{"id":239,"date":"2025-04-09T17:33:04","date_gmt":"2025-04-09T17:33:04","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/6-2-graphs-of-exponential-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-29T17:27:48","modified_gmt":"2025-08-29T17:27:48","slug":"6-2-graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/6-2-graphs-of-exponential-functions\/","title":{"raw":"6.2 Graphs of Exponential Functions","rendered":"6.2 Graphs of Exponential 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_73c684c9-6dae-4f32-a1a1-5208b5bf59c2\" 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 exponential functions.<\/li>\r\n \t<li>Graph exponential functions using transformations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137442020\">As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\r\n\r\n<section id=\"fs-id1165135407520\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Exponential Functions<\/h2>\r\n<p id=\"fs-id1165137592823\">Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form [latex] f(x)=b^x [\/latex] whose base is greater than one. We\u2019ll use the function [latex] f(x)=2^x. [\/latex] Observe how the output values in Table 1 change as the input increases by 1.<\/p>\r\n\r\n<div id=\"Table_04_02_01\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_02_01\"><caption>Table 1<\/caption><colgroup> <col data-width=\"60\" \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -3 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 0 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 3 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x)=2^x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{8} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{4} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{2} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 4 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 8 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nEach output value is the product of the previous output and the base, 2. We call the base 2 the <em data-effect=\"italics\">constant ratio<\/em>. In fact, for any exponential function with the form [latex] f(x)=ab^x, [\/latex] b is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.\r\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\r\n\r\n<ul id=\"fs-id1165137658509\">\r\n \t<li>the output values are positive for all values of <em>x<\/em>;<\/li>\r\n \t<li>as<em> x<\/em> increases, the output values increase without bound; and<\/li>\r\n \t<li>as <em>x<\/em> decreases, the output values grow smaller, approaching zero.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137647215\">Figure 1 shows the exponential growth function [latex] f(x)=2^x. [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_934\" align=\"aligncenter\" width=\"272\"]<img class=\"size-medium wp-image-934\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1-272x300.webp\" alt=\"\" width=\"272\" height=\"300\" \/> Figure 1. Notice that the graph gets close to the x-axis, but never touches it.[\/caption]\r\n<p id=\"fs-id1165137459614\">The domain of [latex] f(x)=2^x [\/latex] is all real numbers, the range is [latex] (0, \\infty), [\/latex] and the horizontal asymptote is [latex] y=0. [\/latex]<\/p>\r\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <strong><span id=\"term-00003\" data-type=\"term\">exponential decay<\/span><\/strong>, we can create a table of values for a function of the form [latex] f(x)=b^x [\/latex] whose base is between zero and one. We\u2019ll use the function [latex] g(x)=\\left(\\frac{1}{2}\\right)^x. [\/latex] Observe how the output values in Table 2 change as the input increases by 1.<\/p>\r\n\r\n<div id=\"Table_04_02_02\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_02_02\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -3 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 0 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 3 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] g(x)=\\left(\\frac{1}{2}\\right)^x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 8 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 4 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{2} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{4} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{8} [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nAgain, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio [latex] \\frac{1}{2}. [\/latex]\r\n<p id=\"fs-id1165137452063\">Notice from the table that<\/p>\r\n\r\n<ul id=\"fs-id1165135499992\">\r\n \t<li>the output values are positive for all values of <em>x<\/em>;<\/li>\r\n \t<li>as <em>x<\/em> increases, the output values grow smaller, approaching zero; and<\/li>\r\n \t<li>as <em>x<\/em> decreases, the output values grow without bound.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137405421\">Figure 2 shows the exponential decay function, [latex] g(x)=\\left(\\frac{1}{2}\\right)^x. [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_935\" align=\"aligncenter\" width=\"281\"]<img class=\"size-medium wp-image-935\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2-281x300.webp\" alt=\"\" width=\"281\" height=\"300\" \/> Figure 2[\/caption]\r\n\r\n<div id=\"page_73c684c9-6dae-4f32-a1a1-5208b5bf59c2\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\"><section data-depth=\"1\">\r\n<p id=\"fs-id1165137723586\">The domain of [latex] g(x)=\\left(\\frac{1}{2}\\right)^x [\/latex] is all real numbers, the range is [latex] (0, \\infty), [\/latex] and the horizontal asymptote is [latex] y=0. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Characteristics of the Graph of the Parent Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAn exponential function with the form [latex] f(x)=b^x, b&gt; 0, b\\not=1, [\/latex]\u00a0has these characteristics:\r\n<ul>\r\n \t<li><span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">one-to-one<\/span> function<\/li>\r\n \t<li>horizontal asymptote: [latex] y=0 [\/latex]<\/li>\r\n \t<li>domain: [latex] (-\\infty, \\infty) [\/latex]<\/li>\r\n \t<li>range: [latex] (0, \\infty) [\/latex]<\/li>\r\n \t<li><em data-effect=\"italics\">x-<\/em>intercept: none<\/li>\r\n \t<li><em data-effect=\"italics\">y-<\/em>intercept: [latex] (0, 1) [\/latex]<\/li>\r\n \t<li>increasing if [latex] b&gt; 1 [\/latex]<\/li>\r\n \t<li>decreasing if [latex] b&lt; 1 [\/latex]<\/li>\r\n<\/ul>\r\nFigure 3 compares the graphs of exponential growth and decay functions.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_936\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-936\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3-300x164.webp\" alt=\"\" width=\"300\" height=\"164\" \/> Figure 3[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-note-body\">\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 an exponential function of the form <\/strong> [latex] f(x)=b^x, [\/latex] graph the function.\r\n<ol>\r\n \t<li>Create a table of points.<\/li>\r\n \t<li>Plot at least 3 points from the table, including the <em data-effect=\"italics\">y<\/em>-intercept [latex] (0, 1). [\/latex]<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>State the domain, [latex] (-\\infty, \\infty), [\/latex] the range, [latex] (0, \\infty), [\/latex] and the horizontal asymptote, [latex] y=0. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Sketching the Graph of an Exponential Function of the Form <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">b<\/em><sup><em data-effect=\"italics\">x<\/em><\/sup><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=0.25^x. [\/latex] State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Before graphing, identify the behavior and create a table of points for the graph.\r\n<ul>\r\n \t<li>Since [latex] b=0.25 [\/latex] is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex] y=0. [\/latex]<\/li>\r\n \t<li>Create a table of points as in Table 3.<\/li>\r\n<\/ul>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">-3<\/td>\r\n<td style=\"width: 12.5%;\">-2<\/td>\r\n<td style=\"width: 12.5%;\">-1<\/td>\r\n<td style=\"width: 12.5%;\">0<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">2<\/td>\r\n<td style=\"width: 12.5%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">64<\/td>\r\n<td style=\"width: 12.5%;\">16<\/td>\r\n<td style=\"width: 12.5%;\">4<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">0.25<\/td>\r\n<td style=\"width: 12.5%;\">0.0625<\/td>\r\n<td style=\"width: 12.5%;\">0.015625<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ul>\r\n \t<li>Plot the <em data-effect=\"italics\">y<\/em>-intercept, [latex] (0, 1), [\/latex] along with two other points. We can use [latex] (-1, 4) [\/latex] and [latex] (1, 0.25). [\/latex]<\/li>\r\n<\/ul>\r\nDraw a smooth curve connecting the points as in Figure 4.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_937\" align=\"aligncenter\" width=\"290\"]<img class=\"size-medium wp-image-937\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/> Figure 4[\/caption]\r\n\r\nThe domain is [latex] (-\\infty, \\infty); [\/latex] the range is [latex] (0, \\infty); [\/latex] the horizontal asymptote is [latex] y=0. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch the graph of [latex]- f(x)=4^x. [\/latex] State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137694074\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Transformations of Exponential Functions<\/h2>\r\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function [latex] f(x)=b^x [\/latex] without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.<\/p>\r\n\r\n<section id=\"fs-id1165134312214\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Vertical Shift<\/h3>\r\n<p id=\"fs-id1165137911544\">The first transformation occurs when we add a constant <em>d<\/em> to the parent function [latex] f(x)=b^x, [\/latex] giving us a vertical shift <em>d<\/em> units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex] f(x)=2^x, [\/latex] we can then graph two vertical shifts alongside it, using [latex] d=3: [\/latex] the upward shift, [latex] g(x)=2^x+3 [\/latex] and the downward shift, [latex] h(x)=2^x-3. [\/latex] Both vertical shifts are shown in Figure 5.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_938\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-938\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5-300x155.webp\" alt=\"\" width=\"300\" height=\"155\" \/> Figure 5[\/caption]\r\n<p id=\"fs-id1165137464499\">Observe the results of shifting [latex] f(x)=2^x [\/latex] vertically:<\/p>\r\n\r\n<ul id=\"fs-id1165135203774\">\r\n \t<li>The domain, [latex] (-\\infty, \\infty) [\/latex] remains unchanged.<\/li>\r\n \t<li>When the function is shifted up 3 units to [latex] g(x)=2^x+3: [\/latex]\r\n<ul id=\"fs-id1165137601587\" data-bullet-style=\"open-circle\">\r\n \t<li>The <em data-effect=\"italics\">y-<\/em>intercept shifts up 3 units to [latex] (0, 4). [\/latex]&gt;<\/li>\r\n \t<li>The asymptote shifts up 3 units to [latex] y=3. [\/latex]<\/li>\r\n \t<li>The range becomes [latex] (3, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>When the function is shifted down 3 units to [latex] h(x)=2^x-3: [\/latex]\r\n<ul id=\"fs-id1165137784817\" data-bullet-style=\"open-circle\">\r\n \t<li>The <em data-effect=\"italics\">y-<\/em>intercept shifts down 3 units to [latex] (0, -2). [\/latex]<\/li>\r\n \t<li>The asymptote also shifts down 3 units to [latex] y=-3. [\/latex]<\/li>\r\n \t<li>The range becomes [latex] (-3, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1165137566517\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Horizontal Shift<\/h3>\r\n<p id=\"fs-id1165137748336\">The next transformation occurs when we add a constant <em>c<\/em> to the input of the parent function [latex] f(x)=b^x, [\/latex] giving us a horizontal shift <em>c<\/em> units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function [latex] f(x)=2^x, [\/latex] we can then graph two horizontal shifts alongside it, using [latex] c=3: [\/latex] the shift left, [latex] g(x)=2^{x+3}. [\/latex] and the shift right, [latex] h(x)=2^{x-3}. [\/latex] Both horizontal shifts are shown in Figure 6.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_939\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-939\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6-300x195.webp\" alt=\"\" width=\"300\" height=\"195\" \/> Figure 6[\/caption]\r\n<p id=\"fs-id1165137411256\">Observe the results of shifting [latex] f(x)=2^x [\/latex] horizontally:<\/p>\r\n\r\n<ul id=\"fs-id1165135187815\">\r\n \t<li>The domain, [latex] (-\\infty, \\infty), [\/latex] remains unchanged.<\/li>\r\n \t<li>The asymptote, [latex] y=0, [\/latex] remains unchanged.<\/li>\r\n \t<li>The <em data-effect=\"italics\">y-<\/em>intercept shifts such that:\r\n<ul id=\"fs-id1165137482879\" data-bullet-style=\"open-circle\">\r\n \t<li>When the function is shifted left 3 units to [latex] g(x)=2^{x+3}, [\/latex] the <em data-effect=\"italics\">y<\/em>-intercept becomes [latex] (0, 8). [\/latex] This is because [latex] 2^{x+3}=(8)2^x, [\/latex] so the initial value of the function is 8.<\/li>\r\n \t<li>When the function is shifted right 3 units to [latex] h(x)=2^{x-3}, [\/latex] the <em data-effect=\"italics\">y<\/em>-intercept becomes [latex] \\left(0, \\frac{1}{8}\\right). [\/latex] Again, see that [latex] 2^{x-3}=\\left(\\frac{1}{8}\\right)2^x, [\/latex] so the initial value of the function is [latex] \\frac{1}{8}. [\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Shifts of the Parents Function [latex] f(x)=b^x [\/latex]<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor any constants <em>c<\/em> and <em>d<\/em> the function [latex] f(x)=b^{x+c}+d [\/latex] shifts the parent function [latex] f(x)=b^x [\/latex]\r\n<ul>\r\n \t<li>vertically <em>d<\/em> units, in the <em data-effect=\"italics\">same<\/em> direction of the sign of <em>d<\/em>.<\/li>\r\n \t<li>horizontally <em>c<\/em> units, in the <em data-effect=\"italics\">opposite<\/em> direction of the sign of<em> c<\/em>.<\/li>\r\n \t<li>The <em data-effect=\"italics\">y<\/em>-intercept becomes [latex] (0, b^c+d). [\/latex]<\/li>\r\n \t<li>The horizontal asymptote becomes [latex] y=d. [\/latex]<\/li>\r\n \t<li>The range becomes [latex] (d, \\infty). [\/latex]<\/li>\r\n \t<li>The domain, [latex] (-\\infty, \\infty), [\/latex] remains unchanged.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\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 an exponential function with the form <\/strong> [latex] f(x)=b^{x+c}+d, [\/latex] graph the translation.\r\n<ol>\r\n \t<li>Draw the horizontal asymptote [latex] y=d. [\/latex]<\/li>\r\n \t<li>Identify the shift as [latex] (-c, d). [\/latex] Shift the graph of [latex] f(x)=b^x [\/latex] left <em>c<\/em> units if <em>c<\/em> is positive, and right <em>c<\/em> units if <em>c<\/em> is negative.<\/li>\r\n \t<li>Shift the graph of [latex] f(x)=b^x [\/latex] up <em>d<\/em> units if <em>d<\/em> is positive, and down <em>d<\/em> units if <em>d<\/em> is negative.<\/li>\r\n \t<li>State the domain, [latex] (-\\infty, \\infty), [\/latex] the range, [latex] (d, \\infty), [\/latex] and the horizontal asymptote [latex] y=d. [\/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 2: Graphing a Shift of an Exponential Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=2^{x+1}-3. [\/latex] State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We have an exponential equation of the form [latex] f(x)=b^{x+c}+d, [\/latex] with [latex] b=2, c=1, [\/latex]\u00a0and [latex] d=-3. [\/latex]\r\n\r\nDraw the horizontal asymptote [latex] y=d, [\/latex] so draw [latex] y=-3. [\/latex]\r\n\r\nIdentify the shift as [latex] (-c, d), [\/latex] so the shift is [latex] (-1, -3). [\/latex]\r\n\r\nShift the graph of [latex] f(x)=b^x [\/latex] left 1 units and down 3 units.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_940\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-940\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7-300x276.webp\" alt=\"\" width=\"300\" height=\"276\" \/> Figure 7[\/caption]\r\n\r\nThe domain is [latex] (-\\infty, \\infty); [\/latex] the range is [latex] (-3, \\infty); [\/latex] the horizontal asymptote is [latex] y=-3. [\/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 #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=2^{x-1}+3. [\/latex] State domain, range, and asymptote.\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 an equation of the form <\/strong> [latex] f(x)=b^{x+c}+d [\/latex] for x, use a graphing calculator to approximate the solution.\r\n<ol>\r\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given exponential equation in the line headed \u201c<strong>Y<sub>1<\/sub>=<\/strong>\u201d.<\/li>\r\n \t<li>Enter the given value for [latex] f(x) [\/latex] in the line headed \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\r\n \t<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em data-effect=\"italics\">y<\/em>-axis so that it includes the value entered for \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\r\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graph of the exponential function along with the line for the specified value of [latex] f(x). [\/latex]<\/li>\r\n \t<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em data-effect=\"italics\">x <\/em>for the indicated value of the function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Approximating the Solution of an Exponential Equation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve [latex] 42=1.2(5)^x+2.8 [\/latex] graphically. Round to the nearest thousandth.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Press <strong>[Y=]<\/strong> and enter [latex] 1.2(5)^x+2.8 [\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values \u20133 to 3 for x and \u20135 to 55 for y. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near [latex] x=2. [\/latex]\r\n\r\nFor a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em data-effect=\"italics\">x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) To the nearest thousandth, [latex] x\\approx 2.166. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/section><\/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\nSolve [latex] 4=7.85(1.15)^x-2.27 [\/latex] graphically. Round to the nearest thousandth.\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137431154\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Stretch or Compression<\/h3>\r\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">stretch<\/span> or <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">compression<\/span> occurs when we multiply the parent function [latex] f(x)=b^x [\/latex] by a constant [latex] [\/latex] |a|&gt; 0.For example, if we begin by graphing the parent function [latex] f(x)=2^x, [\/latex] we can then graph the stretch, using [latex] a=3, [\/latex] to get [latex] g(x)=3(2)^x [\/latex] as shown on the left in Figure 8, and the compression, using [latex] a=\\frac{1}{3}, [\/latex] to get [latex] h(x)=\\frac{1}{3}(2)^x [\/latex] as shown on the right in Figure 8.<\/p>\r\n\r\n[caption id=\"attachment_941\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-941\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-300x137.webp\" alt=\"\" width=\"300\" height=\"137\" \/> Figure 8. (a)[latex] g(x)=3(2)^x [\/latex] stretches the graph of [latex] f(x)=2^x [\/latex] vertically by a factor of 3. (b)[latex] h(x)=\\frac{1}{3}(2)^x [\/latex] compresses the graph of [latex] f(x)=2^x [\/latex] vertically by a factor of [latex] \\frac{1}{3}. [\/latex][\/caption]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Stretches and Compressions of the Parent Function [latex] f(x)=b^x [\/latex] <\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor any factor [latex] a&gt; 0, [\/latex] the function [latex] f(x)=a(b)^x [\/latex]\r\n<ul>\r\n \t<li>is stretched vertically by a factor of <em>a<\/em> if [latex] |a|&gt; 1. [\/latex]<\/li>\r\n \t<li>is compressed vertically by a factor of <em>a<\/em> if [latex] |a|&lt; 1. [\/latex]<\/li>\r\n \t<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex] (0, a). [\/latex]<\/li>\r\n \t<li>has a horizontal asymptote at [latex] y=0, [\/latex] a range of [latex] (0, \\infty), [\/latex] and a domain of [latex] (-\\infty, \\infty), [\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Graphing the Stretch of an Exponential Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=4\\left(\\frac{1}{2}\\right)^x. [\/latex] State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Before graphing, identify the behavior and key points on the graph.\r\n<ul>\r\n \t<li>Since [latex] b=\\frac{1}{2} [\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em> decreases, and the right tail will approach the <em data-effect=\"italics\">x<\/em>-axis as <em>x<\/em> increases.<\/li>\r\n \t<li>Since [latex] a=4, [\/latex] the graph of [latex] f(x)=\\left(\\frac{1}{2}\\right)^x [\/latex] will be stretched by a factor of 4.<\/li>\r\n \t<li>Create a table of points as shown in Table 4.<\/li>\r\n<\/ul>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 4<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">-3<\/td>\r\n<td style=\"width: 12.5%;\">-2<\/td>\r\n<td style=\"width: 12.5%;\">-1<\/td>\r\n<td style=\"width: 12.5%;\">0<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">2<\/td>\r\n<td style=\"width: 12.5%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex] f(x)=4\\left(\\frac{1}{2}\\right)^x [\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">32<\/td>\r\n<td style=\"width: 12.5%;\">16<\/td>\r\n<td style=\"width: 12.5%;\">8<\/td>\r\n<td style=\"width: 12.5%;\">4<\/td>\r\n<td style=\"width: 12.5%;\">2<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">0.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ul>\r\n \t<li>Plot the <em data-effect=\"italics\">y-<\/em>intercept, [latex] (0, 4), [\/latex] along with two other points. We can use [latex] (-1, 8) [\/latex] and [latex] (1, 2). [\/latex]<\/li>\r\n<\/ul>\r\nDraw a smooth curve connecting the points, as shown in Figure 9.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_942\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-942\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9-300x172.webp\" alt=\"\" width=\"300\" height=\"172\" \/> Figure 9[\/caption]\r\n\r\nThe domain is [latex] (-\\infty, \\infty); [\/latex] the range is [latex] (0, \\infty); [\/latex] the horizontal asymptote is [latex] y=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 #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch the graph of [latex] f(x)=\\frac{1}{2}(4)^x. [\/latex] State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135433028\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing Reflections<\/h3>\r\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em data-effect=\"italics\">x<\/em>-axis or the <em data-effect=\"italics\">y<\/em>-axis. When we multiply the parent function [latex] f(x)=b^x [\/latex] by [latex] -1, [\/latex] we get a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When we multiply the input by [latex] -1, [\/latex] we get a reflection about the <em data-effect=\"italics\">y<\/em>-axis. For example, if we begin by graphing the parent function [latex] f(x)=2^x, [\/latex] we can then graph the two reflections alongside it. The reflection about the <em data-effect=\"italics\">x<\/em>-axis, [latex] g(x)=-2^x, [\/latex] is shown on the left side of Figure 10, and the reflection about the <em data-effect=\"italics\">y<\/em>-axis [latex] h(x)=2^{-x}, [\/latex] is shown on the right side of Figure 10.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_943\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-943\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10-300x153.webp\" alt=\"\" width=\"300\" height=\"153\" \/> Figure 10. (a)[latex] g(x)=-2^x [\/latex] reflects the graph of [latex] f(x_=2^x [\/latex] about the x-axis. (b)[latex] g(x)=2^{-x} [\/latex] reflects the graph of [latex] f(x)=2^x [\/latex] about the y-axis.[\/caption]\r\n<div id=\"fs-id1165135477501\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Reflections of the Parent Function [latex] f(x)=b^x [\/latex] <\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe function [latex] f(x)=-b^x [\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex] f(x)=b^x [\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\r\n \t<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex] (0, -1). [\/latex]<\/li>\r\n \t<li>has a range of [latex] (-\\infty, 0). [\/latex]<\/li>\r\n \t<li>has a horizontal asymptote at [latex] y=0 [\/latex] and domain of [latex] (-\\infty, \\infty), [\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex] f(x)=b^{-x} [\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex] f(x)=b^x [\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\r\n \t<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex] ((0, 1), [\/latex] a horizontal asymptote at [latex] y=0, [\/latex] a range of [latex] (0, \\infty), [\/latex] and a domain of [latex] (-\\infty, \\infty), [\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Writing and Graphing the Reflection of an Exponential Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind and graph the equation for a function, [latex] g(x) [\/latex] that reflects [latex] f(x)=\\left(\\frac{1}{4}\\right)^x [\/latex] about the <em data-effect=\"italics\">x<\/em>-axis. State its domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Since we want to reflect the parent function [latex] f(x)=\\left(\\frac{1}{4}\\right)^x [\/latex] about the <em data-effect=\"italics\">x-<\/em>axis, we multiply [latex] f(x) [\/latex] by [latex] -1 [\/latex] to get, [latex] g(x)=-\\left(\\frac{1}{4}\\right)^x. [\/latex] Next we create a table of points as in Table 5.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 5<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">-3<\/td>\r\n<td style=\"width: 12.5%;\">-2<\/td>\r\n<td style=\"width: 12.5%;\">-1<\/td>\r\n<td style=\"width: 12.5%;\">0<\/td>\r\n<td style=\"width: 12.5%;\">1<\/td>\r\n<td style=\"width: 12.5%;\">2<\/td>\r\n<td style=\"width: 12.5%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 12.5%;\">[latex] g(x)=-\\left(\\frac{1}{4}\\right)^x [\/latex]<\/td>\r\n<td style=\"width: 12.5%;\">-64<\/td>\r\n<td style=\"width: 12.5%;\">-16<\/td>\r\n<td style=\"width: 12.5%;\">-4<\/td>\r\n<td style=\"width: 12.5%;\">-1<\/td>\r\n<td style=\"width: 12.5%;\">-0.25<\/td>\r\n<td style=\"width: 12.5%;\">-0.0625<\/td>\r\n<td style=\"width: 12.5%;\">-0.0156<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the <em data-effect=\"italics\">y-<\/em>intercept, [latex] (0, -1), [\/latex] along with two other points. We can use [latex] (-1, -4), [\/latex] and [latex] (1, -0.25). [\/latex] Draw a smooth curve connecting the points:\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_944\" align=\"aligncenter\" width=\"289\"]<img class=\"size-medium wp-image-944\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11-289x300.webp\" alt=\"\" width=\"289\" height=\"300\" \/> Figure 11[\/caption]\r\n\r\nThe domain is [latex] (-\\infty, \\infty); [\/latex] the range is [latex]- (-\\infty, 0); [\/latex] the horizontal asymptote is [latex] y=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\nFind and graph the equation for a function, [latex] g(x), [\/latex] that reflects [latex] f(x)=1.25^x [\/latex] about the <em data-effect=\"italics\">y<\/em>-axis. State its domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135501015\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Summarizing Translations of the Exponential Function<\/h3>\r\n<p id=\"fs-id1165135501021\">Now that we have worked with each type of translation for the exponential function, we can summarize them in Table 6 to arrive at the general equation for translating exponential functions.<\/p>\r\n\r\n<div id=\"Table_04_02_006\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_02_006\"><caption>Table 6<\/caption>\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\" scope=\"colgroup\">Transformations of the Parent Function [latex] f(x)=b^x [\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th scope=\"col\" data-align=\"center\">Transformation<\/th>\r\n<th scope=\"col\" data-align=\"center\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Shift\r\n<ul id=\"fs-id1165137640731\">\r\n \t<li>Horizontally <em>c<\/em> units to the left<\/li>\r\n \t<li>Vertically <em>d<\/em> units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex] f(x)=b^{x+c}+d [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Stretch and Compress\r\n<ul id=\"fs-id1165134074993\">\r\n \t<li>Stretch if [latex] |a|&gt; 1 [\/latex]<\/li>\r\n \t<li>Compression if [latex] |a|&lt; 1 [\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex] f(x)=ab^x [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em data-effect=\"italics\">x<\/em>-axis<\/td>\r\n<td>[latex] f(x)=-b^x [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em data-effect=\"italics\">y<\/em>-axis<\/td>\r\n<td>[latex] f(x)=b^{-x}=\\left(\\frac{1}{b}\\right)^x [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>General equation for all translations<\/td>\r\n<td>[latex] f(x)=ab^{x+c}+d [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Translations of Exponential Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA translation of an exponential function has the form\r\n<p style=\"text-align: center;\">[latex] f(x)=ab^{x+c}+d [\/latex]<\/p>\r\nWhere the parent function, [latex] y=b^x, b&gt; 1, [\/latex] is\r\n<ul>\r\n \t<li>shifted horizontally<em> c<\/em> units to the left.<\/li>\r\n \t<li>stretched vertically by a factor of [latex] |a| [\/latex] if [latex] |a|&gt; 0. [\/latex]<\/li>\r\n \t<li>compressed vertically by a factor of [latex] |a| [\/latex] if [latex] 0&lt; |a|&lt; 1. [\/latex]<\/li>\r\n \t<li>shifted vertically <em>d<\/em> units.<\/li>\r\n \t<li>reflected about the <em data-effect=\"italics\">x-<\/em>axis when [latex] a&lt; 0. [\/latex]<\/li>\r\n<\/ul>\r\nNote the order of the shifts, transformations, and reflections follow the order of operations.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Writing a Function from a Description<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the equation for the function described below. Give the horizontal asymptote, the domain, and the range.\r\n<ul>\r\n \t<li>[latex] f(x)=e^x [\/latex] is vertically stretched by a factor of 2, reflected across the <em data-effect=\"italics\">y<\/em>-axis, and then shifted up 4 units.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We want to find an equation of the general form [latex] f(x)=ab^{x+c}+d. [\/latex] We use the description provided to find a, b, c, and d.\r\n<ul>\r\n \t<li>We are given the parent function [latex] f(x)=e^x, [\/latex] so [latex] b=e. [\/latex]<\/li>\r\n \t<li>The function is stretched by a factor of 2, so [latex] a=2. [\/latex]<\/li>\r\n \t<li>The function is reflected about the <em data-effect=\"italics\">y<\/em>-axis. We replace x with [latex] -x [\/latex] to get: [latex] e^{-x}. [\/latex]<\/li>\r\n \t<li>The graph is shifted vertically 4 units, so [latex] d=4. [\/latex]<\/li>\r\n<\/ul>\r\nSubstituting in the general form we get,\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} f(x) &amp;=&amp; ab^{x+c}+d \\\\ &amp;=&amp; 2e^{x+0}+4 \\\\ &amp;=&amp; 2e^{-x}+4 \\end{array} [\/latex]<\/p>\r\nThe domain is [latex] (-\\infty, \\infty); [\/latex] the range is [latex] (4, \\infty); [\/latex] the horizontal asymptote is [latex] y=4. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the equation for function described below. Give the horizontal asymptote, the domain, and the range.\r\n<ul>\r\n \t<li>[latex] f(x)=e^x [\/latex] is compressed vertically by a factor of [latex] \\frac{1}{3}, [\/latex]\u00a0reflected across the <em data-effect=\"italics\">x<\/em>-axis and then shifted down 2 units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess this online resource for additional instruction and practice with graphing exponential functions.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=7fpazNs1ZRE\">Graph Exponential Functions<\/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\">6.2 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165137634271\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137634275\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165135386454\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135386456\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135386454-solution\">1<\/a><span class=\"os-divider\">. <\/span>What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137724992\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137724994\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137769974\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<div id=\"fs-id1165135696183\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135696185\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135696183-solution\">3<\/a><span class=\"os-divider\">. <\/span>The graph of [latex] f(x)=3^x [\/latex] is reflected about the <em data-effect=\"italics\">y<\/em>-axis and stretched vertically by a factor of 4. What is the equation of the new function, [latex] g(x)? [\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137471014\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137471016\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>The graph of [latex] f(x)=\\left(\\frac{1}{2}\\right)^{-x} [\/latex] is reflected about the <em data-effect=\"italics\">y<\/em>-axis and compressed vertically by a factor of [latex] \\frac{1}{5}. [\/latex] What is the equation of the new function, [latex] g(x)? [\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135459835\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135169299\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135459835-solution\">5<\/a><span class=\"os-divider\">. <\/span>The graph of [latex] f(x)=10^x [\/latex] is reflected about the <em data-effect=\"italics\">x<\/em>-axis and shifted upward 7 units. What is the equation of the new function, [latex] g(x)? [\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135459852\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135459854\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>The graph of [latex] f(x)=(1.68)^x [\/latex] is shifted right 3 units, stretched vertically by a factor of 2, reflected about the <em data-effect=\"italics\">x<\/em>-axis, and then shifted downward 3 units. What is the equation of the new function, [latex] g(x)? [\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept (to the nearest thousandth), domain, and range.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137874974\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137874976\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137874974-solution\">7<\/a><span class=\"os-divider\">. <\/span>The graph of [latex] f(x)=-\\frac{1}{2}\\left(\\frac{1}{4}\\right)^{x-2}+4 [\/latex] is shifted downward 4 units, and then shifted left 2 units, stretched vertically by a factor of 4 and reflected about the <em data-effect=\"italics\">x<\/em>-axis. What is the equation of the new function, [latex] g(x)? [\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137597358\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165137597363\">For the following exercises, graph the function and its reflection about the <em data-effect=\"italics\">y<\/em>-axis on the same axes, and give the <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\r\n\r\n<div id=\"fs-id1165137731815\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137731818\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3\\left(\\frac{1}{3}\\right)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165137758922\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137758924\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137758922-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=-2(0.25)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137556927\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135383139\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=6(1.75)^{-x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137432021\">For the following exercises, graph each set of functions on the same axes.<\/p>\r\n\r\n<div id=\"fs-id1165137579050\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137579053\" class=\"material-set-2\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137579050-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3\\left(\\frac{1}{4}\\right)^x, g(x)=3(2)^x, \\ \\text{and } h(x)=3(4)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137610823\" class=\"material-set-2\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137610825\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{1}{4}(3)^x, g(x)=2(3)^x, \\ \\text{and } h(x)=4(3)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137731675\">For the following exercises, match each function with one of the graphs in Figure 12.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_945\" align=\"aligncenter\" width=\"268\"]<img class=\"size-medium wp-image-945\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-12-268x300.webp\" alt=\"\" width=\"268\" height=\"300\" \/> Figure 12[\/caption]\r\n\r\n<div id=\"fs-id1165137758080\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137758082\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137758080-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2(0.69)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135543453\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135543455\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2(1.28)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137447034\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137762800\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137447034-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2(0.81)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137767457\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135193784\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4(1.28)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137692364\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137692366\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137692364-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2(1.59)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137705261\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137705263\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4(0.69)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\nFor the following exercises, use the graphs shown in Figure 13. All have the form [latex] f(x)=ab^x. [\/latex]\r\n\r\n<section data-depth=\"2\">\r\n\r\n[caption id=\"attachment_946\" align=\"aligncenter\" width=\"274\"]<img class=\"size-medium wp-image-946\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13-274x300.webp\" alt=\"\" width=\"274\" height=\"300\" \/> Figure 13[\/caption]\r\n\r\n<div id=\"fs-id1165137817442\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137817444\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137817442-solution\">19<\/a><span class=\"os-divider\">. <\/span>Which graph has the largest value for<em> b<\/em>?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134040583\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165134040585\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>Which graph has the smallest value for <em>b<\/em>?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137531608\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137531610\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137531608-solution\">21<\/a><span class=\"os-divider\">. <\/span>Which graph has the largest value for <em>a<\/em>?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137836516\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137836518\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22. <\/span>Which graph has the smallest value for <em>a<\/em>?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137936780\">For the following exercises, graph the function and its reflection about the <em data-effect=\"italics\">x<\/em>-axis on the same axes.<\/p>\r\n\r\n<div id=\"fs-id1165137936789\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137936791\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137936789-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{1}{2}(4)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137760861\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137760864\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3(0.75)^x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137727219\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137727221\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137727219-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-4(2)^x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137736387\">For the following exercises, graph the transformation of [latex] f(x)=2^x. [\/latex] Give the horizontal asymptote, the domain, and the range.<\/p>\r\n\r\n<div id=\"fs-id1165135388489\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135388491\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)2^{-x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135700147\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135700149\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135700147-solution\">27<\/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-id1165137849554\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137849556\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2^{x-2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135435788\">For the following exercises, describe the end behavior of the graphs of the functions.<\/p>\r\n\r\n<div id=\"fs-id1165135435791\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135241045\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135435791-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-5(4)^x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137628660\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137628662\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3\\left(\\frac{1}{2}\\right)^x-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137514785\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137514787\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137514785-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3(4)^{-x}+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135417905\">For the following exercises, start with the graph of [latex] f(x)=4^x. [\/latex] Then write a function that results from the given transformation.<\/p>\r\n\r\n<div id=\"fs-id1165135529096\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135529098\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Shift [latex] f(x) [\/latex] 4 units upward\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137652886\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137652888\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137652886-solution\">33<\/a><span class=\"os-divider\">. <\/span>Shift [latex] f(x) [\/latex] 3 units downward\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137762972\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137762974\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Shift [latex] f(x) [\/latex] 2 units left\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135437143\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135437145\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135437143-solution\">35<\/a><span class=\"os-divider\">. <\/span>Shift [latex] f(x) [\/latex] 5 units right\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137526797\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137526799\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>Reflect [latex] f(x) [\/latex] about the <em data-effect=\"italics\">x<\/em>-axis\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135432987\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135432989\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135432987-solution\">37<\/a><span class=\"os-divider\">. <\/span>Reflect [latex] f(x) [\/latex] about the <em data-effect=\"italics\">y<\/em>-axis\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137936583\">For the following exercises, each graph is a transformation of [latex] y=2^x. [\/latex] Write an equation describing the transformation.<\/p>\r\n\r\n<div id=\"fs-id1165137838408\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137838410\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span><span id=\"fs-id1165137838416\" data-type=\"media\" data-alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-947\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135191679\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135191681\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135191679-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1165135536360\" data-type=\"media\" data-alt=\"Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-948\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137692628\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137692630\" 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\">\r\n\r\n<span id=\"fs-id1165137408011\" data-type=\"media\" data-alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-949\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137408027\">For the following exercises, find an exponential equation for the graph.<\/p>\r\n\r\n<div id=\"fs-id1165137550967\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137550969\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137550967-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1165137550975\" data-type=\"media\" data-alt=\"Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-950\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.41-274x300.webp\" alt=\"\" width=\"274\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134341508\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165134341510\" 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\">\r\n\r\n<span id=\"fs-id1165135560715\" data-type=\"media\" data-alt=\"Graph of f(x)=(1\/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-951\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135560731\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165135560673\">For the following exercises, evaluate the exponential functions for the indicated value of x.<\/p>\r\n\r\n<div id=\"fs-id1165135332695\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135332697\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135332695-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=\\frac{1}{3}(7)^{x-2} \\ \\text{for } g(6). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165134313940\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165134313943\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4(2)^{x-1}-2 \\ \\text{for } f(5). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135519341\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135519343\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135519341-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=-\\frac{1}{2}\\left(\\frac{1}{2}\\right)^x+6 \\ \\text{for } h(-7). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135321925\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135321931\">For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.<\/p>\r\n\r\n<div id=\"fs-id1165135190288\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135190290\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] -50=-\\left(\\frac{1}{2}\\right)^{-x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137476627\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137476629\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137476627-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] 116=\\frac{1}{4}\\left(\\frac{1}{8}\\right)^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137838260\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137838262\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] 12=2(3)^x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137838096\" class=\"os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137838098\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137838096-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] 5=3\\left(\\frac{1}{2}\\right)^{x-1}-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137737011\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137737013\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] -30=-4(2)^{x+2}+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137697128\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id1165135187157\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135187160\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135187157-solution\">51<\/a><span class=\"os-divider\">. <\/span>Explore and discuss the graphs of [latex] F(x)=(b)^x [\/latex] and [latex] G(x)=\\left(\\frac{1}{b}\\right)^x. [\/latex] Then make a conjecture about the relationship between the graphs of the functions [latex] b^x [\/latex] and [latex] \\left(\\frac{1}{b}\\right)^x [\/latex] for any real number [latex] b&gt; 0. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135456802\" class=\"material-set-2\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135456804\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Prove the conjecture made in the previous exercise.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135456811\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135456813\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135456811-solution\">53<\/a><span class=\"os-divider\">. <\/span>Explore and discuss the graphs of [latex] f(x)=4^x, g(x)=4^{x-2}, [\/latex] and [latex] h(x)=\\left(\\frac{1}{16}\\right)4^x. [\/latex] Then make a conjecture about the relationship between the graphs of the functions [latex] b^x [\/latex] and [latex] \\left(\\frac{1}{b^n}\\right)b^x [\/latex] for any real number <em data-effect=\"italics\">n <\/em>and real number [latex] b&gt; 0. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137641366\" class=\"material-set-2\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137641369\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Prove the conjecture made in the previous exercise.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_73c684c9-6dae-4f32-a1a1-5208b5bf59c2\" 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 exponential functions.<\/li>\n<li>Graph exponential functions using transformations.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137442020\">As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\n<section id=\"fs-id1165135407520\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Exponential Functions<\/h2>\n<p id=\"fs-id1165137592823\">Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form [latex]f(x)=b^x[\/latex] whose base is greater than one. We\u2019ll use the function [latex]f(x)=2^x.[\/latex] Observe how the output values in Table 1 change as the input increases by 1.<\/p>\n<div id=\"Table_04_02_01\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_02_01\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col data-width=\"60\" \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td data-align=\"center\">[latex]0[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)=2^x[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-align=\"center\">[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>Each output value is the product of the previous output and the base, 2. We call the base 2 the <em data-effect=\"italics\">constant ratio<\/em>. In fact, for any exponential function with the form [latex]f(x)=ab^x,[\/latex] b is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.<\/p>\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\n<ul id=\"fs-id1165137658509\">\n<li>the output values are positive for all values of <em>x<\/em>;<\/li>\n<li>as<em> x<\/em> increases, the output values increase without bound; and<\/li>\n<li>as <em>x<\/em> decreases, the output values grow smaller, approaching zero.<\/li>\n<\/ul>\n<p id=\"fs-id1165137647215\">Figure 1 shows the exponential growth function [latex]f(x)=2^x.[\/latex]<\/p>\n<figure id=\"attachment_934\" aria-describedby=\"caption-attachment-934\" style=\"width: 272px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-934\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1-272x300.webp\" alt=\"\" width=\"272\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1-272x300.webp 272w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1-65x72.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1-225x248.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1-350x386.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-1.webp 360w\" sizes=\"auto, (max-width: 272px) 100vw, 272px\" \/><figcaption id=\"caption-attachment-934\" class=\"wp-caption-text\">Figure 1. Notice that the graph gets close to the x-axis, but never touches it.<\/figcaption><\/figure>\n<p id=\"fs-id1165137459614\">The domain of [latex]f(x)=2^x[\/latex] is all real numbers, the range is [latex](0, \\infty),[\/latex] and the horizontal asymptote is [latex]y=0.[\/latex]<\/p>\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <strong><span id=\"term-00003\" data-type=\"term\">exponential decay<\/span><\/strong>, we can create a table of values for a function of the form [latex]f(x)=b^x[\/latex] whose base is between zero and one. We\u2019ll use the function [latex]g(x)=\\left(\\frac{1}{2}\\right)^x.[\/latex] Observe how the output values in Table 2 change as the input increases by 1.<\/p>\n<div id=\"Table_04_02_02\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_02_02\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td data-align=\"center\">[latex]0[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]g(x)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\n<td data-align=\"center\">[latex]8[\/latex]<\/td>\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio [latex]\\frac{1}{2}.[\/latex]<\/p>\n<p id=\"fs-id1165137452063\">Notice from the table that<\/p>\n<ul id=\"fs-id1165135499992\">\n<li>the output values are positive for all values of <em>x<\/em>;<\/li>\n<li>as <em>x<\/em> increases, the output values grow smaller, approaching zero; and<\/li>\n<li>as <em>x<\/em> decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p id=\"fs-id1165137405421\">Figure 2 shows the exponential decay function, [latex]g(x)=\\left(\\frac{1}{2}\\right)^x.[\/latex]<\/p>\n<figure id=\"attachment_935\" aria-describedby=\"caption-attachment-935\" style=\"width: 281px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-935\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2-281x300.webp\" alt=\"\" width=\"281\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2-281x300.webp 281w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2-65x69.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2-225x240.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2-350x374.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-2.webp 487w\" sizes=\"auto, (max-width: 281px) 100vw, 281px\" \/><figcaption id=\"caption-attachment-935\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<div class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<section data-depth=\"1\">\n<p id=\"fs-id1165137723586\">The domain of [latex]g(x)=\\left(\\frac{1}{2}\\right)^x[\/latex] is all real numbers, the range is [latex](0, \\infty),[\/latex] and the horizontal asymptote is [latex]y=0.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Characteristics of the Graph of the Parent Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>An exponential function with the form [latex]f(x)=b^x, b> 0, b\\not=1,[\/latex]\u00a0has these characteristics:<\/p>\n<ul>\n<li><span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">one-to-one<\/span> function<\/li>\n<li>horizontal asymptote: [latex]y=0[\/latex]<\/li>\n<li>domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li>range: [latex](0, \\infty)[\/latex]<\/li>\n<li><em data-effect=\"italics\">x-<\/em>intercept: none<\/li>\n<li><em data-effect=\"italics\">y-<\/em>intercept: [latex](0, 1)[\/latex]<\/li>\n<li>increasing if [latex]b> 1[\/latex]<\/li>\n<li>decreasing if [latex]b< 1[\/latex]<\/li>\n<\/ul>\n<p>Figure 3 compares the graphs of exponential growth and decay functions.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_936\" aria-describedby=\"caption-attachment-936\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-936\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3-300x164.webp\" alt=\"\" width=\"300\" height=\"164\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3-300x164.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3-65x35.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3-225x123.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3-350x191.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-3.webp 722w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-936\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"os-note-body\">\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 an exponential function of the form <\/strong> [latex]f(x)=b^x,[\/latex] graph the function.<\/p>\n<ol>\n<li>Create a table of points.<\/li>\n<li>Plot at least 3 points from the table, including the <em data-effect=\"italics\">y<\/em>-intercept [latex](0, 1).[\/latex]<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain, [latex](-\\infty, \\infty),[\/latex] the range, [latex](0, \\infty),[\/latex] and the horizontal asymptote, [latex]y=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Sketching the Graph of an Exponential Function of the Form <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">b<\/em><sup><em data-effect=\"italics\">x<\/em><\/sup><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=0.25^x.[\/latex] State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Before graphing, identify the behavior and create a table of points for the graph.<\/p>\n<ul>\n<li>Since [latex]b=0.25[\/latex] is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0.[\/latex]<\/li>\n<li>Create a table of points as in Table 3.<\/li>\n<\/ul>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 12.5%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 12.5%;\">-3<\/td>\n<td style=\"width: 12.5%;\">-2<\/td>\n<td style=\"width: 12.5%;\">-1<\/td>\n<td style=\"width: 12.5%;\">0<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">2<\/td>\n<td style=\"width: 12.5%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 12.5%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 12.5%;\">64<\/td>\n<td style=\"width: 12.5%;\">16<\/td>\n<td style=\"width: 12.5%;\">4<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">0.25<\/td>\n<td style=\"width: 12.5%;\">0.0625<\/td>\n<td style=\"width: 12.5%;\">0.015625<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>Plot the <em data-effect=\"italics\">y<\/em>-intercept, [latex](0, 1),[\/latex] along with two other points. We can use [latex](-1, 4)[\/latex] and [latex](1, 0.25).[\/latex]<\/li>\n<\/ul>\n<p>Draw a smooth curve connecting the points as in Figure 4.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_937\" aria-describedby=\"caption-attachment-937\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-937\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-4.webp 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><figcaption id=\"caption-attachment-937\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p>The domain is [latex](-\\infty, \\infty);[\/latex] the range is [latex](0, \\infty);[\/latex] the horizontal asymptote is [latex]y=0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch the graph of [latex]- f(x)=4^x.[\/latex] State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137694074\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Transformations of Exponential Functions<\/h2>\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function [latex]f(x)=b^x[\/latex] without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.<\/p>\n<section id=\"fs-id1165134312214\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Vertical Shift<\/h3>\n<p id=\"fs-id1165137911544\">The first transformation occurs when we add a constant <em>d<\/em> to the parent function [latex]f(x)=b^x,[\/latex] giving us a vertical shift <em>d<\/em> units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f(x)=2^x,[\/latex] we can then graph two vertical shifts alongside it, using [latex]d=3:[\/latex] the upward shift, [latex]g(x)=2^x+3[\/latex] and the downward shift, [latex]h(x)=2^x-3.[\/latex] Both vertical shifts are shown in Figure 5.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_938\" aria-describedby=\"caption-attachment-938\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-938\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5-300x155.webp\" alt=\"\" width=\"300\" height=\"155\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5-300x155.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5-65x34.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5-225x116.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5-350x180.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-5.webp 584w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-938\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p id=\"fs-id1165137464499\">Observe the results of shifting [latex]f(x)=2^x[\/latex] vertically:<\/p>\n<ul id=\"fs-id1165135203774\">\n<li>The domain, [latex](-\\infty, \\infty)[\/latex] remains unchanged.<\/li>\n<li>When the function is shifted up 3 units to [latex]g(x)=2^x+3:[\/latex]\n<ul id=\"fs-id1165137601587\" data-bullet-style=\"open-circle\">\n<li>The <em data-effect=\"italics\">y-<\/em>intercept shifts up 3 units to [latex](0, 4).[\/latex]&gt;<\/li>\n<li>The asymptote shifts up 3 units to [latex]y=3.[\/latex]<\/li>\n<li>The range becomes [latex](3, \\infty).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>When the function is shifted down 3 units to [latex]h(x)=2^x-3:[\/latex]\n<ul id=\"fs-id1165137784817\" data-bullet-style=\"open-circle\">\n<li>The <em data-effect=\"italics\">y-<\/em>intercept shifts down 3 units to [latex](0, -2).[\/latex]<\/li>\n<li>The asymptote also shifts down 3 units to [latex]y=-3.[\/latex]<\/li>\n<li>The range becomes [latex](-3, \\infty).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1165137566517\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Horizontal Shift<\/h3>\n<p id=\"fs-id1165137748336\">The next transformation occurs when we add a constant <em>c<\/em> to the input of the parent function [latex]f(x)=b^x,[\/latex] giving us a horizontal shift <em>c<\/em> units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function [latex]f(x)=2^x,[\/latex] we can then graph two horizontal shifts alongside it, using [latex]c=3:[\/latex] the shift left, [latex]g(x)=2^{x+3}.[\/latex] and the shift right, [latex]h(x)=2^{x-3}.[\/latex] Both horizontal shifts are shown in Figure 6.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_939\" aria-describedby=\"caption-attachment-939\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-939\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6-300x195.webp\" alt=\"\" width=\"300\" height=\"195\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6-300x195.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6-65x42.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6-225x146.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6-350x227.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-6.webp 414w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-939\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<p id=\"fs-id1165137411256\">Observe the results of shifting [latex]f(x)=2^x[\/latex] horizontally:<\/p>\n<ul id=\"fs-id1165135187815\">\n<li>The domain, [latex](-\\infty, \\infty),[\/latex] remains unchanged.<\/li>\n<li>The asymptote, [latex]y=0,[\/latex] remains unchanged.<\/li>\n<li>The <em data-effect=\"italics\">y-<\/em>intercept shifts such that:\n<ul id=\"fs-id1165137482879\" data-bullet-style=\"open-circle\">\n<li>When the function is shifted left 3 units to [latex]g(x)=2^{x+3},[\/latex] the <em data-effect=\"italics\">y<\/em>-intercept becomes [latex](0, 8).[\/latex] This is because [latex]2^{x+3}=(8)2^x,[\/latex] so the initial value of the function is 8.<\/li>\n<li>When the function is shifted right 3 units to [latex]h(x)=2^{x-3},[\/latex] the <em data-effect=\"italics\">y<\/em>-intercept becomes [latex]\\left(0, \\frac{1}{8}\\right).[\/latex] Again, see that [latex]2^{x-3}=\\left(\\frac{1}{8}\\right)2^x,[\/latex] so the initial value of the function is [latex]\\frac{1}{8}.[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Shifts of the Parents Function [latex]f(x)=b^x[\/latex]<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For any constants <em>c<\/em> and <em>d<\/em> the function [latex]f(x)=b^{x+c}+d[\/latex] shifts the parent function [latex]f(x)=b^x[\/latex]<\/p>\n<ul>\n<li>vertically <em>d<\/em> units, in the <em data-effect=\"italics\">same<\/em> direction of the sign of <em>d<\/em>.<\/li>\n<li>horizontally <em>c<\/em> units, in the <em data-effect=\"italics\">opposite<\/em> direction of the sign of<em> c<\/em>.<\/li>\n<li>The <em data-effect=\"italics\">y<\/em>-intercept becomes [latex](0, b^c+d).[\/latex]<\/li>\n<li>The horizontal asymptote becomes [latex]y=d.[\/latex]<\/li>\n<li>The range becomes [latex](d, \\infty).[\/latex]<\/li>\n<li>The domain, [latex](-\\infty, \\infty),[\/latex] remains unchanged.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\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 an exponential function with the form <\/strong> [latex]f(x)=b^{x+c}+d,[\/latex] graph the translation.<\/p>\n<ol>\n<li>Draw the horizontal asymptote [latex]y=d.[\/latex]<\/li>\n<li>Identify the shift as [latex](-c, d).[\/latex] Shift the graph of [latex]f(x)=b^x[\/latex] left <em>c<\/em> units if <em>c<\/em> is positive, and right <em>c<\/em> units if <em>c<\/em> is negative.<\/li>\n<li>Shift the graph of [latex]f(x)=b^x[\/latex] up <em>d<\/em> units if <em>d<\/em> is positive, and down <em>d<\/em> units if <em>d<\/em> is negative.<\/li>\n<li>State the domain, [latex](-\\infty, \\infty),[\/latex] the range, [latex](d, \\infty),[\/latex] and the horizontal asymptote [latex]y=d.[\/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 2: Graphing a Shift of an Exponential Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=2^{x+1}-3.[\/latex] State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We have an exponential equation of the form [latex]f(x)=b^{x+c}+d,[\/latex] with [latex]b=2, c=1,[\/latex]\u00a0and [latex]d=-3.[\/latex]<\/p>\n<p>Draw the horizontal asymptote [latex]y=d,[\/latex] so draw [latex]y=-3.[\/latex]<\/p>\n<p>Identify the shift as [latex](-c, d),[\/latex] so the shift is [latex](-1, -3).[\/latex]<\/p>\n<p>Shift the graph of [latex]f(x)=b^x[\/latex] left 1 units and down 3 units.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_940\" aria-describedby=\"caption-attachment-940\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-940\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7-300x276.webp\" alt=\"\" width=\"300\" height=\"276\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7-300x276.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7-65x60.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7-225x207.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7-350x322.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-7.webp 404w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-940\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<p>The domain is [latex](-\\infty, \\infty);[\/latex] the range is [latex](-3, \\infty);[\/latex] the horizontal asymptote is [latex]y=-3.[\/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 #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=2^{x-1}+3.[\/latex] State domain, range, and asymptote.<\/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 an equation of the form <\/strong> [latex]f(x)=b^{x+c}+d[\/latex] for x, use a graphing calculator to approximate the solution.<\/p>\n<ol>\n<li>Press <strong>[Y=]<\/strong>. Enter the given exponential equation in the line headed \u201c<strong>Y<sub>1<\/sub>=<\/strong>\u201d.<\/li>\n<li>Enter the given value for [latex]f(x)[\/latex] in the line headed \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em data-effect=\"italics\">y<\/em>-axis so that it includes the value entered for \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graph of the exponential function along with the line for the specified value of [latex]f(x).[\/latex]<\/li>\n<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em data-effect=\"italics\">x <\/em>for the indicated value of the function.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Approximating the Solution of an Exponential Equation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve [latex]42=1.2(5)^x+2.8[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Press <strong>[Y=]<\/strong> and enter [latex]1.2(5)^x+2.8[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values \u20133 to 3 for x and \u20135 to 55 for y. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near [latex]x=2.[\/latex]<\/p>\n<p>For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em data-effect=\"italics\">x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) To the nearest thousandth, [latex]x\\approx 2.166.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/section>\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>Solve [latex]4=7.85(1.15)^x-2.27[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137431154\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Stretch or Compression<\/h3>\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">stretch<\/span> or <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">compression<\/span> occurs when we multiply the parent function [latex]f(x)=b^x[\/latex] by a constant [latex][\/latex] |a|&gt; 0.For example, if we begin by graphing the parent function [latex]f(x)=2^x,[\/latex] we can then graph the stretch, using [latex]a=3,[\/latex] to get [latex]g(x)=3(2)^x[\/latex] as shown on the left in Figure 8, and the compression, using [latex]a=\\frac{1}{3},[\/latex] to get [latex]h(x)=\\frac{1}{3}(2)^x[\/latex] as shown on the right in Figure 8.<\/p>\n<figure id=\"attachment_941\" aria-describedby=\"caption-attachment-941\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-941\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-300x137.webp\" alt=\"\" width=\"300\" height=\"137\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-300x137.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-768x351.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-65x30.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-225x103.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8-350x160.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-8.webp 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-941\" class=\"wp-caption-text\">Figure 8. (a)[latex] g(x)=3(2)^x [\/latex] stretches the graph of [latex] f(x)=2^x [\/latex] vertically by a factor of 3. (b)[latex] h(x)=\\frac{1}{3}(2)^x [\/latex] compresses the graph of [latex] f(x)=2^x [\/latex] vertically by a factor of [latex] \\frac{1}{3}. [\/latex]<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Stretches and Compressions of the Parent Function [latex]f(x)=b^x[\/latex] <\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For any factor [latex]a> 0,[\/latex] the function [latex]f(x)=a(b)^x[\/latex]<\/p>\n<ul>\n<li>is stretched vertically by a factor of <em>a<\/em> if [latex]|a|> 1.[\/latex]<\/li>\n<li>is compressed vertically by a factor of <em>a<\/em> if [latex]|a|< 1.[\/latex]<\/li>\n<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex](0, a).[\/latex]<\/li>\n<li>has a horizontal asymptote at [latex]y=0,[\/latex] a range of [latex](0, \\infty),[\/latex] and a domain of [latex](-\\infty, \\infty),[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Graphing the Stretch of an Exponential Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=4\\left(\\frac{1}{2}\\right)^x.[\/latex] State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Before graphing, identify the behavior and key points on the graph.<\/p>\n<ul>\n<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em> decreases, and the right tail will approach the <em data-effect=\"italics\">x<\/em>-axis as <em>x<\/em> increases.<\/li>\n<li>Since [latex]a=4,[\/latex] the graph of [latex]f(x)=\\left(\\frac{1}{2}\\right)^x[\/latex] will be stretched by a factor of 4.<\/li>\n<li>Create a table of points as shown in Table 4.<\/li>\n<\/ul>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 4<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 12.5%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 12.5%;\">-3<\/td>\n<td style=\"width: 12.5%;\">-2<\/td>\n<td style=\"width: 12.5%;\">-1<\/td>\n<td style=\"width: 12.5%;\">0<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">2<\/td>\n<td style=\"width: 12.5%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 12.5%;\">[latex]f(x)=4\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\n<td style=\"width: 12.5%;\">32<\/td>\n<td style=\"width: 12.5%;\">16<\/td>\n<td style=\"width: 12.5%;\">8<\/td>\n<td style=\"width: 12.5%;\">4<\/td>\n<td style=\"width: 12.5%;\">2<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">0.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>Plot the <em data-effect=\"italics\">y-<\/em>intercept, [latex](0, 4),[\/latex] along with two other points. We can use [latex](-1, 8)[\/latex] and [latex](1, 2).[\/latex]<\/li>\n<\/ul>\n<p>Draw a smooth curve connecting the points, as shown in Figure 9.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_942\" aria-describedby=\"caption-attachment-942\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-942\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9-300x172.webp\" alt=\"\" width=\"300\" height=\"172\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9-300x172.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9-65x37.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9-225x129.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9-350x200.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-9.webp 508w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-942\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<p>The domain is [latex](-\\infty, \\infty);[\/latex] the range is [latex](0, \\infty);[\/latex] the horizontal asymptote is [latex]y=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 #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch the graph of [latex]f(x)=\\frac{1}{2}(4)^x.[\/latex] State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135433028\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing Reflections<\/h3>\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em data-effect=\"italics\">x<\/em>-axis or the <em data-effect=\"italics\">y<\/em>-axis. When we multiply the parent function [latex]f(x)=b^x[\/latex] by [latex]-1,[\/latex] we get a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When we multiply the input by [latex]-1,[\/latex] we get a reflection about the <em data-effect=\"italics\">y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f(x)=2^x,[\/latex] we can then graph the two reflections alongside it. The reflection about the <em data-effect=\"italics\">x<\/em>-axis, [latex]g(x)=-2^x,[\/latex] is shown on the left side of Figure 10, and the reflection about the <em data-effect=\"italics\">y<\/em>-axis [latex]h(x)=2^{-x},[\/latex] is shown on the right side of Figure 10.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_943\" aria-describedby=\"caption-attachment-943\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-943\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10-300x153.webp\" alt=\"\" width=\"300\" height=\"153\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10-300x153.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10-65x33.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10-225x115.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10-350x179.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-10.webp 751w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-943\" class=\"wp-caption-text\">Figure 10. (a)[latex] g(x)=-2^x [\/latex] reflects the graph of [latex] f(x_=2^x [\/latex] about the x-axis. (b)[latex] g(x)=2^{-x} [\/latex] reflects the graph of [latex] f(x)=2^x [\/latex] about the y-axis.<\/figcaption><\/figure>\n<div id=\"fs-id1165135477501\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<div class=\"textbox textbox--examples\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Reflections of the Parent Function [latex]f(x)=b^x[\/latex] <\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The function [latex]f(x)=-b^x[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]f(x)=b^x[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex](0, -1).[\/latex]<\/li>\n<li>has a range of [latex](-\\infty, 0).[\/latex]<\/li>\n<li>has a horizontal asymptote at [latex]y=0[\/latex] and domain of [latex](-\\infty, \\infty),[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>The function [latex]f(x)=b^{-x}[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]f(x)=b^x[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\n<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex]((0, 1),[\/latex] a horizontal asymptote at [latex]y=0,[\/latex] a range of [latex](0, \\infty),[\/latex] and a domain of [latex](-\\infty, \\infty),[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Writing and Graphing the Reflection of an Exponential Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find and graph the equation for a function, [latex]g(x)[\/latex] that reflects [latex]f(x)=\\left(\\frac{1}{4}\\right)^x[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Since we want to reflect the parent function [latex]f(x)=\\left(\\frac{1}{4}\\right)^x[\/latex] about the <em data-effect=\"italics\">x-<\/em>axis, we multiply [latex]f(x)[\/latex] by [latex]-1[\/latex] to get, [latex]g(x)=-\\left(\\frac{1}{4}\\right)^x.[\/latex] Next we create a table of points as in Table 5.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 5<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 12.5%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 12.5%;\">-3<\/td>\n<td style=\"width: 12.5%;\">-2<\/td>\n<td style=\"width: 12.5%;\">-1<\/td>\n<td style=\"width: 12.5%;\">0<\/td>\n<td style=\"width: 12.5%;\">1<\/td>\n<td style=\"width: 12.5%;\">2<\/td>\n<td style=\"width: 12.5%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 12.5%;\">[latex]g(x)=-\\left(\\frac{1}{4}\\right)^x[\/latex]<\/td>\n<td style=\"width: 12.5%;\">-64<\/td>\n<td style=\"width: 12.5%;\">-16<\/td>\n<td style=\"width: 12.5%;\">-4<\/td>\n<td style=\"width: 12.5%;\">-1<\/td>\n<td style=\"width: 12.5%;\">-0.25<\/td>\n<td style=\"width: 12.5%;\">-0.0625<\/td>\n<td style=\"width: 12.5%;\">-0.0156<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the <em data-effect=\"italics\">y-<\/em>intercept, [latex](0, -1),[\/latex] along with two other points. We can use [latex](-1, -4),[\/latex] and [latex](1, -0.25).[\/latex] Draw a smooth curve connecting the points:<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_944\" aria-describedby=\"caption-attachment-944\" style=\"width: 289px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-944\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11-289x300.webp\" alt=\"\" width=\"289\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11-289x300.webp 289w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11-350x363.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-11.webp 358w\" sizes=\"auto, (max-width: 289px) 100vw, 289px\" \/><figcaption id=\"caption-attachment-944\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<p>The domain is [latex](-\\infty, \\infty);[\/latex] the range is [latex]- (-\\infty, 0);[\/latex] the horizontal asymptote is [latex]y=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>Find and graph the equation for a function, [latex]g(x),[\/latex] that reflects [latex]f(x)=1.25^x[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135501015\" data-depth=\"2\">\n<h3 data-type=\"title\">Summarizing Translations of the Exponential Function<\/h3>\n<p id=\"fs-id1165135501021\">Now that we have worked with each type of translation for the exponential function, we can summarize them in Table 6 to arrive at the general equation for translating exponential functions.<\/p>\n<div id=\"Table_04_02_006\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_02_006\">\n<caption>Table 6<\/caption>\n<thead>\n<tr>\n<th colspan=\"2\" scope=\"colgroup\">Transformations of the Parent Function [latex]f(x)=b^x[\/latex]<\/th>\n<\/tr>\n<tr>\n<th scope=\"col\" data-align=\"center\">Transformation<\/th>\n<th scope=\"col\" data-align=\"center\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul id=\"fs-id1165137640731\">\n<li>Horizontally <em>c<\/em> units to the left<\/li>\n<li>Vertically <em>d<\/em> units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f(x)=b^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compress<\/p>\n<ul id=\"fs-id1165134074993\">\n<li>Stretch if [latex]|a|> 1[\/latex]<\/li>\n<li>Compression if [latex]|a|< 1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f(x)=ab^x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td>[latex]f(x)=-b^x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td>[latex]f(x)=b^{-x}=\\left(\\frac{1}{b}\\right)^x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td>[latex]f(x)=ab^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Translations of Exponential Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A translation of an exponential function has the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ab^{x+c}+d[\/latex]<\/p>\n<p>Where the parent function, [latex]y=b^x, b> 1,[\/latex] is<\/p>\n<ul>\n<li>shifted horizontally<em> c<\/em> units to the left.<\/li>\n<li>stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a|> 0.[\/latex]<\/li>\n<li>compressed vertically by a factor of [latex]|a|[\/latex] if [latex]0< |a|< 1.[\/latex]<\/li>\n<li>shifted vertically <em>d<\/em> units.<\/li>\n<li>reflected about the <em data-effect=\"italics\">x-<\/em>axis when [latex]a< 0.[\/latex]<\/li>\n<\/ul>\n<p>Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Writing a Function from a Description<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul>\n<li>[latex]f(x)=e^x[\/latex] is vertically stretched by a factor of 2, reflected across the <em data-effect=\"italics\">y<\/em>-axis, and then shifted up 4 units.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We want to find an equation of the general form [latex]f(x)=ab^{x+c}+d.[\/latex] We use the description provided to find a, b, c, and d.<\/p>\n<ul>\n<li>We are given the parent function [latex]f(x)=e^x,[\/latex] so [latex]b=e.[\/latex]<\/li>\n<li>The function is stretched by a factor of 2, so [latex]a=2.[\/latex]<\/li>\n<li>The function is reflected about the <em data-effect=\"italics\">y<\/em>-axis. We replace x with [latex]-x[\/latex] to get: [latex]e^{-x}.[\/latex]<\/li>\n<li>The graph is shifted vertically 4 units, so [latex]d=4.[\/latex]<\/li>\n<\/ul>\n<p>Substituting in the general form we get,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} f(x) &=& ab^{x+c}+d \\\\ &=& 2e^{x+0}+4 \\\\ &=& 2e^{-x}+4 \\end{array}[\/latex]<\/p>\n<p>The domain is [latex](-\\infty, \\infty);[\/latex] the range is [latex](4, \\infty);[\/latex] the horizontal asymptote is [latex]y=4.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul>\n<li>[latex]f(x)=e^x[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3},[\/latex]\u00a0reflected across the <em data-effect=\"italics\">x<\/em>-axis and then shifted down 2 units.<\/li>\n<\/ul>\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 graphing exponential functions.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=7fpazNs1ZRE\">Graph Exponential Functions<\/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\">6.2 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165137634271\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137634275\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165135386454\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135386456\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135386454-solution\">1<\/a><span class=\"os-divider\">. <\/span>What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137724992\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137724994\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137769974\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<div id=\"fs-id1165135696183\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135696185\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135696183-solution\">3<\/a><span class=\"os-divider\">. <\/span>The graph of [latex]f(x)=3^x[\/latex] is reflected about the <em data-effect=\"italics\">y<\/em>-axis and stretched vertically by a factor of 4. What is the equation of the new function, [latex]g(x)?[\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137471014\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137471016\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>The graph of [latex]f(x)=\\left(\\frac{1}{2}\\right)^{-x}[\/latex] is reflected about the <em data-effect=\"italics\">y<\/em>-axis and compressed vertically by a factor of [latex]\\frac{1}{5}.[\/latex] What is the equation of the new function, [latex]g(x)?[\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135459835\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135169299\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135459835-solution\">5<\/a><span class=\"os-divider\">. <\/span>The graph of [latex]f(x)=10^x[\/latex] is reflected about the <em data-effect=\"italics\">x<\/em>-axis and shifted upward 7 units. What is the equation of the new function, [latex]g(x)?[\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135459852\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135459854\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>The graph of [latex]f(x)=(1.68)^x[\/latex] is shifted right 3 units, stretched vertically by a factor of 2, reflected about the <em data-effect=\"italics\">x<\/em>-axis, and then shifted downward 3 units. What is the equation of the new function, [latex]g(x)?[\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept (to the nearest thousandth), domain, and range.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137874974\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137874976\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137874974-solution\">7<\/a><span class=\"os-divider\">. <\/span>The graph of [latex]f(x)=-\\frac{1}{2}\\left(\\frac{1}{4}\\right)^{x-2}+4[\/latex] is shifted downward 4 units, and then shifted left 2 units, stretched vertically by a factor of 4 and reflected about the <em data-effect=\"italics\">x<\/em>-axis. What is the equation of the new function, [latex]g(x)?[\/latex] State its <em data-effect=\"italics\">y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137597358\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165137597363\">For the following exercises, graph the function and its reflection about the <em data-effect=\"italics\">y<\/em>-axis on the same axes, and give the <em data-effect=\"italics\">y<\/em>-intercept.<\/p>\n<div id=\"fs-id1165137731815\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137731818\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3\\left(\\frac{1}{3}\\right)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137758922\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137758924\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137758922-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=-2(0.25)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137556927\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135383139\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=6(1.75)^{-x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137432021\">For the following exercises, graph each set of functions on the same axes.<\/p>\n<div id=\"fs-id1165137579050\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137579053\" class=\"material-set-2\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137579050-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3\\left(\\frac{1}{4}\\right)^x, g(x)=3(2)^x, \\ \\text{and } h(x)=3(4)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137610823\" class=\"material-set-2\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137610825\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{1}{4}(3)^x, g(x)=2(3)^x, \\ \\text{and } h(x)=4(3)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137731675\">For the following exercises, match each function with one of the graphs in Figure 12.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_945\" aria-describedby=\"caption-attachment-945\" style=\"width: 268px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-945\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-12-268x300.webp\" alt=\"\" width=\"268\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-12-268x300.webp 268w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-12-65x73.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-12-225x252.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-12.webp 347w\" sizes=\"auto, (max-width: 268px) 100vw, 268px\" \/><figcaption id=\"caption-attachment-945\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<div id=\"fs-id1165137758080\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137758082\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137758080-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2(0.69)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135543453\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135543455\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2(1.28)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137447034\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137762800\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137447034-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2(0.81)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137767457\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135193784\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4(1.28)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137692364\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137692366\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137692364-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2(1.59)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137705261\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137705263\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4(0.69)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p>For the following exercises, use the graphs shown in Figure 13. All have the form [latex]f(x)=ab^x.[\/latex]<\/p>\n<section data-depth=\"2\">\n<figure id=\"attachment_946\" aria-describedby=\"caption-attachment-946\" style=\"width: 274px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-946\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13-274x300.webp\" alt=\"\" width=\"274\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13-274x300.webp 274w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13-65x71.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13-225x246.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13-350x383.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2-fig-13.webp 383w\" sizes=\"auto, (max-width: 274px) 100vw, 274px\" \/><figcaption id=\"caption-attachment-946\" class=\"wp-caption-text\">Figure 13<\/figcaption><\/figure>\n<div id=\"fs-id1165137817442\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137817444\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137817442-solution\">19<\/a><span class=\"os-divider\">. <\/span>Which graph has the largest value for<em> b<\/em>?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134040583\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165134040585\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>Which graph has the smallest value for <em>b<\/em>?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137531608\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137531610\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137531608-solution\">21<\/a><span class=\"os-divider\">. <\/span>Which graph has the largest value for <em>a<\/em>?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137836516\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137836518\" data-type=\"problem\">\n<p><span class=\"os-number\">22. <\/span>Which graph has the smallest value for <em>a<\/em>?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137936780\">For the following exercises, graph the function and its reflection about the <em data-effect=\"italics\">x<\/em>-axis on the same axes.<\/p>\n<div id=\"fs-id1165137936789\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137936791\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137936789-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{1}{2}(4)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137760861\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137760864\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3(0.75)^x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137727219\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137727221\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137727219-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-4(2)^x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137736387\">For the following exercises, graph the transformation of [latex]f(x)=2^x.[\/latex] Give the horizontal asymptote, the domain, and the range.<\/p>\n<div id=\"fs-id1165135388489\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135388491\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)2^{-x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135700147\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135700149\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135700147-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=2^x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137849554\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137849556\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2^{x-2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135435788\">For the following exercises, describe the end behavior of the graphs of the functions.<\/p>\n<div id=\"fs-id1165135435791\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135241045\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135435791-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-5(4)^x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137628660\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137628662\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3\\left(\\frac{1}{2}\\right)^x-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137514785\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137514787\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137514785-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3(4)^{-x}+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135417905\">For the following exercises, start with the graph of [latex]f(x)=4^x.[\/latex] Then write a function that results from the given transformation.<\/p>\n<div id=\"fs-id1165135529096\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135529098\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Shift [latex]f(x)[\/latex] 4 units upward<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137652886\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137652888\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137652886-solution\">33<\/a><span class=\"os-divider\">. <\/span>Shift [latex]f(x)[\/latex] 3 units downward<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137762972\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137762974\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Shift [latex]f(x)[\/latex] 2 units left<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135437143\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135437145\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135437143-solution\">35<\/a><span class=\"os-divider\">. <\/span>Shift [latex]f(x)[\/latex] 5 units right<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137526797\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137526799\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>Reflect [latex]f(x)[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135432987\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135432989\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135432987-solution\">37<\/a><span class=\"os-divider\">. <\/span>Reflect [latex]f(x)[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137936583\">For the following exercises, each graph is a transformation of [latex]y=2^x.[\/latex] Write an equation describing the transformation.<\/p>\n<div id=\"fs-id1165137838408\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137838410\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span><span id=\"fs-id1165137838416\" data-type=\"media\" data-alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-947\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.38.webp 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135191679\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135191681\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135191679-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165135536360\" data-type=\"media\" data-alt=\"Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-948\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.39.webp 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137692628\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137692630\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165137408011\" data-type=\"media\" data-alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-949\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.40.webp 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137408027\">For the following exercises, find an exponential equation for the graph.<\/p>\n<div id=\"fs-id1165137550967\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137550969\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137550967-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165137550975\" data-type=\"media\" data-alt=\"Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-950\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.41-274x300.webp\" alt=\"\" width=\"274\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.41-274x300.webp 274w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.41-65x71.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.41-225x246.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.41.webp 341w\" sizes=\"auto, (max-width: 274px) 100vw, 274px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134341508\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165134341510\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165135560715\" data-type=\"media\" data-alt=\"Graph of f(x)=(1\/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-951\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.2.42.webp 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135560731\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165135560673\">For the following exercises, evaluate the exponential functions for the indicated value of x.<\/p>\n<div id=\"fs-id1165135332695\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135332697\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135332695-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=\\frac{1}{3}(7)^{x-2} \\ \\text{for } g(6).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165134313940\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165134313943\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4(2)^{x-1}-2 \\ \\text{for } f(5).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135519341\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135519343\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135519341-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=-\\frac{1}{2}\\left(\\frac{1}{2}\\right)^x+6 \\ \\text{for } h(-7).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135321925\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135321931\">For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.<\/p>\n<div id=\"fs-id1165135190288\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135190290\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]-50=-\\left(\\frac{1}{2}\\right)^{-x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137476627\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137476629\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137476627-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]116=\\frac{1}{4}\\left(\\frac{1}{8}\\right)^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137838260\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137838262\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]12=2(3)^x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137838096\" class=\"os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137838098\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137838096-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]5=3\\left(\\frac{1}{2}\\right)^{x-1}-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137737011\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137737013\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]-30=-4(2)^{x+2}+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137697128\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id1165135187157\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135187160\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135187157-solution\">51<\/a><span class=\"os-divider\">. <\/span>Explore and discuss the graphs of [latex]F(x)=(b)^x[\/latex] and [latex]G(x)=\\left(\\frac{1}{b}\\right)^x.[\/latex] Then make a conjecture about the relationship between the graphs of the functions [latex]b^x[\/latex] and [latex]\\left(\\frac{1}{b}\\right)^x[\/latex] for any real number [latex]b> 0.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135456802\" class=\"material-set-2\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135456804\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135456811\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135456813\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135456811-solution\">53<\/a><span class=\"os-divider\">. <\/span>Explore and discuss the graphs of [latex]f(x)=4^x, g(x)=4^{x-2},[\/latex] and [latex]h(x)=\\left(\\frac{1}{16}\\right)4^x.[\/latex] Then make a conjecture about the relationship between the graphs of the functions [latex]b^x[\/latex] and [latex]\\left(\\frac{1}{b^n}\\right)b^x[\/latex] for any real number <em data-effect=\"italics\">n <\/em>and real number [latex]b> 0.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137641366\" class=\"material-set-2\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137641369\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-239","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/239","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":13,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/239\/revisions"}],"predecessor-version":[{"id":1651,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/239\/revisions\/1651"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/239\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=239"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=239"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=239"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}