairs:\u00a0[latex] (3, 760) [\/latex] and\u00a0[latex] (5, 920). [\/latex] We start by finding the rate of change.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} m &amp;=&amp; \\frac{920-760}{5-3} \\\\ &amp;=&amp; \\frac{\\$160}{2 \\ \\text{policies}} \\\\ &amp;=&amp; \\$80 \\ \\text{per policy} \\end{array} [\/latex]<\/p>\r\nKeeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is $80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold during the week.\r\n\r\nWe can then solve for the initial value.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcll} I(n) &amp;=&amp; 80n+b \\\\ 760 &amp;=&amp; 80(3)+b &amp; \\text{when} \\ n=3, I(3)=760 \\\\ 760 &amp;-&amp; 80(3)=b \\\\ 520 &amp;=&amp; b \\end{array} [\/latex]<\/p>\r\nThe value of\u00a0[latex] b [\/latex] is the starting value for the function and represents Ilya's income when\u00a0[latex] [\/latex] or when no new policies are sold. We can interpret this as Ilya\u2019s base salary for the week, which does not depend upon the number of policies sold.\r\n\r\nWe can now write the final equation.\r\n<p style=\"text-align: center;\">[latex] I(n)=80n+520 [\/latex]<\/p>\r\nOur final interpretation is that Ilya\u2019s base salary is $520 per week and he earns an additional $80 commission for each policy sold.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Using Tabular Form to Write an Equation for a Linear Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nTable 1 relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\"><strong>number of weeks,\u00a0[latex] w [\/latex]<\/strong><\/td>\r\n<td style=\"width: 20%;\">0<\/td>\r\n<td style=\"width: 20%;\">2<\/td>\r\n<td style=\"width: 20%;\">4<\/td>\r\n<td style=\"width: 20%;\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\"><strong>number of rats,\u00a0[latex] P(w) [\/latex]<\/strong><\/td>\r\n<td style=\"width: 20%;\">1000<\/td>\r\n<td style=\"width: 20%;\">1080<\/td>\r\n<td style=\"width: 20%;\">1160<\/td>\r\n<td style=\"width: 20%;\">1240<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can see from the table that the initial value for the number of rats is 100, so [latex] b=1000. [\/latex]\r\n\r\nRather than solving for\u00a0[latex] m, [\/latex] we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.\r\n<p style=\"text-align: center;\">[latex] P(w)=40w+1000 [\/latex]<\/p>\r\nIf we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using\u00a0[latex] (2, 1080) [\/latex] and [latex] (6, 1240). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} m &amp;=&amp; \\frac{1240-1080}{6-2} \\\\ &amp;=&amp; \\frac{160}{4} \\\\ &amp;=&amp; 40 \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Is the initial value always provided in a table of values like Table 1?<\/strong>\r\n\r\n<em>A:\u00a0No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into\u00a0[latex] f(x)=mx+b [\/latex] and solve for [latex] b. [\/latex]<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA new plant food was introduced to a young tree to test its effect on the height of the tree. Table 2 shows the height of the tree, in feet,\u00a0[latex] x [\/latex] months since the measurements began. Write a linear function\u00a0[latex] H(x), [\/latex] where\u00a0[latex] x [\/latex] is the number of months since the start of the experiment.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">0<\/td>\r\n<td style=\"width: 16.6667%;\">2<\/td>\r\n<td style=\"width: 16.6667%;\">4<\/td>\r\n<td style=\"width: 16.6667%;\">8<\/td>\r\n<td style=\"width: 16.6667%;\">12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] H(x) [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">12.5<\/td>\r\n<td style=\"width: 16.6667%;\">13.5<\/td>\r\n<td style=\"width: 16.6667%;\">14.5<\/td>\r\n<td style=\"width: 16.6667%;\">16.5<\/td>\r\n<td style=\"width: 16.6667%;\">18.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id2458614\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Linear Functions<\/h2>\r\n<p id=\"fs-id2458643\">Now that we\u2019ve seen and interpreted graphs of linear functions, let\u2019s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the [latex] y- [\/latex]intercept and slope. And the third method is by using transformations of the identity function [latex] f(x)=x. [\/latex]<\/p>\r\n\r\n<section id=\"fs-id2568421\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Function by Plotting Points<\/h3>\r\n<p id=\"fs-id2237975\">To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function,\u00a0[latex] f(x)=2x, [\/latex] we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point\u00a0[latex] (1, 2). [\/latex] Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point\u00a0[latex] (2, 4). [\/latex] Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a linear function, graph by plotting points.<\/strong>\r\n<ol>\r\n \t<li>Choose a minimum of two input values.<\/li>\r\n \t<li>Evaluate the function at each input value.<\/li>\r\n \t<li>Use the resulting output values to identify coordinate pairs.<\/li>\r\n \t<li>Plot the coordinate pairs on a grid.<\/li>\r\n \t<li>Draw a line through the points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Graphing by Plotting Points<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph\u00a0[latex] f(x)=-\\frac{2}{3}x+5 [\/latex] by plotting points.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Begin\u00a0 by choosing input values. This function includes a fraction with a denominator of 3, so let\u2019s choose multiples of 3 as input values. We will choose 0, 3, and 6.\r\n<p id=\"fs-id1683693\">Evaluate the function at each input value, and use the output value to identify coordinate pairs.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} x=0 &amp;&amp; f(0)=-\\frac{2}{3}(0)+5=5\\rightarrow (0, 5) \\\\ x=3 &amp;&amp; f(3)=-\\frac{2}{3}(3)+5=3\\rightarrow (3, 3) \\\\ x=6 &amp;&amp; f(6)=-\\frac{2}{3}(6)+5=1\\rightarrow (6, 1) \\end{array} [\/latex]<\/p>\r\nPlot the coordinate pairs and draw a line through the points. Figure 11 represents the graph of the function [latex] f(x)=-]frac{2}{3}x+5 [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_661\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-661\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11-300x260.jpeg\" alt=\"\" width=\"300\" height=\"260\" \/> Figure 11. The graph of the linear function [latex] f(x)=-\\frac{2}{3}x+5.[\/latex][\/caption]\r\n<h3>Analysis<\/h3>\r\nThe graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant, which indicates a negative slope. This is also expected from the negative, constant rate of change in the equation for the function.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=-\\frac{3}{4}x+6 [\/latex] by plotting points.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id2654013\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Function Using <em data-effect=\"italics\">y-<\/em>intercept and Slope<\/h3>\r\n<p id=\"fs-id2654022\">Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its <em data-effect=\"italics\">y-<\/em>intercept, which is the point at which the input value is zero. To find the [latex] y- [\/latex]intercept, we can set [latex] x=0 [\/latex] in the equation.<\/p>\r\n<p id=\"fs-id2458574\">The other characteristic of the linear function is its slope<strong>.<\/strong><\/p>\r\n<p id=\"fs-id2458580\">Let\u2019s consider the following function.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=\\frac{1}{2}x+1 [\/latex]<\/p>\r\n<p id=\"fs-id2497289\">The slope is [latex] \\frac{1}{2}. [\/latex] Because the slope is positive, we know the graph will slant upward from left to right. The [latex] y- [\/latex]intercept is the point on the graph when [latex] x=0. [\/latex] The graph crosses the [latex] y- [\/latex]axis at [latex] (0, 1). [\/latex] Now we know the slope and the [latex] y- [\/latex]intercept. We can begin graphing by plotting the point [latex] (0, 1). [\/latex] We know that the slope is the change in the [latex] y- [\/latex]coordinate over the change in the [latex] x- [\/latex]coordinate. This is commonly referred to as rise over run, [latex] m=\\frac{\\text{rise}}{\\text{run}}. [\/latex] From our example, we have [latex] m=\\frac{1}{2}, [\/latex] which means that the rise is 1 and the run is 2. So starting from our [latex] (0, 1), [\/latex]intercept [latex] [\/latex] we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in Figure 12.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_662\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-662\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12-300x200.jpeg\" alt=\"\" width=\"300\" height=\"200\" \/> Figure 12[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Graphical Interpretation of a Linear Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn the equation [latex] f(x)=mx+b [\/latex]\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>[latex] b [\/latex] is the [latex] y- [\/latex]intercept of the graph and indicates the point\u00a0[latex] (0, b) [\/latex] at which the graph crosses the [latex] y- [\/latex]axis.<\/li>\r\n \t<li>[latex] m [\/latex]is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex] m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{x_2-x_1} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Do all linear functions have [latex] y- [\/latex]intercepts?<\/strong>\r\n\r\nA: <em data-effect=\"italics\">Yes. All linear functions cross the [latex] y- [\/latex]axis and therefore have [latex] y- [\/latex]intercepts. (Note: A vertical line is parallel to the [latex] y- [\/latex]axis does not have a [latex] y- [\/latex]intercept, but it is not a function<\/em>.)\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the equation for a linear function, graph the function using the <em data-effect=\"italics\">y<\/em>-intercept and slope.<\/strong>\r\n<ol>\r\n \t<li>Evaluate the function at an input value of zero to find the y-intercept.<\/li>\r\n \t<li>Identify the slope as the rate of change of the input value.<\/li>\r\n \t<li>Plot the point represented by the y-intercept.<\/li>\r\n \t<li>Use\u00a0[latex] \\frac{\\text{rise}}{\\text{run}} [\/latex] to determine at least two more points on the line.<\/li>\r\n \t<li>Sketch the line that passes through the points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 12: Graphing by Using the y-intercept and Slope<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=-\\frac{2}{3}x+5 [\/latex] using the <em>y-<\/em>intercept and slope.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Evaluate the function at\u00a0[latex] x=0 [\/latex] to find the [latex] y- [\/latex]intercept. The output value when\u00a0[latex] x=0 [\/latex] is 5, so the graph will cross the [latex] y- [\/latex]axis at [latex] (0, 5). [\/latex]\r\n\r\nAccording to the equation for the function, the slope of the line is\u00a0[latex] -\\frac{2}{3}. [\/latex] This tells us that for each vertical decrease in the \"rise\" of\u00a0[latex] -2 [\/latex] units, the \"run\" increases by 3 units in the horizontal direction. We can now graph the function by first plotting the [latex] y- [\/latex]intercept on the graph in Figure 13. From the initial value\u00a0[latex] (0, 5) [\/latex] we move down 2 units and to the right 3 units. We can extend the line to the left and right by repeating, and then drawing a line through the points.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_663\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-663\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13-300x229.jpeg\" alt=\"\" width=\"300\" height=\"229\" \/> Figure 13. Graph of [latex] f(x)=-\\frac{2}{3}x+5 [\/latex] and shows how to calculate the rise over run for the slope.[\/caption]\r\n<h3>Analysis<\/h3>\r\nThe graph slants downward from left to right, which means it has a negative slope as expected.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a point on the graph we drew in Example 12 that has a negative <em>x-<\/em>value.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id2239536\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Writing the Equation for a Function from the Graph of a Line<\/h2>\r\n<p id=\"fs-id2239541\">Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at Figure 14. We can see right away that the graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point\u00a0[latex] (0, 4) [\/latex] so this is the <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_665\" align=\"aligncenter\" width=\"293\"]<img class=\"size-medium wp-image-665\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" \/> Figure 14[\/caption]\r\n<p id=\"fs-id2525935\">Then we can calculate the slope by finding the rise and run. We can choose any two points, but let\u2019s look at the point\u00a0[latex] (-2, 0). [\/latex] To get from this point to the <em data-effect=\"italics\">y-<\/em>intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be<\/p>\r\n<p style=\"text-align: center;\">[latex] m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2 [\/latex]<\/p>\r\nSubstituting the slope and\u00a0<em>y-<\/em>intercept into the slope-intercept form of a line gives\r\n<p style=\"text-align: center;\">[latex] y=2x+4 [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a graph of linear function, find the equation to describe the function.<\/strong>\r\n<ol>\r\n \t<li>Identify the<em> y-<\/em>intercept of an equation.<\/li>\r\n \t<li>Choose two points to determine the slope.<\/li>\r\n \t<li>Substitute the <em>y-<\/em>intercept and slope into the slope-intercept form of a line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 13: Matching Linear Functions to their Graphs<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nMatch each equation of the linear functions with one of the lines in Figure 15.\r\n\r\n(a) [latex] f(x)=2x+3 [\/latex]\r\n\r\n(b) [latex] g(x)=2x-3 [\/latex]\r\n\r\n(c) [latex] h(x)=-2x+3 [\/latex]\r\n\r\n(d) [latex] j(x)=\\frac{1}{2}x+3 [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_667\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-667\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15-300x233.jpeg\" alt=\"\" width=\"300\" height=\"233\" \/> Figure 15[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Analyze the information for each function.\r\n\r\n(a) This function has a slope of 2 and a <em data-effect=\"italics\">y<\/em>-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function\u00a0 has the same slope, but a different <em data-effect=\"italics\">y-<\/em>intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through\u00a0[latex] (0, 3) [\/latex] so\u00a0[latex] f [\/latex] must be represented by line I.\r\n\r\n(b) This function also has a slope of 2, but a <em data-effect=\"italics\">y<\/em>-intercept of\u00a0[latex] -3. [\/latex] It must pass through the point\u00a0[latex] (0, 3-) [\/latex] and slant upward from left to right. It must be represented by line III.\r\n\r\n(c) This function has a slope of -2 and a y-intercept of 3. This is the only function listed with a negative slop, so it must be represented by line IV because it slants downward from left to right.\r\n\r\n(d) This function has a slope of\u00a0[latex] \\frac{1}{2} [\/latex] and a y-intercept of 3. It must pass through the point\u00a0[latex] (0, 3) [\/latex] and slant upward from left to right. Lines I and II pass through\u00a0[latex] (0, 3), [\/latex] but the slope of\u00a0[latex] j [\/latex] is less than the slope of\u00a0[latex] f [\/latex] so the line for\u00a0[latex] j [\/latex] must be flatter. This function is represented by Line II.\r\n\r\nNow we can re-label the lines as in Figure 16.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_668\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-668\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16-300x229.jpeg\" alt=\"\" width=\"300\" height=\"229\" \/> Figure 16[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section id=\"fs-id2256536\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Finding the <em data-effect=\"italics\">x<\/em>-intercept of a Line<\/h3>\r\n<p id=\"fs-id2256546\">So far we have been finding the <em data-effect=\"italics\">y-<\/em>intercepts of a function: the point at which the graph of the function crosses the <em data-effect=\"italics\">y<\/em>-axis. Recall that a function may also have an <span id=\"term-00020\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">x<\/em>-intercept<\/span>, which is the <em data-effect=\"italics\">x<\/em>-coordinate of the point where the graph of the function crosses the <em data-effect=\"italics\">x<\/em>-axis. In other words, it is the input value when the output value is zero.<\/p>\r\n<p id=\"fs-id2256573\">To find the <em data-effect=\"italics\">x<\/em>-intercept, set a function\u00a0[latex] f(x) [\/latex] equal to zero and solve for the value of\u00a0[latex] x. [\/latex] For example, consider the function shown.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=3x-6 [\/latex]<\/p>\r\n<p id=\"fs-id2568784\">Set the function equal to 0 and solve for [latex] x. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} 0 &amp;=&amp; 3x-6 \\\\ 6 &amp;=&amp; 3x \\\\ 2 &amp;=&amp; x \\\\ x &amp;=&amp; 2 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2395276\">The graph of the function crosses the <em data-effect=\"italics\">x<\/em>-axis at the point [latex] (2, 0). [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Do all linear functions have <em data-effect=\"italics\">x<\/em>-intercepts?<\/strong>\r\n\r\n<em>A:\u00a0No. However, linear functions of the form\u00a0[latex] y=c, [\/latex] where\u00a0[latex] c [\/latex] is a nonzero real number are the only examples of linear functions with no x-intercept. For example,\u00a0[latex] y=5 [\/latex] is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in Figure 17.<\/em>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_669\" align=\"aligncenter\" width=\"224\"]<img class=\"size-medium wp-image-669\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-17-224x300.jpeg\" alt=\"\" width=\"224\" height=\"300\" \/> Figure 17[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">X-Intercept<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <em>x-<\/em>intercept of the function is value of\u00a0[latex] x [\/latex] when\u00a0[latex] f(x)=0. [\/latex] It can be solved by the equation [latex] 0=mx+b. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 14: Finding an x-intercept<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the x-intercept of [latex] f(x)=\\frac{1}{2}x-3. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Set the function equal to zero to solve for [latex] x. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} 0 &amp;=&amp; \\frac{1}{2}x-3 \\\\ 3 &amp;=&amp; \\frac{1}{2}x \\\\ 6 &amp;=&amp; x \\\\ x &amp;=&amp; 6 \\end{array} [\/latex]<\/p>\r\nThe graph crosses the x-axis at the point [latex] (6, 0). [\/latex]\r\n<h3>Analysis<\/h3>\r\nA graph of the function is shown in Figure 18. We can see that the x-intercept is\u00a0[latex] (6, 0) [\/latex] as we expected.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_670\" align=\"aligncenter\" width=\"293\"]<img class=\"size-medium wp-image-670\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" \/> Figure 18[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div id=\"fs-id1796746\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the x-intercept of [latex] f(x)=\\frac{1}{4}x-4. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<\/section><section id=\"fs-id2403100\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Describing Horizontal and Vertical Lines<\/h3>\r\n<p id=\"fs-id2403105\">There are two special cases of lines on a graph\u2014horizontal and vertical lines. A <span id=\"term-00021\" class=\"no-emphasis\" data-type=\"term\">horizontal line<\/span> indicates a constant output, or <em data-effect=\"italics\">y<\/em>-value. In Figure 19, we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use\u00a0[latex] m=0 [\/latex] in the equation\u00a0[latex] f(x)=mx+b, [\/latex] the equation simplifies to\u00a0[latex] f(x)=b. [\/latex] In other words, the value of the function is a constant. This graph represents the function [latex] f(x)=2. [\/latex]<\/p>\r\n\r\n<\/section><section data-depth=\"2\">[caption id=\"attachment_671\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-671\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19-300x291.jpeg\" alt=\"\" width=\"300\" height=\"291\" \/> Figure 19. A horizontal line representing the function [latex] f(x)=2 [\/latex][\/caption]\r\n<p id=\"fs-id2395091\">A <span id=\"term-00022\" class=\"no-emphasis\" data-type=\"term\">vertical line<\/span> indicates a constant input, or <em data-effect=\"italics\">x<\/em>-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_672\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-672\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20-300x61.jpeg\" alt=\"\" width=\"300\" height=\"61\" \/> Figure 20. Example of how a line has a vertical slope. 0 in the denominator of the slope.[\/caption]\r\n<p id=\"fs-id2228919\">A vertical line, such as the one in Figure 2<strong>1,<\/strong> has an <em data-effect=\"italics\">x<\/em>-intercept, but no <em data-effect=\"italics\">y-<\/em>intercept unless it\u2019s the line\u00a0[latex] x=0. [\/latex] This graph represents the line [latex] x=2. [\/latex]<\/p>\r\n\r\n<\/section><section data-depth=\"2\"><\/section><section data-depth=\"2\">[caption id=\"attachment_673\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-673\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21-300x291.jpeg\" alt=\"\" width=\"300\" height=\"291\" \/> Figure 21. The vertical line,\u00a0[latex] x=2, [\/latex] which does not represent a function[\/caption]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Horizontal and Vertical Lines<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nLines can be horizontal or vertical.\r\n\r\nA\u00a0<strong>horizontal line<\/strong> is a line defined by an equation in the form [latex] f(x)=b. [\/latex]\r\n\r\nA\u00a0<strong>vertical line<\/strong> is a line defined by an equation in the form [latex] x=a. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 15: Writing the Equation of a Horizontal Line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the equation of the line graphed in Figure 22.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_674\" align=\"aligncenter\" width=\"293\"]<img class=\"size-medium wp-image-674\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" \/> Figure 22[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>For any x-value, the y-value is\u00a0[latex] -4, [\/latex] so the equation is [latex] y=-4. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 16: Writing the Equation of a Vertical Line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the equation of the line graphed in Figure 23.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_675\" align=\"aligncenter\" width=\"292\"]<img class=\"size-medium wp-image-675\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23-292x300.jpeg\" alt=\"\" width=\"292\" height=\"300\" \/> Figure 23[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The constant x-value is\u00a0[latex] 7, [\/latex] so the equation is [latex] x=7. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id2239916\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining Whether Lines are Parallel or Perpendicular<\/h2>\r\n<p id=\"fs-id2239921\">The two lines in Figure 24 are <span id=\"term-00025\" class=\"no-emphasis\" data-type=\"term\">parallel lines<\/span>: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the <em data-effect=\"italics\">y<\/em>-intercept. If we shifted one line vertically toward the other, they would become coincident.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_676\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-676\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24-300x253.jpeg\" alt=\"\" width=\"300\" height=\"253\" \/> Figure 24. Parallel lines[\/caption]\r\n\r\n<span class=\"os-title-label\">\u00a0<\/span>\r\n<p id=\"fs-id2239956\">We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the <em data-effect=\"italics\">y<\/em>-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\left. \\begin{array}{11} f(x)=-2x+6 \\\\ f(x)=-2x-4 \\end{array} \\right\\} \\ \\ \\text{parallel}\\hspace{2em}\\left. \\begin{array}{11} f(x)=3x+2\\\\ f(x)=2x+2 \\end{array} \\right\\} \\ \\ \\text{not parallel} [\/latex]<\/p>\r\n<p id=\"fs-id2447804\">Unlike parallel lines, <span id=\"term-00026\" class=\"no-emphasis\" data-type=\"term\">perpendicular lines<\/span> do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in Figure 25 are perpendicular.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_677\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-677\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" \/> Figure 25. Perpendicular lines[\/caption]\r\n<p id=\"fs-id2352289\">Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is 1. So, if\u00a0[latex] m_1 [\/latex] and [latex] m_2 [\/latex] are negative reciprocals of one another, they can be multiplied together to yield [latex] -1. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]m_1m_2=-1 [\/latex]<\/p>\r\n<p id=\"fs-id2395259\">To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is\u00a0[latex] \\frac{1}{8}, [\/latex] and the reciprocal of\u00a0[latex] \\frac{1}{8} [\/latex] is 8. To find the negative reciprocal, first find the reciprocal and then change the sign.<\/p>\r\n<p id=\"fs-id2262400\">As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcll}f(x) &amp;=&amp; \\frac{1}{4}x + 2 &amp; \\quad \\text{negative reciprocal of } \\frac{1}{4} \\text{ is } -4 \\\\f(x) &amp;=&amp; -4x + 3 &amp; \\quad \\text{negative reciprocal of } -4 \\text{ is } \\frac{1}{4}\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2352752\">The product of the slopes is \u20131.<\/p>\r\n<p style=\"text-align: center;\">[latex] -4(\\frac{1}{4})=-1 [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Parallel and Perpendicular Lines<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nTwo lines are\u00a0<strong>parallel lines<\/strong> if they do not intersect. The slopes of the lines are the same.\r\n<p style=\"text-align: center;\">[latex] f(x)=m_1x+b_1 \\ \\ \\text{and} \\ \\ g(x)=m_2x+b_2 \\ \\ \\text{are parallel if and only if} \\ \\ m_1=m_2 [\/latex]<\/p>\r\nIf and only if\u00a0[latex] b_1=b_2 [\/latex] and\u00a0[latex] m_1=m_2, [\/latex] we say the lines coincide. Coincident lines are the same line.\r\n\r\nTwo lines are\u00a0<strong>perpendicular lines<\/strong> if they intersect to form a right angle.\r\n<p style=\"text-align: center;\">[latex] f(x)=m_1x+b_1 \\ \\ \\text{and} \\ \\ g(x)=m_2x+b_2 \\ \\ \\text{are perpendicular if and only if} [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] m_1m_2=-1, \\ \\ \\text{so} \\ \\ m_2=-\\frac{1}{m_1} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17: Identifying Parallel and Perpendicular Lines<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rclrcl}f(x) &amp;=&amp; 2x + 3 &amp; \\quad h(x) &amp;=&amp; -2x + 2 \\\\g(x) &amp;=&amp; \\frac{1}{2}x - 4 &amp; \\quad j(x) &amp;=&amp; 2x - 6\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Parallel lines have the same slope. Because the functions\u00a0[latex] f(x)=2x+3 [\/latex] and\u00a0[latex] j(x)=2x-6 [\/latex] each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because -2 and\u00a0[latex] \\frac{1}{2} [\/latex] are negative reciprocals, the functions\u00a0[latex] g(x)=\\frac{1}{2}x-4 [\/latex] and\u00a0[latex] h(x)=-2x+2 [\/latex] represent perpendicular lines.\r\n<h3>Analysis<\/h3>\r\nA graph of the lines is shown in Figure 26.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_678\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-678\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26-300x207.jpeg\" alt=\"\" width=\"300\" height=\"207\" \/> Figure 26[\/caption]\r\n\r\nThe graph shows that the lines\u00a0[latex] f(x)=2x+3 [\/latex] and\u00a0[latex] j(x)=2x-6 [\/latex] are parallel, and the lines\u00a0[latex] g(x)=\\frac{1}{2}x-4 [\/latex] and\u00a0[latex] h(x)=-2x+2 [\/latex] are perpendicular.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id2365851\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Writing the Equation of a Line Parallel or Perpendicular to a Given Line<\/h2>\r\n<p id=\"fs-id2365857\">If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.<\/p>\r\n\r\n<section id=\"fs-id2365861\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Writing Equations of Parallel Lines<\/h3>\r\n<p id=\"fs-id2365867\">Suppose for example, we are given the equation shown.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=3x+1 [\/latex]<\/p>\r\n<p id=\"fs-id2365903\">We know that the slope of the line formed by the function is 3. We also know that the <em data-effect=\"italics\">y-<\/em>intercept is\u00a0[latex] (0, 1). [\/latex] Any other line with a slope of 3 will be parallel to\u00a0[latex] f(x). [\/latex] So the lines formed by all of the following functions will be parallel to [latex] f(x). [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} g(x) &amp;=&amp; 3x+6 \\\\ h(x) &amp;=&amp; 3x+1 \\\\ p(x) &amp;=&amp; 3x+\\frac{2}{3} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2352637\">Suppose then we want to write the equation of a line that is parallel to\u00a0[latex] f [\/latex] and passes through the point\u00a0[latex] (1, 7). [\/latex] This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of\u00a0[latex] b [\/latex] will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} y-y_1 &amp;=&amp; m(x-x_1) \\\\ y-7 &amp;=&amp; 3(x-1) \\\\ y-1 &amp;=&amp; 3x-3 \\\\ y &amp;=&amp; 3x+4 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1801804\">So\u00a0[latex] g(x)=3x+4 [\/latex] is parallel to\u00a0[latex] f(x)=3x+1 [\/latex] and passes through the point [latex] (1, 7). [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point.<\/strong>\r\n<ol>\r\n \t<li>Find the slope of the function.<\/li>\r\n \t<li>Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 18: Finding a Line Parallel to a Given Line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a line parallel to the graph of\u00a0[latex] f(x)=3x+6 [\/latex] that passes through the point [latex] (3, 0). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The slope of a given line if 3. If we choose the slope-intercept form, we can substitute\u00a0[latex] m=3, x=3, [\/latex] and\u00a0[latex] f(x)=0 [\/latex] into the slope-intercept form to find the <em>y-<\/em>intercept.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} g(x) &amp;=&amp; 3x+b \\\\ 0 &amp;=&amp; 3(3)+b \\\\ b &amp;=&amp; -9 \\end{array} [\/latex]<\/p>\r\nThe line parallel to\u00a0[latex] f(x) [\/latex] that passes through\u00a0[latex] (3, 0) [\/latex] is [latex] g(x)=3x-9. [\/latex]\r\n<h3>Analysis<\/h3>\r\nWe can confirm that the two lines are parallel by graphing then. Figure 27 shows that the two lines will never intersect.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_679\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-679\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/> Figure 27[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id2249992\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Writing Equations of Perpendicular Lines<\/h3>\r\n<p id=\"fs-id2249998\">We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=2x+4 [\/latex]<\/p>\r\n<p id=\"fs-id2496718\">The slope of the line is 2, and its negative reciprocal is\u00a0[latex] -\\frac{1}{2}. [\/latex] Any function with a slope of\u00a0[latex] -\\frac{1}{2} [\/latex] will be perpendicular to\u00a0[latex] f(x). [\/latex] So the lines formed by all of the following functions will be perpendicular to [latex] f(x). [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} g(x) &amp;=&amp; -\\frac{1}{2}x+4 \\\\ h(x) &amp;=&amp; -\\frac{1}{2}x+2 \\\\ p(x) &amp;=&amp; -\\frac{1}{2}x-\\frac{1}{2} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2406643\">As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to\u00a0[latex] f(x) [\/latex] and passes through the point\u00a0[latex] (4, 0). [\/latex] We already know that the slope is\u00a0[latex] -\\frac{1}{2}. [\/latex] Now we can use the point to find the <em data-effect=\"italics\">y<\/em>-intercept by substituting the given values into the slope-intercept form of a line and solving for [latex] b. [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} g(x) &amp;=&amp; mx+b \\\\ 0 &amp;=&amp; -\\frac{1}{2}(4)+b \\\\ 0 &amp;=&amp; -2+b \\\\ 2 &amp;=&amp; b \\\\ b &amp;=&amp; 2 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2253685\">The equation for the function with a slope of\u00a0[latex] -\\frac{1}{2} [\/latex] and a <em data-effect=\"italics\">y-<\/em>intercept of 2 is<\/p>\r\n<p style=\"text-align: center;\">[latex] g(x)=-\\frac{1}{2}x+2 [\/latex]<\/p>\r\n<p id=\"fs-id2270676\">So\u00a0[latex] g(x)=-\\frac{1}{2}x+2 [\/latex] is perpendicular to\u00a0[latex] f(x)=2x+4 [\/latex] and passes through the point\u00a0[latex] (4, 0). [\/latex] Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular, but the product of their slopes is not \u20131. Doesn\u2019t this fact contradict the definition of perpendicular lines?<\/strong>\r\n\r\nA: <em data-effect=\"italics\">No. For two perpendicular linear functions, the product of their slopes is\u00a0[latex] -1. [\/latex] However, a vertical line is not a function so the definition is not contradicted.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the equation of a function and a point through which its graph passes, write the equation of a line perpendicular to the given line.<\/strong>\r\n<ol>\r\n \t<li>Find the slope of the function.<\/li>\r\n \t<li>Determine the negative reciprocal of the slope.<\/li>\r\n \t<li>Substitute the new slope and the values for\u00a0[latex] x [\/latex] and\u00a0[latex] y [\/latex] from the coordinate pair provided into [latex] g(x)=mx+b. [\/latex]<\/li>\r\n \t<li>Solve for [latex] b. [\/latex]<\/li>\r\n \t<li>Write the equation of the line.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 19: Finding the Equation of a Perpendicular Line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the equation of a line perpendicular to\u00a0[latex] f(x)=3x+3 [\/latex] that passes through the point [latex] (3, 0). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The original line has slope\u00a0[latex] m=3, [\/latex] so the slope of the perpendicular line will be its negative reciprocal, or\u00a0[latex] -\\frac{1}{3}. [\/latex] Using this slope and the given point, we can find the equation of the line.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} g(x) &amp;=&amp; -\\frac{1}{3}x+b \\\\ 0 &amp;=&amp; -\\frac{1}{3}(3)+b \\\\ 1 &amp;=&amp; b \\\\ b &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\nThe line perpendicular to\u00a0[latex] f(x) [\/latex] that passes through\u00a0[latex] (3, 0) [\/latex] is [latex] g(x)=-\\frac{1}{3}x+1. [\/latex]\r\n<h3>Analysis<\/h3>\r\nA graph of the two lines in shown in Figure 28.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_680\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-680\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28-300x261.jpeg\" alt=\"\" width=\"300\" height=\"261\" \/> Figure 28[\/caption]\r\n\r\nNote that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div id=\"Example_04_01_20\" class=\"ui-has-child-title\" data-type=\"example\"><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2262199\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1348623\" data-type=\"commentary\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function\u00a0[latex] h(x)=2x-4, [\/latex] write an equation for the line passing through\u00a0[latex] (0,0) [\/latex] that is\r\n\r\n(a) parallel to [latex] h(x) [\/latex]\r\n\r\n(b) perpendicular to [latex] h(x) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given two points on a line and a third point, write the equation of the perpendicular line that passes through the point.<\/strong>\r\n<ol>\r\n \t<li>Determine the slope of the line passing through the points.<\/li>\r\n \t<li>Find the negative reciprocal of the slope.<\/li>\r\n \t<li>Use the slope-intercept form or point-slope form to write the equation by substituting the known values.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20: Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA line passes through the points\u00a0[latex] (-2, 6) [\/latex] and\u00a0[latex] (4, 5). [\/latex] Find the equation of a perpendicular line that passes through the point [latex] (4, 5). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>From the two points of the given line, we can calculate the slope of that line.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} m_1 &amp;=&amp; -\\frac{5-6}{4-(-2)} \\\\ &amp;=&amp; -\\frac{-1}{6} \\\\ &amp;=&amp; -\\frac{1}{6} \\end{array} [\/latex]<\/p>\r\nFind the negative reciprocal of the slope.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} m_2 &amp;=&amp; -\\frac{-1}{-\\frac{1}{6}} \\\\ &amp;=&amp; -1\\left(-\\frac{6}{1}\\right) \\\\ &amp;=&amp; 6 \\end{array} [\/latex]<\/p>\r\nWe can then solve for the y-intercept of the line passing through the point [latex] (4, 5). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} g(x) &amp;=&amp; 6x+b \\\\ 5 &amp;=&amp; 6(4)+b \\\\ 5 &amp;=&amp; 24+b \\\\ -19 &amp;=&amp; b \\\\ b &amp;=&amp; -19 \\end{array} [\/latex]<\/p>\r\nThe equation for the line that is perpendicular to the line passing through the two given points and also passes through the point\u00a0[latex] (4, 5) [\/latex] is\r\n<p style=\"text-align: center;\">[latex] y=6x-19 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA line passes through the points\u00a0[latex] (-2, -15) [\/latex] and\u00a0[latex] (2, -3). [\/latex] Find the equation of a perpendicular line that passes through the point, [latex] (6, 4). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess this online resource for additional instruction and practice with linear functions.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=mzfmuJVI-HA\">Linear Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=_tTdoR1E5fg\">Finding Input of Function from the Output and Graph<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=8KLDGlrjzaw\">Graphing Functions Using Tables<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">4.1 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1799811\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1799815\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1799820\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1799822\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1799820-solution\">1<\/a><span class=\"os-divider\">. <\/span>Terry is skiing down Copper Mountain. Terry's elevation,\u00a0[latex] E(t), [\/latex] in feet after\u00a0[latex] t [\/latex] seconds is given by\u00a0[latex] E(t)=3000-70t. [\/latex] Write a complete sentence describing Terry\u2019s starting elevation and how it is changing over time.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2216172\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2216173\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Jessica is walking home from a friend\u2019s house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2216181\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2216182\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2216181-solution\">3<\/a><span class=\"os-divider\">. <\/span>A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour. Write an equation for the distance of the boat from the reservoir after [latex] t [\/latex] hours.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2216226\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2216227\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the <em data-effect=\"italics\">y<\/em>-intercepts.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2216236\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2216238\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2216236-solution\">5<\/a><span class=\"os-divider\">. <\/span>If a horizontal line has the equation\u00a0[latex] f(x)=a [\/latex] and a vertical line has the equation\u00a0[latex] x=a, [\/latex] what is the point of intersection? Explain why what you found is the point of intersection.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2270562\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id2270567\">For the following exercises, determine whether the equation of the curve can be written as a linear function.<\/p>\r\n\r\n<div id=\"fs-id1798659\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798660\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] y=\\frac{1}{4}x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1798691\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798693\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1798691-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] y=3x-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2496994\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798723\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] y=3x^2-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1798754\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798755\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1798754-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] 3x+5y=15 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1798785\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798786\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] 3x^2+5y=15 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2295720\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2295721\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2295720-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] 3x+5y^2=15 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2295757\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2295758\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] -2x^2+3y^2=6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2295801\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2295802\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2295801-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] -\\frac{x-3}{5}=2y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2295841\">For the following exercises, determine whether each function is increasing or decreasing.<\/p>\r\n\r\n<div id=\"fs-id2295845\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2295846\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1855485\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1855486\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1855485-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=5x+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1855521\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1855522\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] a(x)=5-2x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1855555\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1855556\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1855555-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] b(x)=8-3x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1855591\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1855592\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=-2x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239367\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239368\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239367-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] k(x)=-4x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239406\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239408\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] j(x)=\\frac{1}{2}x-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239447\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239448\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239447-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] p(x)=\\frac{1}{4}x-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239492\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239493\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] n(x)=-\\frac{1}{3}x-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2083727\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2083729\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2083727-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] m(x)=-\\frac{3}{8}x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2083774\">For the following exercises, find the slope of the line that passes through the two given points.<\/p>\r\n\r\n<div id=\"fs-id2083777\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2083778\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] (2, 4) \\ \\ \\text{and} \\ \\ (4, 10) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2083836\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2083837\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2083836-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] (1, 5) \\ \\ \\text{and} \\ \\ (4, 11) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2597471\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2597472\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] (-1, 4) \\ \\ \\text{and} \\ \\ (5, 2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2597529\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2597530\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2597529-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] (8, -2) \\ \\ \\text{and} \\ \\ (4, 6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2597596\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2597597\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] (6, 11) \\ \\ \\text{and} \\ \\ (-4, 3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2498442\">For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.<\/p>\r\n\r\n<div id=\"fs-id2498446\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2498447\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2498446-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] f(-5)=-4, \\ \\ \\text{and} \\ \\ f(5)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2498547\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2498548\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] f(-1)=4, \\ \\ \\text{and} \\ \\ f(5)=1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2653161\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2653162\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2653161-solution\">31<\/a><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex] (2, 4) \\ \\ \\text{and} \\ \\ (4, 10) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2653249\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2653250\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex] (1, 5) \\ \\ \\text{and} \\ \\ (4, 11) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2282224\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2282225\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2282224-solution\">33<\/a><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex] (-1, 4) \\ \\ \\text{and} \\ \\ (5, 2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2282326\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2282327\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex] (-2, 8) \\ \\ \\text{and} \\ \\ (4, 6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2604219\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2604220\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2604219-solution\">35<\/a><span class=\"os-divider\">. <\/span><em data-effect=\"italics\">x<\/em> intercept at\u00a0[latex] (02, 0) [\/latex] and <em data-effect=\"italics\">y<\/em> intercept at [latex] (0, -3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2604318\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2604319\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><em data-effect=\"italics\">x<\/em> intercept at\u00a0[latex] (-5, 0) [\/latex] and <em data-effect=\"italics\">y<\/em> intercept at [latex] (0, 4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2252349\">For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.<\/p>\r\n\r\n<div id=\"fs-id2252353\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2252354\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2252353-solution\">37<\/a><span class=\"os-divider\">.<\/span> [latex] \\begin{array}{rcl} 4x-7y=10 \\\\ 7x+4y=1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2252424\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2252425\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span> [latex] \\begin{array}{rcl} 3y+x=12 \\\\ -y=8x+1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2261858\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261859\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2261858-solution\">39<\/a><span class=\"os-divider\">.<\/span> [latex] \\begin{array}{rcl} 3y+4x=12 \\\\ -6y=8x+1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2261930\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261932\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span> [latex] \\begin{array}{rcl} 6x-9y=10 \\\\ 3x+2y=1 \\end{array} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2445672\">For the following exercises, find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y-<\/em>intercepts of each equation.<\/p>\r\n\r\n<div id=\"fs-id2445686\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2445687\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2445686-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2583719\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2250078\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=2x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2250111\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2250112\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2250111-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=3x-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2479676\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2479677\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] k(x)=-5x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2479711\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2479712\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2479711-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] -2x+5y=20 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239181\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239182\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] 7x+2y=56 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2239207\">For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?<\/p>\r\n\r\n<div id=\"fs-id2239211\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239214\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239211-solution\">47<\/a><span class=\"os-divider\">. <\/span>Line 1: Passes through [latex] (0, 6) \\ \\ \\text{and} \\ \\ (3, -24) [\/latex]\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2239270\">Line 2: Passes through\u00a0[latex] (-1, 19) \\ \\ \\text{and} \\ \\ (8, -71) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1828374\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1828375\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex] (-8, -55) \\ \\ \\text{and} \\ \\ (10, 89) [\/latex]\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1828438\">Line 2: Passes through\u00a0[latex] (9, -44) \\ \\ \\text{and} \\ \\ (4, -14) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1828498\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1828499\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1828498-solution\">49<\/a><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex] (2, 3)\\ \\ \\text{and} \\ \\ (4, -1) [\/latex]\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1797495\">Line 2: Passes through\u00a0[latex] (6, 3) \\ \\ \\text{and} \\ \\ (8, 5) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1797562\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1797563\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex] (1, 7) \\ \\ \\text{and} \\ \\ (5, 5) [\/latex]\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1797619\">Line 2: Passes through\u00a0[latex] (-1, -3) \\ \\ \\text{and} \\ \\ (1, 1) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1797676\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1797677\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1797676-solution\">51<\/a><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex] (2, 5) \\ \\ \\text{and} \\ \\ (5, -1) [\/latex]\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2352442\">Line 2: Passes through\u00a0 [latex] (-3, 7) \\ \\ \\text{and} \\ \\ (3, -5) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2352555\">For the following exercises, write an equation for the line described.<\/p>\r\n\r\n<div id=\"fs-id2352558\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2352559\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Write an equation for a line parallel to\u00a0[latex] f(x)=-5x-3 [\/latex] and passing through the point [latex] (2, -12). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2406979\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2406980\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2406979-solution\">53<\/a><span class=\"os-divider\">. <\/span>Write an equation for a line parallel to\u00a0[latex] g(x)=3x-1 [\/latex] and passing through the point [latex] (4, 9). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2407070\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2407071\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Write an equation for a line perpendicular to\u00a0[latex] h(t)=-2t+4 [\/latex] and passing through the point [latex] (-4, -1). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2407142\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2407143\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2407142-solution\">55<\/a><span class=\"os-divider\">. <\/span>Write an equation for a line perpendicular to\u00a0[latex] p(t)=3t+4 [\/latex] and passing through the point [latex] (3, 1). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2261328\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id2261333\">For the following exercises, find the slope of the line\u00a0graphed.<\/p>\r\n\r\n<div id=\"fs-id2261337\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261338\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261344\" class=\"first-element\" data-type=\"media\" data-alt=\"This is a graph of a decreasing linear function on an x, y coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line passes through points (0, 5) and (4, 0).\">\r\n<img class=\"alignnone size-medium wp-image-681\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2261358\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261359\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2261358-solution\">57<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261364\" class=\"first-element\" data-type=\"media\" data-alt=\"This is a graph of a function on an x, y coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The lines passes through points at (0, -2) and (2, -2).\">\r\n<img class=\"alignnone size-medium wp-image-682\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2261383\">For the following exercises, write an equation for the line graphed.<\/p>\r\n\r\n<div id=\"fs-id2261386\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261387\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261392\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an increasing linear function with points at (0,1) and (3,3)\">\r\n<img class=\"alignnone size-medium wp-image-683\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2261404\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261406\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2261404-solution\">59<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261411\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a decreasing linear function with points (0,5) and (4,0)\">\r\n<img class=\"alignnone size-medium wp-image-684\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2261453\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261454\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261459\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a decreasing linear function with points at (0,3) and (1.5,0)\">\r\n<img class=\"alignnone size-medium wp-image-685\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2605036\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2605037\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2605036-solution\">61<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2605042\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an increasing linear function with points at (1,2) and (0,-2)\">\r\n<img class=\"alignnone size-medium wp-image-686\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2605078\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2605079\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2605084\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function with points at (0, 3) and (3, 3)\">\r\n<img class=\"alignnone size-medium wp-image-687\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2605097\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2605098\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2605097-solution\">63<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2605103\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function with points at (0,-2.5) and (-2.5,-2.5)\">\r\n<img class=\"alignnone size-medium wp-image-688\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-300x298.jpeg\" alt=\"\" width=\"300\" height=\"298\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2605134\">For the following exercises, match the given linear equation with its graph in Figure 29.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_689\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-689\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29-300x296.jpeg\" alt=\"\" width=\"300\" height=\"296\" \/> Figure 29[\/caption]\r\n\r\n<div id=\"fs-id2605162\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2605163\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2605195\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2605196\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2605195-solution\">65<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-3x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2605235\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2385913\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-\\frac{1}{2}x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2354805\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2354806\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2354805-solution\">67<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2354836\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2354837\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2+x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2354867\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2354868\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2354867-solution\">69<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2354905\">For the following exercises, sketch a line with the given features.<\/p>\r\n\r\n<div id=\"fs-id2354908\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2354909\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>An <em data-effect=\"italics\">x<\/em>-intercept of\u00a0[latex] (-4, 0) [\/latex] and <em data-effect=\"italics\">y<\/em>-intercept of [latex] (0, -2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2354975\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2354976\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2354975-solution\">71<\/a><span class=\"os-divider\">. <\/span>An <em data-effect=\"italics\">x<\/em>-intercept of\u00a0[latex] (-2, 0) [\/latex] and <em data-effect=\"italics\">y<\/em>-intercept of [latex] (0, 4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2393349\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393350\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>A <em data-effect=\"italics\">y<\/em>-intercept of\u00a0[latex] (0, 7) [\/latex] and slope [latex] -\\frac{3}{2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2393405\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393406\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2393405-solution\">73<\/a><span class=\"os-divider\">. <\/span>A <em data-effect=\"italics\">y<\/em>-intercept of\u00a0[latex] (0, 3) [\/latex] and slope [latex] \\frac{2}{5} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2393478\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393480\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>Passing through the points\u00a0[latex] (-6, -2) [\/latex] and [latex] (6, -6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239623\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239624\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239623-solution\">75<\/a><span class=\"os-divider\">. <\/span>Passing through the points\u00a0[latex] (-3, -4) [\/latex] and [latex] (3, 0) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2239701\">For the following exercises, sketch the graph of each equation.<\/p>\r\n\r\n<div id=\"fs-id2239704\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239705\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-2x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239740\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239742\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239740-solution\">77<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-3x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239796\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239797\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{1}{3}x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2239837\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2239838\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239837-solution\">79<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{2}{3}x-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2447392\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2447393\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span> [latex] f(t)=3+2t [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2447425\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2447426\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2447425-solution\">81<\/a><span class=\"os-divider\">. <\/span> [latex] p(t)=-2+3t [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2447481\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2447482\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">82<\/span><span class=\"os-divider\">. <\/span> [latex] x=3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2447499\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2447500\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2447499-solution\">83<\/a><span class=\"os-divider\">. <\/span> [latex] x=-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2447539\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2447540\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">84<\/span><span class=\"os-divider\">. <\/span> [latex] r(x)=4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2447566\">For the following exercises, write the equation of the line shown in the graph.<\/p>\r\n\r\n<div id=\"fs-id2447569\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2447570\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2447569-solution\">85<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979120\" class=\"first-element\" data-type=\"media\" data-alt=\"The graph of a line with a slope of 0 and y-intercept at 3.\">\r\n<img class=\"alignnone size-medium wp-image-690\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1979148\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1979149\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">86<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979154\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a line with a slope of 0 and y-intercept at -1.\">\r\n<img class=\"alignnone size-medium wp-image-691\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1979167\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1979168\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1979167-solution\">87<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979173\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a line with an undefined slope and x-intercept at -3.\">\r\n<img class=\"alignnone size-medium wp-image-692\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1979203\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1965935\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">88<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979208\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a line with an undefined slope and x-intercept at 2\">\r\n<img class=\"alignnone size-medium wp-image-693\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1979221\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1979227\">For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.<\/p>\r\n\r\n<div id=\"fs-id1979231\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1979233\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1979231-solution\">89<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_04\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"height: 35px;\" data-id=\"Table_04_01_04\">\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"height: 17px; width: 180.017px;\" data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 68.9167px;\" data-align=\"center\">0<\/td>\r\n<td style=\"height: 17px; width: 118.8px;\" data-align=\"center\">5<\/td>\r\n<td style=\"height: 17px; width: 123.733px;\" data-align=\"center\">10<\/td>\r\n<td style=\"height: 17px; width: 131.2px;\" data-align=\"center\">15<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 180.017px;\" data-align=\"center\">[latex] g(x) [\/latex]<\/td>\r\n<td style=\"height: 18px; width: 68.9167px;\" data-align=\"center\">5<\/td>\r\n<td style=\"height: 18px; width: 118.8px;\" data-align=\"center\">\u201310<\/td>\r\n<td style=\"height: 18px; width: 123.733px;\" data-align=\"center\">\u201325<\/td>\r\n<td style=\"height: 18px; width: 131.2px;\" data-align=\"center\">\u201340<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1798819\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798821\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">90<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_05\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"height: 35px;\" data-id=\"Table_04_01_05\">\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"height: 17px; width: 186.133px;\" data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 72.8833px;\" data-align=\"center\">0<\/td>\r\n<td style=\"height: 17px; width: 104.3px;\" data-align=\"center\">5<\/td>\r\n<td style=\"height: 17px; width: 124.15px;\" data-align=\"center\">10<\/td>\r\n<td style=\"height: 17px; width: 135.2px;\" data-align=\"center\">15<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 186.133px;\" data-align=\"center\">[latex] h(x) [\/latex]<\/td>\r\n<td style=\"height: 18px; width: 72.8833px;\" data-align=\"center\">5<\/td>\r\n<td style=\"height: 18px; width: 104.3px;\" data-align=\"center\">30<\/td>\r\n<td style=\"height: 18px; width: 124.15px;\" data-align=\"center\">105<\/td>\r\n<td style=\"height: 18px; width: 135.2px;\" data-align=\"center\">230<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1798936\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1798937\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1798936-solution\">91<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_06\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"Table_04_01_06\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<td data-align=\"center\">15<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u20135<\/td>\r\n<td data-align=\"center\">20<\/td>\r\n<td data-align=\"center\">45<\/td>\r\n<td data-align=\"center\">70<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1799092\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1799093\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">92<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_07\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"Table_04_01_07\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<td data-align=\"center\">20<\/td>\r\n<td data-align=\"center\">25<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] k(x) [\/latex]<\/td>\r\n<td data-align=\"center\">13<\/td>\r\n<td data-align=\"center\">28<\/td>\r\n<td data-align=\"center\">58<\/td>\r\n<td data-align=\"center\">73<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1828666\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2261900\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1828666-solution\">93<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_08\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"Table_04_01_08\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] g(x) [\/latex]<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">\u201319<\/td>\r\n<td data-align=\"center\">\u201344<\/td>\r\n<td data-align=\"center\">\u201369<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1828835\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1828836\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">94<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_09\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"Table_04_01_09\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] h(x) [\/latex]<\/td>\r\n<td data-align=\"center\">13<\/td>\r\n<td data-align=\"center\">23<\/td>\r\n<td data-align=\"center\">43<\/td>\r\n<td data-align=\"center\">53<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2654316\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2654319\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2654316-solution\">95<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_10\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"Table_04_01_10\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u20134<\/td>\r\n<td data-align=\"center\">16<\/td>\r\n<td data-align=\"center\">36<\/td>\r\n<td data-align=\"center\">56<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2654475\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2654476\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">96<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"Table_04_01_11\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"Table_04_01_11\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] k(x) [\/latex]<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">106<\/td>\r\n<td data-align=\"center\">231<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2353110\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id2353115\">For the following exercises, use a calculator or graphing technology to complete the task.<\/p>\r\n\r\n<div id=\"fs-id2353120\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2353121\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2353120-solution\">97<\/a><span class=\"os-divider\">. <\/span>If\u00a0[latex] f [\/latex] is a linear function,\u00a0[latex] f(0.1)=11.5, [\/latex] and\u00a0[latex] f(0.4-5.9, [\/latex] find an equation for the function.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2353233\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2353234\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">98<\/span><span class=\"os-divider\">. <\/span>Graph the function\u00a0[latex] f [\/latex] on a domain of\u00a0[latex] [-10, 10]: f(x)=0.02x-0.01. [\/latex] Enter the function in a graphing utility. For the viewing window, set the minimum value of\u00a0[latex] x [\/latex] to be\u00a0[latex] -10 [\/latex] and the maximum value of\u00a0[latex] x [\/latex] to be [latex] 10. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2241250\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2241251\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2241250-solution\">99<\/a><span class=\"os-divider\">. <\/span>Graph the function\u00a0[latex] f [\/latex] on a domain of [latex] [-10, 10]: f(x)=2,500x+4,000 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2241349\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2241350\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">100<\/span><span class=\"os-divider\">. <\/span>Table 3 shows the input,\u00a0[latex] w, [\/latex] and output,\u00a0[latex] k, [\/latex] for a linear function [latex] k. [\/latex]\r\n\r\n(a) Fill in the missing values of the table.\r\n\r\n(b) Write the linear function\u00a0[latex] k, [\/latex] round to 3 decimal places.\r\n<div class=\"os-problem-container\">\r\n<div id=\"Table_04_01_12\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_01_12\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] w [\/latex]<\/td>\r\n<td data-align=\"center\">\u201310<\/td>\r\n<td data-align=\"center\">5.5<\/td>\r\n<td data-align=\"center\">67.5<\/td>\r\n<td data-align=\"center\"><em data-effect=\"italics\">b<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] k [\/latex]<\/td>\r\n<td data-align=\"center\">30<\/td>\r\n<td data-align=\"center\">\u201326<\/td>\r\n<td data-align=\"center\"><em data-effect=\"italics\">a<\/em><\/td>\r\n<td data-align=\"center\">\u201344<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2633779\" class=\"os-hasSolution\" data-type=\"exercise\"><header><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2633779-solution\">101<\/a><span class=\"os-divider\">. <\/span>Table 4 shows the input,\u00a0[latex] p, [\/latex] and output,\u00a0[latex] q, [\/latex] for a linear function [latex] q. [\/latex]\r\n(a) Fill in the missing values of the table.\r\n(b) Write the linear function [latex] q. [\/latex]\r\n<div id=\"fs-id2633780\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div id=\"Table_04_01_13\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_01_13\"><caption>Table 4<\/caption>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] p [\/latex]<\/td>\r\n<td data-align=\"center\">0.5<\/td>\r\n<td data-align=\"center\">0.8<\/td>\r\n<td data-align=\"center\">12<\/td>\r\n<td data-align=\"center\"><em data-effect=\"italics\">b<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] q [\/latex]<\/td>\r\n<td data-align=\"center\">400<\/td>\r\n<td data-align=\"center\">700<\/td>\r\n<td data-align=\"center\"><em data-effect=\"italics\">a<\/em><\/td>\r\n<td data-align=\"center\">1,000,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id2633952\" data-type=\"exercise\"><header><span class=\"os-number\">102<\/span><span class=\"os-divider\">. <\/span>Graph the linear function\u00a0[latex] f [\/latex] on a domain of\u00a0[latex] [-10, 10] [\/latex] for the function whose slope is\u00a0[latex] \\frac{1}{8} [\/latex] and <em data-effect=\"italics\">y<\/em>-intercept is\u00a0[latex] \\frac{31}{16}. [\/latex] Label the points for the input values of\u00a0[latex] -10 [\/latex] and [latex] 10. [\/latex]<\/header><\/div>\r\n<div id=\"fs-id2511972\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2511973\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2511972-solution\">103<\/a><span class=\"os-divider\">. <\/span>Graph the linear function\u00a0[latex] f [\/latex] on a domain of\u00a0[latex] [-0.1, 0.1] [\/latex] for the function whose slope is 75 and <em data-effect=\"italics\">y<\/em>-intercept is\u00a0[latex] -22.5. [\/latex] Label the points for the input values of\u00a0[latex] -0.1 [\/latex] and [latex] 0.1. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2512070\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2512071\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">104<\/span><span class=\"os-divider\">. <\/span>Graph the linear function\u00a0[latex] f [\/latex] where\u00a0[latex] f(x)=ax+b [\/latex] on the same set of axes on a domain of\u00a0[latex] [-4, 4] [\/latex] for the following values of\u00a0[latex] a [\/latex] and [latex] b [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(a) [latex] a=2; b=3 [\/latex]\r\n\r\n(b) [latex] a=2; b=4 [\/latex]\r\n\r\n(c) [latex] a=2; b=-4 [\/latex]\r\n\r\n(d) [latex] a=2; b=-5 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id2266929\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id2266935\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2266936\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2266935-solution\">105<\/a><span class=\"os-divider\">. <\/span>Find the value of\u00a0[latex] x [\/latex] if a linear function goes through the following points and has the following slope: [latex] (x, 2), (-4, 6), m=3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2267009\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2267010\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">106<\/span><span class=\"os-divider\">. <\/span>Find the value of [latex] y [\/latex] if a linear function goes through the following points and has the following slope: [latex] (10, y), (25, 100), m=-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2267082\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2267083\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2267082-solution\">107<\/a><span class=\"os-divider\">. <\/span>Find the equation of the line that passes through the following points:\u00a0[latex] (a, b) \\ \\ \\text{and} \\ \\ (a, b+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2267152\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2267153\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">108<\/span><span class=\"os-divider\">. <\/span>Find the equation of the line that passes through the following points:\u00a0[latex] (2a, b) \\ \\ \\text{and} \\ \\ (a, b+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2762578\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2762579\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2762578-solution\">109<\/a><span class=\"os-divider\">. <\/span>Find the equation of the line that passes through the following points:\u00a0[latex] (a, 0) \\ \\ \\text{and} \\ \\ (c, d) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2762640\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2762641\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">110. <\/span>Find the equation of the line parallel to the line\u00a0[latex] g(x)=-0.01x+2.01 [\/latex] through the point [latex] (1, 2). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2762721\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2762722\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2762721-solution\">111<\/a><span class=\"os-divider\">. <\/span>Find the equation of the line perpendicular to the line\u00a0[latex] g(x)=-0.01x+2.01 [\/latex] through the point [latex] (1, 2). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2762790\">For the following exercises, use the functions\u00a0[latex] f(x)=-0.1x+200 [\/latex] and [latex] g(x)=20x+0.1. [\/latex]<\/p>\r\n\r\n<div id=\"fs-id2393555\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393556\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">112<\/span><span class=\"os-divider\">. <\/span>Find the point of intersection of the lines\u00a0[latex] f [\/latex] and [latex] g [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2393649\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393650\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2393649-solution\">113<\/a><span class=\"os-divider\">. <\/span>Where is\u00a0[latex] f(x) [\/latex] greater than\u00a0[latex] g(x)? [\/latex] Where is [latex] g(x) [\/latex] greater than [latex] f(x)? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2393754\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id2393760\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393761\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">114<\/span><span class=\"os-divider\">. <\/span>At noon, a barista notices that they have $20 in their tip jar. If the barista makes an average of $0.50 from each customer, how much will they have in the tip jar if they serve\u00a0[latex] n [\/latex] more customers during the shift?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2393799\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393800\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2393799-solution\">115<\/a><span class=\"os-divider\">. <\/span>At VASA fitness in Aurora, CO, a gym membership with two personal training sessions costs $125, while gym membership with five personal training sessions costs $260. What is cost per session?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2393805\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2393806\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">116<\/span><span class=\"os-divider\">. <\/span>Aspen Blue Boutique in Centennial, CO, finds there is a linear relationship between the number of shirts,\u00a0[latex] n, [\/latex] it can sell and the price,\u00a0[latex] p, [\/latex] it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of\u00a0[latex] \\$30 [\/latex] while 3,000 shirts can be sold at a price of $22. Find a linear equation in the form\u00a0[latex] p(n)=mn+b [\/latex] that gives the price\u00a0[latex] p [\/latex] they can charge for\u00a0 shirts.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2249337\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2249338\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2249337-solution\">117<\/a><span class=\"os-divider\">. <\/span>T-Mobile charges for service according to the formula:\u00a0[latex] C(n)=24+0.1n, [\/latex] where\u00a0[latex] n [\/latex] is the number of minutes talked, and\u00a0[latex] C(n) [\/latex] is the monthly charge, in dollars. Find and interpret the rate of change and initial value.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2249416\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2249417\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">118<\/span><span class=\"os-divider\">. <\/span>A farmer finds there is a linear relationship between the number of bean stalks,\u00a0[latex] n, [\/latex] she plants and the yield,\u00a0[latex] y, [\/latex] each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form\u00a0[latex] y=mn+b [\/latex] that gives the yield when\u00a0[latex] n [\/latex] stalks are planted.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2249513\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2249514\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2249513-solution\">119<\/a><span class=\"os-divider\">. <\/span> Colorado\u2019s population in the year 1980 was 2,889,735. In 2000, the population was 4,301,261. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2249522\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2249523\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">120<\/span><span class=\"os-divider\">. <\/span>A town\u2019s population has been growing linearly. In 2003, the population was 45,000, and the population has been growing by 1,700 people each year. Write an equation,\u00a0[latex] P(t), [\/latex] for the population\u00a0 years after 2003.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2631646\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2631648\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2631646-solution\">121<\/a><span class=\"os-divider\">. <\/span>Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function:\u00a0[latex] I(x)=1054x+23,286, [\/latex] where\u00a0[latex] x [\/latex] is the number of years after 1990. Which of the following interprets the slope in the context of the problem?\r\n\r\n(a) As of 1990, average annual income was $23,286.\r\n\r\n(b) In the ten-year period from 1990\u20131999, average annual income increased by a total of $1,054.\r\n\r\n(c) Each year in the decade of the 1990s, average annual income increased by $1,054.\r\n\r\n(d) Average annual income rose to a level of $23,286 by the end of 1999.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2631728\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2631729\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">122<\/span><span class=\"os-divider\">. <\/span>When temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of\u00a0[latex] C, [\/latex] the Celsius temperature, [latex] F(C). [\/latex]\r\n\r\n(a) Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius.\r\n\r\n(b) Find and interpret [latex] F(28) [\/latex]\r\n\r\n(c) Find and interpret [latex] F(-40) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_f48a9644-4329-4387-9b38-4ac039f12570\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Represent a linear function.<\/li>\n<li>Determine whether a linear function is increasing, decreasing, or constant.<\/li>\n<li>Interpret slope as a rate of change.<\/li>\n<li>Write and interpret an equation for a linear function.<\/li>\n<li>Graph linear functions.<\/li>\n<li>Determine whether lines are parallel or perpendicular.<\/li>\n<li>Write the equation of a line parallel or perpendicular to a given line.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section>\n<ul id=\"list-00001\"><\/ul>\n<\/section>\n<\/div>\n<figure id=\"attachment_647\" aria-describedby=\"caption-attachment-647\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-647\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-300x169.jpg\" alt=\"\" width=\"300\" height=\"169\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-300x169.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-1024x576.jpg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-768x432.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-65x37.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-225x127.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1-350x197.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-1.jpg 1200w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-647\" class=\"wp-caption-text\">Figure 1. &#8220;RTD Line Train 305 southbound at Broadway Station&#8221; by FoamingInDenver is licensed under CC0 1.0<\/figcaption><\/figure>\n<div>\n<div>\n<p>Suppose the Denver RTD train travels at a speed of 79 mph for a period of time once it is 3 miles from the station.\u00a0 How can we analyze the train\u2019s distance from the station over a period of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train\u2019s distance from the station at a given point in time.<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id2374595\" data-depth=\"1\">\n<h2 data-type=\"title\">Representing Linear Functions<\/h2>\n<p id=\"fs-id2673768\">The function describing the train\u2019s motion is a <span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">linear function<\/span>, which is defined as a function with a constant rate of change. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train\u2019s motion as a function using each method.<\/p>\n<section id=\"fs-id2052037\" data-depth=\"2\">\n<h3 data-type=\"title\">Representing a Linear Function in Word Form<\/h3>\n<p id=\"fs-id1723576\">Let\u2019s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.<\/p>\n<ul id=\"fs-id2157898\">\n<li><em data-effect=\"italics\">The train\u2019s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.<\/em><\/li>\n<\/ul>\n<p id=\"fs-id1502969\">The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. <span class=\"TextRun SCXW99595078 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW99595078 BCX2\">As<\/span><span class=\"NormalTextRun SCXW99595078 BCX2\"> the time(input) increases by 1 hour, the corresponding distance (output)<\/span> <span class=\"NormalTextRun SCXW99595078 BCX2\">increases by 79 miles.\u00a0 <\/span><span class=\"NormalTextRun SCXW99595078 BCX2\">The train began moving at this constant speed at a distance of 3 miles from the station.<\/span><\/span><\/p>\n<\/section>\n<section id=\"fs-id2216588\" data-depth=\"2\">\n<h3 data-type=\"title\">Representing a Linear Function in Function Notation<\/h3>\n<p id=\"fs-id1701341\">Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">slope-intercept form<\/span> of a line, where\u00a0[latex]x[\/latex] is the input value,\u00a0[latex]m[\/latex] is the rate of change, and\u00a0[latex]b[\/latex] is the initial value of the dependent variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\text{Equation form} \\quad & y = mx + b \\\\\\text{Function notation} \\quad & f(x) = mx + b\\end{array}[\/latex]<\/p>\n<p>In the example of the Denver RTD, we might use the notation\u00a0[latex]D(t)[\/latex] where the total distance [latex]D[\/latex] is a function of the time\u00a0[latex]t.[\/latex] The rate [latex]m[\/latex] is 79 miles per hour. The initial value of the dependent variable\u00a0[latex]b[\/latex] is the original distance from the station, 3 miles.\u00a0 We can write a generalized equation to represent the motion of the train.<\/p>\n<p style=\"text-align: center;\">[latex]D(t)=79t+3[\/latex]<\/p>\n<\/section>\n<\/section>\n<\/div>\n<section id=\"fs-id1349346\" data-depth=\"2\">\n<h3 data-type=\"title\">Representing a Linear Function in Tabular Form<\/h3>\n<p id=\"fs-id2040470\">A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in Figure 2. From the table, we can see that the distance changes by 79 miles for every 1 hour increase in time.<\/p>\n<figure id=\"attachment_1255\" aria-describedby=\"caption-attachment-1255\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1255\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2-300x184.png\" alt=\"\" width=\"300\" height=\"184\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2-300x184.png 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2-65x40.png 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2-225x138.png 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2-350x214.png 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-2.png 438w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1255\" class=\"wp-caption-text\">Figure 2. Tabular representation of the function D showing selected input and output values.<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Can the input in the previous example be any real number?<\/strong><\/p>\n<p>A: <em data-effect=\"italics\">No. The input represents time so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1726229\" data-depth=\"2\">\n<h3 data-type=\"title\">Representing a Linear Function in Graphical Form<\/h3>\n<p id=\"fs-id1630800\">Another way to represent linear functions is visually, using a graph. We can use the function relationship [latex]D(t)=83t+250,[\/latex] to draw a graph as represented in Figure 3. Notice the graph is a line<strong>.<\/strong> When we plot a linear function, the graph is always a line.<\/p>\n<p id=\"fs-id2059569\">The rate of change, which is constant, determines the slant, or <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">slope<\/span> of the line. The point at which the input value is zero is the vertical intercept, or [latex]y-[\/latex]intercept, of the line. We can see from the graph that the [latex]y-[\/latex]intercept in the train example we just saw is\u00a0[latex](0, 250)[\/latex] and represents the distance of the train from the station when it began moving at a constant speed.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_649\" aria-describedby=\"caption-attachment-649\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-649\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3-300x178.jpeg\" alt=\"\" width=\"300\" height=\"178\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3-300x178.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3-225x134.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3-350x208.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-3.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-649\" class=\"wp-caption-text\">Figure 3. The graph of\u00a0[latex] D9t)=83t+250. [\/latex] Graphs of linear functions are lines because the rate of change is constant.<\/figcaption><\/figure>\n<p id=\"fs-id1843158\">Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line [latex]f(x)=2x+1.[\/latex] Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Linear Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>linear <span id=\"term-00008\" data-type=\"term\">function<\/span><\/strong> is a function whose graph is a line. Linear functions can be written in the <strong><span id=\"term-00009\" data-type=\"term\">slope-intercept form<\/span> <\/strong>of a line<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=mx+b[\/latex]<\/p>\n<p>where\u00a0[latex]b[\/latex] is the initial or starting value of the function (when input,\u00a0[latex]x=0[\/latex]) and\u00a0[latex]m[\/latex] is the constant rate of change, or slope of the function. The [latex]y-[\/latex]intercept is at [latex](0, b).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Using a linear function to find the cost of ordering Tamales at the Mexican restaurant, Santiago\u2019s<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div>\n<div>\n<p>The cost,\u00a0[latex]C,[\/latex] in dollars of ordering Tamales depends on the number of tamales\u00a0[latex]t[\/latex] and this relationship may be modeled by the equation\u00a0[latex]C(t)=3t+6.99,[\/latex] where\u00a0[latex]\\$3[\/latex] is the cost per tamale and [latex]\\$6.99[\/latex] is the standard delivery charge.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To restate the function in words, we need to describe each part of the equation. The cost as a function of the number of tamales equals 3 times the number of tamales ordered plus the delivery charge of $6.99.<\/p>\n<h3>Analysis<\/h3>\n<p>The initial value, $6.99, is the standard delivery fee. The rate of change, or slope, is $3 per tamale. This tells us that the cost of the order increases $3 for each extra tamale ordered.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1630015\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining Whether a Linear Function Is Increasing, Decreasing, or Constant<\/h2>\n<p id=\"fs-id2209400\">The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">increasing function<\/span>, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in Figure 5<strong>(a)<\/strong>. For a <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">decreasing function<\/span>, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in Figure 5<strong>(b)<\/strong>. If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in Figure 5<strong>(c)<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_650\" aria-describedby=\"caption-attachment-650\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-650\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-300x115.jpeg\" alt=\"\" width=\"300\" height=\"115\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-300x115.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-768x295.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-65x25.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-225x87.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5-350x135.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-5.jpeg 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-650\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Increasing and Decreasing Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The slope determines if the function is an <strong>increasing linear function<\/strong>, a <strong>decreasing linear function<\/strong>, or a constant function.<\/p>\n<ul>\n<li>[latex]f(x)=mx+b[\/latex] is an increasing function if [latex]m>0.[\/latex]<\/li>\n<li>[latex]f(x)=mx+b[\/latex] is a decreasing function if [latex]m<0.[\/latex]<\/li>\n<li>[latex]f(x)=mx+b[\/latex] is a constant function if [latex]m=0.[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Deciding Whether a Function Is Increasing, Decreasing, or Constant<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Studies from the early 2010s indicated that teens sent about 60 texts a day, while more recent data indicates much higher messaging rates among all users, particularly considering the various apps with which people can communicate.<sup id=\"footnote-ref2\" data-type=\"footnote-number\"><a role=\"doc-noteref\" href=\"4-1-linear-functions#fs-id1759513\" data-type=\"footnote-link\">3<\/a><\/sup>. For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.<\/p>\n<p>(a) The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.<\/p>\n<p>(b) Lewis has a limit of 500 texts per month in their data plan. The input is the number of days, and output is the total number of texts remaining for the month.<\/p>\n<p>(c) Sofia has an unlimited number of texts in their data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Analyze each function.<\/p>\n<p>(a) The function can be represented as\u00a0[latex]f(x)=60x[\/latex] where\u00a0[latex]x[\/latex] is the number of days. The slope, 60, is positive so the function is increasing. This makes sense because the total number of texts increases with each day.<\/p>\n<p>(b) The function can be represented as\u00a0[latex]f(x)=500-60x[\/latex] where\u00a0[latex]x[\/latex] is the number of days. In this case, the slope is negative so the function is decreasing. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after\u00a0[latex]x[\/latex] days<\/p>\n<p>(c) The cost function can be represented as\u00a0[latex]f(x)=50[\/latex] because the number of days does not affect the total cost. The slope is 0 so the function is constant.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section><\/section>\n<\/section>\n<section id=\"fs-id2407166\" data-depth=\"1\">\n<h2 data-type=\"title\">Interpreting Slope as a Rate of Change<\/h2>\n<p id=\"fs-id1716534\">In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input,\u00a0[latex]x_1[\/latex] and\u00a0[latex]x_2,[\/latex] and two corresponding values for the output,\u00a0[latex]y_1[\/latex] and\u00a0[latex]y_2[\/latex] \u2014 which can be represented by a set of points,\u00a0[latex](x_1, y_1)[\/latex] and\u00a0[latex](x_2, y_2)[\/latex] \u2014 we can calculate the slope [latex]m.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{x_2-x_1}[\/latex]<\/p>\n<p id=\"fs-id1853016\">Note that in function notation we can obtain two corresponding values for the output\u00a0[latex]y_1[\/latex] and\u00a0[latex]y_2[\/latex] for the function\u00a0[latex]f, y_1=f(x_1)[\/latex] and\u00a0[latex]y_2=f(x_2),[\/latex] so we could equivalently write<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{f(x_2)-f(x_1)}{x_2-x_1}[\/latex]<\/p>\n<p id=\"fs-id1503277\">Figure 6 indicates how the slope of the line between the points,\u00a0[latex](x_1, y_1)[\/latex] and\u00a0[latex](x_2, y_2),[\/latex] is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_651\" aria-describedby=\"caption-attachment-651\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-651\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6-300x247.jpeg\" alt=\"\" width=\"300\" height=\"247\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6-300x247.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6-65x53.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6-225x185.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6-350x288.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-6.jpeg 467w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-651\" class=\"wp-caption-text\">Figure 6. The slope of a function is calculated by the change in [latex] y [\/latex] divided by the change in [latex] x. [\/latex] It does not matter which coordinate is used as the [latex] (x_2, y_2) [\/latex] and which is the [latex] (x_1, y_1), [\/latex] as long as each calculation is started with the elements from the same coordinate pair.<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Are the units for slope always<\/strong> [latex]\\frac{\\text{units for the output}}{\\text{units for the input}}?[\/latex]<\/p>\n<p>A: <em data-effect=\"italics\">Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.<\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Calculate Slope<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The slope, or rate of change, of a function\u00a0[latex]m[\/latex] can be calculated according to the following:<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{x_2-x_1}[\/latex]<\/p>\n<p>where\u00a0[latex]x_1[\/latex] and\u00a0[latex]x_2[\/latex] are input values,\u00a0[latex]y_1[\/latex] and\u00a0[latex]y_2[\/latex] are output values.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given two points from a linear function, calculate and interpret the slope.<\/strong><\/p>\n<ol>\n<li>Determine the units for output and input values.<\/li>\n<li>Calculate the change of output values and change of input values.<\/li>\n<li>Interpret the slope as the change in output values per unit of the input value.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Finding the Slope of a Linear Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If\u00a0[latex]f(x)[\/latex] is a linear function, and\u00a0[latex](3, -2)[\/latex] and\u00a0[latex](8, 1)[\/latex] are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The coordinate pairs are\u00a0[latex](3, -2)[\/latex] and\u00a0[latex](8, 1).[\/latex] To find the rate of change, we divide the change in output by the change in input.<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-(-2)}{8-3}=\\frac{3}{5}[\/latex]<\/p>\n<p>We could also write the slope as\u00a0[latex]m=0.6.[\/latex] The function is increasing because [latex]m>0.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or [latex]y-[\/latex]coordinate, used corresponds with the first input value, or [latex]x-[\/latex]coordinate, used. Note that if we had reversed them, we would have obtained the same slope.<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{(-2)-(1)}{3-8}=\\frac{-3}{-5}=\\frac{3}{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If\u00a0[latex]f(x)[\/latex] is a linear function, and\u00a0[latex](2, 3)[\/latex] and\u00a0[latex](0, 4)[\/latex] are points on the line, find the slope. Is this function increasing or decreasing?<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Finding the Population Change from a Linear Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The population of Aurora increased from 338,774 in 2010 to 381,057 in 2020. Find the change of population per year if we assume the change was constant from 2010 to 2020.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The rate of change relates to the population to the change in time. The population increased by\u00a0[latex]381,057-338,774=42,283[\/latex] people over the 10-year interval.\u00a0 To find the rate of change, divide the change in the number of people by the number of years.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{42,483 \\ \\ \\text{people}}{10 \\ \\ \\text{years}}\\approx\\frac{4,228}{\\text{year}}[\/latex]<\/p>\n<p>So, the population increased by approximately 4,228 people per year.<\/p>\n<h3>Analysis<\/h3>\n<p>Because we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1700480\" data-depth=\"1\">\n<h2 data-type=\"title\">Writing and Interpreting an Equation for a Linear Function<\/h2>\n<p id=\"fs-id1688492\">Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form and the point-slope form. Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function\u00a0[latex]f[\/latex] in Figure 7.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_653\" aria-describedby=\"caption-attachment-653\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-653\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7-300x214.jpeg\" alt=\"\" width=\"300\" height=\"214\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7-300x214.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7-65x46.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7-225x160.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7-350x249.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-7.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-653\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<p id=\"fs-id1308976\">We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let\u2019s choose\u00a0[latex](0, 7)[\/latex] and [latex](4, 4).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}m &=& \\frac{y_2 - y_1}{x_2 - x_1} \\\\&=& \\frac{4 - 7}{4 - 0} \\\\&=& -\\frac{3}{4}\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1574658\">Now we can substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}y - y_1 &=& m(x - x_1) \\\\\\quad y - 4 &=& -\\frac{3}{4}(x - 4)\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1591011\">If we want to rewrite the equation in the slope-intercept form, we would find<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}y - 4 &=& -\\frac{3}{4}(x - 4) \\\\y - 4 &=& -\\frac{3}{4}x + 3 \\\\y &=& -\\frac{3}{4}x + 7\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1804015\">If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the [latex]y-[\/latex]axis when the output value is 7. Therefore,\u00a0[latex]b=7.[\/latex] We now have the initial value\u00a0[latex]b[\/latex] and the slope\u00a0[latex]m[\/latex] so we can substitute\u00a0[latex]m[\/latex] and\u00a0[latex]b[\/latex] into the slope-intercept form of a line.<br \/>\n<span id=\"fs-id1109555\" data-type=\"media\" data-alt=\"This image shows the equation f of x equals m times x plus b. It shows that m is the value negative three fourths and b is 7. It then shows the equation rewritten as f of x equals negative three fourths times x plus 7.\"><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-654 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx-300x95.jpeg\" alt=\"\" width=\"300\" height=\"95\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx-300x95.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx-65x21.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx-225x72.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx-350x111.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fx.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"fs-id2065131\">So the function is\u00a0[latex]f(x)=-\\frac{3}{4}x+7,[\/latex] and the linear equation would be [latex]y=-\\frac{3}{4}x+7.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the graph of a linear function, write an equation to represent the function.<\/strong><\/p>\n<ol>\n<li>Identify two points on the line.<\/li>\n<li>Use the two points to calculate the slope.<\/li>\n<li>Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.<\/li>\n<li>Substitute the slope and y-intercept into the slope-intercept form of a line equation.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Writing an Equation for a Linear Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write an equation for a linear function given a graph of\u00a0[latex]f[\/latex] shown in Figure 8.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_655\" aria-describedby=\"caption-attachment-655\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-655\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8-293x300.jpeg 293w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8-225x230.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8-350x359.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-8.jpeg 369w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><figcaption id=\"caption-attachment-655\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Identify two points on the line, such as\u00a0[latex](0, 2)[\/latex] and\u00a0[latex](-2, -4).[\/latex] Use the points to calculate the slope.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}m &=& \\frac{y_2 - y_1}{x_2 - x_1} \\\\&=& \\frac{-4 - 2}{-2 - 0} \\\\&=& \\frac{-6}{-2} \\\\&=& 3\\end{array}[\/latex]<\/p>\n<p>Substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}y - y_1 &=& m(x - x_1) \\\\y - (-4) &=& 3(x - (-2)) \\\\y + 4 &=& 3(x + 2)\\end{array}[\/latex]<\/p>\n<p>We can use algebra to rewrite the equation in the slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}y + 4 &=& 3(x + 2) \\\\y + 4 &=& 3x + 6 \\\\y &=& 3x + 2\\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>This makes sense because we can see from Figure 9 that the line crosses the [latex]y-[\/latex]axis at the point\u00a0[latex](0, 2),[\/latex] which is the [latex]y-[\/latex]intercept, so [latex]b=2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_656\" aria-describedby=\"caption-attachment-656\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-656\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9-293x300.jpeg 293w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9-225x230.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9-350x359.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-9.jpeg 369w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><figcaption id=\"caption-attachment-656\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Writing an Equation for a Linear Cost Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Suppose Bailey starts a company in which they incur a fixed cost of $1,250 per month for the overhead, which includes their office rent. Their production costs are $37.50 per item. Write a linear function\u00a0[latex]C[\/latex] where\u00a0[latex]C(x)[\/latex] is the cost for\u00a0[latex]x[\/latex] items produced in a given month.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is $37.50. The variable cost, called the marginal cost, is represented by\u00a0[latex]37.5[\/latex] The cost Bailey incurs is the sum of these two costs, represented by [latex]C(x)=1250+37.5x.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>If Bailey produces 100 items in a month, their monthly cost is found by substituting 100 for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}C(100) &=& 1250 + 37.5(100) \\\\&=& 5000\\end{array}[\/latex]<\/p>\n<p>So their monthly cost would be $5,000.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Writing an Equation for a Linear Function Given Two Points<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If\u00a0[latex]f[\/latex] is a linear function, with\u00a0[latex]f(3)=-2,[\/latex] and\u00a0[latex]f(8)=1,[\/latex] find an equation for the function in slope-intercept form.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can write the given points using the coordinates.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}f(3) &=& -2 \\rightarrow (3, -2) \\\\f(8) &=& 1 \\rightarrow (8, 1)\\end{array}[\/latex]<\/p>\n<p>We can then use the points to calculate the slope.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} m &=& \\frac{y_2-y_1}{x_2-x_1} \\\\ &=& \\frac{1-(-2)}{8-3} \\\\ &=& \\frac{3}{5} \\end{array}[\/latex]<\/p>\n<p>Substitute the slope and the coordinates of one of the points into the point-slope form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} y-y_1 &=& m(x-x_1) \\\\ y-(-2) &=& \\frac{3}{5}(x-3) \\end{array}[\/latex]<\/p>\n<p>We can use algebra to rewrite the equation in the slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} y+2 &=& \\frac{3}{5}(x-3) \\\\ y+2 &=& \\frac{3}{5}x-\\frac{9}{5} \\\\ y &=& \\frac{3}{5}x-\\frac{19}{5} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If\u00a0[latex]f(x)[\/latex] is a linear function, with\u00a0[latex]f(2)=-11[\/latex] and\u00a0[latex]f(4)=-25,[\/latex] write an equation for the function in slope-intercept form.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1577178\" data-depth=\"1\">\n<h2 data-type=\"title\">Modeling Real-World Problems with Linear Functions<\/h2>\n<p id=\"fs-id1421845\">In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a linear function\u00a0[latex]f[\/latex] and the initial value and rate of change, evaluate [latex]f(c).[\/latex]<\/strong><\/p>\n<ol>\n<li>Determine the initial value and the rate of change (slope).<\/li>\n<li>Substitute the values into [latex]f(x)=mx+b.[\/latex]<\/li>\n<li>Evaluate the function at [latex]x=c.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Using a Linear Function to Determine the Number of Songs in a Music Collection<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Emma currently has 200 songs in her music collection. Every month, she adds 15 new songs. Write a formula for the number of songs,\u00a0[latex]N,[\/latex] in her collection as a function of time,\u00a0[latex]t,[\/latex] the number of months. How many songs will she own at the end of one year?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The initial value for this function is 200 because she currently owns 200 songs, so\u00a0[latex]N(0)=200,[\/latex] which means that [latex]b=200.[\/latex]<\/p>\n<p>The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that\u00a0[latex]m=15.[\/latex] We can substitute the initial value and the rate of change into the slope-intercept form of a line.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_658\" aria-describedby=\"caption-attachment-658\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-658\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10-300x81.jpeg\" alt=\"\" width=\"300\" height=\"81\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10-300x81.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10-65x17.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10-225x61.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10-350x94.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-10.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-658\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<p>We can write this formula [latex]N(t)=15t+200.[\/latex]<\/p>\n<p>With this formula, we can then predict how many songs Emma will have at the end of one year (12 months). In other words, we can evaluate the function at [latex]t=12.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} N(12) &=& 15(12)+200 \\\\ &=& 180+200 \\\\ &=& 380 \\end{array}[\/latex]<\/p>\n<p>Emma will have 380 songs in 12 months.<\/p>\n<h3>Analysis<\/h3>\n<p>Notice that N is an increasing linear function. As the input (the number of months) increases, the output (number of songs) increases as well.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Using a Linear Function to Calculate Salary Based on Commission<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya&#8217;s weekly income\u00a0[latex]I,[\/latex] depends on the number of new policies,\u00a0[latex]n,[\/latex] he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for\u00a0[latex]I(n),[\/latex] and interpret the meaning of the components of the equation.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The given information gives us two input-output pairs:\u00a0[latex](3, 760)[\/latex] and\u00a0[latex](5, 920).[\/latex] We start by finding the rate of change.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} m &=& \\frac{920-760}{5-3} \\\\ &=& \\frac{\\$160}{2 \\ \\text{policies}} \\\\ &=& \\$80 \\ \\text{per policy} \\end{array}[\/latex]<\/p>\n<p>Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is $80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold during the week.<\/p>\n<p>We can then solve for the initial value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcll} I(n) &=& 80n+b \\\\ 760 &=& 80(3)+b & \\text{when} \\ n=3, I(3)=760 \\\\ 760 &-& 80(3)=b \\\\ 520 &=& b \\end{array}[\/latex]<\/p>\n<p>The value of\u00a0[latex]b[\/latex] is the starting value for the function and represents Ilya&#8217;s income when\u00a0[latex][\/latex] or when no new policies are sold. We can interpret this as Ilya\u2019s base salary for the week, which does not depend upon the number of policies sold.<\/p>\n<p>We can now write the final equation.<\/p>\n<p style=\"text-align: center;\">[latex]I(n)=80n+520[\/latex]<\/p>\n<p>Our final interpretation is that Ilya\u2019s base salary is $520 per week and he earns an additional $80 commission for each policy sold.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Using Tabular Form to Write an Equation for a Linear Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Table 1 relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\"><strong>number of weeks,\u00a0[latex]w[\/latex]<\/strong><\/td>\n<td style=\"width: 20%;\">0<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\"><strong>number of rats,\u00a0[latex]P(w)[\/latex]<\/strong><\/td>\n<td style=\"width: 20%;\">1000<\/td>\n<td style=\"width: 20%;\">1080<\/td>\n<td style=\"width: 20%;\">1160<\/td>\n<td style=\"width: 20%;\">1240<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can see from the table that the initial value for the number of rats is 100, so [latex]b=1000.[\/latex]<\/p>\n<p>Rather than solving for\u00a0[latex]m,[\/latex] we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.<\/p>\n<p style=\"text-align: center;\">[latex]P(w)=40w+1000[\/latex]<\/p>\n<p>If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using\u00a0[latex](2, 1080)[\/latex] and [latex](6, 1240).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} m &=& \\frac{1240-1080}{6-2} \\\\ &=& \\frac{160}{4} \\\\ &=& 40 \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Is the initial value always provided in a table of values like Table 1?<\/strong><\/p>\n<p><em>A:\u00a0No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into\u00a0[latex]f(x)=mx+b[\/latex] and solve for [latex]b.[\/latex]<\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A new plant food was introduced to a young tree to test its effect on the height of the tree. Table 2 shows the height of the tree, in feet,\u00a0[latex]x[\/latex] months since the measurements began. Write a linear function\u00a0[latex]H(x),[\/latex] where\u00a0[latex]x[\/latex] is the number of months since the start of the experiment.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">0<\/td>\n<td style=\"width: 16.6667%;\">2<\/td>\n<td style=\"width: 16.6667%;\">4<\/td>\n<td style=\"width: 16.6667%;\">8<\/td>\n<td style=\"width: 16.6667%;\">12<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]H(x)[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">12.5<\/td>\n<td style=\"width: 16.6667%;\">13.5<\/td>\n<td style=\"width: 16.6667%;\">14.5<\/td>\n<td style=\"width: 16.6667%;\">16.5<\/td>\n<td style=\"width: 16.6667%;\">18.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2458614\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Linear Functions<\/h2>\n<p id=\"fs-id2458643\">Now that we\u2019ve seen and interpreted graphs of linear functions, let\u2019s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the [latex]y-[\/latex]intercept and slope. And the third method is by using transformations of the identity function [latex]f(x)=x.[\/latex]<\/p>\n<section id=\"fs-id2568421\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Function by Plotting Points<\/h3>\n<p id=\"fs-id2237975\">To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function,\u00a0[latex]f(x)=2x,[\/latex] we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point\u00a0[latex](1, 2).[\/latex] Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point\u00a0[latex](2, 4).[\/latex] Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a linear function, graph by plotting points.<\/strong><\/p>\n<ol>\n<li>Choose a minimum of two input values.<\/li>\n<li>Evaluate the function at each input value.<\/li>\n<li>Use the resulting output values to identify coordinate pairs.<\/li>\n<li>Plot the coordinate pairs on a grid.<\/li>\n<li>Draw a line through the points.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Graphing by Plotting Points<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph\u00a0[latex]f(x)=-\\frac{2}{3}x+5[\/latex] by plotting points.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Begin\u00a0 by choosing input values. This function includes a fraction with a denominator of 3, so let\u2019s choose multiples of 3 as input values. We will choose 0, 3, and 6.<\/p>\n<p id=\"fs-id1683693\">Evaluate the function at each input value, and use the output value to identify coordinate pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} x=0 && f(0)=-\\frac{2}{3}(0)+5=5\\rightarrow (0, 5) \\\\ x=3 && f(3)=-\\frac{2}{3}(3)+5=3\\rightarrow (3, 3) \\\\ x=6 && f(6)=-\\frac{2}{3}(6)+5=1\\rightarrow (6, 1) \\end{array}[\/latex]<\/p>\n<p>Plot the coordinate pairs and draw a line through the points. Figure 11 represents the graph of the function [latex]f(x)=-]frac{2}{3}x+5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_661\" aria-describedby=\"caption-attachment-661\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-661\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11-300x260.jpeg\" alt=\"\" width=\"300\" height=\"260\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11-300x260.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11-65x56.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11-225x195.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11-350x304.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-11.jpeg 400w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-661\" class=\"wp-caption-text\">Figure 11. The graph of the linear function [latex] f(x)=-\\frac{2}{3}x+5.[\/latex]<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>The graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant, which indicates a negative slope. This is also expected from the negative, constant rate of change in the equation for the function.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=-\\frac{3}{4}x+6[\/latex] by plotting points.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2654013\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Function Using <em data-effect=\"italics\">y-<\/em>intercept and Slope<\/h3>\n<p id=\"fs-id2654022\">Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its <em data-effect=\"italics\">y-<\/em>intercept, which is the point at which the input value is zero. To find the [latex]y-[\/latex]intercept, we can set [latex]x=0[\/latex] in the equation.<\/p>\n<p id=\"fs-id2458574\">The other characteristic of the linear function is its slope<strong>.<\/strong><\/p>\n<p id=\"fs-id2458580\">Let\u2019s consider the following function.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\frac{1}{2}x+1[\/latex]<\/p>\n<p id=\"fs-id2497289\">The slope is [latex]\\frac{1}{2}.[\/latex] Because the slope is positive, we know the graph will slant upward from left to right. The [latex]y-[\/latex]intercept is the point on the graph when [latex]x=0.[\/latex] The graph crosses the [latex]y-[\/latex]axis at [latex](0, 1).[\/latex] Now we know the slope and the [latex]y-[\/latex]intercept. We can begin graphing by plotting the point [latex](0, 1).[\/latex] We know that the slope is the change in the [latex]y-[\/latex]coordinate over the change in the [latex]x-[\/latex]coordinate. This is commonly referred to as rise over run, [latex]m=\\frac{\\text{rise}}{\\text{run}}.[\/latex] From our example, we have [latex]m=\\frac{1}{2},[\/latex] which means that the rise is 1 and the run is 2. So starting from our [latex](0, 1),[\/latex]intercept [latex][\/latex] we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in Figure 12.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_662\" aria-describedby=\"caption-attachment-662\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-662\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12-300x200.jpeg\" alt=\"\" width=\"300\" height=\"200\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12-300x200.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12-65x43.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12-225x150.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12-350x233.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-12.jpeg 400w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-662\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Graphical Interpretation of a Linear Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In the equation [latex]f(x)=mx+b[\/latex]<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>[latex]b[\/latex] is the [latex]y-[\/latex]intercept of the graph and indicates the point\u00a0[latex](0, b)[\/latex] at which the graph crosses the [latex]y-[\/latex]axis.<\/li>\n<li>[latex]m[\/latex]is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\frac{\\Delta y}{\\Delta x}=\\frac{y_2-y_1}{x_2-x_1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Do all linear functions have [latex]y-[\/latex]intercepts?<\/strong><\/p>\n<p>A: <em data-effect=\"italics\">Yes. All linear functions cross the [latex]y-[\/latex]axis and therefore have [latex]y-[\/latex]intercepts. (Note: A vertical line is parallel to the [latex]y-[\/latex]axis does not have a [latex]y-[\/latex]intercept, but it is not a function<\/em>.)<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the equation for a linear function, graph the function using the <em data-effect=\"italics\">y<\/em>-intercept and slope.<\/strong><\/p>\n<ol>\n<li>Evaluate the function at an input value of zero to find the y-intercept.<\/li>\n<li>Identify the slope as the rate of change of the input value.<\/li>\n<li>Plot the point represented by the y-intercept.<\/li>\n<li>Use\u00a0[latex]\\frac{\\text{rise}}{\\text{run}}[\/latex] to determine at least two more points on the line.<\/li>\n<li>Sketch the line that passes through the points.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 12: Graphing by Using the y-intercept and Slope<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=-\\frac{2}{3}x+5[\/latex] using the <em>y-<\/em>intercept and slope.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Evaluate the function at\u00a0[latex]x=0[\/latex] to find the [latex]y-[\/latex]intercept. The output value when\u00a0[latex]x=0[\/latex] is 5, so the graph will cross the [latex]y-[\/latex]axis at [latex](0, 5).[\/latex]<\/p>\n<p>According to the equation for the function, the slope of the line is\u00a0[latex]-\\frac{2}{3}.[\/latex] This tells us that for each vertical decrease in the &#8220;rise&#8221; of\u00a0[latex]-2[\/latex] units, the &#8220;run&#8221; increases by 3 units in the horizontal direction. We can now graph the function by first plotting the [latex]y-[\/latex]intercept on the graph in Figure 13. From the initial value\u00a0[latex](0, 5)[\/latex] we move down 2 units and to the right 3 units. We can extend the line to the left and right by repeating, and then drawing a line through the points.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_663\" aria-describedby=\"caption-attachment-663\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-663\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13-300x229.jpeg\" alt=\"\" width=\"300\" height=\"229\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13-300x229.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13-65x50.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13-225x172.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13-350x267.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-13.jpeg 405w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-663\" class=\"wp-caption-text\">Figure 13. Graph of [latex] f(x)=-\\frac{2}{3}x+5 [\/latex] and shows how to calculate the rise over run for the slope.<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>The graph slants downward from left to right, which means it has a negative slope as expected.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a point on the graph we drew in Example 12 that has a negative <em>x-<\/em>value.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id2239536\" data-depth=\"1\">\n<h2 data-type=\"title\">Writing the Equation for a Function from the Graph of a Line<\/h2>\n<p id=\"fs-id2239541\">Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at Figure 14. We can see right away that the graph crosses the <em data-effect=\"italics\">y<\/em>-axis at the point\u00a0[latex](0, 4)[\/latex] so this is the <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_665\" aria-describedby=\"caption-attachment-665\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-665\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14-293x300.jpeg 293w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14-225x230.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14-350x359.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-14.jpeg 369w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><figcaption id=\"caption-attachment-665\" class=\"wp-caption-text\">Figure 14<\/figcaption><\/figure>\n<p id=\"fs-id2525935\">Then we can calculate the slope by finding the rise and run. We can choose any two points, but let\u2019s look at the point\u00a0[latex](-2, 0).[\/latex] To get from this point to the <em data-effect=\"italics\">y-<\/em>intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be<\/p>\n<p style=\"text-align: center;\">[latex]m=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{2}=2[\/latex]<\/p>\n<p>Substituting the slope and\u00a0<em>y-<\/em>intercept into the slope-intercept form of a line gives<\/p>\n<p style=\"text-align: center;\">[latex]y=2x+4[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a graph of linear function, find the equation to describe the function.<\/strong><\/p>\n<ol>\n<li>Identify the<em> y-<\/em>intercept of an equation.<\/li>\n<li>Choose two points to determine the slope.<\/li>\n<li>Substitute the <em>y-<\/em>intercept and slope into the slope-intercept form of a line.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 13: Matching Linear Functions to their Graphs<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Match each equation of the linear functions with one of the lines in Figure 15.<\/p>\n<p>(a) [latex]f(x)=2x+3[\/latex]<\/p>\n<p>(b) [latex]g(x)=2x-3[\/latex]<\/p>\n<p>(c) [latex]h(x)=-2x+3[\/latex]<\/p>\n<p>(d) [latex]j(x)=\\frac{1}{2}x+3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_667\" aria-describedby=\"caption-attachment-667\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-667\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15-300x233.jpeg\" alt=\"\" width=\"300\" height=\"233\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15-300x233.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15-65x50.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15-225x175.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15-350x272.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-15.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-667\" class=\"wp-caption-text\">Figure 15<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Analyze the information for each function.<\/p>\n<p>(a) This function has a slope of 2 and a <em data-effect=\"italics\">y<\/em>-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function\u00a0 has the same slope, but a different <em data-effect=\"italics\">y-<\/em>intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through\u00a0[latex](0, 3)[\/latex] so\u00a0[latex]f[\/latex] must be represented by line I.<\/p>\n<p>(b) This function also has a slope of 2, but a <em data-effect=\"italics\">y<\/em>-intercept of\u00a0[latex]-3.[\/latex] It must pass through the point\u00a0[latex](0, 3-)[\/latex] and slant upward from left to right. It must be represented by line III.<\/p>\n<p>(c) This function has a slope of -2 and a y-intercept of 3. This is the only function listed with a negative slop, so it must be represented by line IV because it slants downward from left to right.<\/p>\n<p>(d) This function has a slope of\u00a0[latex]\\frac{1}{2}[\/latex] and a y-intercept of 3. It must pass through the point\u00a0[latex](0, 3)[\/latex] and slant upward from left to right. Lines I and II pass through\u00a0[latex](0, 3),[\/latex] but the slope of\u00a0[latex]j[\/latex] is less than the slope of\u00a0[latex]f[\/latex] so the line for\u00a0[latex]j[\/latex] must be flatter. This function is represented by Line II.<\/p>\n<p>Now we can re-label the lines as in Figure 16.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_668\" aria-describedby=\"caption-attachment-668\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-668\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16-300x229.jpeg\" alt=\"\" width=\"300\" height=\"229\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16-300x229.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16-65x50.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16-225x172.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16-350x268.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-16.jpeg 489w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-668\" class=\"wp-caption-text\">Figure 16<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<section id=\"fs-id2256536\" data-depth=\"2\">\n<h3 data-type=\"title\">Finding the <em data-effect=\"italics\">x<\/em>-intercept of a Line<\/h3>\n<p id=\"fs-id2256546\">So far we have been finding the <em data-effect=\"italics\">y-<\/em>intercepts of a function: the point at which the graph of the function crosses the <em data-effect=\"italics\">y<\/em>-axis. Recall that a function may also have an <span id=\"term-00020\" class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">x<\/em>-intercept<\/span>, which is the <em data-effect=\"italics\">x<\/em>-coordinate of the point where the graph of the function crosses the <em data-effect=\"italics\">x<\/em>-axis. In other words, it is the input value when the output value is zero.<\/p>\n<p id=\"fs-id2256573\">To find the <em data-effect=\"italics\">x<\/em>-intercept, set a function\u00a0[latex]f(x)[\/latex] equal to zero and solve for the value of\u00a0[latex]x.[\/latex] For example, consider the function shown.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=3x-6[\/latex]<\/p>\n<p id=\"fs-id2568784\">Set the function equal to 0 and solve for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} 0 &=& 3x-6 \\\\ 6 &=& 3x \\\\ 2 &=& x \\\\ x &=& 2 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id2395276\">The graph of the function crosses the <em data-effect=\"italics\">x<\/em>-axis at the point [latex](2, 0).[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Do all linear functions have <em data-effect=\"italics\">x<\/em>-intercepts?<\/strong><\/p>\n<p><em>A:\u00a0No. However, linear functions of the form\u00a0[latex]y=c,[\/latex] where\u00a0[latex]c[\/latex] is a nonzero real number are the only examples of linear functions with no x-intercept. For example,\u00a0[latex]y=5[\/latex] is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in Figure 17.<\/em><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_669\" aria-describedby=\"caption-attachment-669\" style=\"width: 224px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-669\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-17-224x300.jpeg\" alt=\"\" width=\"224\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-17-224x300.jpeg 224w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-17-65x87.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-17-225x301.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-17.jpeg 300w\" sizes=\"auto, (max-width: 224px) 100vw, 224px\" \/><figcaption id=\"caption-attachment-669\" class=\"wp-caption-text\">Figure 17<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">X-Intercept<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <em>x-<\/em>intercept of the function is value of\u00a0[latex]x[\/latex] when\u00a0[latex]f(x)=0.[\/latex] It can be solved by the equation [latex]0=mx+b.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 14: Finding an x-intercept<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the x-intercept of [latex]f(x)=\\frac{1}{2}x-3.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Set the function equal to zero to solve for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} 0 &=& \\frac{1}{2}x-3 \\\\ 3 &=& \\frac{1}{2}x \\\\ 6 &=& x \\\\ x &=& 6 \\end{array}[\/latex]<\/p>\n<p>The graph crosses the x-axis at the point [latex](6, 0).[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>A graph of the function is shown in Figure 18. We can see that the x-intercept is\u00a0[latex](6, 0)[\/latex] as we expected.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_670\" aria-describedby=\"caption-attachment-670\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-670\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18-293x300.jpeg 293w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18-225x230.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18-350x359.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-18.jpeg 369w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><figcaption id=\"caption-attachment-670\" class=\"wp-caption-text\">Figure 18<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1796746\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<div class=\"textbox textbox--exercises\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the x-intercept of [latex]f(x)=\\frac{1}{4}x-4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2403100\" data-depth=\"2\">\n<h3 data-type=\"title\">Describing Horizontal and Vertical Lines<\/h3>\n<p id=\"fs-id2403105\">There are two special cases of lines on a graph\u2014horizontal and vertical lines. A <span id=\"term-00021\" class=\"no-emphasis\" data-type=\"term\">horizontal line<\/span> indicates a constant output, or <em data-effect=\"italics\">y<\/em>-value. In Figure 19, we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use\u00a0[latex]m=0[\/latex] in the equation\u00a0[latex]f(x)=mx+b,[\/latex] the equation simplifies to\u00a0[latex]f(x)=b.[\/latex] In other words, the value of the function is a constant. This graph represents the function [latex]f(x)=2.[\/latex]<\/p>\n<\/section>\n<section data-depth=\"2\">\n<figure id=\"attachment_671\" aria-describedby=\"caption-attachment-671\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-671\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19-300x291.jpeg\" alt=\"\" width=\"300\" height=\"291\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19-300x291.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19-65x63.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19-225x219.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19-350x340.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-19.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-671\" class=\"wp-caption-text\">Figure 19. A horizontal line representing the function [latex] f(x)=2 [\/latex]<\/figcaption><\/figure>\n<p id=\"fs-id2395091\">A <span id=\"term-00022\" class=\"no-emphasis\" data-type=\"term\">vertical line<\/span> indicates a constant input, or <em data-effect=\"italics\">x<\/em>-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_672\" aria-describedby=\"caption-attachment-672\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-672\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20-300x61.jpeg\" alt=\"\" width=\"300\" height=\"61\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20-300x61.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20-65x13.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20-225x46.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20-350x71.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-20.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-672\" class=\"wp-caption-text\">Figure 20. Example of how a line has a vertical slope. 0 in the denominator of the slope.<\/figcaption><\/figure>\n<p id=\"fs-id2228919\">A vertical line, such as the one in Figure 2<strong>1,<\/strong> has an <em data-effect=\"italics\">x<\/em>-intercept, but no <em data-effect=\"italics\">y-<\/em>intercept unless it\u2019s the line\u00a0[latex]x=0.[\/latex] This graph represents the line [latex]x=2.[\/latex]<\/p>\n<\/section>\n<section data-depth=\"2\"><\/section>\n<section data-depth=\"2\">\n<figure id=\"attachment_673\" aria-describedby=\"caption-attachment-673\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-673\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21-300x291.jpeg\" alt=\"\" width=\"300\" height=\"291\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21-300x291.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21-65x63.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21-225x219.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21-350x340.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-21.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-673\" class=\"wp-caption-text\">Figure 21. The vertical line,\u00a0[latex] x=2, [\/latex] which does not represent a function<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Horizontal and Vertical Lines<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Lines can be horizontal or vertical.<\/p>\n<p>A\u00a0<strong>horizontal line<\/strong> is a line defined by an equation in the form [latex]f(x)=b.[\/latex]<\/p>\n<p>A\u00a0<strong>vertical line<\/strong> is a line defined by an equation in the form [latex]x=a.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 15: Writing the Equation of a Horizontal Line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the equation of the line graphed in Figure 22.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_674\" aria-describedby=\"caption-attachment-674\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-674\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22-293x300.jpeg 293w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22-225x230.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22-350x359.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-22.jpeg 369w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><figcaption id=\"caption-attachment-674\" class=\"wp-caption-text\">Figure 22<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>For any x-value, the y-value is\u00a0[latex]-4,[\/latex] so the equation is [latex]y=-4.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 16: Writing the Equation of a Vertical Line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the equation of the line graphed in Figure 23.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_675\" aria-describedby=\"caption-attachment-675\" style=\"width: 292px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-675\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23-292x300.jpeg\" alt=\"\" width=\"292\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23-292x300.jpeg 292w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23-225x231.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23-350x360.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-23.jpeg 364w\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" \/><figcaption id=\"caption-attachment-675\" class=\"wp-caption-text\">Figure 23<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The constant x-value is\u00a0[latex]7,[\/latex] so the equation is [latex]x=7.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id2239916\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining Whether Lines are Parallel or Perpendicular<\/h2>\n<p id=\"fs-id2239921\">The two lines in Figure 24 are <span id=\"term-00025\" class=\"no-emphasis\" data-type=\"term\">parallel lines<\/span>: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the <em data-effect=\"italics\">y<\/em>-intercept. If we shifted one line vertically toward the other, they would become coincident.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_676\" aria-describedby=\"caption-attachment-676\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-676\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24-300x253.jpeg\" alt=\"\" width=\"300\" height=\"253\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24-300x253.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24-65x55.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24-225x189.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24-350x295.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-24.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-676\" class=\"wp-caption-text\">Figure 24. Parallel lines<\/figcaption><\/figure>\n<p><span class=\"os-title-label\">\u00a0<\/span><\/p>\n<p id=\"fs-id2239956\">We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the <em data-effect=\"italics\">y<\/em>-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.<\/p>\n<p style=\"text-align: center;\">[latex]\\left. \\begin{array}{11} f(x)=-2x+6 \\\\ f(x)=-2x-4 \\end{array} \\right\\} \\ \\ \\text{parallel}\\hspace{2em}\\left. \\begin{array}{11} f(x)=3x+2\\\\ f(x)=2x+2 \\end{array} \\right\\} \\ \\ \\text{not parallel}[\/latex]<\/p>\n<p id=\"fs-id2447804\">Unlike parallel lines, <span id=\"term-00026\" class=\"no-emphasis\" data-type=\"term\">perpendicular lines<\/span> do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in Figure 25 are perpendicular.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_677\" aria-describedby=\"caption-attachment-677\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-677\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25-300x272.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25-65x59.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25-225x204.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25-350x317.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-25.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-677\" class=\"wp-caption-text\">Figure 25. Perpendicular lines<\/figcaption><\/figure>\n<p id=\"fs-id2352289\">Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is 1. So, if\u00a0[latex]m_1[\/latex] and [latex]m_2[\/latex] are negative reciprocals of one another, they can be multiplied together to yield [latex]-1.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]m_1m_2=-1[\/latex]<\/p>\n<p id=\"fs-id2395259\">To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is\u00a0[latex]\\frac{1}{8},[\/latex] and the reciprocal of\u00a0[latex]\\frac{1}{8}[\/latex] is 8. To find the negative reciprocal, first find the reciprocal and then change the sign.<\/p>\n<p id=\"fs-id2262400\">As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcll}f(x) &=& \\frac{1}{4}x + 2 & \\quad \\text{negative reciprocal of } \\frac{1}{4} \\text{ is } -4 \\\\f(x) &=& -4x + 3 & \\quad \\text{negative reciprocal of } -4 \\text{ is } \\frac{1}{4}\\end{array}[\/latex]<\/p>\n<p id=\"fs-id2352752\">The product of the slopes is \u20131.<\/p>\n<p style=\"text-align: center;\">[latex]-4(\\frac{1}{4})=-1[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Parallel and Perpendicular Lines<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Two lines are\u00a0<strong>parallel lines<\/strong> if they do not intersect. The slopes of the lines are the same.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=m_1x+b_1 \\ \\ \\text{and} \\ \\ g(x)=m_2x+b_2 \\ \\ \\text{are parallel if and only if} \\ \\ m_1=m_2[\/latex]<\/p>\n<p>If and only if\u00a0[latex]b_1=b_2[\/latex] and\u00a0[latex]m_1=m_2,[\/latex] we say the lines coincide. Coincident lines are the same line.<\/p>\n<p>Two lines are\u00a0<strong>perpendicular lines<\/strong> if they intersect to form a right angle.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=m_1x+b_1 \\ \\ \\text{and} \\ \\ g(x)=m_2x+b_2 \\ \\ \\text{are perpendicular if and only if}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]m_1m_2=-1, \\ \\ \\text{so} \\ \\ m_2=-\\frac{1}{m_1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17: Identifying Parallel and Perpendicular Lines<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rclrcl}f(x) &=& 2x + 3 & \\quad h(x) &=& -2x + 2 \\\\g(x) &=& \\frac{1}{2}x - 4 & \\quad j(x) &=& 2x - 6\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Parallel lines have the same slope. Because the functions\u00a0[latex]f(x)=2x+3[\/latex] and\u00a0[latex]j(x)=2x-6[\/latex] each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because -2 and\u00a0[latex]\\frac{1}{2}[\/latex] are negative reciprocals, the functions\u00a0[latex]g(x)=\\frac{1}{2}x-4[\/latex] and\u00a0[latex]h(x)=-2x+2[\/latex] represent perpendicular lines.<\/p>\n<h3>Analysis<\/h3>\n<p>A graph of the lines is shown in Figure 26.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_678\" aria-describedby=\"caption-attachment-678\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-678\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26-300x207.jpeg\" alt=\"\" width=\"300\" height=\"207\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26-300x207.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26-65x45.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26-225x155.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26-350x241.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-26.jpeg 540w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-678\" class=\"wp-caption-text\">Figure 26<\/figcaption><\/figure>\n<p>The graph shows that the lines\u00a0[latex]f(x)=2x+3[\/latex] and\u00a0[latex]j(x)=2x-6[\/latex] are parallel, and the lines\u00a0[latex]g(x)=\\frac{1}{2}x-4[\/latex] and\u00a0[latex]h(x)=-2x+2[\/latex] are perpendicular.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2365851\" data-depth=\"1\">\n<h2 data-type=\"title\">Writing the Equation of a Line Parallel or Perpendicular to a Given Line<\/h2>\n<p id=\"fs-id2365857\">If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.<\/p>\n<section id=\"fs-id2365861\" data-depth=\"2\">\n<h3 data-type=\"title\">Writing Equations of Parallel Lines<\/h3>\n<p id=\"fs-id2365867\">Suppose for example, we are given the equation shown.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=3x+1[\/latex]<\/p>\n<p id=\"fs-id2365903\">We know that the slope of the line formed by the function is 3. We also know that the <em data-effect=\"italics\">y-<\/em>intercept is\u00a0[latex](0, 1).[\/latex] Any other line with a slope of 3 will be parallel to\u00a0[latex]f(x).[\/latex] So the lines formed by all of the following functions will be parallel to [latex]f(x).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} g(x) &=& 3x+6 \\\\ h(x) &=& 3x+1 \\\\ p(x) &=& 3x+\\frac{2}{3} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id2352637\">Suppose then we want to write the equation of a line that is parallel to\u00a0[latex]f[\/latex] and passes through the point\u00a0[latex](1, 7).[\/latex] This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of\u00a0[latex]b[\/latex] will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} y-y_1 &=& m(x-x_1) \\\\ y-7 &=& 3(x-1) \\\\ y-1 &=& 3x-3 \\\\ y &=& 3x+4 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1801804\">So\u00a0[latex]g(x)=3x+4[\/latex] is parallel to\u00a0[latex]f(x)=3x+1[\/latex] and passes through the point [latex](1, 7).[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point.<\/strong><\/p>\n<ol>\n<li>Find the slope of the function.<\/li>\n<li>Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 18: Finding a Line Parallel to a Given Line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a line parallel to the graph of\u00a0[latex]f(x)=3x+6[\/latex] that passes through the point [latex](3, 0).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The slope of a given line if 3. If we choose the slope-intercept form, we can substitute\u00a0[latex]m=3, x=3,[\/latex] and\u00a0[latex]f(x)=0[\/latex] into the slope-intercept form to find the <em>y-<\/em>intercept.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} g(x) &=& 3x+b \\\\ 0 &=& 3(3)+b \\\\ b &=& -9 \\end{array}[\/latex]<\/p>\n<p>The line parallel to\u00a0[latex]f(x)[\/latex] that passes through\u00a0[latex](3, 0)[\/latex] is [latex]g(x)=3x-9.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We can confirm that the two lines are parallel by graphing then. Figure 27 shows that the two lines will never intersect.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_679\" aria-describedby=\"caption-attachment-679\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-679\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27-350x351.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-27.jpeg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-679\" class=\"wp-caption-text\">Figure 27<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2249992\" data-depth=\"2\">\n<h3 data-type=\"title\">Writing Equations of Perpendicular Lines<\/h3>\n<p id=\"fs-id2249998\">We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=2x+4[\/latex]<\/p>\n<p id=\"fs-id2496718\">The slope of the line is 2, and its negative reciprocal is\u00a0[latex]-\\frac{1}{2}.[\/latex] Any function with a slope of\u00a0[latex]-\\frac{1}{2}[\/latex] will be perpendicular to\u00a0[latex]f(x).[\/latex] So the lines formed by all of the following functions will be perpendicular to [latex]f(x).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} g(x) &=& -\\frac{1}{2}x+4 \\\\ h(x) &=& -\\frac{1}{2}x+2 \\\\ p(x) &=& -\\frac{1}{2}x-\\frac{1}{2} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id2406643\">As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to\u00a0[latex]f(x)[\/latex] and passes through the point\u00a0[latex](4, 0).[\/latex] We already know that the slope is\u00a0[latex]-\\frac{1}{2}.[\/latex] Now we can use the point to find the <em data-effect=\"italics\">y<\/em>-intercept by substituting the given values into the slope-intercept form of a line and solving for [latex]b.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} g(x) &=& mx+b \\\\ 0 &=& -\\frac{1}{2}(4)+b \\\\ 0 &=& -2+b \\\\ 2 &=& b \\\\ b &=& 2 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id2253685\">The equation for the function with a slope of\u00a0[latex]-\\frac{1}{2}[\/latex] and a <em data-effect=\"italics\">y-<\/em>intercept of 2 is<\/p>\n<p style=\"text-align: center;\">[latex]g(x)=-\\frac{1}{2}x+2[\/latex]<\/p>\n<p id=\"fs-id2270676\">So\u00a0[latex]g(x)=-\\frac{1}{2}x+2[\/latex] is perpendicular to\u00a0[latex]f(x)=2x+4[\/latex] and passes through the point\u00a0[latex](4, 0).[\/latex] Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular, but the product of their slopes is not \u20131. Doesn\u2019t this fact contradict the definition of perpendicular lines?<\/strong><\/p>\n<p>A: <em data-effect=\"italics\">No. For two perpendicular linear functions, the product of their slopes is\u00a0[latex]-1.[\/latex] However, a vertical line is not a function so the definition is not contradicted.<\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the equation of a function and a point through which its graph passes, write the equation of a line perpendicular to the given line.<\/strong><\/p>\n<ol>\n<li>Find the slope of the function.<\/li>\n<li>Determine the negative reciprocal of the slope.<\/li>\n<li>Substitute the new slope and the values for\u00a0[latex]x[\/latex] and\u00a0[latex]y[\/latex] from the coordinate pair provided into [latex]g(x)=mx+b.[\/latex]<\/li>\n<li>Solve for [latex]b.[\/latex]<\/li>\n<li>Write the equation of the line.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 19: Finding the Equation of a Perpendicular Line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the equation of a line perpendicular to\u00a0[latex]f(x)=3x+3[\/latex] that passes through the point [latex](3, 0).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The original line has slope\u00a0[latex]m=3,[\/latex] so the slope of the perpendicular line will be its negative reciprocal, or\u00a0[latex]-\\frac{1}{3}.[\/latex] Using this slope and the given point, we can find the equation of the line.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} g(x) &=& -\\frac{1}{3}x+b \\\\ 0 &=& -\\frac{1}{3}(3)+b \\\\ 1 &=& b \\\\ b &=& 1 \\end{array}[\/latex]<\/p>\n<p>The line perpendicular to\u00a0[latex]f(x)[\/latex] that passes through\u00a0[latex](3, 0)[\/latex] is [latex]g(x)=-\\frac{1}{3}x+1.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>A graph of the two lines in shown in Figure 28.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_680\" aria-describedby=\"caption-attachment-680\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-680\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28-300x261.jpeg\" alt=\"\" width=\"300\" height=\"261\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28-300x261.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28-65x57.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28-225x196.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28-350x305.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-28.jpeg 427w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-680\" class=\"wp-caption-text\">Figure 28<\/figcaption><\/figure>\n<p>Note that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"Example_04_01_20\" class=\"ui-has-child-title\" data-type=\"example\">\n<section>\n<div class=\"body\">\n<div id=\"fs-id2262199\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1348623\" data-type=\"commentary\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function\u00a0[latex]h(x)=2x-4,[\/latex] write an equation for the line passing through\u00a0[latex](0,0)[\/latex] that is<\/p>\n<p>(a) parallel to [latex]h(x)[\/latex]<\/p>\n<p>(b) perpendicular to [latex]h(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given two points on a line and a third point, write the equation of the perpendicular line that passes through the point.<\/strong><\/p>\n<ol>\n<li>Determine the slope of the line passing through the points.<\/li>\n<li>Find the negative reciprocal of the slope.<\/li>\n<li>Use the slope-intercept form or point-slope form to write the equation by substituting the known values.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20: Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A line passes through the points\u00a0[latex](-2, 6)[\/latex] and\u00a0[latex](4, 5).[\/latex] Find the equation of a perpendicular line that passes through the point [latex](4, 5).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>From the two points of the given line, we can calculate the slope of that line.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} m_1 &=& -\\frac{5-6}{4-(-2)} \\\\ &=& -\\frac{-1}{6} \\\\ &=& -\\frac{1}{6} \\end{array}[\/latex]<\/p>\n<p>Find the negative reciprocal of the slope.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} m_2 &=& -\\frac{-1}{-\\frac{1}{6}} \\\\ &=& -1\\left(-\\frac{6}{1}\\right) \\\\ &=& 6 \\end{array}[\/latex]<\/p>\n<p>We can then solve for the y-intercept of the line passing through the point [latex](4, 5).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} g(x) &=& 6x+b \\\\ 5 &=& 6(4)+b \\\\ 5 &=& 24+b \\\\ -19 &=& b \\\\ b &=& -19 \\end{array}[\/latex]<\/p>\n<p>The equation for the line that is perpendicular to the line passing through the two given points and also passes through the point\u00a0[latex](4, 5)[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]y=6x-19[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A line passes through the points\u00a0[latex](-2, -15)[\/latex] and\u00a0[latex](2, -3).[\/latex] Find the equation of a perpendicular line that passes through the point, [latex](6, 4).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access this online resource for additional instruction and practice with linear functions.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=mzfmuJVI-HA\">Linear Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=_tTdoR1E5fg\">Finding Input of Function from the Output and Graph<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=8KLDGlrjzaw\">Graphing Functions Using Tables<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">4.1 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1799811\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1799815\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1799820\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1799822\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1799820-solution\">1<\/a><span class=\"os-divider\">. <\/span>Terry is skiing down Copper Mountain. Terry&#8217;s elevation,\u00a0[latex]E(t),[\/latex] in feet after\u00a0[latex]t[\/latex] seconds is given by\u00a0[latex]E(t)=3000-70t.[\/latex] Write a complete sentence describing Terry\u2019s starting elevation and how it is changing over time.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2216172\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2216173\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>Jessica is walking home from a friend\u2019s house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2216181\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2216182\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2216181-solution\">3<\/a><span class=\"os-divider\">. <\/span>A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour. Write an equation for the distance of the boat from the reservoir after [latex]t[\/latex] hours.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2216226\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2216227\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the <em data-effect=\"italics\">y<\/em>-intercepts.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2216236\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2216238\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2216236-solution\">5<\/a><span class=\"os-divider\">. <\/span>If a horizontal line has the equation\u00a0[latex]f(x)=a[\/latex] and a vertical line has the equation\u00a0[latex]x=a,[\/latex] what is the point of intersection? Explain why what you found is the point of intersection.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2270562\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id2270567\">For the following exercises, determine whether the equation of the curve can be written as a linear function.<\/p>\n<div id=\"fs-id1798659\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798660\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]y=\\frac{1}{4}x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1798691\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798693\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1798691-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]y=3x-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2496994\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798723\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]y=3x^2-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1798754\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798755\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1798754-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]3x+5y=15[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1798785\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798786\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]3x^2+5y=15[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2295720\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2295721\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2295720-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]3x+5y^2=15[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2295757\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2295758\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]-2x^2+3y^2=6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2295801\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2295802\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2295801-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]-\\frac{x-3}{5}=2y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2295841\">For the following exercises, determine whether each function is increasing or decreasing.<\/p>\n<div id=\"fs-id2295845\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2295846\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1855485\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1855486\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1855485-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=5x+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1855521\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1855522\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]a(x)=5-2x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1855555\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1855556\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1855555-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]b(x)=8-3x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1855591\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1855592\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=-2x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239367\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239368\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239367-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]k(x)=-4x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239406\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239408\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]j(x)=\\frac{1}{2}x-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239447\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239448\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239447-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]p(x)=\\frac{1}{4}x-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239492\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239493\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]n(x)=-\\frac{1}{3}x-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2083727\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2083729\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2083727-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]m(x)=-\\frac{3}{8}x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2083774\">For the following exercises, find the slope of the line that passes through the two given points.<\/p>\n<div id=\"fs-id2083777\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2083778\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex](2, 4) \\ \\ \\text{and} \\ \\ (4, 10)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2083836\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2083837\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2083836-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex](1, 5) \\ \\ \\text{and} \\ \\ (4, 11)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2597471\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2597472\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex](-1, 4) \\ \\ \\text{and} \\ \\ (5, 2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2597529\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2597530\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2597529-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex](8, -2) \\ \\ \\text{and} \\ \\ (4, 6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2597596\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2597597\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex](6, 11) \\ \\ \\text{and} \\ \\ (-4, 3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2498442\">For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.<\/p>\n<div id=\"fs-id2498446\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2498447\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2498446-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]f(-5)=-4, \\ \\ \\text{and} \\ \\ f(5)=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2498547\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2498548\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]f(-1)=4, \\ \\ \\text{and} \\ \\ f(5)=1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2653161\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2653162\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2653161-solution\">31<\/a><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex](2, 4) \\ \\ \\text{and} \\ \\ (4, 10)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2653249\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2653250\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex](1, 5) \\ \\ \\text{and} \\ \\ (4, 11)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2282224\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2282225\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2282224-solution\">33<\/a><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex](-1, 4) \\ \\ \\text{and} \\ \\ (5, 2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2282326\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2282327\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Passes through\u00a0[latex](-2, 8) \\ \\ \\text{and} \\ \\ (4, 6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2604219\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2604220\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2604219-solution\">35<\/a><span class=\"os-divider\">. <\/span><em data-effect=\"italics\">x<\/em> intercept at\u00a0[latex](02, 0)[\/latex] and <em data-effect=\"italics\">y<\/em> intercept at [latex](0, -3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2604318\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2604319\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><em data-effect=\"italics\">x<\/em> intercept at\u00a0[latex](-5, 0)[\/latex] and <em data-effect=\"italics\">y<\/em> intercept at [latex](0, 4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2252349\">For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.<\/p>\n<div id=\"fs-id2252353\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2252354\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2252353-solution\">37<\/a><span class=\"os-divider\">.<\/span> [latex]\\begin{array}{rcl} 4x-7y=10 \\\\ 7x+4y=1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2252424\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2252425\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span> [latex]\\begin{array}{rcl} 3y+x=12 \\\\ -y=8x+1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2261858\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261859\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2261858-solution\">39<\/a><span class=\"os-divider\">.<\/span> [latex]\\begin{array}{rcl} 3y+4x=12 \\\\ -6y=8x+1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2261930\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261932\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span> [latex]\\begin{array}{rcl} 6x-9y=10 \\\\ 3x+2y=1 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2445672\">For the following exercises, find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y-<\/em>intercepts of each equation.<\/p>\n<div id=\"fs-id2445686\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2445687\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2445686-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2583719\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2250078\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=2x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2250111\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2250112\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2250111-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=3x-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2479676\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2479677\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]k(x)=-5x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2479711\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2479712\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2479711-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]-2x+5y=20[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239181\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239182\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]7x+2y=56[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2239207\">For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?<\/p>\n<div id=\"fs-id2239211\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239214\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239211-solution\">47<\/a><span class=\"os-divider\">. <\/span>Line 1: Passes through [latex](0, 6) \\ \\ \\text{and} \\ \\ (3, -24)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2239270\">Line 2: Passes through\u00a0[latex](-1, 19) \\ \\ \\text{and} \\ \\ (8, -71)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1828374\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1828375\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex](-8, -55) \\ \\ \\text{and} \\ \\ (10, 89)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1828438\">Line 2: Passes through\u00a0[latex](9, -44) \\ \\ \\text{and} \\ \\ (4, -14)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1828498\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1828499\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1828498-solution\">49<\/a><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex](2, 3)\\ \\ \\text{and} \\ \\ (4, -1)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1797495\">Line 2: Passes through\u00a0[latex](6, 3) \\ \\ \\text{and} \\ \\ (8, 5)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1797562\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1797563\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex](1, 7) \\ \\ \\text{and} \\ \\ (5, 5)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1797619\">Line 2: Passes through\u00a0[latex](-1, -3) \\ \\ \\text{and} \\ \\ (1, 1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1797676\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1797677\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1797676-solution\">51<\/a><span class=\"os-divider\">. <\/span>Line 1: Passes through\u00a0[latex](2, 5) \\ \\ \\text{and} \\ \\ (5, -1)[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2352442\">Line 2: Passes through\u00a0 [latex](-3, 7) \\ \\ \\text{and} \\ \\ (3, -5)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2352555\">For the following exercises, write an equation for the line described.<\/p>\n<div id=\"fs-id2352558\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2352559\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Write an equation for a line parallel to\u00a0[latex]f(x)=-5x-3[\/latex] and passing through the point [latex](2, -12).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2406979\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2406980\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2406979-solution\">53<\/a><span class=\"os-divider\">. <\/span>Write an equation for a line parallel to\u00a0[latex]g(x)=3x-1[\/latex] and passing through the point [latex](4, 9).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2407070\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2407071\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Write an equation for a line perpendicular to\u00a0[latex]h(t)=-2t+4[\/latex] and passing through the point [latex](-4, -1).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2407142\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2407143\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2407142-solution\">55<\/a><span class=\"os-divider\">. <\/span>Write an equation for a line perpendicular to\u00a0[latex]p(t)=3t+4[\/latex] and passing through the point [latex](3, 1).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2261328\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id2261333\">For the following exercises, find the slope of the line\u00a0graphed.<\/p>\n<div id=\"fs-id2261337\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261338\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261344\" class=\"first-element\" data-type=\"media\" data-alt=\"This is a graph of a decreasing linear function on an x, y coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line passes through points (0, 5) and (4, 0).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-681\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56-350x351.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.56.jpeg 437w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2261358\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261359\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2261358-solution\">57<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261364\" class=\"first-element\" data-type=\"media\" data-alt=\"This is a graph of a function on an x, y coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The lines passes through points at (0, -2) and (2, -2).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-682\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-225x225.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57-350x350.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.57.jpeg 437w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2261383\">For the following exercises, write an equation for the line graphed.<\/p>\n<div id=\"fs-id2261386\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261387\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261392\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an increasing linear function with points at (0,1) and (3,3)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-683\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58-350x351.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.58.jpeg 437w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2261404\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261406\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2261404-solution\">59<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261411\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a decreasing linear function with points (0,5) and (4,0)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-684\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-225x225.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59-350x350.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.59.jpeg 438w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2261453\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261454\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2261459\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a decreasing linear function with points at (0,3) and (1.5,0)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-685\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60-225x224.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.60.jpeg 313w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2605036\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2605037\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2605036-solution\">61<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2605042\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an increasing linear function with points at (1,2) and (0,-2)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-686\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-225x225.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61-350x350.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.61.jpeg 438w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2605078\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2605079\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2605084\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function with points at (0, 3) and (3, 3)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-687\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62-225x224.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.62.jpeg 439w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2605097\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2605098\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2605097-solution\">63<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2605103\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function with points at (0,-2.5) and (-2.5,-2.5)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-688\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-300x298.jpeg\" alt=\"\" width=\"300\" height=\"298\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-300x298.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-225x223.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63-350x348.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.63.jpeg 440w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2605134\">For the following exercises, match the given linear equation with its graph in Figure 29.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_689\" aria-describedby=\"caption-attachment-689\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-689\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29-300x296.jpeg\" alt=\"\" width=\"300\" height=\"296\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29-300x296.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29-65x64.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29-225x222.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29-350x345.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1-fig-29.jpeg 425w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-689\" class=\"wp-caption-text\">Figure 29<\/figcaption><\/figure>\n<div id=\"fs-id2605162\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2605163\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2605195\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2605196\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2605195-solution\">65<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-3x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2605235\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2385913\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-\\frac{1}{2}x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2354805\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2354806\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2354805-solution\">67<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2354836\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2354837\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2+x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2354867\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2354868\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2354867-solution\">69<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2354905\">For the following exercises, sketch a line with the given features.<\/p>\n<div id=\"fs-id2354908\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2354909\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>An <em data-effect=\"italics\">x<\/em>-intercept of\u00a0[latex](-4, 0)[\/latex] and <em data-effect=\"italics\">y<\/em>-intercept of [latex](0, -2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2354975\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2354976\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2354975-solution\">71<\/a><span class=\"os-divider\">. <\/span>An <em data-effect=\"italics\">x<\/em>-intercept of\u00a0[latex](-2, 0)[\/latex] and <em data-effect=\"italics\">y<\/em>-intercept of [latex](0, 4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2393349\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393350\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>A <em data-effect=\"italics\">y<\/em>-intercept of\u00a0[latex](0, 7)[\/latex] and slope [latex]-\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2393405\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393406\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2393405-solution\">73<\/a><span class=\"os-divider\">. <\/span>A <em data-effect=\"italics\">y<\/em>-intercept of\u00a0[latex](0, 3)[\/latex] and slope [latex]\\frac{2}{5}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2393478\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393480\" data-type=\"problem\">\n<p><span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>Passing through the points\u00a0[latex](-6, -2)[\/latex] and [latex](6, -6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239623\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239624\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239623-solution\">75<\/a><span class=\"os-divider\">. <\/span>Passing through the points\u00a0[latex](-3, -4)[\/latex] and [latex](3, 0)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2239701\">For the following exercises, sketch the graph of each equation.<\/p>\n<div id=\"fs-id2239704\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239705\" data-type=\"problem\">\n<p><span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-2x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239740\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239742\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239740-solution\">77<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-3x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239796\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239797\" data-type=\"problem\">\n<p><span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{1}{3}x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2239837\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2239838\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2239837-solution\">79<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{2}{3}x-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2447392\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2447393\" data-type=\"problem\">\n<p><span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span> [latex]f(t)=3+2t[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2447425\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2447426\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2447425-solution\">81<\/a><span class=\"os-divider\">. <\/span> [latex]p(t)=-2+3t[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2447481\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2447482\" data-type=\"problem\">\n<p><span class=\"os-number\">82<\/span><span class=\"os-divider\">. <\/span> [latex]x=3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2447499\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2447500\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2447499-solution\">83<\/a><span class=\"os-divider\">. <\/span> [latex]x=-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2447539\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2447540\" data-type=\"problem\">\n<p><span class=\"os-number\">84<\/span><span class=\"os-divider\">. <\/span> [latex]r(x)=4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2447566\">For the following exercises, write the equation of the line shown in the graph.<\/p>\n<div id=\"fs-id2447569\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2447570\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2447569-solution\">85<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979120\" class=\"first-element\" data-type=\"media\" data-alt=\"The graph of a line with a slope of 0 and y-intercept at 3.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-690\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.85.jpeg 436w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1979148\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1979149\" data-type=\"problem\">\n<p><span class=\"os-number\">86<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979154\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a line with a slope of 0 and y-intercept at -1.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-691\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-225x225.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86-350x350.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.86.jpeg 438w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1979167\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1979168\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1979167-solution\">87<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979173\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a line with an undefined slope and x-intercept at -3.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-692\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-225x225.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87-350x350.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.87.jpeg 438w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1979203\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1965935\" data-type=\"problem\">\n<p><span class=\"os-number\">88<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1979208\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a line with an undefined slope and x-intercept at 2\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-693\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/4.1.88.jpeg 314w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1979221\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1979227\">For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.<\/p>\n<div id=\"fs-id1979231\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1979233\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1979231-solution\">89<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_04\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"height: 35px;\" data-id=\"Table_04_01_04\">\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"height: 17px; width: 180.017px;\" data-align=\"center\">[latex]x[\/latex]<\/td>\n<td style=\"height: 17px; width: 68.9167px;\" data-align=\"center\">0<\/td>\n<td style=\"height: 17px; width: 118.8px;\" data-align=\"center\">5<\/td>\n<td style=\"height: 17px; width: 123.733px;\" data-align=\"center\">10<\/td>\n<td style=\"height: 17px; width: 131.2px;\" data-align=\"center\">15<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 180.017px;\" data-align=\"center\">[latex]g(x)[\/latex]<\/td>\n<td style=\"height: 18px; width: 68.9167px;\" data-align=\"center\">5<\/td>\n<td style=\"height: 18px; width: 118.8px;\" data-align=\"center\">\u201310<\/td>\n<td style=\"height: 18px; width: 123.733px;\" data-align=\"center\">\u201325<\/td>\n<td style=\"height: 18px; width: 131.2px;\" data-align=\"center\">\u201340<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1798819\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798821\" data-type=\"problem\">\n<p><span class=\"os-number\">90<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_05\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"height: 35px;\" data-id=\"Table_04_01_05\">\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"height: 17px; width: 186.133px;\" data-align=\"center\">[latex]x[\/latex]<\/td>\n<td style=\"height: 17px; width: 72.8833px;\" data-align=\"center\">0<\/td>\n<td style=\"height: 17px; width: 104.3px;\" data-align=\"center\">5<\/td>\n<td style=\"height: 17px; width: 124.15px;\" data-align=\"center\">10<\/td>\n<td style=\"height: 17px; width: 135.2px;\" data-align=\"center\">15<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 186.133px;\" data-align=\"center\">[latex]h(x)[\/latex]<\/td>\n<td style=\"height: 18px; width: 72.8833px;\" data-align=\"center\">5<\/td>\n<td style=\"height: 18px; width: 104.3px;\" data-align=\"center\">30<\/td>\n<td style=\"height: 18px; width: 124.15px;\" data-align=\"center\">105<\/td>\n<td style=\"height: 18px; width: 135.2px;\" data-align=\"center\">230<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1798936\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1798937\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1798936-solution\">91<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_06\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"Table_04_01_06\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">10<\/td>\n<td data-align=\"center\">15<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)[\/latex]<\/td>\n<td data-align=\"center\">\u20135<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">45<\/td>\n<td data-align=\"center\">70<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1799092\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1799093\" data-type=\"problem\">\n<p><span class=\"os-number\">92<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_07\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"Table_04_01_07\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">10<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">25<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]k(x)[\/latex]<\/td>\n<td data-align=\"center\">13<\/td>\n<td data-align=\"center\">28<\/td>\n<td data-align=\"center\">58<\/td>\n<td data-align=\"center\">73<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1828666\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2261900\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id1828666-solution\">93<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_08\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"Table_04_01_08\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">6<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]g(x)[\/latex]<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">\u201319<\/td>\n<td data-align=\"center\">\u201344<\/td>\n<td data-align=\"center\">\u201369<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1828835\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1828836\" data-type=\"problem\">\n<p><span class=\"os-number\">94<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_09\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"Table_04_01_09\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">10<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]h(x)[\/latex]<\/td>\n<td data-align=\"center\">13<\/td>\n<td data-align=\"center\">23<\/td>\n<td data-align=\"center\">43<\/td>\n<td data-align=\"center\">53<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2654316\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2654319\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2654316-solution\">95<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_10\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"Table_04_01_10\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)[\/latex]<\/td>\n<td data-align=\"center\">\u20134<\/td>\n<td data-align=\"center\">16<\/td>\n<td data-align=\"center\">36<\/td>\n<td data-align=\"center\">56<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2654475\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2654476\" data-type=\"problem\">\n<p><span class=\"os-number\">96<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"Table_04_01_11\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"Table_04_01_11\">\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]k(x)[\/latex]<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">106<\/td>\n<td data-align=\"center\">231<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2353110\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id2353115\">For the following exercises, use a calculator or graphing technology to complete the task.<\/p>\n<div id=\"fs-id2353120\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2353121\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2353120-solution\">97<\/a><span class=\"os-divider\">. <\/span>If\u00a0[latex]f[\/latex] is a linear function,\u00a0[latex]f(0.1)=11.5,[\/latex] and\u00a0[latex]f(0.4-5.9,[\/latex] find an equation for the function.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2353233\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2353234\" data-type=\"problem\">\n<p><span class=\"os-number\">98<\/span><span class=\"os-divider\">. <\/span>Graph the function\u00a0[latex]f[\/latex] on a domain of\u00a0[latex][-10, 10]: f(x)=0.02x-0.01.[\/latex] Enter the function in a graphing utility. For the viewing window, set the minimum value of\u00a0[latex]x[\/latex] to be\u00a0[latex]-10[\/latex] and the maximum value of\u00a0[latex]x[\/latex] to be [latex]10.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2241250\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2241251\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2241250-solution\">99<\/a><span class=\"os-divider\">. <\/span>Graph the function\u00a0[latex]f[\/latex] on a domain of [latex][-10, 10]: f(x)=2,500x+4,000[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2241349\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2241350\" data-type=\"problem\">\n<p><span class=\"os-number\">100<\/span><span class=\"os-divider\">. <\/span>Table 3 shows the input,\u00a0[latex]w,[\/latex] and output,\u00a0[latex]k,[\/latex] for a linear function [latex]k.[\/latex]<\/p>\n<p>(a) Fill in the missing values of the table.<\/p>\n<p>(b) Write the linear function\u00a0[latex]k,[\/latex] round to 3 decimal places.<\/p>\n<div class=\"os-problem-container\">\n<div id=\"Table_04_01_12\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_01_12\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]w[\/latex]<\/td>\n<td data-align=\"center\">\u201310<\/td>\n<td data-align=\"center\">5.5<\/td>\n<td data-align=\"center\">67.5<\/td>\n<td data-align=\"center\"><em data-effect=\"italics\">b<\/em><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]k[\/latex]<\/td>\n<td data-align=\"center\">30<\/td>\n<td data-align=\"center\">\u201326<\/td>\n<td data-align=\"center\"><em data-effect=\"italics\">a<\/em><\/td>\n<td data-align=\"center\">\u201344<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2633779\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2633779-solution\">101<\/a><span class=\"os-divider\">. <\/span>Table 4 shows the input,\u00a0[latex]p,[\/latex] and output,\u00a0[latex]q,[\/latex] for a linear function [latex]q.[\/latex]<br \/>\n(a) Fill in the missing values of the table.<br \/>\n(b) Write the linear function [latex]q.[\/latex]<\/p>\n<div id=\"fs-id2633780\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div id=\"Table_04_01_13\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_01_13\">\n<caption>Table 4<\/caption>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]p[\/latex]<\/td>\n<td data-align=\"center\">0.5<\/td>\n<td data-align=\"center\">0.8<\/td>\n<td data-align=\"center\">12<\/td>\n<td data-align=\"center\"><em data-effect=\"italics\">b<\/em><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]q[\/latex]<\/td>\n<td data-align=\"center\">400<\/td>\n<td data-align=\"center\">700<\/td>\n<td data-align=\"center\"><em data-effect=\"italics\">a<\/em><\/td>\n<td data-align=\"center\">1,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id2633952\" data-type=\"exercise\">\n<header><span class=\"os-number\">102<\/span><span class=\"os-divider\">. <\/span>Graph the linear function\u00a0[latex]f[\/latex] on a domain of\u00a0[latex][-10, 10][\/latex] for the function whose slope is\u00a0[latex]\\frac{1}{8}[\/latex] and <em data-effect=\"italics\">y<\/em>-intercept is\u00a0[latex]\\frac{31}{16}.[\/latex] Label the points for the input values of\u00a0[latex]-10[\/latex] and [latex]10.[\/latex]<\/header>\n<\/div>\n<div id=\"fs-id2511972\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2511973\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2511972-solution\">103<\/a><span class=\"os-divider\">. <\/span>Graph the linear function\u00a0[latex]f[\/latex] on a domain of\u00a0[latex][-0.1, 0.1][\/latex] for the function whose slope is 75 and <em data-effect=\"italics\">y<\/em>-intercept is\u00a0[latex]-22.5.[\/latex] Label the points for the input values of\u00a0[latex]-0.1[\/latex] and [latex]0.1.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2512070\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2512071\" data-type=\"problem\">\n<p><span class=\"os-number\">104<\/span><span class=\"os-divider\">. <\/span>Graph the linear function\u00a0[latex]f[\/latex] where\u00a0[latex]f(x)=ax+b[\/latex] on the same set of axes on a domain of\u00a0[latex][-4, 4][\/latex] for the following values of\u00a0[latex]a[\/latex] and [latex]b[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(a) [latex]a=2; b=3[\/latex]<\/p>\n<p>(b) [latex]a=2; b=4[\/latex]<\/p>\n<p>(c) [latex]a=2; b=-4[\/latex]<\/p>\n<p>(d) [latex]a=2; b=-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id2266929\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id2266935\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2266936\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2266935-solution\">105<\/a><span class=\"os-divider\">. <\/span>Find the value of\u00a0[latex]x[\/latex] if a linear function goes through the following points and has the following slope: [latex](x, 2), (-4, 6), m=3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2267009\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2267010\" data-type=\"problem\">\n<p><span class=\"os-number\">106<\/span><span class=\"os-divider\">. <\/span>Find the value of [latex]y[\/latex] if a linear function goes through the following points and has the following slope: [latex](10, y), (25, 100), m=-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2267082\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2267083\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2267082-solution\">107<\/a><span class=\"os-divider\">. <\/span>Find the equation of the line that passes through the following points:\u00a0[latex](a, b) \\ \\ \\text{and} \\ \\ (a, b+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2267152\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2267153\" data-type=\"problem\">\n<p><span class=\"os-number\">108<\/span><span class=\"os-divider\">. <\/span>Find the equation of the line that passes through the following points:\u00a0[latex](2a, b) \\ \\ \\text{and} \\ \\ (a, b+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2762578\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2762579\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2762578-solution\">109<\/a><span class=\"os-divider\">. <\/span>Find the equation of the line that passes through the following points:\u00a0[latex](a, 0) \\ \\ \\text{and} \\ \\ (c, d)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2762640\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2762641\" data-type=\"problem\">\n<p><span class=\"os-number\">110. <\/span>Find the equation of the line parallel to the line\u00a0[latex]g(x)=-0.01x+2.01[\/latex] through the point [latex](1, 2).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2762721\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2762722\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2762721-solution\">111<\/a><span class=\"os-divider\">. <\/span>Find the equation of the line perpendicular to the line\u00a0[latex]g(x)=-0.01x+2.01[\/latex] through the point [latex](1, 2).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2762790\">For the following exercises, use the functions\u00a0[latex]f(x)=-0.1x+200[\/latex] and [latex]g(x)=20x+0.1.[\/latex]<\/p>\n<div id=\"fs-id2393555\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393556\" data-type=\"problem\">\n<p><span class=\"os-number\">112<\/span><span class=\"os-divider\">. <\/span>Find the point of intersection of the lines\u00a0[latex]f[\/latex] and [latex]g[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2393649\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393650\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2393649-solution\">113<\/a><span class=\"os-divider\">. <\/span>Where is\u00a0[latex]f(x)[\/latex] greater than\u00a0[latex]g(x)?[\/latex] Where is [latex]g(x)[\/latex] greater than [latex]f(x)?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2393754\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id2393760\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393761\" data-type=\"problem\">\n<p><span class=\"os-number\">114<\/span><span class=\"os-divider\">. <\/span>At noon, a barista notices that they have $20 in their tip jar. If the barista makes an average of $0.50 from each customer, how much will they have in the tip jar if they serve\u00a0[latex]n[\/latex] more customers during the shift?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2393799\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393800\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2393799-solution\">115<\/a><span class=\"os-divider\">. <\/span>At VASA fitness in Aurora, CO, a gym membership with two personal training sessions costs $125, while gym membership with five personal training sessions costs $260. What is cost per session?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2393805\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2393806\" data-type=\"problem\">\n<p><span class=\"os-number\">116<\/span><span class=\"os-divider\">. <\/span>Aspen Blue Boutique in Centennial, CO, finds there is a linear relationship between the number of shirts,\u00a0[latex]n,[\/latex] it can sell and the price,\u00a0[latex]p,[\/latex] it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of\u00a0[latex]\\$30[\/latex] while 3,000 shirts can be sold at a price of $22. Find a linear equation in the form\u00a0[latex]p(n)=mn+b[\/latex] that gives the price\u00a0[latex]p[\/latex] they can charge for\u00a0 shirts.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2249337\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2249338\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2249337-solution\">117<\/a><span class=\"os-divider\">. <\/span>T-Mobile charges for service according to the formula:\u00a0[latex]C(n)=24+0.1n,[\/latex] where\u00a0[latex]n[\/latex] is the number of minutes talked, and\u00a0[latex]C(n)[\/latex] is the monthly charge, in dollars. Find and interpret the rate of change and initial value.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2249416\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2249417\" data-type=\"problem\">\n<p><span class=\"os-number\">118<\/span><span class=\"os-divider\">. <\/span>A farmer finds there is a linear relationship between the number of bean stalks,\u00a0[latex]n,[\/latex] she plants and the yield,\u00a0[latex]y,[\/latex] each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form\u00a0[latex]y=mn+b[\/latex] that gives the yield when\u00a0[latex]n[\/latex] stalks are planted.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2249513\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2249514\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2249513-solution\">119<\/a><span class=\"os-divider\">. <\/span> Colorado\u2019s population in the year 1980 was 2,889,735. In 2000, the population was 4,301,261. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2249522\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2249523\" data-type=\"problem\">\n<p><span class=\"os-number\">120<\/span><span class=\"os-divider\">. <\/span>A town\u2019s population has been growing linearly. In 2003, the population was 45,000, and the population has been growing by 1,700 people each year. Write an equation,\u00a0[latex]P(t),[\/latex] for the population\u00a0 years after 2003.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2631646\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2631648\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-4\" data-page-slug=\"chapter-4\" data-page-uuid=\"9b243f17-7ca7-5ccb-a050-461bbd5bf001\" data-page-fragment=\"fs-id2631646-solution\">121<\/a><span class=\"os-divider\">. <\/span>Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function:\u00a0[latex]I(x)=1054x+23,286,[\/latex] where\u00a0[latex]x[\/latex] is the number of years after 1990. Which of the following interprets the slope in the context of the problem?<\/p>\n<p>(a) As of 1990, average annual income was $23,286.<\/p>\n<p>(b) In the ten-year period from 1990\u20131999, average annual income increased by a total of $1,054.<\/p>\n<p>(c) Each year in the decade of the 1990s, average annual income increased by $1,054.<\/p>\n<p>(d) Average annual income rose to a level of $23,286 by the end of 1999.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2631728\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2631729\" data-type=\"problem\">\n<p><span class=\"os-number\">122<\/span><span class=\"os-divider\">. <\/span>When temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of\u00a0[latex]C,[\/latex] the Celsius temperature, [latex]F(C).[\/latex]<\/p>\n<p>(a) Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius.<\/p>\n<p>(b) Find and interpret [latex]F(28)[\/latex]<\/p>\n<p>(c) Find and interpret [latex]F(-40)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-290","chapter","type-chapter","status-publish","hentry"],"part":156,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/290","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":26,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/290\/revisions"}],"predecessor-version":[{"id":1509,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/290\/revisions\/1509"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/156"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/290\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=290"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=290"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=290"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}