{"id":136,"date":"2025-04-09T17:15:45","date_gmt":"2025-04-09T17:15:45","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-5-transformation-of-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-19T18:57:05","modified_gmt":"2025-08-19T18:57:05","slug":"3-5-transformation-of-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-5-transformation-of-functions\/","title":{"raw":"3.5 Transformation of Functions","rendered":"3.5 Transformation of 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_5f6ff02a-1000-410d-b034-af26fbd86d0b\" 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>Graph functions using vertical and horizontal shifts.<\/li>\r\n \t<li>Graph functions using reflections about the x-axis and the y-axis.<\/li>\r\n \t<li>Determine whether a function is even, odd, or neither from its graph.<\/li>\r\n \t<li>Graph functions using compressions and stretches.<\/li>\r\n \t<li>Combine transformations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_557\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-557\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1-300x200.jpeg\" alt=\"\" width=\"300\" height=\"200\" \/> Figure 1 (credit: \"Misko\"\/Flickr)[\/caption]\r\n<p id=\"fs-id1165137742090\">We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.<\/p>\r\n\r\n<section id=\"fs-id1165137827988\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Functions Using Vertical and Horizontal Shifts<\/h2>\r\n<p id=\"fs-id1165137654715\">Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.<\/p>\r\n\r\n<section id=\"fs-id1165137535664\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Identifying Vertical Shifts<\/h3>\r\n<p id=\"fs-id1165135503932\">One simple kind of <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">transformation<\/span> involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a <strong>vertical shift<\/strong>, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function\u00a0[latex] g(x)=f(x)+k, [\/latex] the function\u00a0[latex] f(x) [\/latex] is shifted vertically\u00a0[latex] k [\/latex] units. See <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_002\">Figure 2<\/a> for an example.<\/p>\r\n\r\n[caption id=\"attachment_558\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-558 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2-300x180.jpeg\" alt=\"\" width=\"300\" height=\"180\" \/> Figure 2 Vertical shift by\u00a0[latex] k=1 [\/latex] of the cube root function [latex] f(x)=\\sqrt[3]{x}. [\/latex][\/caption]\r\n<div id=\"Figure_01_05_002\" class=\"os-figure\"><\/div>\r\n<p id=\"fs-id1165137439125\">To help you visualize the concept of a vertical shift, consider that\u00a0[latex] y=f(x). [\/latex] Therefore,\u00a0[latex] f(x)+k [\/latex] is equivalent to\u00a0[latex] y+k. [\/latex] Every unit of\u00a0[latex] y [\/latex] is replaced by\u00a0[latex] y+k, [\/latex] so the <em data-effect=\"italics\">y<\/em>-value increases or decreases depending on the value of\u00a0[latex] k. [\/latex] The result is a shift upward or downward.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Vertical Shift<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven a function\u00a0[latex] f(x), [\/latex] a new function\u00a0[latex] g(x)=f(x)+k, [\/latex] where\u00a0[latex] k [\/latex] is a constant, is a <strong>vertical shift<\/strong> of the function\u00a0[latex] f(x). [\/latex] All the output values change by\u00a0[latex] k [\/latex] units. If\u00a0[latex] k [\/latex] is positive, the graph will shift up. If\u00a0[latex] k [\/latex] is negative, the graphs will shift down.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-note-body\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Adding a Constant to a Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nTo regulate temperature in CCA's classroom building, airflow vents near the roof open and close throughout the day. Figure 3 shows the area of open vents\u00a0[latex] V [\/latex] (in square feet) throughout the day in hours after midnight,\u00a0[latex] t. [\/latex] During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_559\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-559\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3-300x201.jpeg\" alt=\"\" width=\"300\" height=\"201\" \/> Figure 3[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_004\">Figure 4<\/a>.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_560\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-560\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4-300x203.jpeg\" alt=\"\" width=\"300\" height=\"203\" \/> Figure 4[\/caption]\r\n\r\nNotice that in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_004\">Figure 4<\/a>, for each input value, the output value has increased by 20, so if we call the new function\u00a0[latex] S(t), [\/latex] we could write\r\n<p style=\"text-align: center;\">[latex] S(t)=V(t)+20 [\/latex]<\/p>\r\nThis notation tells us that for any value of\u00a0[latex] t, S(t) [\/latex] can be found by evaluating the function\u00a0[latex] V [\/latex] at the same input and then adding 20 to the result. This defines\u00a0[latex] S [\/latex] as a transformation of the function\u00a0[latex] V, [\/latex] in this case a vertical shift up 20 unites. Notice that, with a vertical shift, the input values stay the same and only the output values change. See Table 1.\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: 14.2857%;\">[latex] t [\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">0<\/td>\r\n<td style=\"width: 14.2857%;\">8<\/td>\r\n<td style=\"width: 14.2857%;\">10<\/td>\r\n<td style=\"width: 14.2857%;\">17<\/td>\r\n<td style=\"width: 14.2857%;\">19<\/td>\r\n<td style=\"width: 14.2857%;\">24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">[latex] V(t) [\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">0<\/td>\r\n<td style=\"width: 14.2857%;\">0<\/td>\r\n<td style=\"width: 14.2857%;\">220<\/td>\r\n<td style=\"width: 14.2857%;\">220<\/td>\r\n<td style=\"width: 14.2857%;\">0<\/td>\r\n<td style=\"width: 14.2857%;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 14.2857%;\">[latex] S(t) [\/latex]<\/td>\r\n<td style=\"width: 14.2857%;\">20<\/td>\r\n<td style=\"width: 14.2857%;\">20<\/td>\r\n<td style=\"width: 14.2857%;\">240<\/td>\r\n<td style=\"width: 14.2857%;\">240<\/td>\r\n<td style=\"width: 14.2857%;\">20<\/td>\r\n<td style=\"width: 14.2857%;\">20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/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 a tabular function, create a new row to represent a vertical shift.<\/strong>\r\n<ol>\r\n \t<li>Identify the output row or column.<\/li>\r\n \t<li>Determine the magnitude of the shift.<\/li>\r\n \t<li>Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/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 2: Shifting a Tabular Function Vertically<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f(x) [\/latex] is given in Table 2. Create a table for the function [latex] g(x)=f(x)-3. [\/latex]\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: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">1<\/td>\r\n<td style=\"width: 20%;\">3<\/td>\r\n<td style=\"width: 20%;\">7<\/td>\r\n<td style=\"width: 20%;\">11<\/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>The formula\u00a0[latex] g(x)=f(x)-3 [\/latex] tells us that we can find the output values of\u00a0[latex] g [\/latex] by subtracting 3 from the output values of\u00a0[latex] f. [\/latex] For example:\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcll} f(2) &amp;=&amp; 1 &amp; \\text{Given} \\\\ g(x) &amp;=&amp; f(x)-3 &amp; \\text{Given transformation} \\\\ g(2) &amp;=&amp; f(2)-3 \\\\ &amp;=&amp; 1-3 \\\\ &amp;=&amp; -2 \\end{array} [\/latex]<\/p>\r\nSubtracting 3 from each\u00a0[latex] f(x) [\/latex] value, we can complete a table of values for\u00a0[latex] g(x) [\/latex] as shown in Table 3.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 51px;\" border=\"0\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">2<\/td>\r\n<td style=\"width: 20%; height: 17px;\">4<\/td>\r\n<td style=\"width: 20%; height: 17px;\">6<\/td>\r\n<td style=\"width: 20%; height: 17px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">1<\/td>\r\n<td style=\"width: 20%; height: 17px;\">3<\/td>\r\n<td style=\"width: 20%; height: 17px;\">7<\/td>\r\n<td style=\"width: 20%; height: 17px;\">11<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] g(x) [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">-2<\/td>\r\n<td style=\"width: 20%; height: 17px;\">0<\/td>\r\n<td style=\"width: 20%; height: 17px;\">4<\/td>\r\n<td style=\"width: 20%; height: 17px;\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Analysis<\/h3>\r\nAs with the earlier vertical shift, notice the input values stay the same and only the output values change.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_02\" class=\"ui-has-child-title\" data-type=\"example\"><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165135581153\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137737963\" data-type=\"commentary\">\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\nThe function\u00a0[latex] h(t)=-4.9t^2+30t [\/latex] gives the height\u00a0[latex] h [\/latex] of a ball (in meters) thrown upward from the ground after\u00a0[latex] t [\/latex] seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function\u00a0[latex] b(t) [\/latex] to\u00a0[latex] h(t) [\/latex] and then find a formula for [latex] b(t). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137597159\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Identifying Horizontal Shifts<\/h3>\r\n<p id=\"fs-id1165137404493\">We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a <strong>horizontal shift<\/strong>, shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_005\">Figure 5<\/a>.<\/p>\r\n\r\n[caption id=\"attachment_561\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-561 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5-300x177.jpeg\" alt=\"\" width=\"300\" height=\"177\" \/> Figure 5. Horizontal shift of the function [latex] f(x)=\\sqrt[3]{x}. [\/latex] Notice that [latex] (x+1) [\/latex] means [latex] h=-1, [\/latex] which shifts the graph to the left, towards <em>negative values of<\/em> [latex] x. [\/latex][\/caption]\r\n<p id=\"eip-884\">For example, if\u00a0[latex] f(x)=x^2, [\/latex] then\u00a0[latex] g(x)=(x-2)^2 [\/latex] is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in [latex] f. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Horizontal Shift<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven a function\u00a0[latex] f, [\/latex] a new function\u00a0[latex] g(x)=f(x-h), [\/latex] where\u00a0[latex] h [\/latex] is a constant, is a <strong>horizontal shift<\/strong> of the function\u00a0[latex] f. [\/latex] If\u00a0[latex] h [\/latex] is positive, the graph will shift right. If\u00a0[latex] h [\/latex] is negative, the graph will shift left.\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 3: Adding a Constant to an Input<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nReturning to our building airflow example from Figure 3, suppose that in Fall the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can set\u00a0[latex] V(t) [\/latex] to be the original program and\u00a0[latex] F(t) [\/latex] to be the revised program.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl} V(t) &amp;=&amp; \\text{the original venting plan} \\\\ F(t) &amp;=&amp; \\text{starting 2 hrs sooner} \\end{array} [\/latex]<\/p>\r\nIn the new graph, at each time, the airflow is the same as the original function\u00a0[latex] V [\/latex] was 2 hours later. For example, in the original function\u00a0[latex] V, [\/latex] the airflow starts to change at 8 a.m., whereas for the function\u00a0[latex] F, [\/latex] the airflow starts to change at 6 a.m. The comparable function values are\u00a0[latex] V(8)=F(6). [\/latex] See Figure 6. Notice also that the vents first opened to\u00a0[latex] 220\\hspace{0.5em}\\text{ft}^2 [\/latex] at 10 a.m. under the original plan, while under the new plan the vents reach\u00a0[latex] 220\\hspace{0.5em}\\text{ft}^2 [\/latex] at 8 a.m., so [latex] V(10)=F(8). [\/latex]\r\n\r\nIn both cases, we see that, because\u00a0[latex] F(t) [\/latex] starts 2 hours sooner,\u00a0[latex] h=-2. [\/latex] That means that the same output values are reached when [latex] F(t)=V(t-(-2))=V(t+2). [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_563\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-563\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6-300x203.jpeg\" alt=\"\" width=\"300\" height=\"203\" \/> Figure 6[\/caption]\r\n<h3>Analysis<\/h3>\r\nNote that\u00a0[latex] V(t+2) [\/latex] has the effect of shifting the graph to the <em>left.<\/em>\r\n\r\nHorizontal change or \u201cinside changes\u201d affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function [latex] F(t) [\/latex] uses the same outputs as [latex] V(t), [\/latex] but matches those outputs to inputs 2 hours earlier than those of [latex] V(t). [\/latex] Said another way, we must add 2 hours to the input of [latex] V [\/latex] to find the corresponding output for [latex] F: F(t)=V(t+2). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\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 tabular function, create a new row to represent a horizontal shift.<\/strong>\r\n<ol>\r\n \t<li>Identify the input row or column.<\/li>\r\n \t<li>Determine the magnitude of the shift.<\/li>\r\n \t<li>Add the shift to the value of each input cell.<\/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 4: Shifting a Tabular Function Horizontally<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f(x) [\/latex] is given in Table 4. Create a table for the function [latex] g(x)=f(x-3). [\/latex]\r\n<table class=\"grid\" border=\"0\"><caption>Table 4<\/caption>\r\n<tbody>\r\n<tr>\r\n<td>[latex] x [\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x) [\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/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>The formula\u00a0[latex] g(x)=f(x-3) [\/latex] tells us that the output values of\u00a0[latex] g [\/latex] are the same as the output value of\u00a0[latex] f [\/latex] when the input value is 3 less than the original value. For example, we know that\u00a0[latex] f(2)=1. [\/latex] To get the same output from the function\u00a0[latex] g, [\/latex] we will need an input value that is 3 <em>larger<\/em>. We input a value that is 3 larger for\u00a0[latex] g(x) [\/latex] because the function takes 3 away before evaluating the function [latex] f. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl} g(5) &amp;=&amp; f(5-3) \\\\ &amp;=&amp; f(2) \\\\ &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\nWe continue with the other values to create Table 5.\r\n<table class=\"grid\" border=\"0\"><caption>Table 5<\/caption>\r\n<tbody>\r\n<tr>\r\n<td>[latex] x [\/latex]<\/td>\r\n<td>5<\/td>\r\n<td>7<\/td>\r\n<td>9<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] x-3 [\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] f(x-3) [\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] g(x) [\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe result is that the function\u00a0[latex] g(x) [\/latex] has been shifted to the right by 3. Notice the output values for\u00a0[latex] g(x) [\/latex] remain the same as the output values for\u00a0[latex] f(x), [\/latex] but the corresponding input values, [latex] x, [\/latex] have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.\r\n<h3>Analysis<\/h3>\r\nFigure 7 represents both of the functions. We can see the horizontal shift in each point.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_564\" align=\"aligncenter\" width=\"265\"]<img class=\"size-medium wp-image-564\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7-265x300.png\" alt=\"\" width=\"265\" height=\"300\" \/> Figure 7[\/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 5: Identifying a Horizontal Shift of a Toolkit Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFigure 9 represents a transformation of the toolkit function\u00a0[latex] f(x)=x^2. [\/latex] Relate this new function\u00a0[latex] g(x) [\/latex] to\u00a0[latex] f(x), [\/latex] and then find a formula for [latex] g(x). [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_568\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-568\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8-300x202.jpeg\" alt=\"\" width=\"300\" height=\"202\" \/> Figure 8[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Notice that the graph is identical in shape to the\u00a0[latex] f(x)=x^2 [\/latex] function, but the <em data-effect=\"italics\">x-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so\r\n<p style=\"text-align: center;\">[latex] g(x)=f(x-2) [\/latex]<\/p>\r\nNotice how we must input the value\u00a0[latex] x=2 [\/latex] to get the output value\u00a0[latex] y=0; [\/latex] the <em data-effect=\"italics\">x<\/em>-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the\u00a0[latex] f(x) [\/latex] function to write a formula for\u00a0[latex] g(x) [\/latex] by evaluating [latex] f(x-2). [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcll} f(x) &amp;=&amp; x^2 \\\\ g(x) &amp;=&amp; f(x-2) \\\\ g(x) &amp;=&amp; f(x-2) &amp;=&amp; (x-2)^2 \\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nTo determine whether the shift is\u00a0[latex] +2 [\/latex] or\u00a0[latex] -2 [\/latex] consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function,\u00a0[latex] f(0)-0. [\/latex] In our shifted function\u00a0[latex] g(2)=0. [\/latex] To obtain the output value of 0 from the function\u00a0[latex] f, [\/latex] we need to decide whether a plus or a minus sign will work to satisfy\u00a0[latex] g(2)=f(x-2)=f(0)=0. [\/latex] For this to work, we will need to <em data-effect=\"italics\">subtract<\/em> 2 units from our input values.\r\n<div id=\"fs-id1165133349274\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><\/div>\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: Interpreting Horizontal versus Vertical Shifts<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe function\u00a0[latex] G(m) [\/latex] gives the number of gallons of gas required to drive\u00a0[latex] m [\/latex] miles. Interpret\u00a0[latex] G(m)+10 [\/latex] and [latex] G(m+10). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>[latex] G(m)+10 [\/latex] can be interpreted as adding 10 to the output, gallons. This is the gas required to drive\u00a0[latex] m [\/latex] miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.\r\n[latex] G(m+10) [\/latex] can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than\u00a0[latex] m [\/latex] miles. The graph would indicate a horizontal shift.\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\nGiven the function\u00a0[latex] f(x)=\\sqrt{x}, [\/latex] graph the original function\u00a0[latex] f(x) [\/latex] and the transformation\u00a0[latex] g(x)=f(x+2) [\/latex] on the same axes. Is this a horizontal or a vertical shift? Which was is the graph shifted and by how many units?\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135250592\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Combining Vertical and Horizontal Shifts<\/h3>\r\n<p id=\"fs-id1165137676099\">Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output (<em data-effect=\"italics\">y<\/em>-) values and shift the function up or down. Horizontal shifts are inside changes that affect the input (<em data-effect=\"italics\">x<\/em>-) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down <em data-effect=\"italics\">and<\/em> left or right.<\/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 function and both a vertical and a horizontal shift, sketch the graph.<\/strong>\r\n<ol>\r\n \t<li>Identify the vertical and horizontal shifts from the formula.<\/li>\r\n \t<li>The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.<\/li>\r\n \t<li>The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.<\/li>\r\n \t<li>Apply the shifts to the graph in either order.<\/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 7: Graphing Combined Vertical and Horizontal Shifts<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven\u00a0[latex] f(x)=|x|, [\/latex] sketch a graph of [latex] h(x)=f(x+1)-3. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The function\u00a0[latex] f [\/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of\u00a0[latex] h [\/latex] has transformed\u00a0[latex] f [\/latex] in two ways:\u00a0[latex] f(x+1) [\/latex] is a change on the inside of the function, giving a horizontal shift led by 1, and the subtraction by 3 in\u00a0[latex] f(x+1)-3 [\/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in Figure 9.\r\n\r\nLet us follow one point of the graph of [latex] f(x)=|x|. [\/latex]\r\n<ul>\r\n \t<li>The point\u00a0[latex] (0, 0) [\/latex] is transformed first by shifting left 1 unit: [latex] (0, 0)\\rightarrow (-1, 0) [\/latex]<\/li>\r\n \t<li>The point\u00a0[latex] (-1, 0) [\/latex] is transformed next by shifting down 3 units: [latex] (-1, 0)\\rightarrow (-1, -3) [\/latex]<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_569\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-569\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9-300x238.jpeg\" alt=\"\" width=\"300\" height=\"238\" \/> Figure 9[\/caption]\r\n\r\nFigure 10 shows the graph of [latex] h. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_570\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-570\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 10[\/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\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven\u00a0[latex] f(x)=|x|, [\/latex] sketch a graph of [latex] h(x)=f(x-2)+4. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div id=\"fs-id1165137400636\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137400639\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Identifying Combined Vertical and Horizontal Shifts<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite a formula for the graph shown in Figure 11, which is a transformation of the toolkit square root function.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_571\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-571\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11-300x180.jpeg\" alt=\"\" width=\"300\" height=\"180\" \/> Figure 11[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as\r\n<p style=\"text-align: center;\">[latex] h(x)=f(x-1)+2 [\/latex]<\/p>\r\nUsing the formula for the square root function, we can write\r\n<p style=\"text-align: center;\">[latex] h(x)=\\sqrt{x-1}+2 [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNote that this transformation has changed the domain and range of the function. This new graph has domain\u00a0[latex] [1, \\infty) [\/latex] and range [latex] [2, \\infty). [\/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\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite a formula for a transformation of the toolkit reciprocal function\u00a0[latex] f(x)=\\frac{1}{x} [\/latex] that shifts the function's graph one unit to the right and one unit up.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id1165137600415\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Functions Using Reflections about the Axes<\/h2>\r\n<p id=\"fs-id1165137772409\">Another transformation that can be applied to a function is a reflection over the <em data-effect=\"italics\">x<\/em>- or <em data-effect=\"italics\">y<\/em>-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the <em data-effect=\"italics\">x<\/em>-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the <em data-effect=\"italics\">y<\/em>-axis. The reflections are shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_013\">Figure 12<\/a>.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_572\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-572\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" \/> Figure 12 Vertical and horizontal reflections of a function.[\/caption]\r\n<p id=\"fs-id1165137642152\">Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the <em data-effect=\"italics\">x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Reflections<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven a function\u00a0[latex] f(x), [\/latex] a new function\u00a0[latex] g(x)=-f(x) [\/latex] is a <strong>vertical reflection<\/strong> of the function\u00a0[latex] f(x), [\/latex] sometimes called a reflection about (or over, or through) the x-axis.\r\n\r\nGiven a function\u00a0[latex] f(x), [\/latex] a new function\u00a0[latex] g(x)=f(-x) [\/latex] is a <strong>horizontal reflection<\/strong> of the function\u00a0[latex] f(x) [\/latex] sometimes called a reflection about the y-axis.,\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 a function, reflect the graph both vertically and horizontally.<\/strong>\r\n<ol>\r\n \t<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\r\n \t<li>Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the <em data-effect=\"italics\">y<\/em>-axis.<\/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 9: Reflecting a Graph Horizontally and Vertically<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nReflect the graph of\u00a0[latex] s(t)=\\sqrt{t} [\/latex] (a) vertically and (b) horizontally.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) Reflecting the graph vertically means that each output value will be reflected over the horizontal t-axis shown in Figure 13.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_573\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-573\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-300x136.jpeg\" alt=\"\" width=\"300\" height=\"136\" \/> Figure 13 Vertical reflection of the square root function[\/caption]\r\n\r\nBecause each output value is the opposite of the original output value, we can write\r\n<p style=\"text-align: center;\">[latex] V9t)=-s(t) \\ \\ \\text{or} \\ \\ V(t)=-\\sqrt{t} [\/latex]<\/p>\r\nNotice that this is an outside change, or vertical shift, that affects the output\u00a0[latex] s(t) [\/latex] values, so the negative sign belongs outside of the function.\r\n\r\n(b) Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 14.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_574\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-574\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-300x142.jpeg\" alt=\"\" width=\"300\" height=\"142\" \/> Figure 14 Horizontal reflection of the square root function[\/caption]\r\n\r\nBecause each input value is the opposite of the original input value, we can write\r\n<p style=\"text-align: center;\">[latex] H(t)=s(-1) \\ \\ \\text{or} \\ \\ H(t)=\\sqrt{-t} [\/latex]<\/p>\r\nNotice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.\r\n\r\nNote that these transformations can affect the domain and range of the functions. While the original square root function has domain\u00a0[latex] [0, \\infty) [\/latex] and range\u00a0[latex] [0, \\infty), [\/latex] the vertical reflection gives the\u00a0[latex] V(t) [\/latex] function the range\u00a0[latex] (-\\infty, 0] [\/latex] and the horizontal reflection gives the\u00a0[latex] H(t) [\/latex] function the domain [latex] (-\\infty, 0]. [\/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\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nReflect the graph of\u00a0[latex] f(x)=|x-1| [\/latex] (a) vertically and (b) horizontally.\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 10: Reflecting a Tabular Function Horizontally and Vertically<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f(x) [\/latex] is given as Table 6. Create a table for the functions below.\r\n\r\n(a) [latex] g(x)=-f(x) [\/latex]\r\n\r\n(b) [latex] h(x)=f(-x) [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 6<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">1<\/td>\r\n<td style=\"width: 20%;\">3<\/td>\r\n<td style=\"width: 20%;\">7<\/td>\r\n<td style=\"width: 20%;\">11<\/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>(a) For\u00a0[latex] g(x) [\/latex] the negative sign outside the function indicates a vertical reflection, so the x-values stay the same and each output value will be the opposite of the original output value. See Table 7.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 7<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] g(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">-1<\/td>\r\n<td style=\"width: 20%;\">-3<\/td>\r\n<td style=\"width: 20%;\">-7<\/td>\r\n<td style=\"width: 20%;\">-11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n(b) For\u00a0[latex] h(x) [\/latex] the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the\u00a0[latex] h(x) [\/latex] values stay the same as the\u00a0[latex] f(x) [\/latex] values. See Table 8.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 32px;\" border=\"0\"><caption>Table 8<\/caption>\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">-2<\/td>\r\n<td style=\"width: 20%; height: 17px;\">-4<\/td>\r\n<td style=\"width: 20%; height: 17px;\">-6<\/td>\r\n<td style=\"width: 20%; height: 17px;\">-8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 20%; height: 15px;\">[latex] h(x) [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">1<\/td>\r\n<td style=\"width: 20%; height: 15px;\">3<\/td>\r\n<td style=\"width: 20%; height: 15px;\">7<\/td>\r\n<td style=\"width: 20%; height: 15px;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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\nA function\u00a0[latex] f(x) [\/latex] is given as Table 9. Create a table for the functions below.\r\n\r\n(a) [latex] g(x)=-f(x) [\/latex]\r\n\r\n(b) [latex] h(x)=f(-x) [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 9<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%;\">-2<\/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<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">5<\/td>\r\n<td style=\"width: 20%;\">10<\/td>\r\n<td style=\"width: 20%;\">15<\/td>\r\n<td style=\"width: 20%;\">20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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: Applying a Learning Model Equation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA common model for learning has an equation similar to\u00a0[latex] k(t)=-2^{-t}+1, [\/latex] where\u00a0[latex] k [\/latex] is the percentage of mastery that can be achieved after\u00a0[latex] t [\/latex] practice sessions. This is a transformation of the function\u00a0[latex] f(t)=2^t [\/latex] shown in Figure 15. Sketch a graph of [latex] k(t). [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_578\" align=\"aligncenter\" width=\"285\"]<img class=\"size-medium wp-image-578\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15-285x300.jpeg\" alt=\"\" width=\"285\" height=\"300\" \/> Figure 15[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>This equation combines three transformations into one equation.\r\n<ul>\r\n \t<li>A horizontal reflection: [latex] f(-t)=2^{-t} [\/latex]<\/li>\r\n \t<li>A vertical reflection: [latex] -f(-t)=-2^{-t} [\/latex]<\/li>\r\n \t<li>A vertical shift: [latex] -f(-t)+1=-2^{-t}+1 [\/latex]<\/li>\r\n<\/ul>\r\nWe can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points\u00a0[latex] (0, 1) [\/latex] and [latex] (1, 2). [\/latex]\r\n<ol>\r\n \t<li>First, we apply a horizontal reflection: [latex] (0, 1) (-1, 2). [\/latex]<\/li>\r\n \t<li>Then, we apply a vertical reflection: [latex] (0, -1) (-1, -2) [\/latex]<\/li>\r\n \t<li>Finally, we apply a vertical shift: [latex] (0, 0) (-1, -1) [\/latex]<\/li>\r\n<\/ol>\r\nThis means that the original points,\u00a0[latex] (0, 1) [\/latex] and\u00a0[latex] (1, 2) [\/latex] become\u00a0[latex] (0, 0) [\/latex] and\u00a0[latex]- (-1, -1) [\/latex] after we apply the transformations.\r\n\r\nIn Figure 16, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_579\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-579\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-300x135.jpeg\" alt=\"\" width=\"300\" height=\"135\" \/> Figure 16[\/caption]\r\n<h3>Analysis<\/h3>\r\nAs a model for learning, this function would be limited to a domain of\u00a0[latex] t\\ge0, [\/latex] with corresponding range [latex] [0, 1). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165137657438\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135209709\" data-type=\"commentary\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the toolkit function\u00a0[latex] f(x)=x^2, [\/latex] graph\u00a0[latex] g(x)=-f(x) [\/latex] and\u00a0[latex] h(x)=f(-x). [\/latex] Take note of any surprising behavior for these functions.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135182974\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining Even and Odd Functions<\/h2>\r\n<p id=\"fs-id1165135532474\">Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions\u00a0[latex] f(x)=x^2 [\/latex] or\u00a0[latex] f(x)=|x| [\/latex] will result in the original graph. We say that these types of graphs are symmetric about the <em data-effect=\"italics\">y<\/em>-axis. A function whose graph is symmetric about the <em data-effect=\"italics\">y<\/em>-axis is called an <strong>even function.<\/strong><\/p>\r\n<p id=\"fs-id1165137939530\">If the graphs of\u00a0[latex] f(x)=x^3 [\/latex] or\u00a0[latex] f(x)=\\frac{1}{x} [\/latex] were reflected over <em data-effect=\"italics\">both<\/em> axes, the result would be the original graph, as shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_022\">Figure 17<\/a>.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_580\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-580\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-300x109.jpeg\" alt=\"\" width=\"300\" height=\"109\" \/> Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.[\/caption]\r\n<p id=\"fs-id1165137406881\">We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\r\n<p id=\"fs-id1165134573214\">Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example,\u00a0[latex] f(x)=2^x [\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex] f(x)=0. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Even and Odd Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function is called an <strong>even function<\/strong> if for every input [latex] x [\/latex]\r\n<p style=\"text-align: center;\">[latex] f(x)=f(-x) [\/latex]<\/p>\r\nThe graph of an even function is symmetric about the [latex] y- [\/latex]axis.\r\n\r\nA function is called an <strong>odd function<\/strong> if for every input [latex] x [\/latex]\r\n<p style=\"text-align: center;\">[latex] f(x)=-f(-x) [\/latex]<\/p>\r\nThe graph of an odd function is symmetric about the origin.\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 formula for a function, determine if the function is even, odd, or neither.<\/strong>\r\n<ol>\r\n \t<li>Determine whether the function satisfies\u00a0[latex] f(x)=f(-x). [\/latex] If it does, it is even.<\/li>\r\n \t<li>Determine whether the function satisfies\u00a0[latex] f(x)=-f(-x). [\/latex] If is does, it is odd.<\/li>\r\n \t<li>If the function does not satisfy either rule, it is neither even nor odd.<\/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: Determining whether a Function is Even, Odd, or Neither<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIs the function\u00a0[latex] f(x)=x^3+2x [\/latex] even, odd, or neither?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.\r\n<p style=\"text-align: center;\">[latex] f(-x)=(-x)^3+2(-x)=-x^3-2x [\/latex]<\/p>\r\nThis does not return us to the original function, so this function is not even. We can now test the rule for odd functions.\r\n<p style=\"text-align: center;\">[latex] -f(-x)=-(-x^3-2x)=x^2+2x [\/latex]<\/p>\r\nBecause\u00a0[latex] -f(-x)=f(x), [\/latex] this is an odd function.\r\n<h3>Analysis<\/h3>\r\nConsider the graph\u00a0[latex] f [\/latex] of in Figure 18. Notice that the graph is symmetric about the origin. For every point\u00a0[latex] (x, y) [\/latex] on the graph, the corresponding point\u00a0[latex] (-x, -y) [\/latex] is also on the graph. For example, [latex] (1, 3) [\/latex] is on the graph of\u00a0[latex] f [\/latex] and the corresponding point\u00a0[latex] (-1, -3) [\/latex] is also on the graph.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_581\" align=\"aligncenter\" width=\"291\"]<img class=\"size-medium wp-image-581\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18-291x300.jpeg\" alt=\"\" width=\"291\" height=\"300\" \/> Figure 18[\/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\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIs the function\u00a0[latex] f(s)=s^4+3s^2+7 [\/latex] even, odd, or neither?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137654768\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Functions Using Stretches and Compressions<\/h2>\r\n<p id=\"fs-id1165137654773\">Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\r\n<p id=\"fs-id1165137675403\">We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\r\n\r\n<section id=\"fs-id1165137793506\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Vertical Stretches and Compressions<\/h3>\r\n<p id=\"fs-id1165137461225\">When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a<strong> vertical compression<\/strong>. <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_025\">Figure 19<\/a> shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_582\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-582\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19-300x201.jpeg\" alt=\"\" width=\"300\" height=\"201\" \/> Figure 19 Vertical stretch and compression[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Vertical Stretches and Compressions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven a function\u00a0[latex] f(x), [\/latex] a new function\u00a0[latex] g(x)=af(x), [\/latex] where\u00a0[latex] a [\/latex] is a constant, is a <strong>vertical stretch<\/strong> or\u00a0<strong>vertical compression<\/strong> of the function [latex] f(x). [\/latex]\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>If\u00a0[latex] a&gt;1, [\/latex] then the graph will be stretched.<\/li>\r\n \t<li>If\u00a0[latex] 0&lt;a&lt;1, [\/latex] then the graph will be compressed.<\/li>\r\n \t<li>If\u00a0[latex] a&lt;0, [\/latex] then there will be a combination of a vertical stretch or compression with a vertical reflection.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\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 a function, graph its vertical stretch.<\/strong>\r\n<ol>\r\n \t<li>Identify the value of [latex] a. [\/latex]<\/li>\r\n \t<li>Multiply all range values by [latex] a. [\/latex]<\/li>\r\n \t<li>If [latex] a&gt;1, [\/latex] the graph is stretched by a factor of [latex] a. [\/latex]<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 40px;\">If [latex] 0&lt; a&lt;1, [\/latex] the graph is compressed by a factor of [latex] a. [\/latex]<\/p>\r\n<p style=\"padding-left: 40px;\">If\u00a0[latex] a&lt;0, [\/latex] the graph is either stretched or compressed and also reflection about the x-axis.<\/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 13: Graphing a Vertical Stretch<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] P(t) [\/latex] models the population of fruit flies. The graph is shown in Figure 20.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_583\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-583\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20-300x226.jpeg\" alt=\"\" width=\"300\" height=\"226\" \/> Figure 20[\/caption]\r\n\r\nA biology student at CCA is comparing this population to another population,\u00a0[latex] Q, [\/latex] whose growth follows the same pattern, but is twice as large. Sketch the graph of this population.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values. Graphically, this is shown in Figure 21.\r\n\r\nIf we choose four reference points,\u00a0[latex] (0, 1), (3, 3), (6, 2) [\/latex] and\u00a0[latex] (7, 0) [\/latex] we will multiply all of the outputs by 2.\r\n\r\nThe following shows where the new points for the new graph will be located.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} (0, 1) &amp;\\rightarrow &amp; (0, 2) \\\\ (3, 3) &amp;\\rightarrow &amp; (3, 6) \\\\ (6, 2) &amp;\\rightarrow &amp; (6, 4) \\\\ (7, 0) &amp;\\rightarrow &amp; (7, 0) \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_584\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-584\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21-300x226.jpeg\" alt=\"\" width=\"300\" height=\"226\" \/> Figure 21[\/caption]\r\n\r\nSymbolically, this relationship is written as\r\n<p style=\"text-align: center;\">[latex] Q(t)=2P(t) [\/latex]<\/p>\r\nThis means that for any input\u00a0[latex] t [\/latex] the value of the function\u00a0[latex] Q [\/latex] is twice the value of the function\u00a0[latex] P. [\/latex] Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values,\u00a0[latex] t, [\/latex] stay the same while the output values are twice as large as before.\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\">How Tow<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<\/strong>\r\n<ol>\r\n \t<li>Determine the value of [latex] a. [\/latex]<\/li>\r\n \t<li>Multiply all of the output values by [latex] a. [\/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 14: Finding a Vertical Compression of a Tabular Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f [\/latex] is given as Table 10. Create a table for the function [latex] g(x)=\\frac{1}{2}f(x). [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 10<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">1<\/td>\r\n<td style=\"width: 20%;\">3<\/td>\r\n<td style=\"width: 20%;\">7<\/td>\r\n<td style=\"width: 20%;\">11<\/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>The formula\u00a0[latex] g(x)=\\frac{1}{2}f(x) [\/latex] tells us that the output values of\u00a0[latex] g [\/latex] are half of the output values of\u00a0[latex] f [\/latex] with the same inputs. For example, we know that\u00a0[latex] f(4)=3. [\/latex] Then\r\n<p style=\"text-align: center;\">[latex] g(4)=\\frac{1}{2}f(4)=\\frac{1}{2}(3)=\\frac{3}{2} [\/latex]<\/p>\r\nWe do the same for the other values to produce Table 11.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 30px;\" border=\"0\"><caption>Table 11<\/caption>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 20%; height: 15px;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">2<\/td>\r\n<td style=\"width: 20%; height: 15px;\">4<\/td>\r\n<td style=\"width: 20%; height: 15px;\">6<\/td>\r\n<td style=\"width: 20%; height: 15px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 20%; height: 15px;\">[latex] g(x) [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">[latex] \\frac{1}{2} [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">[latex] \\frac{3}{2} [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">[latex] \\frac{7}{2} [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">[latex] \\frac{11}{2} [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Analysis<\/h3>\r\nThe result is that the function\u00a0[latex] g(x) [\/latex] has been compressed vertically by\u00a0[latex] \\frac{1}{2}. [\/latex] Each output value is divided in half, so the graph is half the original height.\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 #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f [\/latex] is given as Table 12. Create a table for the function [latex] g(x)=\\frac{3}{4}f(x). [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 12<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">12<\/td>\r\n<td style=\"width: 20%;\">16<\/td>\r\n<td style=\"width: 20%;\">20<\/td>\r\n<td style=\"width: 20%;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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: Recognizing a Vertical Stretch<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe graph in Figure 22 is a transformation of the toolkit function\u00a0[latex] f(x)=x^3. [\/latex] Relate this new function\u00a0[latex] g(x) [\/latex] to\u00a0[latex] f(x) [\/latex] and then find a formula for [latex] g(x). [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_585\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-585\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" \/> Figure 22[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that\u00a0[latex] g(2)=2. [\/latex] With the basic cubic function at the same input,\u00a0[latex] f(2)=2^3=8. [\/latex] Based on that, it appears that the outputs of\u00a0[latex] g [\/latex] are\u00a0[latex] \\frac{1}{4} [\/latex] the outputs of the function\u00a0[latex] f [\/latex] because\u00a0[latex] g(2)=\\frac{1}{4}f(2). [\/latex] From this we can fairly safely conclude that [latex] g(x)=\\frac{1}{4}f(x). [\/latex]\r\n\r\nWe can write a formula for\u00a0[latex] g [\/latex] by using the definition of the function [latex] f. [\/latex]\r\n<p style=\"text-align: center;\">[latex] g(x)=\\frac{1}{4}f(x)=\\frac{1}{4}x^3 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135344103\" data-depth=\"2\">\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\nWrite the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<h3 data-type=\"title\">Horizontal Stretches and Compressions<\/h3>\r\n<p id=\"fs-id1165133167751\">Now we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_586\" align=\"aligncenter\" width=\"284\"]<img class=\"size-medium wp-image-586\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23-284x300.jpeg\" alt=\"\" width=\"284\" height=\"300\" \/> Figure 23[\/caption]\r\n<p id=\"fs-id1165133366207\">Given a function\u00a0[latex] y=f(x), [\/latex] the form\u00a0[latex] y=f(bx) [\/latex] results in a horizontal stretch or compression. Consider the function\u00a0[latex] y=x^2. [\/latex] Observe <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_029\">Figure 23<\/a>. The graph of\u00a0[latex] y=(0.5x)^2 [\/latex] is a horizontal stretch of the graph of the function\u00a0[latex] y=x^2 [\/latex] by a factor of 2. The graph of\u00a0[latex] y=(2x)^2 [\/latex] is a horizontal compression of the graph of the function\u00a0[latex] y=x^2 [\/latex] by a factor of [latex] \\frac{1}{2} [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Horizontal Stretches and Compressions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven a function\u00a0[latex] f(x), [\/latex] a new function\u00a0[latex] g(x)=f(bx), [\/latex] where\u00a0[latex] b [\/latex] is a constant, is a <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex] f(x). [\/latex]\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>If\u00a0[latex] b&gt;1 [\/latex] then the graph will be compressed by [latex] b. [\/latex]<\/li>\r\n \t<li>If\u00a0[latex] 0&lt;b&lt;1, [\/latex] then the graph will be stretched by [latex] \\frac{1}{b}. [\/latex]<\/li>\r\n \t<li>If\u00a0[latex] b&lt;0, [\/latex] then there will be a combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\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 a description of a function, sketch a horizontal compression or stretch.<\/strong>\r\n<ol>\r\n \t<li>Write a formula to represent the function.<\/li>\r\n \t<li>Set\u00a0[latex] g(x)=f(bx) [\/latex] where\u00a0[latex] b&gt;1 [\/latex] for a compression or\u00a0[latex] 0&lt;b&lt;1 [\/latex] for a stretch.<\/li>\r\n \t<li style=\"list-style-type: none;\"><\/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 16: Graphing a Horizontal Compression<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSuppose a biology student is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population,\u00a0[latex] R, [\/latex] will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Symbolically, we could write\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcll} R(1) &amp;=&amp; P(2), \\\\ R(2) &amp;=&amp; P(4), &amp; \\text{and in general,} \\\\ R(t) &amp;=&amp; P(2t). \\end{array} [\/latex]<\/p>\r\nSee Figure 24 for a graphical comparison of the original population and the compressed population.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_587\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-587\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-300x123.jpeg\" alt=\"\" width=\"300\" height=\"123\" \/> Figure 24 (a) Original population graph (b) Compressed population graph[\/caption]\r\n<h3>Analysis<\/h3>\r\nNote that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.\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 17: Finding a Horizontal Stretch for a Tabular Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f(x) [\/latex] is given as Table 13. Create a table for the function [latex] g(x)=f(\\frac{1}{2}x). [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 32px;\" border=\"0\"><caption>Table 13<\/caption>\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">2<\/td>\r\n<td style=\"width: 20%; height: 17px;\">4<\/td>\r\n<td style=\"width: 20%; height: 17px;\">6<\/td>\r\n<td style=\"width: 20%; height: 17px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 20%; height: 15px;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 15px;\">1<\/td>\r\n<td style=\"width: 20%; height: 15px;\">3<\/td>\r\n<td style=\"width: 20%; height: 15px;\">7<\/td>\r\n<td style=\"width: 20%; height: 15px;\">11<\/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>The formula [latex] g(x)=f(\\frac{1}{2}x) [\/latex] tells us that the output values for [latex] g [\/latex] are the same as the output values for the function [latex] f [\/latex] at an input half the size. Notice that we do not have enough information to determine [latex] g(2) [\/latex] because [latex] g(2)=f(\\frac{1}{2}\\cdot 2)=f(1), [\/latex] and we do not have a value for [latex] f(1) [\/latex] in our table. Our input values to [latex] g [\/latex] will need to be twice as large to get inputs for [latex] f [\/latex] that we can evaluate. For example, we can determine [latex] g(4). [\/latex]\r\n<p style=\"text-align: center;\">[latex] g(4)=f(\\frac{1}{2}\\cdot 4)=f(2)=1 [\/latex]<\/p>\r\nWe do the same for the other values to produce Table 14.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 14<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%;\">4<\/td>\r\n<td style=\"width: 20%;\">8<\/td>\r\n<td style=\"width: 20%;\">12<\/td>\r\n<td style=\"width: 20%;\">16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] g(x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">1<\/td>\r\n<td style=\"width: 20%;\">3<\/td>\r\n<td style=\"width: 20%;\">7<\/td>\r\n<td style=\"width: 20%;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFigure 25 shows the graphs of both of these sets of points.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_588\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-588\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-300x102.jpeg\" alt=\"\" width=\"300\" height=\"102\" \/> Figure 25[\/caption]\r\n<h3>Analysis<\/h3>\r\nBecause each input value has been doubled, the result is that the function\u00a0[latex] g(x) [\/latex] has been stretched horizontally by a factor of 2.\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 18: Recognizing a Horizontal Compression on a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nRelate the function\u00a0[latex] g(x) [\/latex] to\u00a0[latex] f(x) [\/latex] in Figure 26.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_589\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-589\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26-300x179.jpeg\" alt=\"\" width=\"300\" height=\"179\" \/> Figure 26[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The graph of [latex]\u00a0g(x) [\/latex] looks like the graph of[latex] f(x) [\/latex]\u00a0 horizontally compressed. Because [latex] f(x) [\/latex] ends at [latex] (6, 4) [\/latex] and [latex] g(x) [\/latex] ends at [latex] (2, 4) [\/latex] we can see that the values have been compressed by [latex] \\frac{1}{3}, [\/latex] because [latex] 6(\\frac{1}{3})=2. [\/latex] We might also notice that [latex] g(2)=f(6) [\/latex] and [latex] g(1)=f(3). [\/latex] Either way, we can describe this relationship as [latex] g(x)=f(3x). [\/latex] This is a horizontal compression by [latex] \\frac{1}{3}. [\/latex]\r\n<h3>Analysis<\/h3>\r\nNotice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of\u00a0[latex] \\frac{1}{4} [\/latex] in our function:\u00a0[latex] f(\\frac{1}{4}x). [\/latex] This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.\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 #11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite a formula for the toolkit square root function horizontally stretched by a factor of 3.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id1165137676302\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Performing a Sequence of Transformations<\/h2>\r\n<p id=\"fs-id1165137387533\">When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.<\/p>\r\n<p id=\"fs-id1165137387540\">When we see an expression such as [latex] 2f(x)+3, [\/latex] which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of [latex] f(x), [\/latex] we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.<\/p>\r\n<p id=\"fs-id1165137483273\">Horizontal transformations are a little trickier to think about. When we write\u00a0[latex] g(x)=f(2x+3), [\/latex] for example, we have to think about how the inputs to the function\u00a0[latex] g [\/latex] relate to the inputs to the function\u00a0[latex] f. [\/latex] Suppose we know\u00a0[latex] f(7)=12. [\/latex] What input to\u00a0[latex] g [\/latex] would produce that output? In other words, what value of\u00a0[latex] x [\/latex] will allow\u00a0[latex] g(x)=f(2x+3)=12? [\/latex] We would need\u00a0[latex] 2x+3=7. [\/latex] To solve for\u00a0[latex] x, [\/latex] we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.<\/p>\r\n<p id=\"fs-id1165137580228\">This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(bx+p)=f(b(x+\\frac{p}{b})) [\/latex]<\/p>\r\n<p id=\"fs-id1165137427526\">Let\u2019s work through an example.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=(2x+4)^2 [\/latex]<\/p>\r\n<p id=\"fs-id1165137438427\">We can factor out a 2.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=(2(x+2))^2 [\/latex]<\/p>\r\n<p id=\"fs-id1165135349189\">Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Combining Transformations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhen combining vertical transformations written in the form\u00a0[latex] af(x)_k, [\/latex] first vertically stretch by\u00a0[latex] a [\/latex] and then vertically stretch by [latex] k. [\/latex]\r\n\r\nWhen combining horizontal transformations written in the form\u00a0[latex] f(bx-h), [\/latex] first horizontally shift by\u00a0[latex] h [\/latex] and then horizontally stretch by [latex] 1b. [\/latex]\r\n\r\nWhen combining horizontal transformations written in the form\u00a0[latex] f(b(x-h)), [\/latex] first horizontally stretch by\u00a0[latex] \\frac{1}{b}, [\/latex] and then horizontally shift by [latex] h. [\/latex]\r\n\r\nHorizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.\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 19: Finding a Triple Transformation of a Tabular Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven Table 15 for the function\u00a0[latex] f(x), [\/latex] create a table of values for the function [latex] g(x)=2f(3x)+1. [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 34px;\" border=\"0\"><caption>Table 15<\/caption>\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">6<\/td>\r\n<td style=\"width: 20%; height: 17px;\">12<\/td>\r\n<td style=\"width: 20%; height: 17px;\">18<\/td>\r\n<td style=\"width: 20%; height: 17px;\">24<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"width: 20%; height: 17px;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 20%; height: 17px;\">10<\/td>\r\n<td style=\"width: 20%; height: 17px;\">14<\/td>\r\n<td style=\"width: 20%; height: 17px;\">15<\/td>\r\n<td style=\"width: 20%; height: 17px;\">17<\/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>There are three steps to the transformation, and we will work from the inside out. Starting with the horizontal transformations,\u00a0[latex] f(3x) [\/latex] is a horizontal compression by\u00a0[latex] \\frac{1}{3}, [\/latex] which means we multiply each [latex] x- [\/latex]value by\u00a0[latex] \\frac{1}{3}. [\/latex] See Table 16.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 16<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] f(3x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">10<\/td>\r\n<td style=\"width: 20%;\">14<\/td>\r\n<td style=\"width: 20%;\">15<\/td>\r\n<td style=\"width: 20%;\">17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLooking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See Table 17.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 17<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] 2f(3x) [\/latex]<\/td>\r\n<td style=\"width: 20%;\">20<\/td>\r\n<td style=\"width: 20%;\">28<\/td>\r\n<td style=\"width: 20%;\">30<\/td>\r\n<td style=\"width: 20%;\">34<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFinally, we can apply the vertical shift, which will add 1 to all the output values. See Table 18.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 18<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] x [\/latex]<\/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<td style=\"width: 20%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\">[latex] g(x)=2f(3x)+1 [\/latex]<\/td>\r\n<td style=\"width: 20%;\">21<\/td>\r\n<td style=\"width: 20%;\">29<\/td>\r\n<td style=\"width: 20%;\">31<\/td>\r\n<td style=\"width: 20%;\">35<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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 20: Finding a Triple Transformation of a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the graph of\u00a0[latex] f(x) [\/latex] in Figure 27 to sketch a graph of [latex] k(x)=f(\\frac{1}{2}x+1)-3. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_593\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-593\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 27[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To simplify, let's start by factoring out the inside of the function.\r\n<p style=\"text-align: center;\">[latex] f(\\frac{1}{2}x+1)-3=f(\\frac{1}{2}(x=2))-3 [\/latex]<\/p>\r\nBy factoring the inside, we can first horizontally stretch by 2, as indicated by the\u00a0[latex] \\frac{1}{2} [\/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point\u00a0[latex] (0, 2) [\/latex] remains at\u00a0[latex] (0, 2) [\/latex] while the point\u00a0[latex] (2, 0) [\/latex] will stretch to\u00a0[latex] (4, 0). [\/latex] See Figure 28.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_594\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-594\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 28[\/caption]\r\n\r\nNext, we horizontally shift left by 2 units, as indicated by\u00a0[latex] x+2. [\/latex] See Figure 29.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_595\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-595\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 29[\/caption]\r\n\r\nLast, we vertically shift down by 3 to complete our sketch, as indicated by the\u00a0[latex] -3 [\/latex] on the outside of the function. See Figure 30.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_596\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-596\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 30[\/caption]\r\n\r\n<\/details><\/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 transformation of functions.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=An29CALYjAA\">Function Transformations<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/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\">3.5 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165137436217\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137728393\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137728398\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135180434\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137728398-solution\">1<\/a><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137550351\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137550353\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137922588\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137922590\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137922588-solution\">3<\/a><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134227890\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134227892\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the <em data-effect=\"italics\">x<\/em>-axis from a reflection with respect to the <em data-effect=\"italics\">y<\/em>-axis?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137734659\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137734661\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137734659-solution\">5<\/a><span class=\"os-divider\">. <\/span>How can you determine whether a function is odd or even from the formula of the function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137454081\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"eip-375\">For the following exercises, write a formula for the function obtained when the graph is shifted as described.<\/p>\r\n\r\n<div id=\"fs-id1165135168321\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135168323\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">.\u00a0[latex] f(x)=\\sqrt{x} [\/latex] <\/span>is shifted up 1 unit and to the left 2 units.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134041416\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134041418\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134041416-solution\">7<\/a><span class=\"os-divider\">.\u00a0[latex] f(x)=|x| [\/latex] <\/span>is shifted down 3 units and to the right 1 unit.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134061972\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134061974\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">.\u00a0[latex] f(x)=\\frac{1}{x} [\/latex] <\/span>is shifted down 4 units and to the right 3 units.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137805973\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137805975\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137805973-solution\">9<\/a><span class=\"os-divider\">.\u00a0[latex] f(x)=\\frac{1}{x^2} [\/latex] <\/span>is shifted up 2 units and to the left 4 units.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137407590\">For the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex] f. [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135397258\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133111635\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10.<\/span> [latex] y=f(x-49) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135193434\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135193436\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135193434-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] y=f(x+43) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135571667\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135571670\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] y=f(x+3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135575988\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135575991\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135575988-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] y=f(x-4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135701452\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137551379\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] y=f(x)+5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137936723\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137936725\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137936723-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] y=f(x)+8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135369401\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135369403\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] y=f(x)-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137454950\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132945534\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137454950-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] y=f(x)-7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135205732\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135205734\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] y=f(x-2)+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134220843\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134220845\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134220843-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] y=f(x+4)-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137896305\">For the following exercises, determine the interval(s) on which the function is increasing and decreasing.<\/p>\r\n\r\n<div id=\"fs-id1165137896310\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137896312\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">.<\/span> [latex] f(x)=4(x+1)^2-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135547247\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135547250\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135547247-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=5(x+3)^2-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137679200\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137679202\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] a(x)=\\sqrt{-x+4} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135650778\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135250825\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135650778-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] k(x)=\\sqrt[-3]{x}-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135403290\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165137694193\">For the following exercises, use the graph of\u00a0[latex] f(x)=2^x [\/latex] shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_201\">Figure 31<\/a> to sketch a graph of each transformation of [latex] f(x). [\/latex]<\/p>\r\n\r\n<\/section><section data-depth=\"2\">\r\n<div id=\"fs-id1165135394223\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135394226\" data-type=\"problem\">\r\n\r\n[caption id=\"attachment_597\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-597\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 31[\/caption]\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=2^x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137887426\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137887428\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137887426-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=2^x-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135436604\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137724122\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] w(x)=2^{x-1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137448386\">For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.<\/p>\r\n\r\n<div id=\"fs-id1165137448391\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137448393\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137448391-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] f(t)=(t+1)^2-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137932662\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135209555\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=|x-1|+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135421533\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135421535\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135421533-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] k(x)=(x+2)^3-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137424880\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137424883\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] m(t)=3+\\sqrt{t+2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137464226\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<div id=\"fs-id1165137681998\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137682000\" class=\"material-set-2\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137681998-solution\">31<\/a><span class=\"os-divider\">. <\/span>Tabular representations for the functions\u00a0[latex] f, g, [\/latex] and\u00a0[latex] h [\/latex] are given below. Write\u00a0[latex] g(x) [\/latex] and\u00a0[latex] h(x) [\/latex] as transformations of [latex] f(x). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n&nbsp;\r\n<div id=\"fs-id1165137432561\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"fs-id1165137432561\" data-label=\"\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">\u22123<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135634096\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"fs-id1165135634096\" data-label=\"\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] g(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">\u22123<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135330589\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"fs-id1165135330589\" data-label=\"\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] h(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">3<\/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-id1165137734475\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137734477\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Tabular representations for the functions\u00a0[latex] f, g, [\/latex] and\u00a0[latex] h [\/latex] are given below. Write\u00a0[latex] g(x) [\/latex] and\u00a0[latex] h(x) [\/latex] as transformations of [latex] f(x). [\/latex]\r\n<div class=\"os-problem-container\">\r\n<div id=\"fs-id1165134558032\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"fs-id1165134558032\" data-label=\"\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">\u22123<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165134380916\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"fs-id1165134380916\" data-label=\"\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">\u22123<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] g(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">\u22123<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137894261\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"fs-id1165137894261\" data-label=\"\"><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22121<\/td>\r\n<td data-align=\"center\">0<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] h(x) [\/latex]<\/td>\r\n<td data-align=\"center\">\u22122<\/td>\r\n<td data-align=\"center\">\u22124<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">0<\/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<p id=\"fs-id1165137570566\">For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/p>\r\n\r\n<div id=\"fs-id1165137431229\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137431231\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137431229-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135543438\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-598\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.33-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135481230\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135481232\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137851362\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-599\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137817635\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137817637\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137817635-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165133341017\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-600\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.35-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132929618\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132929620\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135203675\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-601\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36-275x300.jpeg\" alt=\"\" width=\"275\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135487204\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135487206\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135487204-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135535017\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-602\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.37-165x300.jpeg\" alt=\"\" width=\"165\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135190411\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135190413\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165133277626\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img class=\"alignnone size-full wp-image-603\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.38.jpeg\" alt=\"\" width=\"285\" height=\"226\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133103936\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133103938\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133103936-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134362846\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-604\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39-297x300.jpeg\" alt=\"\" width=\"297\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134043550\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043552\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<img class=\"alignnone size-medium wp-image-605\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n\r\n<span id=\"fs-id1165134036728\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><\/span>For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134187277\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134187279\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134187277-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137834957\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img class=\"alignnone size-full wp-image-606\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.41.jpeg\" alt=\"\" width=\"225\" height=\"225\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137930320\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137930323\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135487154\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img class=\"alignnone size-full wp-image-607\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.42.jpeg\" alt=\"\" width=\"215\" height=\"222\" \/><\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134177109\">For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.<\/p>\r\n\r\n<div id=\"fs-id1165134177113\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137806559\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134177113-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137806566\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-608\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.43-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135384400\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137693606\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137693612\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a cubic function.\" data-display=\"block\"><img class=\"alignnone size-full wp-image-609\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.44.jpeg\" alt=\"\" width=\"252\" height=\"251\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137892243\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134352554\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137892243-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134352561\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img class=\"alignnone size-full wp-image-610\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.45.jpeg\" alt=\"\" width=\"184\" height=\"191\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135411377\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135411379\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165133155251\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img class=\"alignnone size-full wp-image-611\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.46.jpeg\" alt=\"\" width=\"285\" height=\"285\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165132924966\">For the following exercises, determine whether the function is odd, even, or neither.<\/p>\r\n\r\n<div id=\"fs-id1165132924969\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132924971\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165132924969-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3x^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135671514\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137761968\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=\\sqrt{x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137828008\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137828010\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137828008-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=\\frac{1}{x}+3x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134187165\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134187167\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=(x-2)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134039317\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134039319\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134039317-solution\">51<\/a><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-id1165135630957\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135630959\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=2x-x^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137571611\">For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex] f. [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137599981\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137599983\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137599981-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=-f(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133065712\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133065714\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=f(-x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135412892\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135412894\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135412892-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=4f(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137939898\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137939900\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=6f(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135440224\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135440226\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135440224-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=f(5x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135364548\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135364550\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=f(2x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133307633\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133307635\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133307633-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=f(\\frac{1}{3}x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137854807\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137854809\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=f(\\frac{1}{5}x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137664915\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137664917\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137664915-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=3f(-x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135192324\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135192326\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=-f(3x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135637428\">For the following exercises, write a formula for the function\u00a0[latex] g [\/latex] that results when the graph of a given toolkit function is transformed as described.<\/p>\r\n\r\n<div id=\"fs-id1165135195127\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135195130\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135195127-solution\">63<\/a><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex] f(x)=|x| [\/latex] is reflected over the [latex] y- [\/latex]axis and horizontally compressed by a factor of [latex] \\frac{1}{4}. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135404231\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135404233\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex] f(x)=\\sqrt{x} [\/latex] is reflected over the [latex] x- [\/latex]axis and horizontally stretched by a factor of 2.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137634443\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137634445\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137634443-solution\">65<\/a><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex] f(x)=\\frac{1}{x^2} [\/latex] is vertically compressed by a factor of\u00a0[latex] \\frac{1}{3}, [\/latex] then shifted to the left 2 units and down 3 units.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137731439\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137731442\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex] f(x)=\\frac{1]{x} [\/latex] is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137642586\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137642588\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137642586-solution\">67<\/a><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex] f(x)=x^2 [\/latex] is vertically compressed by a factor of\u00a0[latex] \\frac{1}{2}, [\/latex] then shifted to the right 5 units and up 1 unit.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137653191\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137653193\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex] f(x)=x^2 [\/latex] is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137668699\">For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/p>\r\n\r\n<div id=\"fs-id1165137668704\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137668706\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137668704-solution\">69<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=4(x+1)^2-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135199465\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135199468\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=f(x+3)^2-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134069304\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134069306\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134069304-solution\">71<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=-2|x-4|+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137758532\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137758534\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span> [latex] k(x)=\\sqrt[-3]{x}-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135351654\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135351656\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135351654-solution\">73<\/a><span class=\"os-divider\">. <\/span> [latex] m(x)=\\frac{1}{2}x^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137832423\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137832425\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">74<\/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-id1165134155168\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134155170\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134155168-solution\">75<\/a><span class=\"os-divider\">. <\/span> [latex] p(x)=(\\frac{1}{3}x)^3-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137898977\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898979\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span> [latex] q(x)=(\\frac{1}{4}x)^3+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137861993\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137861995\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137861993-solution\">77<\/a><span class=\"os-divider\">. <\/span> [latex] a(x)=\\sqrt{-x+4} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134338807\">For the following exercises, use the graph in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_233\">Figure 32<\/a> to sketch the given transformations.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_612\" align=\"alignnone\" width=\"298\"]<img class=\"size-medium wp-image-612\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/> Figure 32[\/caption]\r\n\r\n<div id=\"fs-id1165135706785\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135208393\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=f(x)-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135432954\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135432956\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135432954-solution\">79<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=-f(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137722436\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137722438\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=f(x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134269560\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134269563\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134269560-solution\">81<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=f(x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_5f6ff02a-1000-410d-b034-af26fbd86d0b\" 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>Graph functions using vertical and horizontal shifts.<\/li>\n<li>Graph functions using reflections about the x-axis and the y-axis.<\/li>\n<li>Determine whether a function is even, odd, or neither from its graph.<\/li>\n<li>Graph functions using compressions and stretches.<\/li>\n<li>Combine transformations.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"attachment_557\" aria-describedby=\"caption-attachment-557\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-557\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1-300x200.jpeg\" alt=\"\" width=\"300\" height=\"200\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1-300x200.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1-65x43.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1-225x150.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1-350x233.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-1.jpeg 488w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-557\" class=\"wp-caption-text\">Figure 1 (credit: &#8220;Misko&#8221;\/Flickr)<\/figcaption><\/figure>\n<p id=\"fs-id1165137742090\">We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.<\/p>\n<section id=\"fs-id1165137827988\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Functions Using Vertical and Horizontal Shifts<\/h2>\n<p id=\"fs-id1165137654715\">Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.<\/p>\n<section id=\"fs-id1165137535664\" data-depth=\"2\">\n<h3 data-type=\"title\">Identifying Vertical Shifts<\/h3>\n<p id=\"fs-id1165135503932\">One simple kind of <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">transformation<\/span> involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a <strong>vertical shift<\/strong>, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function\u00a0[latex]g(x)=f(x)+k,[\/latex] the function\u00a0[latex]f(x)[\/latex] is shifted vertically\u00a0[latex]k[\/latex] units. See <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_002\">Figure 2<\/a> for an example.<\/p>\n<figure id=\"attachment_558\" aria-describedby=\"caption-attachment-558\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-558 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2-300x180.jpeg\" alt=\"\" width=\"300\" height=\"180\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2-300x180.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2-225x135.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2-350x210.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-2.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-558\" class=\"wp-caption-text\">Figure 2 Vertical shift by\u00a0[latex] k=1 [\/latex] of the cube root function [latex] f(x)=\\sqrt[3]{x}. [\/latex]<\/figcaption><\/figure>\n<div id=\"Figure_01_05_002\" class=\"os-figure\"><\/div>\n<p id=\"fs-id1165137439125\">To help you visualize the concept of a vertical shift, consider that\u00a0[latex]y=f(x).[\/latex] Therefore,\u00a0[latex]f(x)+k[\/latex] is equivalent to\u00a0[latex]y+k.[\/latex] Every unit of\u00a0[latex]y[\/latex] is replaced by\u00a0[latex]y+k,[\/latex] so the <em data-effect=\"italics\">y<\/em>-value increases or decreases depending on the value of\u00a0[latex]k.[\/latex] The result is a shift upward or downward.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Vertical Shift<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given a function\u00a0[latex]f(x),[\/latex] a new function\u00a0[latex]g(x)=f(x)+k,[\/latex] where\u00a0[latex]k[\/latex] is a constant, is a <strong>vertical shift<\/strong> of the function\u00a0[latex]f(x).[\/latex] All the output values change by\u00a0[latex]k[\/latex] units. If\u00a0[latex]k[\/latex] is positive, the graph will shift up. If\u00a0[latex]k[\/latex] is negative, the graphs will shift down.<\/p>\n<\/div>\n<\/div>\n<div class=\"os-note-body\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Adding a Constant to a Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>To regulate temperature in CCA&#8217;s classroom building, airflow vents near the roof open and close throughout the day. Figure 3 shows the area of open vents\u00a0[latex]V[\/latex] (in square feet) throughout the day in hours after midnight,\u00a0[latex]t.[\/latex] During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_559\" aria-describedby=\"caption-attachment-559\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-559\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3-300x201.jpeg\" alt=\"\" width=\"300\" height=\"201\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3-300x201.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3-225x151.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3-350x234.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-3.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-559\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_004\">Figure 4<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_560\" aria-describedby=\"caption-attachment-560\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-560\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4-300x203.jpeg\" alt=\"\" width=\"300\" height=\"203\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4-300x203.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4-225x152.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4-350x236.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-4.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-560\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p>Notice that in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_004\">Figure 4<\/a>, for each input value, the output value has increased by 20, so if we call the new function\u00a0[latex]S(t),[\/latex] we could write<\/p>\n<p style=\"text-align: center;\">[latex]S(t)=V(t)+20[\/latex]<\/p>\n<p>This notation tells us that for any value of\u00a0[latex]t, S(t)[\/latex] can be found by evaluating the function\u00a0[latex]V[\/latex] at the same input and then adding 20 to the result. This defines\u00a0[latex]S[\/latex] as a transformation of the function\u00a0[latex]V,[\/latex] in this case a vertical shift up 20 unites. Notice that, with a vertical shift, the input values stay the same and only the output values change. See Table 1.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 14.2857%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<td style=\"width: 14.2857%;\">8<\/td>\n<td style=\"width: 14.2857%;\">10<\/td>\n<td style=\"width: 14.2857%;\">17<\/td>\n<td style=\"width: 14.2857%;\">19<\/td>\n<td style=\"width: 14.2857%;\">24<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">[latex]V(t)[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<td style=\"width: 14.2857%;\">220<\/td>\n<td style=\"width: 14.2857%;\">220<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<td style=\"width: 14.2857%;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;\">[latex]S(t)[\/latex]<\/td>\n<td style=\"width: 14.2857%;\">20<\/td>\n<td style=\"width: 14.2857%;\">20<\/td>\n<td style=\"width: 14.2857%;\">240<\/td>\n<td style=\"width: 14.2857%;\">240<\/td>\n<td style=\"width: 14.2857%;\">20<\/td>\n<td style=\"width: 14.2857%;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\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 a tabular function, create a new row to represent a vertical shift.<\/strong><\/p>\n<ol>\n<li>Identify the output row or column.<\/li>\n<li>Determine the magnitude of the shift.<\/li>\n<li>Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Shifting a Tabular Function Vertically<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f(x)[\/latex] is given in Table 2. Create a table for the function [latex]g(x)=f(x)-3.[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">1<\/td>\n<td style=\"width: 20%;\">3<\/td>\n<td style=\"width: 20%;\">7<\/td>\n<td style=\"width: 20%;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The formula\u00a0[latex]g(x)=f(x)-3[\/latex] tells us that we can find the output values of\u00a0[latex]g[\/latex] by subtracting 3 from the output values of\u00a0[latex]f.[\/latex] For example:<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcll} f(2) &=& 1 & \\text{Given} \\\\ g(x) &=& f(x)-3 & \\text{Given transformation} \\\\ g(2) &=& f(2)-3 \\\\ &=& 1-3 \\\\ &=& -2 \\end{array}[\/latex]<\/p>\n<p>Subtracting 3 from each\u00a0[latex]f(x)[\/latex] value, we can complete a table of values for\u00a0[latex]g(x)[\/latex] as shown in Table 3.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 51px;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">2<\/td>\n<td style=\"width: 20%; height: 17px;\">4<\/td>\n<td style=\"width: 20%; height: 17px;\">6<\/td>\n<td style=\"width: 20%; height: 17px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">1<\/td>\n<td style=\"width: 20%; height: 17px;\">3<\/td>\n<td style=\"width: 20%; height: 17px;\">7<\/td>\n<td style=\"width: 20%; height: 17px;\">11<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]g(x)[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">-2<\/td>\n<td style=\"width: 20%; height: 17px;\">0<\/td>\n<td style=\"width: 20%; height: 17px;\">4<\/td>\n<td style=\"width: 20%; height: 17px;\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Analysis<\/h3>\n<p>As with the earlier vertical shift, notice the input values stay the same and only the output values change.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_02\" class=\"ui-has-child-title\" data-type=\"example\">\n<section>\n<div class=\"body\">\n<div id=\"fs-id1165135581153\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137737963\" data-type=\"commentary\">\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>The function\u00a0[latex]h(t)=-4.9t^2+30t[\/latex] gives the height\u00a0[latex]h[\/latex] of a ball (in meters) thrown upward from the ground after\u00a0[latex]t[\/latex] seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function\u00a0[latex]b(t)[\/latex] to\u00a0[latex]h(t)[\/latex] and then find a formula for [latex]b(t).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137597159\" data-depth=\"2\">\n<h3 data-type=\"title\">Identifying Horizontal Shifts<\/h3>\n<p id=\"fs-id1165137404493\">We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a <strong>horizontal shift<\/strong>, shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_005\">Figure 5<\/a>.<\/p>\n<figure id=\"attachment_561\" aria-describedby=\"caption-attachment-561\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-561 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5-300x177.jpeg\" alt=\"\" width=\"300\" height=\"177\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5-300x177.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5-65x38.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5-225x133.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5-350x207.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-5.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-561\" class=\"wp-caption-text\">Figure 5. Horizontal shift of the function [latex] f(x)=\\sqrt[3]{x}. [\/latex] Notice that [latex] (x+1) [\/latex] means [latex] h=-1, [\/latex] which shifts the graph to the left, towards <em>negative values of<\/em> [latex] x. [\/latex]<\/figcaption><\/figure>\n<p id=\"eip-884\">For example, if\u00a0[latex]f(x)=x^2,[\/latex] then\u00a0[latex]g(x)=(x-2)^2[\/latex] is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in [latex]f.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Horizontal Shift<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given a function\u00a0[latex]f,[\/latex] a new function\u00a0[latex]g(x)=f(x-h),[\/latex] where\u00a0[latex]h[\/latex] is a constant, is a <strong>horizontal shift<\/strong> of the function\u00a0[latex]f.[\/latex] If\u00a0[latex]h[\/latex] is positive, the graph will shift right. If\u00a0[latex]h[\/latex] is negative, the graph will shift left.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Adding a Constant to an Input<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Returning to our building airflow example from Figure 3, suppose that in Fall the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can set\u00a0[latex]V(t)[\/latex] to be the original program and\u00a0[latex]F(t)[\/latex] to be the revised program.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl} V(t) &=& \\text{the original venting plan} \\\\ F(t) &=& \\text{starting 2 hrs sooner} \\end{array}[\/latex]<\/p>\n<p>In the new graph, at each time, the airflow is the same as the original function\u00a0[latex]V[\/latex] was 2 hours later. For example, in the original function\u00a0[latex]V,[\/latex] the airflow starts to change at 8 a.m., whereas for the function\u00a0[latex]F,[\/latex] the airflow starts to change at 6 a.m. The comparable function values are\u00a0[latex]V(8)=F(6).[\/latex] See Figure 6. Notice also that the vents first opened to\u00a0[latex]220\\hspace{0.5em}\\text{ft}^2[\/latex] at 10 a.m. under the original plan, while under the new plan the vents reach\u00a0[latex]220\\hspace{0.5em}\\text{ft}^2[\/latex] at 8 a.m., so [latex]V(10)=F(8).[\/latex]<\/p>\n<p>In both cases, we see that, because\u00a0[latex]F(t)[\/latex] starts 2 hours sooner,\u00a0[latex]h=-2.[\/latex] That means that the same output values are reached when [latex]F(t)=V(t-(-2))=V(t+2).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_563\" aria-describedby=\"caption-attachment-563\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-563\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6-300x203.jpeg\" alt=\"\" width=\"300\" height=\"203\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6-300x203.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6-225x152.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6-350x236.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-6.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-563\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>Note that\u00a0[latex]V(t+2)[\/latex] has the effect of shifting the graph to the <em>left.<\/em><\/p>\n<p>Horizontal change or \u201cinside changes\u201d affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function [latex]F(t)[\/latex] uses the same outputs as [latex]V(t),[\/latex] but matches those outputs to inputs 2 hours earlier than those of [latex]V(t).[\/latex] Said another way, we must add 2 hours to the input of [latex]V[\/latex] to find the corresponding output for [latex]F: F(t)=V(t+2).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\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 tabular function, create a new row to represent a horizontal shift.<\/strong><\/p>\n<ol>\n<li>Identify the input row or column.<\/li>\n<li>Determine the magnitude of the shift.<\/li>\n<li>Add the shift to the value of each input cell.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Shifting a Tabular Function Horizontally<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f(x)[\/latex] is given in Table 4. Create a table for the function [latex]g(x)=f(x-3).[\/latex]<\/p>\n<table class=\"grid\">\n<caption>Table 4<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x)[\/latex]<\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The formula\u00a0[latex]g(x)=f(x-3)[\/latex] tells us that the output values of\u00a0[latex]g[\/latex] are the same as the output value of\u00a0[latex]f[\/latex] when the input value is 3 less than the original value. For example, we know that\u00a0[latex]f(2)=1.[\/latex] To get the same output from the function\u00a0[latex]g,[\/latex] we will need an input value that is 3 <em>larger<\/em>. We input a value that is 3 larger for\u00a0[latex]g(x)[\/latex] because the function takes 3 away before evaluating the function [latex]f.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl} g(5) &=& f(5-3) \\\\ &=& f(2) \\\\ &=& 1 \\end{array}[\/latex]<\/p>\n<p>We continue with the other values to create Table 5.<\/p>\n<table class=\"grid\">\n<caption>Table 5<\/caption>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>9<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>[latex]x-3[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]f(x-3)[\/latex]<\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>[latex]g(x)[\/latex]<\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The result is that the function\u00a0[latex]g(x)[\/latex] has been shifted to the right by 3. Notice the output values for\u00a0[latex]g(x)[\/latex] remain the same as the output values for\u00a0[latex]f(x),[\/latex] but the corresponding input values, [latex]x,[\/latex] have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.<\/p>\n<h3>Analysis<\/h3>\n<p>Figure 7 represents both of the functions. We can see the horizontal shift in each point.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_564\" aria-describedby=\"caption-attachment-564\" style=\"width: 265px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-564\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7-265x300.png\" alt=\"\" width=\"265\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7-265x300.png 265w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7-65x74.png 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7-225x255.png 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7-350x396.png 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-7.png 357w\" sizes=\"auto, (max-width: 265px) 100vw, 265px\" \/><figcaption id=\"caption-attachment-564\" class=\"wp-caption-text\">Figure 7<\/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 5: Identifying a Horizontal Shift of a Toolkit Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Figure 9 represents a transformation of the toolkit function\u00a0[latex]f(x)=x^2.[\/latex] Relate this new function\u00a0[latex]g(x)[\/latex] to\u00a0[latex]f(x),[\/latex] and then find a formula for [latex]g(x).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_568\" aria-describedby=\"caption-attachment-568\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-568\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8-300x202.jpeg\" alt=\"\" width=\"300\" height=\"202\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8-300x202.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8-225x152.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8-350x236.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-8.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-568\" 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>Notice that the graph is identical in shape to the\u00a0[latex]f(x)=x^2[\/latex] function, but the <em data-effect=\"italics\">x-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so<\/p>\n<p style=\"text-align: center;\">[latex]g(x)=f(x-2)[\/latex]<\/p>\n<p>Notice how we must input the value\u00a0[latex]x=2[\/latex] to get the output value\u00a0[latex]y=0;[\/latex] the <em data-effect=\"italics\">x<\/em>-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the\u00a0[latex]f(x)[\/latex] function to write a formula for\u00a0[latex]g(x)[\/latex] by evaluating [latex]f(x-2).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcll} f(x) &=& x^2 \\\\ g(x) &=& f(x-2) \\\\ g(x) &=& f(x-2) &=& (x-2)^2 \\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>To determine whether the shift is\u00a0[latex]+2[\/latex] or\u00a0[latex]-2[\/latex] consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function,\u00a0[latex]f(0)-0.[\/latex] In our shifted function\u00a0[latex]g(2)=0.[\/latex] To obtain the output value of 0 from the function\u00a0[latex]f,[\/latex] we need to decide whether a plus or a minus sign will work to satisfy\u00a0[latex]g(2)=f(x-2)=f(0)=0.[\/latex] For this to work, we will need to <em data-effect=\"italics\">subtract<\/em> 2 units from our input values.<\/p>\n<div id=\"fs-id1165133349274\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><\/div>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Interpreting Horizontal versus Vertical Shifts<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The function\u00a0[latex]G(m)[\/latex] gives the number of gallons of gas required to drive\u00a0[latex]m[\/latex] miles. Interpret\u00a0[latex]G(m)+10[\/latex] and [latex]G(m+10).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>[latex]G(m)+10[\/latex] can be interpreted as adding 10 to the output, gallons. This is the gas required to drive\u00a0[latex]m[\/latex] miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.<br \/>\n[latex]G(m+10)[\/latex] can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than\u00a0[latex]m[\/latex] miles. The graph would indicate a horizontal shift.<\/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>Given the function\u00a0[latex]f(x)=\\sqrt{x},[\/latex] graph the original function\u00a0[latex]f(x)[\/latex] and the transformation\u00a0[latex]g(x)=f(x+2)[\/latex] on the same axes. Is this a horizontal or a vertical shift? Which was is the graph shifted and by how many units?<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135250592\" data-depth=\"2\">\n<h3 data-type=\"title\">Combining Vertical and Horizontal Shifts<\/h3>\n<p id=\"fs-id1165137676099\">Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output (<em data-effect=\"italics\">y<\/em>-) values and shift the function up or down. Horizontal shifts are inside changes that affect the input (<em data-effect=\"italics\">x<\/em>-) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down <em data-effect=\"italics\">and<\/em> left or right.<\/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 function and both a vertical and a horizontal shift, sketch the graph.<\/strong><\/p>\n<ol>\n<li>Identify the vertical and horizontal shifts from the formula.<\/li>\n<li>The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.<\/li>\n<li>The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.<\/li>\n<li>Apply the shifts to the graph in either order.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Graphing Combined Vertical and Horizontal Shifts<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given\u00a0[latex]f(x)=|x|,[\/latex] sketch a graph of [latex]h(x)=f(x+1)-3.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The function\u00a0[latex]f[\/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of\u00a0[latex]h[\/latex] has transformed\u00a0[latex]f[\/latex] in two ways:\u00a0[latex]f(x+1)[\/latex] is a change on the inside of the function, giving a horizontal shift led by 1, and the subtraction by 3 in\u00a0[latex]f(x+1)-3[\/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in Figure 9.<\/p>\n<p>Let us follow one point of the graph of [latex]f(x)=|x|.[\/latex]<\/p>\n<ul>\n<li>The point\u00a0[latex](0, 0)[\/latex] is transformed first by shifting left 1 unit: [latex](0, 0)\\rightarrow (-1, 0)[\/latex]<\/li>\n<li>The point\u00a0[latex](-1, 0)[\/latex] is transformed next by shifting down 3 units: [latex](-1, 0)\\rightarrow (-1, -3)[\/latex]<\/li>\n<\/ul>\n<figure id=\"attachment_569\" aria-describedby=\"caption-attachment-569\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-569\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9-300x238.jpeg\" alt=\"\" width=\"300\" height=\"238\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9-300x238.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9-65x52.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9-225x178.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9-350x278.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-9.jpeg 469w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-569\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<p>Figure 10 shows the graph of [latex]h.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_570\" aria-describedby=\"caption-attachment-570\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-570\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-10.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-570\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\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>Given\u00a0[latex]f(x)=|x|,[\/latex] sketch a graph of [latex]h(x)=f(x-2)+4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div id=\"fs-id1165137400636\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137400639\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Identifying Combined Vertical and Horizontal Shifts<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write a formula for the graph shown in Figure 11, which is a transformation of the toolkit square root function.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_571\" aria-describedby=\"caption-attachment-571\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-571\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11-300x180.jpeg\" alt=\"\" width=\"300\" height=\"180\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11-300x180.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11-225x135.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11-350x210.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-11.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-571\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as<\/p>\n<p style=\"text-align: center;\">[latex]h(x)=f(x-1)+2[\/latex]<\/p>\n<p>Using the formula for the square root function, we can write<\/p>\n<p style=\"text-align: center;\">[latex]h(x)=\\sqrt{x-1}+2[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Note that this transformation has changed the domain and range of the function. This new graph has domain\u00a0[latex][1, \\infty)[\/latex] and range [latex][2, \\infty).[\/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 #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write a formula for a transformation of the toolkit reciprocal function\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] that shifts the function&#8217;s graph one unit to the right and one unit up.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1165137600415\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Functions Using Reflections about the Axes<\/h2>\n<p id=\"fs-id1165137772409\">Another transformation that can be applied to a function is a reflection over the <em data-effect=\"italics\">x<\/em>&#8211; or <em data-effect=\"italics\">y<\/em>-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the <em data-effect=\"italics\">x<\/em>-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the <em data-effect=\"italics\">y<\/em>-axis. The reflections are shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_013\">Figure 12<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_572\" aria-describedby=\"caption-attachment-572\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-572\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12-300x272.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12-65x59.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12-225x204.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12-350x318.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-12.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-572\" class=\"wp-caption-text\">Figure 12 Vertical and horizontal reflections of a function.<\/figcaption><\/figure>\n<p id=\"fs-id1165137642152\">Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the <em data-effect=\"italics\">x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Reflections<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given a function\u00a0[latex]f(x),[\/latex] a new function\u00a0[latex]g(x)=-f(x)[\/latex] is a <strong>vertical reflection<\/strong> of the function\u00a0[latex]f(x),[\/latex] sometimes called a reflection about (or over, or through) the x-axis.<\/p>\n<p>Given a function\u00a0[latex]f(x),[\/latex] a new function\u00a0[latex]g(x)=f(-x)[\/latex] is a <strong>horizontal reflection<\/strong> of the function\u00a0[latex]f(x)[\/latex] sometimes called a reflection about the y-axis.,<\/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 a function, reflect the graph both vertically and horizontally.<\/strong><\/p>\n<ol>\n<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n<li>Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Reflecting a Graph Horizontally and Vertically<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Reflect the graph of\u00a0[latex]s(t)=\\sqrt{t}[\/latex] (a) vertically and (b) horizontally.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) Reflecting the graph vertically means that each output value will be reflected over the horizontal t-axis shown in Figure 13.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_573\" aria-describedby=\"caption-attachment-573\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-573\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-300x136.jpeg\" alt=\"\" width=\"300\" height=\"136\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-300x136.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-768x348.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-65x29.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-225x102.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13-350x159.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-13.jpeg 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-573\" class=\"wp-caption-text\">Figure 13 Vertical reflection of the square root function<\/figcaption><\/figure>\n<p>Because each output value is the opposite of the original output value, we can write<\/p>\n<p style=\"text-align: center;\">[latex]V9t)=-s(t) \\ \\ \\text{or} \\ \\ V(t)=-\\sqrt{t}[\/latex]<\/p>\n<p>Notice that this is an outside change, or vertical shift, that affects the output\u00a0[latex]s(t)[\/latex] values, so the negative sign belongs outside of the function.<\/p>\n<p>(b) Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 14.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_574\" aria-describedby=\"caption-attachment-574\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-574\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-300x142.jpeg\" alt=\"\" width=\"300\" height=\"142\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-300x142.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-768x363.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-65x31.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-225x106.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14-350x166.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-14.jpeg 782w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-574\" class=\"wp-caption-text\">Figure 14 Horizontal reflection of the square root function<\/figcaption><\/figure>\n<p>Because each input value is the opposite of the original input value, we can write<\/p>\n<p style=\"text-align: center;\">[latex]H(t)=s(-1) \\ \\ \\text{or} \\ \\ H(t)=\\sqrt{-t}[\/latex]<\/p>\n<p>Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.<\/p>\n<p>Note that these transformations can affect the domain and range of the functions. While the original square root function has domain\u00a0[latex][0, \\infty)[\/latex] and range\u00a0[latex][0, \\infty),[\/latex] the vertical reflection gives the\u00a0[latex]V(t)[\/latex] function the range\u00a0[latex](-\\infty, 0][\/latex] and the horizontal reflection gives the\u00a0[latex]H(t)[\/latex] function the domain [latex](-\\infty, 0].[\/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 #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Reflect the graph of\u00a0[latex]f(x)=|x-1|[\/latex] (a) vertically and (b) horizontally.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Reflecting a Tabular Function Horizontally and Vertically<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f(x)[\/latex] is given as Table 6. Create a table for the functions below.<\/p>\n<p>(a) [latex]g(x)=-f(x)[\/latex]<\/p>\n<p>(b) [latex]h(x)=f(-x)[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 6<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">1<\/td>\n<td style=\"width: 20%;\">3<\/td>\n<td style=\"width: 20%;\">7<\/td>\n<td style=\"width: 20%;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) For\u00a0[latex]g(x)[\/latex] the negative sign outside the function indicates a vertical reflection, so the x-values stay the same and each output value will be the opposite of the original output value. See Table 7.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 7<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]g(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">-1<\/td>\n<td style=\"width: 20%;\">-3<\/td>\n<td style=\"width: 20%;\">-7<\/td>\n<td style=\"width: 20%;\">-11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(b) For\u00a0[latex]h(x)[\/latex] the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the\u00a0[latex]h(x)[\/latex] values stay the same as the\u00a0[latex]f(x)[\/latex] values. See Table 8.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 32px;\">\n<caption>Table 8<\/caption>\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">-2<\/td>\n<td style=\"width: 20%; height: 17px;\">-4<\/td>\n<td style=\"width: 20%; height: 17px;\">-6<\/td>\n<td style=\"width: 20%; height: 17px;\">-8<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 20%; height: 15px;\">[latex]h(x)[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">1<\/td>\n<td style=\"width: 20%; height: 15px;\">3<\/td>\n<td style=\"width: 20%; height: 15px;\">7<\/td>\n<td style=\"width: 20%; height: 15px;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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>A function\u00a0[latex]f(x)[\/latex] is given as Table 9. Create a table for the functions below.<\/p>\n<p>(a) [latex]g(x)=-f(x)[\/latex]<\/p>\n<p>(b) [latex]h(x)=f(-x)[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 9<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">-2<\/td>\n<td style=\"width: 20%;\">0<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">5<\/td>\n<td style=\"width: 20%;\">10<\/td>\n<td style=\"width: 20%;\">15<\/td>\n<td style=\"width: 20%;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Applying a Learning Model Equation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A common model for learning has an equation similar to\u00a0[latex]k(t)=-2^{-t}+1,[\/latex] where\u00a0[latex]k[\/latex] is the percentage of mastery that can be achieved after\u00a0[latex]t[\/latex] practice sessions. This is a transformation of the function\u00a0[latex]f(t)=2^t[\/latex] shown in Figure 15. Sketch a graph of [latex]k(t).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_578\" aria-describedby=\"caption-attachment-578\" style=\"width: 285px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-578\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15-285x300.jpeg\" alt=\"\" width=\"285\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15-285x300.jpeg 285w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15-225x237.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15-350x369.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-15.jpeg 353w\" sizes=\"auto, (max-width: 285px) 100vw, 285px\" \/><figcaption id=\"caption-attachment-578\" 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>This equation combines three transformations into one equation.<\/p>\n<ul>\n<li>A horizontal reflection: [latex]f(-t)=2^{-t}[\/latex]<\/li>\n<li>A vertical reflection: [latex]-f(-t)=-2^{-t}[\/latex]<\/li>\n<li>A vertical shift: [latex]-f(-t)+1=-2^{-t}+1[\/latex]<\/li>\n<\/ul>\n<p>We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points\u00a0[latex](0, 1)[\/latex] and [latex](1, 2).[\/latex]<\/p>\n<ol>\n<li>First, we apply a horizontal reflection: [latex](0, 1) (-1, 2).[\/latex]<\/li>\n<li>Then, we apply a vertical reflection: [latex](0, -1) (-1, -2)[\/latex]<\/li>\n<li>Finally, we apply a vertical shift: [latex](0, 0) (-1, -1)[\/latex]<\/li>\n<\/ol>\n<p>This means that the original points,\u00a0[latex](0, 1)[\/latex] and\u00a0[latex](1, 2)[\/latex] become\u00a0[latex](0, 0)[\/latex] and\u00a0[latex]- (-1, -1)[\/latex] after we apply the transformations.<\/p>\n<p>In Figure 16, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_579\" aria-describedby=\"caption-attachment-579\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-579\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-300x135.jpeg\" alt=\"\" width=\"300\" height=\"135\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-300x135.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-768x345.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-65x29.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-225x101.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16-350x157.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-16.jpeg 914w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-579\" class=\"wp-caption-text\">Figure 16<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>As a model for learning, this function would be limited to a domain of\u00a0[latex]t\\ge0,[\/latex] with corresponding range [latex][0, 1).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165137657438\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135209709\" data-type=\"commentary\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the toolkit function\u00a0[latex]f(x)=x^2,[\/latex] graph\u00a0[latex]g(x)=-f(x)[\/latex] and\u00a0[latex]h(x)=f(-x).[\/latex] Take note of any surprising behavior for these functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135182974\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining Even and Odd Functions<\/h2>\n<p id=\"fs-id1165135532474\">Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions\u00a0[latex]f(x)=x^2[\/latex] or\u00a0[latex]f(x)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the <em data-effect=\"italics\">y<\/em>-axis. A function whose graph is symmetric about the <em data-effect=\"italics\">y<\/em>-axis is called an <strong>even function.<\/strong><\/p>\n<p id=\"fs-id1165137939530\">If the graphs of\u00a0[latex]f(x)=x^3[\/latex] or\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] were reflected over <em data-effect=\"italics\">both<\/em> axes, the result would be the original graph, as shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_022\">Figure 17<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_580\" aria-describedby=\"caption-attachment-580\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-580\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-300x109.jpeg\" alt=\"\" width=\"300\" height=\"109\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-300x109.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-1024x371.jpeg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-768x278.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-65x24.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-225x82.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17-350x127.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-17.jpeg 1142w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-580\" class=\"wp-caption-text\">Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.<\/figcaption><\/figure>\n<p id=\"fs-id1165137406881\">We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\n<p id=\"fs-id1165134573214\">Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example,\u00a0[latex]f(x)=2^x[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f(x)=0.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Even and Odd Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=f(-x)[\/latex]<\/p>\n<p>The graph of an even function is symmetric about the [latex]y-[\/latex]axis.<\/p>\n<p>A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=-f(-x)[\/latex]<\/p>\n<p>The graph of an odd function is symmetric about the origin.<\/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 formula for a function, determine if the function is even, odd, or neither.<\/strong><\/p>\n<ol>\n<li>Determine whether the function satisfies\u00a0[latex]f(x)=f(-x).[\/latex] If it does, it is even.<\/li>\n<li>Determine whether the function satisfies\u00a0[latex]f(x)=-f(-x).[\/latex] If is does, it is odd.<\/li>\n<li>If the function does not satisfy either rule, it is neither even nor odd.<\/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: Determining whether a Function is Even, Odd, or Neither<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Is the function\u00a0[latex]f(x)=x^3+2x[\/latex] even, odd, or neither?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\n<p style=\"text-align: center;\">[latex]f(-x)=(-x)^3+2(-x)=-x^3-2x[\/latex]<\/p>\n<p>This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\n<p style=\"text-align: center;\">[latex]-f(-x)=-(-x^3-2x)=x^2+2x[\/latex]<\/p>\n<p>Because\u00a0[latex]-f(-x)=f(x),[\/latex] this is an odd function.<\/p>\n<h3>Analysis<\/h3>\n<p>Consider the graph\u00a0[latex]f[\/latex] of in Figure 18. Notice that the graph is symmetric about the origin. For every point\u00a0[latex](x, y)[\/latex] on the graph, the corresponding point\u00a0[latex](-x, -y)[\/latex] is also on the graph. For example, [latex](1, 3)[\/latex] is on the graph of\u00a0[latex]f[\/latex] and the corresponding point\u00a0[latex](-1, -3)[\/latex] is also on the graph.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_581\" aria-describedby=\"caption-attachment-581\" style=\"width: 291px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-581\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18-291x300.jpeg\" alt=\"\" width=\"291\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18-291x300.jpeg 291w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18-225x232.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18-350x361.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-18.jpeg 362w\" sizes=\"auto, (max-width: 291px) 100vw, 291px\" \/><figcaption id=\"caption-attachment-581\" class=\"wp-caption-text\">Figure 18<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Is the function\u00a0[latex]f(s)=s^4+3s^2+7[\/latex] even, odd, or neither?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137654768\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Functions Using Stretches and Compressions<\/h2>\n<p id=\"fs-id1165137654773\">Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\n<p id=\"fs-id1165137675403\">We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\n<section id=\"fs-id1165137793506\" data-depth=\"2\">\n<h3 data-type=\"title\">Vertical Stretches and Compressions<\/h3>\n<p id=\"fs-id1165137461225\">When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a<strong> vertical compression<\/strong>. <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_025\">Figure 19<\/a> shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_582\" aria-describedby=\"caption-attachment-582\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-582\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19-300x201.jpeg\" alt=\"\" width=\"300\" height=\"201\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19-300x201.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19-225x151.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19-350x234.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-19.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-582\" class=\"wp-caption-text\">Figure 19 Vertical stretch and compression<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Vertical Stretches and Compressions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given a function\u00a0[latex]f(x),[\/latex] a new function\u00a0[latex]g(x)=af(x),[\/latex] where\u00a0[latex]a[\/latex] is a constant, is a <strong>vertical stretch<\/strong> or\u00a0<strong>vertical compression<\/strong> of the function [latex]f(x).[\/latex]<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>If\u00a0[latex]a>1,[\/latex] then the graph will be stretched.<\/li>\n<li>If\u00a0[latex]0<a<1,[\/latex] then the graph will be compressed.<\/li>\n<li>If\u00a0[latex]a<0,[\/latex] then there will be a combination of a vertical stretch or compression with a vertical reflection.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\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 a function, graph its vertical stretch.<\/strong><\/p>\n<ol>\n<li>Identify the value of [latex]a.[\/latex]<\/li>\n<li>Multiply all range values by [latex]a.[\/latex]<\/li>\n<li>If [latex]a>1,[\/latex] the graph is stretched by a factor of [latex]a.[\/latex]<\/li>\n<\/ol>\n<p style=\"padding-left: 40px;\">If [latex]0< a<1,[\/latex] the graph is compressed by a factor of [latex]a.[\/latex]<\/p>\n<p style=\"padding-left: 40px;\">If\u00a0[latex]a<0,[\/latex] the graph is either stretched or compressed and also reflection about the x-axis.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 13: Graphing a Vertical Stretch<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]P(t)[\/latex] models the population of fruit flies. The graph is shown in Figure 20.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_583\" aria-describedby=\"caption-attachment-583\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-583\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20-300x226.jpeg\" alt=\"\" width=\"300\" height=\"226\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20-300x226.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20-65x49.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20-225x170.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20-350x264.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-20.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-583\" class=\"wp-caption-text\">Figure 20<\/figcaption><\/figure>\n<p>A biology student at CCA is comparing this population to another population,\u00a0[latex]Q,[\/latex] whose growth follows the same pattern, but is twice as large. Sketch the graph of this population.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values. Graphically, this is shown in Figure 21.<\/p>\n<p>If we choose four reference points,\u00a0[latex](0, 1), (3, 3), (6, 2)[\/latex] and\u00a0[latex](7, 0)[\/latex] we will multiply all of the outputs by 2.<\/p>\n<p>The following shows where the new points for the new graph will be located.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} (0, 1) &\\rightarrow & (0, 2) \\\\ (3, 3) &\\rightarrow & (3, 6) \\\\ (6, 2) &\\rightarrow & (6, 4) \\\\ (7, 0) &\\rightarrow & (7, 0) \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_584\" aria-describedby=\"caption-attachment-584\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-584\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21-300x226.jpeg\" alt=\"\" width=\"300\" height=\"226\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21-300x226.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21-65x49.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21-225x170.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21-350x264.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-21.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-584\" class=\"wp-caption-text\">Figure 21<\/figcaption><\/figure>\n<p>Symbolically, this relationship is written as<\/p>\n<p style=\"text-align: center;\">[latex]Q(t)=2P(t)[\/latex]<\/p>\n<p>This means that for any input\u00a0[latex]t[\/latex] the value of the function\u00a0[latex]Q[\/latex] is twice the value of the function\u00a0[latex]P.[\/latex] Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values,\u00a0[latex]t,[\/latex] stay the same while the output values are twice as large as before.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How Tow<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<\/strong><\/p>\n<ol>\n<li>Determine the value of [latex]a.[\/latex]<\/li>\n<li>Multiply all of the output values by [latex]a.[\/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 14: Finding a Vertical Compression of a Tabular Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f[\/latex] is given as Table 10. Create a table for the function [latex]g(x)=\\frac{1}{2}f(x).[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 10<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">1<\/td>\n<td style=\"width: 20%;\">3<\/td>\n<td style=\"width: 20%;\">7<\/td>\n<td style=\"width: 20%;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The formula\u00a0[latex]g(x)=\\frac{1}{2}f(x)[\/latex] tells us that the output values of\u00a0[latex]g[\/latex] are half of the output values of\u00a0[latex]f[\/latex] with the same inputs. For example, we know that\u00a0[latex]f(4)=3.[\/latex] Then<\/p>\n<p style=\"text-align: center;\">[latex]g(4)=\\frac{1}{2}f(4)=\\frac{1}{2}(3)=\\frac{3}{2}[\/latex]<\/p>\n<p>We do the same for the other values to produce Table 11.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 30px;\">\n<caption>Table 11<\/caption>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 20%; height: 15px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">2<\/td>\n<td style=\"width: 20%; height: 15px;\">4<\/td>\n<td style=\"width: 20%; height: 15px;\">6<\/td>\n<td style=\"width: 20%; height: 15px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 20%; height: 15px;\">[latex]g(x)[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">[latex]\\frac{3}{2}[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">[latex]\\frac{7}{2}[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">[latex]\\frac{11}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Analysis<\/h3>\n<p>The result is that the function\u00a0[latex]g(x)[\/latex] has been compressed vertically by\u00a0[latex]\\frac{1}{2}.[\/latex] Each output value is divided in half, so the graph is half the original height.<\/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 #9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f[\/latex] is given as Table 12. Create a table for the function [latex]g(x)=\\frac{3}{4}f(x).[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 12<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">12<\/td>\n<td style=\"width: 20%;\">16<\/td>\n<td style=\"width: 20%;\">20<\/td>\n<td style=\"width: 20%;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 15: Recognizing a Vertical Stretch<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The graph in Figure 22 is a transformation of the toolkit function\u00a0[latex]f(x)=x^3.[\/latex] Relate this new function\u00a0[latex]g(x)[\/latex] to\u00a0[latex]f(x)[\/latex] and then find a formula for [latex]g(x).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_585\" aria-describedby=\"caption-attachment-585\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-585\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22-300x272.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22-65x59.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22-225x204.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22-350x318.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-22.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-585\" 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>When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that\u00a0[latex]g(2)=2.[\/latex] With the basic cubic function at the same input,\u00a0[latex]f(2)=2^3=8.[\/latex] Based on that, it appears that the outputs of\u00a0[latex]g[\/latex] are\u00a0[latex]\\frac{1}{4}[\/latex] the outputs of the function\u00a0[latex]f[\/latex] because\u00a0[latex]g(2)=\\frac{1}{4}f(2).[\/latex] From this we can fairly safely conclude that [latex]g(x)=\\frac{1}{4}f(x).[\/latex]<\/p>\n<p>We can write a formula for\u00a0[latex]g[\/latex] by using the definition of the function [latex]f.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]g(x)=\\frac{1}{4}f(x)=\\frac{1}{4}x^3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135344103\" data-depth=\"2\">\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>Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.<\/p>\n<\/div>\n<\/div>\n<h3 data-type=\"title\">Horizontal Stretches and Compressions<\/h3>\n<p id=\"fs-id1165133167751\">Now we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_586\" aria-describedby=\"caption-attachment-586\" style=\"width: 284px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-586\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23-284x300.jpeg\" alt=\"\" width=\"284\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23-284x300.jpeg 284w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23-65x69.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23-225x237.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23-350x369.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-23.jpeg 487w\" sizes=\"auto, (max-width: 284px) 100vw, 284px\" \/><figcaption id=\"caption-attachment-586\" class=\"wp-caption-text\">Figure 23<\/figcaption><\/figure>\n<p id=\"fs-id1165133366207\">Given a function\u00a0[latex]y=f(x),[\/latex] the form\u00a0[latex]y=f(bx)[\/latex] results in a horizontal stretch or compression. Consider the function\u00a0[latex]y=x^2.[\/latex] Observe <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_029\">Figure 23<\/a>. The graph of\u00a0[latex]y=(0.5x)^2[\/latex] is a horizontal stretch of the graph of the function\u00a0[latex]y=x^2[\/latex] by a factor of 2. The graph of\u00a0[latex]y=(2x)^2[\/latex] is a horizontal compression of the graph of the function\u00a0[latex]y=x^2[\/latex] by a factor of [latex]\\frac{1}{2}[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Horizontal Stretches and Compressions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given a function\u00a0[latex]f(x),[\/latex] a new function\u00a0[latex]g(x)=f(bx),[\/latex] where\u00a0[latex]b[\/latex] is a constant, is a <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f(x).[\/latex]<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>If\u00a0[latex]b>1[\/latex] then the graph will be compressed by [latex]b.[\/latex]<\/li>\n<li>If\u00a0[latex]0<b<1,[\/latex] then the graph will be stretched by [latex]\\frac{1}{b}.[\/latex]<\/li>\n<li>If\u00a0[latex]b<0,[\/latex] then there will be a combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\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 a description of a function, sketch a horizontal compression or stretch.<\/strong><\/p>\n<ol>\n<li>Write a formula to represent the function.<\/li>\n<li>Set\u00a0[latex]g(x)=f(bx)[\/latex] where\u00a0[latex]b>1[\/latex] for a compression or\u00a0[latex]0<b<1[\/latex] for a stretch.<\/li>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 16: Graphing a Horizontal Compression<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Suppose a biology student is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population,\u00a0[latex]R,[\/latex] will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Symbolically, we could write<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcll} R(1) &=& P(2), \\\\ R(2) &=& P(4), & \\text{and in general,} \\\\ R(t) &=& P(2t). \\end{array}[\/latex]<\/p>\n<p>See Figure 24 for a graphical comparison of the original population and the compressed population.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_587\" aria-describedby=\"caption-attachment-587\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-587\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-300x123.jpeg\" alt=\"\" width=\"300\" height=\"123\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-300x123.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-768x316.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-65x27.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-225x92.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24-350x144.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-24.jpeg 976w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-587\" class=\"wp-caption-text\">Figure 24 (a) Original population graph (b) Compressed population graph<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17: Finding a Horizontal Stretch for a Tabular Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f(x)[\/latex] is given as Table 13. Create a table for the function [latex]g(x)=f(\\frac{1}{2}x).[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 32px;\">\n<caption>Table 13<\/caption>\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">2<\/td>\n<td style=\"width: 20%; height: 17px;\">4<\/td>\n<td style=\"width: 20%; height: 17px;\">6<\/td>\n<td style=\"width: 20%; height: 17px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 20%; height: 15px;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%; height: 15px;\">1<\/td>\n<td style=\"width: 20%; height: 15px;\">3<\/td>\n<td style=\"width: 20%; height: 15px;\">7<\/td>\n<td style=\"width: 20%; height: 15px;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The formula [latex]g(x)=f(\\frac{1}{2}x)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g(2)[\/latex] because [latex]g(2)=f(\\frac{1}{2}\\cdot 2)=f(1),[\/latex] and we do not have a value for [latex]f(1)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g(4).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]g(4)=f(\\frac{1}{2}\\cdot 4)=f(2)=1[\/latex]<\/p>\n<p>We do the same for the other values to produce Table 14.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 14<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<td style=\"width: 20%;\">12<\/td>\n<td style=\"width: 20%;\">16<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]g(x)[\/latex]<\/td>\n<td style=\"width: 20%;\">1<\/td>\n<td style=\"width: 20%;\">3<\/td>\n<td style=\"width: 20%;\">7<\/td>\n<td style=\"width: 20%;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Figure 25 shows the graphs of both of these sets of points.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_588\" aria-describedby=\"caption-attachment-588\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-588\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-300x102.jpeg\" alt=\"\" width=\"300\" height=\"102\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-300x102.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-768x262.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-65x22.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-225x77.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25-350x120.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-25.jpeg 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-588\" class=\"wp-caption-text\">Figure 25<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>Because each input value has been doubled, the result is that the function\u00a0[latex]g(x)[\/latex] has been stretched horizontally by a factor of 2.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 18: Recognizing a Horizontal Compression on a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Relate the function\u00a0[latex]g(x)[\/latex] to\u00a0[latex]f(x)[\/latex] in Figure 26.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_589\" aria-describedby=\"caption-attachment-589\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-589\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26-300x179.jpeg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26-300x179.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26-225x134.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26-350x209.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-26.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-589\" class=\"wp-caption-text\">Figure 26<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The graph of [latex]\u00a0g(x)[\/latex] looks like the graph of[latex]f(x)[\/latex]\u00a0 horizontally compressed. Because [latex]f(x)[\/latex] ends at [latex](6, 4)[\/latex] and [latex]g(x)[\/latex] ends at [latex](2, 4)[\/latex] we can see that the values have been compressed by [latex]\\frac{1}{3},[\/latex] because [latex]6(\\frac{1}{3})=2.[\/latex] We might also notice that [latex]g(2)=f(6)[\/latex] and [latex]g(1)=f(3).[\/latex] Either way, we can describe this relationship as [latex]g(x)=f(3x).[\/latex] This is a horizontal compression by [latex]\\frac{1}{3}.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of\u00a0[latex]\\frac{1}{4}[\/latex] in our function:\u00a0[latex]f(\\frac{1}{4}x).[\/latex] This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.<\/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 #11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1165137676302\" data-depth=\"1\">\n<h2 data-type=\"title\">Performing a Sequence of Transformations<\/h2>\n<p id=\"fs-id1165137387533\">When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.<\/p>\n<p id=\"fs-id1165137387540\">When we see an expression such as [latex]2f(x)+3,[\/latex] which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of [latex]f(x),[\/latex] we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.<\/p>\n<p id=\"fs-id1165137483273\">Horizontal transformations are a little trickier to think about. When we write\u00a0[latex]g(x)=f(2x+3),[\/latex] for example, we have to think about how the inputs to the function\u00a0[latex]g[\/latex] relate to the inputs to the function\u00a0[latex]f.[\/latex] Suppose we know\u00a0[latex]f(7)=12.[\/latex] What input to\u00a0[latex]g[\/latex] would produce that output? In other words, what value of\u00a0[latex]x[\/latex] will allow\u00a0[latex]g(x)=f(2x+3)=12?[\/latex] We would need\u00a0[latex]2x+3=7.[\/latex] To solve for\u00a0[latex]x,[\/latex] we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.<\/p>\n<p id=\"fs-id1165137580228\">This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.<\/p>\n<p style=\"text-align: center;\">[latex]f(bx+p)=f(b(x+\\frac{p}{b}))[\/latex]<\/p>\n<p id=\"fs-id1165137427526\">Let\u2019s work through an example.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(2x+4)^2[\/latex]<\/p>\n<p id=\"fs-id1165137438427\">We can factor out a 2.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(2(x+2))^2[\/latex]<\/p>\n<p id=\"fs-id1165135349189\">Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Combining Transformations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>When combining vertical transformations written in the form\u00a0[latex]af(x)_k,[\/latex] first vertically stretch by\u00a0[latex]a[\/latex] and then vertically stretch by [latex]k.[\/latex]<\/p>\n<p>When combining horizontal transformations written in the form\u00a0[latex]f(bx-h),[\/latex] first horizontally shift by\u00a0[latex]h[\/latex] and then horizontally stretch by [latex]1b.[\/latex]<\/p>\n<p>When combining horizontal transformations written in the form\u00a0[latex]f(b(x-h)),[\/latex] first horizontally stretch by\u00a0[latex]\\frac{1}{b},[\/latex] and then horizontally shift by [latex]h.[\/latex]<\/p>\n<p>Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 19: Finding a Triple Transformation of a Tabular Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given Table 15 for the function\u00a0[latex]f(x),[\/latex] create a table of values for the function [latex]g(x)=2f(3x)+1.[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 34px;\">\n<caption>Table 15<\/caption>\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">6<\/td>\n<td style=\"width: 20%; height: 17px;\">12<\/td>\n<td style=\"width: 20%; height: 17px;\">18<\/td>\n<td style=\"width: 20%; height: 17px;\">24<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 20%; height: 17px;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 20%; height: 17px;\">10<\/td>\n<td style=\"width: 20%; height: 17px;\">14<\/td>\n<td style=\"width: 20%; height: 17px;\">15<\/td>\n<td style=\"width: 20%; height: 17px;\">17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>There are three steps to the transformation, and we will work from the inside out. Starting with the horizontal transformations,\u00a0[latex]f(3x)[\/latex] is a horizontal compression by\u00a0[latex]\\frac{1}{3},[\/latex] which means we multiply each [latex]x-[\/latex]value by\u00a0[latex]\\frac{1}{3}.[\/latex] See Table 16.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 16<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]f(3x)[\/latex]<\/td>\n<td style=\"width: 20%;\">10<\/td>\n<td style=\"width: 20%;\">14<\/td>\n<td style=\"width: 20%;\">15<\/td>\n<td style=\"width: 20%;\">17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See Table 17.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 17<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]2f(3x)[\/latex]<\/td>\n<td style=\"width: 20%;\">20<\/td>\n<td style=\"width: 20%;\">28<\/td>\n<td style=\"width: 20%;\">30<\/td>\n<td style=\"width: 20%;\">34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Finally, we can apply the vertical shift, which will add 1 to all the output values. See Table 18.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 18<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 20%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 20%;\">2<\/td>\n<td style=\"width: 20%;\">4<\/td>\n<td style=\"width: 20%;\">6<\/td>\n<td style=\"width: 20%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\">[latex]g(x)=2f(3x)+1[\/latex]<\/td>\n<td style=\"width: 20%;\">21<\/td>\n<td style=\"width: 20%;\">29<\/td>\n<td style=\"width: 20%;\">31<\/td>\n<td style=\"width: 20%;\">35<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20: Finding a Triple Transformation of a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the graph of\u00a0[latex]f(x)[\/latex] in Figure 27 to sketch a graph of [latex]k(x)=f(\\frac{1}{2}x+1)-3.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_593\" aria-describedby=\"caption-attachment-593\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-593\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-27.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-593\" class=\"wp-caption-text\">Figure 27<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To simplify, let&#8217;s start by factoring out the inside of the function.<\/p>\n<p style=\"text-align: center;\">[latex]f(\\frac{1}{2}x+1)-3=f(\\frac{1}{2}(x=2))-3[\/latex]<\/p>\n<p>By factoring the inside, we can first horizontally stretch by 2, as indicated by the\u00a0[latex]\\frac{1}{2}[\/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point\u00a0[latex](0, 2)[\/latex] remains at\u00a0[latex](0, 2)[\/latex] while the point\u00a0[latex](2, 0)[\/latex] will stretch to\u00a0[latex](4, 0).[\/latex] See Figure 28.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_594\" aria-describedby=\"caption-attachment-594\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-594\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-28.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-594\" class=\"wp-caption-text\">Figure 28<\/figcaption><\/figure>\n<p>Next, we horizontally shift left by 2 units, as indicated by\u00a0[latex]x+2.[\/latex] See Figure 29.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_595\" aria-describedby=\"caption-attachment-595\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-595\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-29.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-595\" class=\"wp-caption-text\">Figure 29<\/figcaption><\/figure>\n<p>Last, we vertically shift down by 3 to complete our sketch, as indicated by the\u00a0[latex]-3[\/latex] on the outside of the function. See Figure 30.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_596\" aria-describedby=\"caption-attachment-596\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-596\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-30.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-596\" class=\"wp-caption-text\">Figure 30<\/figcaption><\/figure>\n<\/details>\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 transformation of functions.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=An29CALYjAA\">Function Transformations<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\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\">3.5 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165137436217\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137728393\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137728398\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135180434\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137728398-solution\">1<\/a><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137550351\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137550353\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137922588\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137922590\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137922588-solution\">3<\/a><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134227890\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134227892\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the <em data-effect=\"italics\">x<\/em>-axis from a reflection with respect to the <em data-effect=\"italics\">y<\/em>-axis?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137734659\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137734661\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137734659-solution\">5<\/a><span class=\"os-divider\">. <\/span>How can you determine whether a function is odd or even from the formula of the function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137454081\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"eip-375\">For the following exercises, write a formula for the function obtained when the graph is shifted as described.<\/p>\n<div id=\"fs-id1165135168321\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135168323\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">.\u00a0[latex]f(x)=\\sqrt{x}[\/latex] <\/span>is shifted up 1 unit and to the left 2 units.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134041416\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134041418\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134041416-solution\">7<\/a><span class=\"os-divider\">.\u00a0[latex]f(x)=|x|[\/latex] <\/span>is shifted down 3 units and to the right 1 unit.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134061972\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134061974\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">.\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] <\/span>is shifted down 4 units and to the right 3 units.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137805973\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137805975\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137805973-solution\">9<\/a><span class=\"os-divider\">.\u00a0[latex]f(x)=\\frac{1}{x^2}[\/latex] <\/span>is shifted up 2 units and to the left 4 units.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137407590\">For the following exercises, describe how the graph of the function is a transformation of the graph of the original function [latex]f.[\/latex]<\/p>\n<div id=\"fs-id1165135397258\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133111635\" data-type=\"problem\">\n<p><span class=\"os-number\">10.<\/span> [latex]y=f(x-49)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135193434\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135193436\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135193434-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]y=f(x+43)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135571667\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135571670\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]y=f(x+3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135575988\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135575991\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135575988-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]y=f(x-4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135701452\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137551379\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]y=f(x)+5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137936723\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137936725\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137936723-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]y=f(x)+8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135369401\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135369403\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]y=f(x)-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137454950\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132945534\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137454950-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]y=f(x)-7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135205732\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135205734\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]y=f(x-2)+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134220843\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134220845\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134220843-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]y=f(x+4)-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137896305\">For the following exercises, determine the interval(s) on which the function is increasing and decreasing.<\/p>\n<div id=\"fs-id1165137896310\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137896312\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">.<\/span> [latex]f(x)=4(x+1)^2-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135547247\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135547250\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135547247-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=5(x+3)^2-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137679200\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137679202\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]a(x)=\\sqrt{-x+4}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135650778\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135250825\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135650778-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]k(x)=\\sqrt[-3]{x}-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135403290\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165137694193\">For the following exercises, use the graph of\u00a0[latex]f(x)=2^x[\/latex] shown in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_201\">Figure 31<\/a> to sketch a graph of each transformation of [latex]f(x).[\/latex]<\/p>\n<\/section>\n<section data-depth=\"2\">\n<div id=\"fs-id1165135394223\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135394226\" data-type=\"problem\">\n<figure id=\"attachment_597\" aria-describedby=\"caption-attachment-597\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-597\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5-fig-31.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-597\" class=\"wp-caption-text\">Figure 31<\/figcaption><\/figure>\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=2^x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137887426\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137887428\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137887426-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=2^x-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135436604\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137724122\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]w(x)=2^{x-1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137448386\">For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.<\/p>\n<div id=\"fs-id1165137448391\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137448393\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137448391-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]f(t)=(t+1)^2-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137932662\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135209555\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=|x-1|+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135421533\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135421535\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135421533-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]k(x)=(x+2)^3-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137424880\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137424883\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]m(t)=3+\\sqrt{t+2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137464226\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<div id=\"fs-id1165137681998\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137682000\" class=\"material-set-2\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137681998-solution\">31<\/a><span class=\"os-divider\">. <\/span>Tabular representations for the functions\u00a0[latex]f, g,[\/latex] and\u00a0[latex]h[\/latex] are given below. Write\u00a0[latex]g(x)[\/latex] and\u00a0[latex]h(x)[\/latex] as transformations of [latex]f(x).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>&nbsp;<\/p>\n<div id=\"fs-id1165137432561\" class=\"os-table\">\n<table class=\"grid\" data-id=\"fs-id1165137432561\" data-label=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">\u22123<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135634096\" class=\"os-table\">\n<table class=\"grid\" data-id=\"fs-id1165135634096\" data-label=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">3<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]g(x)[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">\u22123<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135330589\" class=\"os-table\">\n<table class=\"grid\" data-id=\"fs-id1165135330589\" data-label=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]h(x)[\/latex]<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137734475\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137734477\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Tabular representations for the functions\u00a0[latex]f, g,[\/latex] and\u00a0[latex]h[\/latex] are given below. Write\u00a0[latex]g(x)[\/latex] and\u00a0[latex]h(x)[\/latex] as transformations of [latex]f(x).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<div id=\"fs-id1165134558032\" class=\"os-table\">\n<table class=\"grid\" data-id=\"fs-id1165134558032\" data-label=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)[\/latex]<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">\u22123<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165134380916\" class=\"os-table\">\n<table class=\"grid\" data-id=\"fs-id1165134380916\" data-label=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">\u22123<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">1<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]g(x)[\/latex]<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">\u22123<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137894261\" class=\"os-table\">\n<table class=\"grid\" data-id=\"fs-id1165137894261\" data-label=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22121<\/td>\n<td data-align=\"center\">0<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]h(x)[\/latex]<\/td>\n<td data-align=\"center\">\u22122<\/td>\n<td data-align=\"center\">\u22124<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137570566\">For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/p>\n<div id=\"fs-id1165137431229\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137431231\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137431229-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135543438\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-598\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.33-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.33-295x300.jpeg 295w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.33-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.33-225x229.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.33.jpeg 312w\" sizes=\"auto, (max-width: 295px) 100vw, 295px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135481230\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135481232\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137851362\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-599\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.34.jpeg 312w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137817635\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137817637\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137817635-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165133341017\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-600\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.35-295x300.jpeg\" alt=\"\" width=\"295\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.35-295x300.jpeg 295w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.35-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.35-225x229.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.35.jpeg 312w\" sizes=\"auto, (max-width: 295px) 100vw, 295px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132929618\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132929620\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135203675\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-601\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36-275x300.jpeg\" alt=\"\" width=\"275\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36-275x300.jpeg 275w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36-65x71.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36-225x245.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36-350x382.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.36.jpeg 375w\" sizes=\"auto, (max-width: 275px) 100vw, 275px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135487204\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135487206\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135487204-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135535017\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-602\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.37-165x300.jpeg\" alt=\"\" width=\"165\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.37-165x300.jpeg 165w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.37-65x118.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.37.jpeg 225w\" sizes=\"auto, (max-width: 165px) 100vw, 165px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135190411\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135190413\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165133277626\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-603\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.38.jpeg\" alt=\"\" width=\"285\" height=\"226\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.38.jpeg 285w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.38-65x52.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.38-225x178.jpeg 225w\" sizes=\"auto, (max-width: 285px) 100vw, 285px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133103936\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133103938\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133103936-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134362846\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-604\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39-297x300.jpeg\" alt=\"\" width=\"297\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39-297x300.jpeg 297w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39-225x227.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39-350x354.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.39.jpeg 375w\" sizes=\"auto, (max-width: 297px) 100vw, 297px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134043550\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043552\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-605\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-225x227.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40-350x353.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.40.jpeg 375w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<p><span id=\"fs-id1165134036728\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><\/span>For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134187277\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134187279\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134187277-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137834957\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-606\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.41.jpeg\" alt=\"\" width=\"225\" height=\"225\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.41.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.41-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.41-65x65.jpeg 65w\" sizes=\"auto, (max-width: 225px) 100vw, 225px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137930320\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137930323\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135487154\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-607\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.42.jpeg\" alt=\"\" width=\"215\" height=\"222\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.42.jpeg 215w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.42-65x67.jpeg 65w\" sizes=\"auto, (max-width: 215px) 100vw, 215px\" \/><\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134177109\">For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.<\/p>\n<div id=\"fs-id1165134177113\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137806559\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134177113-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137806566\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-608\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.43-300x272.jpeg\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.43-300x272.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.43-65x59.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.43-225x204.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.43.jpeg 312w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135384400\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137693606\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137693612\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a cubic function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-609\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.44.jpeg\" alt=\"\" width=\"252\" height=\"251\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.44.jpeg 252w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.44-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.44-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.44-225x224.jpeg 225w\" sizes=\"auto, (max-width: 252px) 100vw, 252px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137892243\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134352554\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137892243-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134352561\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a square root function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-610\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.45.jpeg\" alt=\"\" width=\"184\" height=\"191\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.45.jpeg 184w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.45-65x67.jpeg 65w\" sizes=\"auto, (max-width: 184px) 100vw, 184px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135411377\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135411379\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165133155251\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-611\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.46.jpeg\" alt=\"\" width=\"285\" height=\"285\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.46.jpeg 285w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.46-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.46-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.5.46-225x225.jpeg 225w\" sizes=\"auto, (max-width: 285px) 100vw, 285px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165132924966\">For the following exercises, determine whether the function is odd, even, or neither.<\/p>\n<div id=\"fs-id1165132924969\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132924971\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165132924969-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135671514\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137761968\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137828008\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137828010\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137828008-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=\\frac{1}{x}+3x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134187165\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134187167\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=(x-2)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134039317\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134039319\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134039317-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=2x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135630957\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135630959\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=2x-x^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137571611\">For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f.[\/latex]<\/p>\n<div id=\"fs-id1165137599981\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137599983\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137599981-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=-f(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133065712\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133065714\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=f(-x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135412892\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135412894\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135412892-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=4f(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137939898\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137939900\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=6f(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135440224\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135440226\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135440224-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=f(5x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135364548\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135364550\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=f(2x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133307633\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133307635\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133307633-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=f(\\frac{1}{3}x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137854807\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137854809\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=f(\\frac{1}{5}x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137664915\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137664917\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137664915-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=3f(-x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135192324\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135192326\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=-f(3x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135637428\">For the following exercises, write a formula for the function\u00a0[latex]g[\/latex] that results when the graph of a given toolkit function is transformed as described.<\/p>\n<div id=\"fs-id1165135195127\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135195130\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135195127-solution\">63<\/a><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex]f(x)=|x|[\/latex] is reflected over the [latex]y-[\/latex]axis and horizontally compressed by a factor of [latex]\\frac{1}{4}.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135404231\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135404233\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex]f(x)=\\sqrt{x}[\/latex] is reflected over the [latex]x-[\/latex]axis and horizontally stretched by a factor of 2.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137634443\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137634445\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137634443-solution\">65<\/a><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex]f(x)=\\frac{1}{x^2}[\/latex] is vertically compressed by a factor of\u00a0[latex]\\frac{1}{3},[\/latex] then shifted to the left 2 units and down 3 units.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137731439\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137731442\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex]f(x)=\\frac{1]{x}[\/latex] is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137642586\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137642588\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137642586-solution\">67<\/a><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex]f(x)=x^2[\/latex] is vertically compressed by a factor of\u00a0[latex]\\frac{1}{2},[\/latex] then shifted to the right 5 units and up 1 unit.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137653191\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137653193\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>The graph of\u00a0[latex]f(x)=x^2[\/latex] is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137668699\">For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/p>\n<div id=\"fs-id1165137668704\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137668706\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137668704-solution\">69<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=4(x+1)^2-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135199465\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135199468\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=f(x+3)^2-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134069304\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134069306\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134069304-solution\">71<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=-2|x-4|+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137758532\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137758534\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span> [latex]k(x)=\\sqrt[-3]{x}-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135351654\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135351656\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135351654-solution\">73<\/a><span class=\"os-divider\">. <\/span> [latex]m(x)=\\frac{1}{2}x^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137832423\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137832425\" data-type=\"problem\">\n<p><span class=\"os-number\">74<\/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-id1165134155168\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134155170\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134155168-solution\">75<\/a><span class=\"os-divider\">. <\/span> [latex]p(x)=(\\frac{1}{3}x)^3-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137898977\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898979\" data-type=\"problem\">\n<p><span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span> [latex]q(x)=(\\frac{1}{4}x)^3+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137861993\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137861995\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137861993-solution\">77<\/a><span class=\"os-divider\">. <\/span> [latex]a(x)=\\sqrt{-x+4}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134338807\">For the following exercises, use the graph in <a class=\"autogenerated-content\" href=\"3-5-transformation-of-functions#Figure_01_05_233\">Figure 32<\/a> to sketch the given transformations.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_612\" aria-describedby=\"caption-attachment-612\" style=\"width: 298px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-612\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32-350x352.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/83.5-fig-32.jpeg 459w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><figcaption id=\"caption-attachment-612\" class=\"wp-caption-text\">Figure 32<\/figcaption><\/figure>\n<div id=\"fs-id1165135706785\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135208393\" data-type=\"problem\">\n<p><span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=f(x)-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135432954\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135432956\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135432954-solution\">79<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=-f(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137722436\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137722438\" data-type=\"problem\">\n<p><span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=f(x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134269560\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134269563\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134269560-solution\">81<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=f(x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-136","chapter","type-chapter","status-publish","hentry"],"part":105,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/136","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":23,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/136\/revisions"}],"predecessor-version":[{"id":1500,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/136\/revisions\/1500"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/136\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=136"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=136"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=136"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}