{"id":295,"date":"2019-02-08T21:05:21","date_gmt":"2019-02-08T21:05:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-exponential-functions\/"},"modified":"2025-03-31T20:33:33","modified_gmt":"2025-03-31T20:33:33","slug":"graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-exponential-functions\/","title":{"raw":"2.3 Graphs of Exponential Functions","rendered":"2.3 Graphs of Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Graph exponential functions.<\/li>\r\n \t<li>Determine the end behavior and horizontal asymptotes of exponential functions.<\/li>\r\n \t<li>Graph exponential functions using transformations.<\/li>\r\n<\/ul>\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<div id=\"fs-id1165135407520\" class=\"bc-section section\">\r\n<h3>Graphing Exponential Functions<\/h3>\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\\left(x\\right)={b}^{x}[\/latex] whose base is greater than one. We\u2019ll use the function [latex]f\\left(x\\right)={2}^{x}.[\/latex] Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_01\">Table 1<\/a>\u00a0change as the input increases by [latex]1.[\/latex]<\/p>\r\n\r\n<table id=\"Table_04_02_01\" style=\"height: 38px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=2^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8).\"><caption>Table 1<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"width: 120.5px; height: 12px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 83.5px; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; height: 12px; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; height: 12px; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td class=\"border\" style=\"width: 120.5px; height: 26px; text-align: center;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 83.5px; height: 26px; text-align: center;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; height: 26px; text-align: center;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; height: 26px; text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137432031\">Each output value is the product of the previous output and the base, [latex]2.[\/latex] We call the base [latex]2[\/latex] the <strong><em>constant ratio<\/em><\/strong> or <strong><em>growth factor<\/em><\/strong>. In fact, for any exponential function with the form [latex]f\\left(x\\right)=a{b}^{x},[\/latex] [latex]b[\/latex] is the constant ratio of the function. This means that each time we increase the input by 1, we multiply the output by [latex]b[\/latex]. Notice from the table that the output values are positive for all values of [latex]x.[\/latex]<\/p>\r\n\r\n<h3>End Behavior of [latex]f\\left(x\\right)=ab^x[\/latex]<\/h3>\r\nOften we want to know what happens to the output value of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] moves far to the left or far to the right.\u00a0 This is known as the <strong>end behavior<\/strong>\u00a0or <strong>long term behavior<\/strong> of the function.\u00a0 We will continue to study the function [latex]f\\left(x\\right)=2^x[\/latex] and determine its end behavior.\r\n\r\nBegin by looking at two tables of values.\u00a0 Table 2 shows function values as x moves far to the left.\u00a0 We choose x values of -10, -100 and -250 and evaluate [latex]f\\left(x\\right)[\/latex] at these values so we can observe what is happening to the output values to the far left.\u00a0 Note that -250 is not considered long term behavior for most functions but for an exponential function it is about the limit of what our current technology can compute.\u00a0 In Table 3, we move far toward the right choosing our x values to be 10, 100, and 250 and evaluate the function so we can observe what happens to the output when x gets large.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-10<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-100<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-250<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]f\\left(x\\right)=2^x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">9.77E-4<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">7.89E-31<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">5.53E-76<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">100<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">250<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]f\\left(x\\right)=2^x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1024<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1.27E30<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1.81E75<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRecognize that scientific notation is being used in Table 2 and Table 3 for the output values.\u00a0 In Table 2, we observe that\u00a0as [latex]x[\/latex] decreases or becomes more and more negative, the output values get closer and closer to zero from above.\u00a0 We capture this idea using arrow notation and write as [latex]x\\to-\\infty,f\\left(x\\right)\\to0.[\/latex] This is read, \"As x decreases without bound, [latex]f[\/latex] of x goes to zero.\" When we are studying end behavior and we observe that the output is getting closer and closer to a value, we say that there is a horizontal asymptote.\u00a0 In this example, [latex]y=0[\/latex] is the horizontal asymptote on the left hand side.\r\n\r\nFurther, Table 3 shows that as [latex]x[\/latex] increases or becomes larger and larger, the output values also become larger and larger or increase without bound.\u00a0 We write as [latex]x\\to\\infty,f\\left(x\\right)\\to\\infty.[\/latex] Since these output values increase without bound, there is not a horizontal asymptote in this direction.\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137782455\">A <strong>horizontal asymptote<\/strong> of a graph is a horizontal line [latex]y=b[\/latex] where the graph approaches the line as the inputs increase or decrease without bound. We write as [latex]x\\to \\infty \\textrm{ or }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to b.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137647215\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_001\">Figure 1<\/a>\u00a0shows the exponential growth function [latex]f\\left(x\\right)={2}^{x}.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_001\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"295\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210258\/CNX_Precalc_Figure_04_02_001.jpg\" alt=\"Graph of the exponential function, 2^(x), with labeled points at (-3, 1\/8), (-2, \u00bc), (-1, \u00bd), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.\" width=\"295\" height=\"315\" \/> Figure 1: Notice that the graph gets close to the x-axis, but never touches it.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137459614\">We observe in the graph above that as x becomes more negative, the graph is getting closer to the x-axis but never touches it demonstrating the horizontal asymptote of [latex]y=0.[\/latex]\u00a0 Other characteristics of the graph can also be observed.\u00a0 The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers, and the range is [latex]\\left(0,\\infty \\right).[\/latex]\u00a0 Notice that the function is also increasing and concave up on [latex]\\left(-\\infty,\\infty \\right).[\/latex]<\/p>\r\n\r\n<h3>Exponential Decay Graphically<\/h3>\r\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <span class=\"no-emphasis\">exponential decay<\/span>, we can create a table of values for a function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex] whose base is between zero and one. We\u2019ll use the function [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.[\/latex] Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_02\">Table 4<\/a>\u00a0change as the input increases by 1.<\/p>\r\n\r\n<table id=\"Table_04_02_02\" summary=\"Two rows and eight columns. The first row is labeled, \u201cf(x)=2^x\u201d, with the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, \u201cg(x)=log_2(x)\u201d, with the following values: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\"><caption>Table 4<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 178.5px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 178.5px; text-align: center;\">[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135347846\">Again, notice that each time the input is increased by 2, the output is multiplied by the base, or constant ratio [latex]b=\\frac{1}{2}.[\/latex]<\/p>\r\nTo look at the end behavior of the exponential decay function, we again create tables with input values to the far left and right.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\"><caption>Table 5<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-10<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-100<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-250<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1024<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1.27E30<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">\u00a01.81E75<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\"><caption>Table 6<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">100<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">250<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">9.77E-4<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">7.89E-31<\/td>\r\n<td class=\"border\" style=\"width: 25%; text-align: center;\">5.53E-76<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137452063\">Notice from the tables above that:<\/p>\r\n\r\n<ul id=\"fs-id1165135499992\">\r\n \t<li>the output values are positive for all values of [latex]x.[\/latex]<\/li>\r\n \t<li>as [latex]x[\/latex] decreases, the output values grow without bound so as [latex]x\\to-\\infty,g\\left(x\\right)\\to\\infty.[\/latex]<\/li>\r\n \t<li>as [latex]x[\/latex] increases without bound, the output values approach zero from above so as [latex]x\\to\\infty,g\\left(x\\right)\\to0.[\/latex] The horizontal asymptote is [latex]y=0[\/latex] on the right hand side.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137405421\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_002\">Figure 2<\/a>\u00a0shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_002\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"350\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210301\/CNX_Precalc_Figure_04_02_002.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"350\" height=\"374\" \/> Figure 2 The graph shows that as x gets larger, the output gets close to zero.[\/caption]\r\n\r\n<\/div>\r\nAgain we observe the end behavior and see that as x increases, the graph approaches the x-axis and there is a horizontal asymptote of [latex]y=0.[\/latex]\u00a0 Other characteristics can also be observed from the graph.\u00a0 The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers, and the range is [latex]\\left(0,\\infty \\right).[\/latex]\u00a0 Notice that the function is decreasing and concave up on [latex]\\left(-\\infty,\\infty \\right).[\/latex]\r\n<div id=\"fs-id1165135571835\">\r\n<h3>Characteristics of the Graph of the Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1165137848929\">An exponential function with the form [latex]f\\left(x\\right)={b}^{x},[\/latex] [latex]b&gt;0,[\/latex] [latex]b\\ne 1,[\/latex] has these characteristics:<\/p>\r\n\r\n<ul>\r\n \t<li><span class=\"no-emphasis\">one-to-one<\/span> function<\/li>\r\n \t<li>horizontal asymptote: [latex]y=0[\/latex] on one side<\/li>\r\n \t<li>domain: [latex]\\left(\u2013\\infty , \\infty \\right)[\/latex]<\/li>\r\n \t<li>range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\r\n \t<li><em>x-<\/em>intercept: none<\/li>\r\n \t<li><em>y-<\/em>intercept: [latex]\\left(0,1\\right)[\/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 \t<li>concave up<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137471878\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_003\">Figure 3<\/a>\u00a0compares the graphs of <span class=\"no-emphasis\">exponential growth<\/span> and decay functions.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_003\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"533\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210304\/CNX_Precalc_Figure_04_02_003new.jpg\" alt=\"\" width=\"533\" height=\"297\" \/> Figure 3[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134195243\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135194093\"><strong>Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x},[\/latex] graph the function.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137435782\" type=\"1\">\r\n \t<li>Create a table of points.<\/li>\r\n \t<li>Plot at least [latex]3[\/latex] points from the table, including the vertical intercept [latex]\\left(0,1\\right).[\/latex]<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] the range, [latex]\\left(0,\\infty \\right),[\/latex] and on which side the horizontal asymptote, [latex]y=0[\/latex] occurs.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_02_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135208984\">\r\n<div id=\"fs-id1165137453336\">\r\n<h3>Example 1:\u00a0 Sketching the Graph of an Exponential Function of the Form <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165137767671\">Sketch a graph of [latex]f\\left(x\\right)={0.25}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135696740\">[reveal-answer q=\"fs-id1165135696740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135696740\"]\r\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\r\n\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 end behavior of the graph will increase without bound, and the right end behavior will approach the horizontal asymptote [latex]y=0.[\/latex]<\/li>\r\n \t<li>Create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_03\">Table 7<\/a>.\r\n<table id=\"Table_04_02_03\" style=\"height: 34px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, 0.015625).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 108.656px; text-align: center;\" colspan=\"6\"><strong>Table 7<\/strong><\/td>\r\n<td style=\"width: 100.656px;\"><\/td>\r\n<td style=\"width: 113.656px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td class=\"border\" style=\"height: 11px; width: 108.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 77.6562px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 77.6562px; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 75.6562px; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 71.6562px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 87.6562px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 100.656px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 11px; width: 113.656px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px;\">\r\n<td class=\"border\" style=\"height: 23px; width: 108.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 77.6562px; text-align: center;\">[latex]64[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 77.6562px; text-align: center;\">[latex]16[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 75.6562px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 71.6562px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 87.6562px; text-align: center;\">[latex]0.25[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 100.656px; text-align: center;\">[latex]0.0625[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 23px; width: 113.656px; text-align: center;\">[latex]0.015625[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Plot the <em>y<\/em>-intercept, [latex]\\left(0,1\\right),[\/latex] along with two other points. We can use [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(1,0.25\\right).[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_004\">Figure 4<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_004\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210308\/CNX_Precalc_Figure_04_02_004.jpg\" alt=\"Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).\" width=\"487\" height=\"332\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137548870\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135499977\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_04_02_01\">\r\n<div id=\"fs-id1165137761245\">\r\n<p id=\"fs-id1165137548853\">Sketch the graph of [latex]f\\left(x\\right)={4}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137731723\">[reveal-answer q=\"fs-id1165137731723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137731723\"]\r\n<p id=\"fs-id1165137500954\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the left side.<\/p>\r\n<span id=\"fs-id1165137437648\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210311\/CNX_Precalc_Figure_04_02_005.jpg\" alt=\"Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137694074\" class=\"bc-section section\">\r\n<h3>Graphing Transformations of Exponential Functions<\/h3>\r\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with our toolkit functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the exponential function [latex]f\\left(x\\right)={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<strong>NOTE:<\/strong> In this section we will be using different notation for horizontal and vertical shifts. Recall that in section 1.6, we considered functions in the form [latex]g\\left(x\\right)=af\\left(b\\left(x-h\\right)\\right)+k.[\/latex] In this notation, [latex]k[\/latex] indicated the vertical shift, [latex]h[\/latex] indicated the horizontal shift,\u00a0[latex]a[\/latex] indicated the vertical stretch\/compression, and\u00a0[latex]b[\/latex] indicated the horizontal stretch\/compression.\u00a0 When we study exponential functions, we have already designated\u00a0[latex]k[\/latex] to indicate continuous growth.\u00a0 Therefore, we will modify our notation and use\u00a0[latex]c[\/latex] to represent horizontal shifts and [latex]d[\/latex] to represent vertical shifts. Vertical stretches\/compressions will still be represented by [latex]a.[\/latex] The variable [latex]b[\/latex] will represent the base of the exponential function and not represent a horizontal stretch or compression.\r\n<div id=\"fs-id1165134312214\" class=\"bc-section section\">\r\n<h4>Graphing a Vertical Shift<\/h4>\r\n<p id=\"fs-id1165137911544\">The first transformation occurs when we add a constant [latex]d[\/latex] to the exponential function [latex]f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">vertical shift\u00a0<\/span>[latex]d[\/latex] units in the same direction as the sign.\u00a0For example, if we begin by graphing the function, [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph two vertical shifts alongside it, using [latex]d=3[\/latex] and [latex]d=-3:[\/latex] the upward shift, [latex]g\\left(x\\right)=f\\left(x\\right)+3={2}^{x}+3[\/latex] and the downward shift, [latex]h\\left(x\\right)=f\\left(x\\right)-3={2}^{x}-3.[\/latex] Both vertical shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_006\">Figure 5<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_006\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"316\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210315\/CNX_Precalc_Figure_04_02_006.jpg\" alt=\"Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions\u2019 transformations are described in the text.\" width=\"316\" height=\"408\" \/> Figure 5[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137464499\">Observe the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically:<\/p>\r\n\r\n<ul id=\"fs-id1165135203774\">\r\n \t<li>The domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex] remains unchanged.<\/li>\r\n \t<li>When the function is shifted up [latex]3[\/latex] units to [latex]g\\left(x\\right)={2}^{x}+3:[\/latex]\r\n<ul id=\"fs-id1165137601587\">\r\n \t<li>The <em>y-<\/em>intercept shifts up [latex]3[\/latex] units to [latex]\\left(0,4\\right).[\/latex]<\/li>\r\n \t<li>The horizontal asymptote shifts up [latex]3[\/latex] units to [latex]y=3[\/latex] on the left side.<\/li>\r\n \t<li>The range becomes [latex]\\left(3,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>When the function is shifted down [latex]3[\/latex] units to [latex]h\\left(x\\right)={2}^{x}-3:[\/latex]\r\n<ul id=\"fs-id1165137784817\">\r\n \t<li>The <em>y-<\/em>intercept shifts down [latex]3[\/latex] units to [latex]\\left(0,-2\\right).[\/latex]<\/li>\r\n \t<li>The horizontal asymptote also shifts down [latex]3[\/latex] units to [latex]y=-3[\/latex] on the left side.<\/li>\r\n \t<li>The range becomes [latex]\\left(-3,\\infty \\right).[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137566517\" class=\"bc-section section\">\r\n<h4>Graphing a Horizontal Shift<\/h4>\r\n<p id=\"fs-id1165137748336\">The next transformation occurs when we subtract a constant [latex]c[\/latex] from the input of the exponential function [latex]f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">horizontal shift\u00a0<\/span>[latex]c[\/latex] units in the direction of the sign of [latex]c[\/latex]. The equation is given by [latex]f\\left(x-c\\right)=b^{x-c}.[\/latex] For example, if we begin by graphing the function [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph two horizontal shifts alongside it, using<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]c=-3:[\/latex] the shift left, [latex]g\\left(x\\right)=f\\left(x+3\\right)={2}^{x+3},[\/latex] and using<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]c=3:[\/latex] the shift right, [latex]h\\left(x\\right)=f\\left(x-3\\right)={2}^{x-3}.[\/latex]<\/p>\r\nBoth horizontal shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_007\">Figure 6<\/a>.\r\n<div id=\"CNX_Precalc_Figure_04_02_007\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"450\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210318\/CNX_Precalc_Figure_04_02_007.jpg\" alt=\"Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions\u2019 asymptotes are at y=0Note that each functions\u2019 transformations are described in the text.\" width=\"450\" height=\"294\" \/> Figure 6[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137411256\">Observe the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] horizontally:<\/p>\r\n\r\n<ul id=\"fs-id1165135187815\">\r\n \t<li>The domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] remains unchanged.<\/li>\r\n \t<li>The horizontal asymptote, [latex]y=0,[\/latex] remains unchanged.<\/li>\r\n \t<li>The vertical intercept shifts such that:\r\n<ul id=\"fs-id1165137482879\">\r\n \t<li>When the function is shifted left [latex]3[\/latex] units to [latex]g\\left(x\\right)={2}^{x+3},[\/latex] the vertical intercept becomes [latex]\\left(0,8\\right).[\/latex] This is because [latex]{2}^{x+3}={2}^{x}{2}^{3}=\\left(8\\right){2}^{x}[\/latex] using the rules of exponents, so the initial value of the function is [latex]8.[\/latex]<\/li>\r\n \t<li>When the function is shifted right [latex]3[\/latex] units to [latex]h\\left(x\\right)={2}^{x-3},[\/latex] the vertical intercept becomes [latex]\\left(0,\\frac{1}{8}\\right).[\/latex] Again, see that [latex]{2}^{x-3}={2}^{x}{2}^{-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 id=\"fs-id1165134042183\">\r\n<h3>Shifts of the Function y\u00a0= <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165134037589\">For any constants [latex]c[\/latex] and [latex]d,[\/latex] the function [latex]f\\left(x\\right)={b}^{x-c}+d[\/latex] shifts the exponential function [latex]y={b}^{x}[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li>vertically [latex]d[\/latex] units, in the direction of the sign of [latex]d,[\/latex] and<\/li>\r\n \t<li>horizontally [latex]c[\/latex] units, in the direction of the sign of [latex]c.[\/latex]<\/li>\r\n \t<li>The vertical intercept becomes [latex]\\left(0,{b}^{-c}+d\\right).[\/latex]<\/li>\r\n \t<li>The horizontal asymptote becomes [latex]y=d[\/latex] on the same side.<\/li>\r\n \t<li>The range becomes [latex]\\left(d,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] remains unchanged.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135500732\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135500706\"><strong>Given an exponential function with the form [latex]f\\left(x\\right)={b}^{x-c}+d,[\/latex] graph the translation.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137767676\" type=\"1\">\r\n \t<li>Draw the horizontal asymptote [latex]y=d.[\/latex]<\/li>\r\n \t<li>Identify [latex]c[\/latex] and [latex]d[\/latex].\u00a0 Shift the graph of [latex]y={b}^{x}[\/latex] right [latex]c[\/latex] units if [latex]c[\/latex] is positive, and left [latex]c[\/latex] units if [latex]c[\/latex] is negative.<\/li>\r\n \t<li>Shift the graph of [latex]y={b}^{x}[\/latex] up [latex]d[\/latex] units if [latex]d[\/latex] is positive, and down [latex]d[\/latex] units if [latex]d[\/latex] is negative.<\/li>\r\n \t<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] the range, [latex]\\left(d,\\infty \\right),[\/latex] and the horizontal asymptote [latex]y=d.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_02_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137834201\">\r\n<div id=\"fs-id1165137416701\">\r\n<h3>Example 2:\u00a0 Graphing a Shift of an Exponential Function<\/h3>\r\n<p id=\"fs-id1165137563667\">Graph [latex]f\\left(x\\right)={2}^{x+1}-3.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135175234\">[reveal-answer q=\"fs-id1165135175234\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135175234\"]\r\n<p id=\"fs-id1165137923482\">We have an exponential equation of the form [latex]f\\left(x\\right)={b}^{x-c}+d,[\/latex] with [latex]b=2,[\/latex] [latex]c=-1,[\/latex] and [latex]d=-3.[\/latex]<\/p>\r\n<p id=\"fs-id1165137469681\">Draw the horizontal asymptote [latex]y=d[\/latex], so draw [latex]y=-3.[\/latex]\u00a0 Shift the graph of [latex]y={2}^{x}[\/latex] left 1 unit and down 3 units.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"278\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210320\/CNX_Precalc_Figure_04_02_008.jpg\" alt=\"Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).\" width=\"278\" height=\"296\" \/> Figure 7[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134199602\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(-3,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=-3[\/latex] on the left side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135241073\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_04_02_02\">\r\n<div id=\"fs-id1165137805939\">\r\n<p id=\"fs-id1165137805941\">Graph [latex]f\\left(x\\right)={2}^{x-1}+3.[\/latex] State domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137731918\">[reveal-answer q=\"fs-id1165137731918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137731918\"]\r\n<p id=\"fs-id1165135513714\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(3,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=3[\/latex] on the left side.<\/p>\r\n<span id=\"fs-id1165137628194\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210323\/CNX_Precalc_Figure_04_02_009.jpg\" alt=\"Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).\" width=\"319\" height=\"321\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137431154\" class=\"bc-section section\">\r\n<h4>Graphing a Vertical Stretch or Compression<\/h4>\r\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <span class=\"no-emphasis\">stretch<\/span> or <span class=\"no-emphasis\">compression<\/span> occurs when we multiply the exponential function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|&gt;0.[\/latex] For example, if we begin by graphing the function [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph the vertical stretch, using [latex]a=3,[\/latex] to get [latex]g\\left(x\\right)=3f\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] as shown on the left in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">Figure 8a<\/a>, and the vertical compression, using [latex]a=\\frac{1}{3},[\/latex] to get [latex]h\\left(x\\right)=\\frac{1}{3}f\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] as shown on the right in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">Figure 8b<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_010\" class=\"wp-caption aligncenter\" style=\"width: 957px;\">[caption id=\"\" align=\"alignnone\" width=\"671\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210327\/CNX_Precalc_Figure_04_02_010.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" width=\"671\" height=\"306\" \/> Figure 8. (a) [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] stretches the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]3.[\/latex] (b) [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] compresses the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]\\frac{1}{3}.[\/latex][\/caption]<\/div>\r\n<div id=\"fs-id1165137627908\">\r\n<h3>Stretches and Compressions of the Function <i>y<\/i>\u00a0= <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165137696285\">For any factor [latex]a&gt;0,[\/latex] the function [latex]f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137476370\">\r\n \t<li>stretches [latex]y={b}^{x}[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|&gt;1.[\/latex]<\/li>\r\n \t<li>compresses [latex]y={b}^{x}[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|&lt;1.[\/latex]<\/li>\r\n \t<li>has a vertical intercept of [latex]\\left(0,a\\right).[\/latex]<\/li>\r\n \t<li>has a horizontal asymptote at [latex]y=0,[\/latex] a range of [latex]\\left(0,\\infty \\right),[\/latex] and a domain of [latex]\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from [latex]y={b}^{x}.[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135528997\">\r\n<div id=\"fs-id1165135656098\">\r\n<h3 id=\"fs-id1165135656100\">Example 3:\u00a0 Graphing the Stretch of an Exponential Function<\/h3>\r\n<p id=\"fs-id1165135656104\">Sketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137657436\">[reveal-answer q=\"fs-id1165137657436\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137657436\"]\r\n<div>\r\n\r\nBefore graphing, identify the behavior and key points on the graph.\r\n<ul id=\"fs-id1165137657441\">\r\n \t<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left end behavior of the graph will increase without bound as [latex]x[\/latex] decreases without bound, and the right end behavior will approach the <em>x<\/em>-axis as [latex]x[\/latex] increases without bound.<\/li>\r\n \t<li>Since [latex]a=4,[\/latex] the graph of [latex]y={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched by a factor of [latex]4.[\/latex]<\/li>\r\n \t<li>Create a table of points as shown in <a class=\"autogenerated-content\" href=\"#Table_04_02_04\">Table 8<\/a>.\r\n<table id=\"Table_04_02_04\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=4(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 32), (-2, 16), (-1, 8), (0, 4), (1, 2), (2, 1), and (3, 0.5).\"><caption>Table 8<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 163.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 163.656px; text-align: center;\">[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]32[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]16[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]0.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Plot the vertical intercept, [latex]\\left(0,4\\right),[\/latex] along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right).[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135319502\">Draw a smooth curve connecting the points, as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_011\">Figure 9<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_011\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"296\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210330\/CNX_Precalc_Figure_04_02_011.jpg\" alt=\"Graph of the function, f(x) = 4(1\/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).\" width=\"296\" height=\"293\" \/> Figure 9[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137442037\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135541809\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_04_02_04\">\r\n<div>\r\n<p id=\"fs-id1165137452032\">Sketch the graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137694067\">[reveal-answer q=\"fs-id1165137694067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137694067\"]\r\n<p id=\"fs-id1165137653325\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the left side.<\/p>\r\n<span id=\"fs-id1165135417835\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210333\/CNX_Precalc_Figure_04_02_012.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"407\" height=\"245\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135433028\" class=\"bc-section section\">\r\n<h4>Graphing Reflections<\/h4>\r\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect an exponential function about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the exponential function [latex]f\\left(x\\right)={b}^{x}[\/latex] by [latex]-1,[\/latex] we get a vertical reflection about the <em>x<\/em>-axis. When we multiply the input by [latex]-1,[\/latex] we get a <span class=\"no-emphasis\">reflection<\/span> about the <em>y<\/em>-axis. For example, if we begin by graphing the function [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x},[\/latex] is shown on the left side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">Figure 10a<\/a>, and the reflection about the <em>y<\/em>-axis [latex]h\\left(x\\right)={2}^{-x},[\/latex] is shown on the right side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">Figure 10b<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_013\" class=\"wp-caption aligncenter\" style=\"width: 966px;\">[caption id=\"\" align=\"aligncenter\" width=\"612\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210337\/CNX_Precalc_Figure_04_02_013.jpg\" alt=\"Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.\" width=\"612\" height=\"394\" \/> Figure 10 (a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]g\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.[\/caption]<\/div>\r\n<div id=\"fs-id1165135477501\">\r\n<h3>Reflections of the\u00a0 Function <i>y<\/i>\u00a0= f(x) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165137455888\">The function [latex]g\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li>reflects the function [latex]y={b}^{x}[\/latex] over the <em>x<\/em>-axis.<\/li>\r\n \t<li>has a vertical intercept of [latex]\\left(0,-1\\right).[\/latex]<\/li>\r\n \t<li>has a range of [latex]\\left(-\\infty ,0\\right).[\/latex]<\/li>\r\n \t<li>has a horizontal asymptote at [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from the function [latex]y={b}^{x}.[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137742185\">The function [latex]h\\left(x\\right)={b}^{-x}[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137551240\">\r\n \t<li>reflects the\u00a0 function [latex]y={b}^{x}[\/latex] over the <em>y<\/em>-axis.<\/li>\r\n \t<li>has a vertical intercept of [latex]\\left(0,1\\right),[\/latex] a horizontal asymptote at [latex]y=0,[\/latex] a range of [latex]\\left(0,\\infty \\right),[\/latex] and a domain of [latex]\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from the function [latex]y={b}^{x}.[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137406134\">\r\n<div id=\"fs-id1165137406136\">\r\n<h3>Example 4:\u00a0 Writing and Graphing the Reflection of an Exponential Function<\/h3>\r\n<p id=\"fs-id1165137896193\">Find and graph the equation for a function, [latex]g\\left(x\\right),[\/latex] that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] over the <em>x<\/em>-axis. State its domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137937537\">[reveal-answer q=\"fs-id1165137937537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137937537\"]\r\n<p id=\"fs-id1165137937539\">Since we want to reflect the function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by [latex]-1[\/latex] to get, [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}.[\/latex] Next we create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_005\">Table 9<\/a>.<\/p>\r\n\r\n<table id=\"Table_04_02_005\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=-(1\/4)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, -64), (-2, -16), (-1, -4), (0, -1), (1, -0.25), (2, -0.0625), and (3, -0.0156).\"><caption>Table 9<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 105.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 91.6563px; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 105.656px; text-align: center;\"><strong>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-64[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-16[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-4[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 91.6563px; text-align: center;\">[latex]-0.25[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]-0.0625[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]-0.0156[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"eip-id1167546794019\">Plot the <em>y-<\/em>intercept, [latex]\\left(0,-1\\right),[\/latex] along with two other points. We can use [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(1,-0.25\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135369275\">Draw a smooth curve connecting the points:<\/p>\r\n\r\n<div id=\"CNX_Precalc_Figure_04_02_014\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"308\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210340\/CNX_Precalc_Figure_04_02_014.jpg\" alt=\"Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).\" width=\"308\" height=\"257\" \/> Figure 11.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137828154\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(-\\infty ,0\\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135205992\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_04_02_05\">\r\n<div id=\"fs-id1165135254653\">\r\n<p id=\"fs-id1165135254655\">Find and graph the equation for a function, [latex]g\\left(x\\right),[\/latex] that reflects [latex]f\\left(x\\right)={1.25}^{x}[\/latex] over the <em>y<\/em>-axis. State its domain, range, and horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135368458\">[reveal-answer q=\"fs-id1165135368458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135368458\"]\r\n<p id=\"fs-id1165135368461\">The function is [latex]g\\left(x\\right)={1.25}^{-x}={0.8}^{x}.[\/latex] The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\r\n<span id=\"fs-id1165137828034\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210343\/CNX_Precalc_Figure_04_02_015.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" width=\"452\" height=\"298\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135501015\" class=\"bc-section section\">\r\n<h4>Summarizing Translations of the Exponential Function<\/h4>\r\nNow that we have worked with each type of translation for the exponential function, we can summarize them in <a class=\"autogenerated-content\" href=\"#Table_04_02_006\">Table 10<\/a>\u00a0to arrive at the general equation for translating exponential functions.\r\n<table id=\"Table_04_02_006\"><caption>Table 10<\/caption>\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Translations of the Function [latex]y={b}^{x}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th class=\"border\">Translation<\/th>\r\n<th class=\"border\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Shift\r\n<ul id=\"fs-id1165137640731\">\r\n \t<li>Horizontally [latex]c[\/latex] units to the left or right<\/li>\r\n \t<li>Vertically [latex]d[\/latex] units up or down<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td class=\"border\">[latex]f\\left(x\\right)={b}^{x-c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Vertical Stretch and Compression\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]0&lt;|a|&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td class=\"border\">[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Reflect about the <em>x<\/em>-axis<\/td>\r\n<td class=\"border\">[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Reflect about the <em>y<\/em>-axis<\/td>\r\n<td class=\"border\">[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">General equation for all translations<\/td>\r\n<td class=\"border\">[latex]f\\left(x\\right)=a{b}^{x-c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Q&amp;A<\/h3>\r\n<strong>Why isn't there a discussion on horizontal stretches and compressions?<\/strong>\r\n\r\n<em>Recall the exponential rule [latex]b^{mn}={\\left(b^m\\right)}^n.[\/latex] Essentially a horizontal compression would be a change in the base of the function.\u00a0 For example,\u00a0[latex]b^{3x}={\\left(b^3\\right)}^x.[\/latex] The original base is [latex]b[\/latex] with a horizontal compression by a factor of [latex]\\frac{1}{3}[\/latex], but we can also simply consider this as a function with the new base [latex]b^3[\/latex].\u00a0\u00a0<\/em>\r\n\r\n<em>Think of [latex]f\\left(x\\right)=2^{3x}.[\/latex] We can think of this as a horizontal compression by a factor of [latex]\\frac{1}{3}[\/latex] of the function [latex]y=2^{x}.[\/latex] The point (1,2) will be compressed to the point [latex]\\left(\\frac{1}{3},2\\right).[\/latex] Notice that if we used the function [latex]g\\left(x\\right)=\\left(2^{3}\\right)^{x}\\text{ or }g\\left(x\\right)=8^{x},[\/latex] we would also see the point\u00a0[latex]\\left(\\frac{1}{3},2\\right).[\/latex] This helps us see that we can achieve the same results as horizontal compressions by rewriting the function with a different base.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_04_02_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137937623\">\r\n<div id=\"fs-id1165135250578\">\r\n<h3 id=\"fs-id1165135250580\">Example 5:\u00a0 Writing a Function from a Description<\/h3>\r\n<p id=\"fs-id1165135250584\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\r\n\r\n<ul id=\"fs-id1165137724821\">\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of [latex]2[\/latex], reflected across the <em>y<\/em>-axis, and then shifted up [latex]4[\/latex] units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135532412\">[reveal-answer q=\"fs-id1165135532412\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135532412\"]\r\n<p id=\"fs-id1165135532414\">We want to find an equation of the general form [latex]g\\left(x\\right)=a{b}^{x-c}+d.[\/latex] We use the description provided to find [latex]a,[\/latex] [latex]b,[\/latex] [latex]c,[\/latex] and [latex]d.[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137807102\">\r\n \t<li>We are given the function [latex]f\\left(x\\right)={e}^{x},[\/latex] so [latex]b=e.[\/latex]<\/li>\r\n \t<li>The function is stretched by a factor of [latex]2[\/latex], so [latex]a=2.[\/latex]<\/li>\r\n \t<li>The function is reflected about the <em>y<\/em>-axis. We replace [latex]x[\/latex] 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\n<p id=\"fs-id1165137634849\">Substituting in the general form we get,<\/p>\r\n\r\n<div id=\"eip-id1165137832492\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*} g\\left(x\\right)&amp;=a{b}^{x-c}+d\\hfill\\\\ \\hfill&amp;=2{e}^{-x-0}+4\\hfill\\\\ \\hfill&amp;=2{e}^{-x}+4\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137665666\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(4,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=4[\/latex] on the right side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137553895\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"ti_04_02_06\">\r\n<div id=\"fs-id1165137724079\">\r\n<p id=\"fs-id1165137724081\">Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\r\n\r\n<ul id=\"fs-id1165137539693\">\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3},[\/latex] reflected across the <em>x<\/em>-axis and then shifted down [latex]2[\/latex] units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137724110\">[reveal-answer q=\"fs-id1165137724110\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137724110\"]\r\n<p id=\"fs-id1165137724112\">[latex]g\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2;[\/latex] the domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(-\\infty ,-2\\right);[\/latex] the horizontal asymptote is [latex]y=-2[\/latex] on the right side.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135560786\" class=\"precalculus media\">\r\n<div id=\"fs-id1165137566517\" class=\"bc-section section\">\r\n<h3>Approximating Solutions to an Exponential Equation with the Calculator<\/h3>\r\nSometimes we want to find out when an exponential function will have a particular output value.\u00a0 The next sections will focus on being able to do this algebraically.\u00a0 Currently, we can use technology to determine what input will give a particular output for the transformed exponential function.\r\n<div id=\"fs-id1165137639988\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137756810\"><strong>Given an equation of the form [latex]y=a{b}^{x-c}+d,[\/latex] use a graphing calculator to approximate the solution.<\/strong><\/p>\r\n\r\n<ul id=\"fs-id1165137842461\">\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]y[\/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>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]y.[\/latex]<\/li>\r\n \t<li>To find the value of [latex]x,[\/latex] 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 [latex]x[\/latex] for the indicated output value of the function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137618985\">\r\n<div id=\"fs-id1165137618987\">\r\n<h3>Example 6:\u00a0 Approximating the Solution of an Exponential Equation<\/h3>\r\n<p id=\"fs-id1165135449598\">Solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically. Round to the nearest thousandth.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137653309\">[reveal-answer q=\"fs-id1165137653309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137653309\"]\r\n<p id=\"fs-id1165137737383\">Press <strong>[Y=]<\/strong> and enter [latex]1.2{\\left(5\\right)}^{x}+2.8[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong>Y2=<\/strong>. For a window, use the values \u20133 to 3 for [latex]x[\/latex] and \u20135 to 55 for [latex]y.[\/latex] Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near [latex]x=2.[\/latex]<\/p>\r\n<p id=\"fs-id1165137460953\">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>x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be slightly 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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135545893\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"ti_04_02_03\">\r\n<div id=\"fs-id1165137838712\">\r\n<p id=\"fs-id1165137838714\">Solve [latex]4=7.85{\\left(1.15\\right)}^{x}-2.27[\/latex] graphically. Round to the nearest thousandth.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137854192\">[reveal-answer q=\"fs-id1165137854192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137854192\"]\r\n<p id=\"fs-id1165137854194\">[latex]x\\approx -1.608[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137785000\">Access this online resource for additional instruction and practice with graphing exponential functions.<\/p>\r\n\r\n<ul id=\"fs-id1165137785004\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/graphexpfunc\">Graph Exponential Functions<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137661989\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"fs-id2055298\" summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 414px;\">General Form for the Translation of the Function [latex]y={b}^{x}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 225px;\">[latex]f\\left(x\\right)=a{b}^{x-c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137447701\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165137447708\">\r\n \t<li>The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a <em>y-<\/em>intercept at [latex]\\left(0, 1\\right),[\/latex] domain [latex]\\left(-\\infty , \\infty \\right),[\/latex] range[latex]\\left(0, \\infty \\right),[\/latex] and horizontal asymptote [latex]y=0.[\/latex]<\/li>\r\n \t<li>End behavior describes what happens to the output if you go very far to the left or right.\r\n<ul>\r\n \t<li>If [latex]b&gt;1,[\/latex] the function is increasing. The left end behavior of the graph will approach the horizontal asymptote [latex]y=0,[\/latex] and the right end behavior will increase without bound.<\/li>\r\n \t<li>If [latex]0 \\lt b \\lt 1,[\/latex] the function is decreasing. The left end behavior of the graph will increase without bound, and the right end behavior will approach the horizontal asymptote [latex]y=0.[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the exponential function [latex]y={b}^{x}.[\/latex]<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={b}^{x-c}[\/latex] represents a horizontal shift of the exponential function [latex]y={b}^{x}.[\/latex]<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)=a{b}^{x},[\/latex] where [latex]a&gt;0,[\/latex] represents a vertical stretch if [latex]|a|&gt;1[\/latex] or compression if [latex]0&lt;|a|&lt;1[\/latex] of the exponential function [latex]y={b}^{x}.[\/latex]<\/li>\r\n \t<li>When the exponential function [latex]y={b}^{x}[\/latex] is multiplied by [latex]-1,[\/latex]\u00a0 the result, [latex]g\\left(x\\right)=-{b}^{x},[\/latex] is a reflection about the <em>x<\/em>-axis. When the input is multiplied by [latex]-1,[\/latex] the result, [latex]h\\left(x\\right)={b}^{-x},[\/latex] is a reflection about the <em>y<\/em>-axis.<\/li>\r\n \t<li>All translations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x-c}+d.[\/latex]<\/li>\r\n \t<li>Using the general equation [latex]f\\left(x\\right)=a{b}^{x-c}+d,[\/latex] we can write the equation of a function given its description.<\/li>\r\n \t<li>Approximate solutions of the equation [latex]y={b}^{x-c}+d[\/latex] can be found using a graphing calculator.<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Graph exponential functions.<\/li>\n<li>Determine the end behavior and horizontal asymptotes of exponential functions.<\/li>\n<li>Graph exponential functions using transformations.<\/li>\n<\/ul>\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<div id=\"fs-id1165135407520\" class=\"bc-section section\">\n<h3>Graphing Exponential Functions<\/h3>\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\\left(x\\right)={b}^{x}[\/latex] whose base is greater than one. We\u2019ll use the function [latex]f\\left(x\\right)={2}^{x}.[\/latex] Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_01\">Table 1<\/a>\u00a0change as the input increases by [latex]1.[\/latex]<\/p>\n<table id=\"Table_04_02_01\" style=\"height: 38px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=2^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8).\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"width: 120.5px; height: 12px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 83.5px; height: 12px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; height: 12px; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; height: 12px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 12px; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td class=\"border\" style=\"width: 120.5px; height: 26px; text-align: center;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 83.5px; height: 26px; text-align: center;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; height: 26px; text-align: center;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; height: 26px; text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; height: 26px; text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137432031\">Each output value is the product of the previous output and the base, [latex]2.[\/latex] We call the base [latex]2[\/latex] the <strong><em>constant ratio<\/em><\/strong> or <strong><em>growth factor<\/em><\/strong>. In fact, for any exponential function with the form [latex]f\\left(x\\right)=a{b}^{x},[\/latex] [latex]b[\/latex] is the constant ratio of the function. This means that each time we increase the input by 1, we multiply the output by [latex]b[\/latex]. Notice from the table that the output values are positive for all values of [latex]x.[\/latex]<\/p>\n<h3>End Behavior of [latex]f\\left(x\\right)=ab^x[\/latex]<\/h3>\n<p>Often we want to know what happens to the output value of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] moves far to the left or far to the right.\u00a0 This is known as the <strong>end behavior<\/strong>\u00a0or <strong>long term behavior<\/strong> of the function.\u00a0 We will continue to study the function [latex]f\\left(x\\right)=2^x[\/latex] and determine its end behavior.<\/p>\n<p>Begin by looking at two tables of values.\u00a0 Table 2 shows function values as x moves far to the left.\u00a0 We choose x values of -10, -100 and -250 and evaluate [latex]f\\left(x\\right)[\/latex] at these values so we can observe what is happening to the output values to the far left.\u00a0 Note that -250 is not considered long term behavior for most functions but for an exponential function it is about the limit of what our current technology can compute.\u00a0 In Table 3, we move far toward the right choosing our x values to be 10, 100, and 250 and evaluate the function so we can observe what happens to the output when x gets large.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-10<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-100<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-250<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]f\\left(x\\right)=2^x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">9.77E-4<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">7.89E-31<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">5.53E-76<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">10<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">100<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">250<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]f\\left(x\\right)=2^x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1024<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1.27E30<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1.81E75<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Recognize that scientific notation is being used in Table 2 and Table 3 for the output values.\u00a0 In Table 2, we observe that\u00a0as [latex]x[\/latex] decreases or becomes more and more negative, the output values get closer and closer to zero from above.\u00a0 We capture this idea using arrow notation and write as [latex]x\\to-\\infty,f\\left(x\\right)\\to0.[\/latex] This is read, &#8220;As x decreases without bound, [latex]f[\/latex] of x goes to zero.&#8221; When we are studying end behavior and we observe that the output is getting closer and closer to a value, we say that there is a horizontal asymptote.\u00a0 In this example, [latex]y=0[\/latex] is the horizontal asymptote on the left hand side.<\/p>\n<p>Further, Table 3 shows that as [latex]x[\/latex] increases or becomes larger and larger, the output values also become larger and larger or increase without bound.\u00a0 We write as [latex]x\\to\\infty,f\\left(x\\right)\\to\\infty.[\/latex] Since these output values increase without bound, there is not a horizontal asymptote in this direction.<\/p>\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137782455\">A <strong>horizontal asymptote<\/strong> of a graph is a horizontal line [latex]y=b[\/latex] where the graph approaches the line as the inputs increase or decrease without bound. We write as [latex]x\\to \\infty \\textrm{ or }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to b.[\/latex]<\/p>\n<\/div>\n<p id=\"fs-id1165137647215\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_001\">Figure 1<\/a>\u00a0shows the exponential growth function [latex]f\\left(x\\right)={2}^{x}.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_001\" class=\"small\">\n<div style=\"width: 305px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210258\/CNX_Precalc_Figure_04_02_001.jpg\" alt=\"Graph of the exponential function, 2^(x), with labeled points at (-3, 1\/8), (-2, \u00bc), (-1, \u00bd), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.\" width=\"295\" height=\"315\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1: Notice that the graph gets close to the x-axis, but never touches it.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137459614\">We observe in the graph above that as x becomes more negative, the graph is getting closer to the x-axis but never touches it demonstrating the horizontal asymptote of [latex]y=0.[\/latex]\u00a0 Other characteristics of the graph can also be observed.\u00a0 The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers, and the range is [latex]\\left(0,\\infty \\right).[\/latex]\u00a0 Notice that the function is also increasing and concave up on [latex]\\left(-\\infty,\\infty \\right).[\/latex]<\/p>\n<h3>Exponential Decay Graphically<\/h3>\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <span class=\"no-emphasis\">exponential decay<\/span>, we can create a table of values for a function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex] whose base is between zero and one. We\u2019ll use the function [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.[\/latex] Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_02\">Table 4<\/a>\u00a0change as the input increases by 1.<\/p>\n<table id=\"Table_04_02_02\" summary=\"Two rows and eight columns. The first row is labeled, \u201cf(x)=2^x\u201d, with the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, \u201cg(x)=log_2(x)\u201d, with the following values: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\n<caption>Table 4<\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 178.5px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 178.5px; text-align: center;\">[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]8[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"width: 83.5px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 79.5px; text-align: center;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135347846\">Again, notice that each time the input is increased by 2, the output is multiplied by the base, or constant ratio [latex]b=\\frac{1}{2}.[\/latex]<\/p>\n<p>To look at the end behavior of the exponential decay function, we again create tables with input values to the far left and right.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 5<\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-10<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-100<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">-250<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1024<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">1.27E30<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">\u00a01.81E75<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 6<\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">10<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">100<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">250<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^x[\/latex]<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">9.77E-4<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">7.89E-31<\/td>\n<td class=\"border\" style=\"width: 25%; text-align: center;\">5.53E-76<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137452063\">Notice from the tables above that:<\/p>\n<ul id=\"fs-id1165135499992\">\n<li>the output values are positive for all values of [latex]x.[\/latex]<\/li>\n<li>as [latex]x[\/latex] decreases, the output values grow without bound so as [latex]x\\to-\\infty,g\\left(x\\right)\\to\\infty.[\/latex]<\/li>\n<li>as [latex]x[\/latex] increases without bound, the output values approach zero from above so as [latex]x\\to\\infty,g\\left(x\\right)\\to0.[\/latex] The horizontal asymptote is [latex]y=0[\/latex] on the right hand side.<\/li>\n<\/ul>\n<p id=\"fs-id1165137405421\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_002\">Figure 2<\/a>\u00a0shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_002\" class=\"small\">\n<div style=\"width: 360px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210301\/CNX_Precalc_Figure_04_02_002.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"350\" height=\"374\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2 The graph shows that as x gets larger, the output gets close to zero.<\/p>\n<\/div>\n<\/div>\n<p>Again we observe the end behavior and see that as x increases, the graph approaches the x-axis and there is a horizontal asymptote of [latex]y=0.[\/latex]\u00a0 Other characteristics can also be observed from the graph.\u00a0 The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers, and the range is [latex]\\left(0,\\infty \\right).[\/latex]\u00a0 Notice that the function is decreasing and concave up on [latex]\\left(-\\infty,\\infty \\right).[\/latex]<\/p>\n<div id=\"fs-id1165135571835\">\n<h3>Characteristics of the Graph of the Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<div class=\"textbox\">\n<p id=\"fs-id1165137848929\">An exponential function with the form [latex]f\\left(x\\right)={b}^{x},[\/latex] [latex]b>0,[\/latex] [latex]b\\ne 1,[\/latex] has these characteristics:<\/p>\n<ul>\n<li><span class=\"no-emphasis\">one-to-one<\/span> function<\/li>\n<li>horizontal asymptote: [latex]y=0[\/latex] on one side<\/li>\n<li>domain: [latex]\\left(\u2013\\infty , \\infty \\right)[\/latex]<\/li>\n<li>range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n<li><em>x-<\/em>intercept: none<\/li>\n<li><em>y-<\/em>intercept: [latex]\\left(0,1\\right)[\/latex]<\/li>\n<li>increasing if [latex]b>1[\/latex]<\/li>\n<li>decreasing if [latex]b<1[\/latex]<\/li>\n<li>concave up<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137471878\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_003\">Figure 3<\/a>\u00a0compares the graphs of <span class=\"no-emphasis\">exponential growth<\/span> and decay functions.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_003\" class=\"medium\">\n<div style=\"width: 543px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210304\/CNX_Precalc_Figure_04_02_003new.jpg\" alt=\"\" width=\"533\" height=\"297\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134195243\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135194093\"><strong>Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x},[\/latex] graph the function.<\/strong><\/p>\n<ol id=\"fs-id1165137435782\" type=\"1\">\n<li>Create a table of points.<\/li>\n<li>Plot at least [latex]3[\/latex] points from the table, including the vertical intercept [latex]\\left(0,1\\right).[\/latex]<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] the range, [latex]\\left(0,\\infty \\right),[\/latex] and on which side the horizontal asymptote, [latex]y=0[\/latex] occurs.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_02_01\" class=\"textbox examples\">\n<div id=\"fs-id1165135208984\">\n<div id=\"fs-id1165137453336\">\n<h3>Example 1:\u00a0 Sketching the Graph of an Exponential Function of the Form <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137767671\">Sketch a graph of [latex]f\\left(x\\right)={0.25}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135696740\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135696740\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135696740\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137734539\">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 end behavior of the graph will increase without bound, and the right end behavior will approach the horizontal asymptote [latex]y=0.[\/latex]<\/li>\n<li>Create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_03\">Table 7<\/a>.<br \/>\n<table id=\"Table_04_02_03\" style=\"height: 34px;\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, 0.015625).\">\n<tbody>\n<tr>\n<td style=\"width: 108.656px; text-align: center;\" colspan=\"6\"><strong>Table 7<\/strong><\/td>\n<td style=\"width: 100.656px;\"><\/td>\n<td style=\"width: 113.656px;\"><\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td class=\"border\" style=\"height: 11px; width: 108.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 11px; width: 77.6562px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"height: 11px; width: 77.6562px; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"height: 11px; width: 75.6562px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"height: 11px; width: 71.6562px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"height: 11px; width: 87.6562px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"height: 11px; width: 100.656px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"height: 11px; width: 113.656px; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td class=\"border\" style=\"height: 23px; width: 108.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 23px; width: 77.6562px; text-align: center;\">[latex]64[\/latex]<\/td>\n<td class=\"border\" style=\"height: 23px; width: 77.6562px; text-align: center;\">[latex]16[\/latex]<\/td>\n<td class=\"border\" style=\"height: 23px; width: 75.6562px; text-align: center;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"height: 23px; width: 71.6562px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"height: 23px; width: 87.6562px; text-align: center;\">[latex]0.25[\/latex]<\/td>\n<td class=\"border\" style=\"height: 23px; width: 100.656px; text-align: center;\">[latex]0.0625[\/latex]<\/td>\n<td class=\"border\" style=\"height: 23px; width: 113.656px; text-align: center;\">[latex]0.015625[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y<\/em>-intercept, [latex]\\left(0,1\\right),[\/latex] along with two other points. We can use [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(1,0.25\\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_004\">Figure 4<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_004\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210308\/CNX_Precalc_Figure_04_02_004.jpg\" alt=\"Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).\" width=\"487\" height=\"332\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137548870\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135499977\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_04_02_01\">\n<div id=\"fs-id1165137761245\">\n<p id=\"fs-id1165137548853\">Sketch the graph of [latex]f\\left(x\\right)={4}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137731723\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137731723\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137731723\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137500954\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the left side.<\/p>\n<p><span id=\"fs-id1165137437648\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210311\/CNX_Precalc_Figure_04_02_005.jpg\" alt=\"Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137694074\" class=\"bc-section section\">\n<h3>Graphing Transformations of Exponential Functions<\/h3>\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with our toolkit functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the exponential function [latex]f\\left(x\\right)={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<p><strong>NOTE:<\/strong> In this section we will be using different notation for horizontal and vertical shifts. Recall that in section 1.6, we considered functions in the form [latex]g\\left(x\\right)=af\\left(b\\left(x-h\\right)\\right)+k.[\/latex] In this notation, [latex]k[\/latex] indicated the vertical shift, [latex]h[\/latex] indicated the horizontal shift,\u00a0[latex]a[\/latex] indicated the vertical stretch\/compression, and\u00a0[latex]b[\/latex] indicated the horizontal stretch\/compression.\u00a0 When we study exponential functions, we have already designated\u00a0[latex]k[\/latex] to indicate continuous growth.\u00a0 Therefore, we will modify our notation and use\u00a0[latex]c[\/latex] to represent horizontal shifts and [latex]d[\/latex] to represent vertical shifts. Vertical stretches\/compressions will still be represented by [latex]a.[\/latex] The variable [latex]b[\/latex] will represent the base of the exponential function and not represent a horizontal stretch or compression.<\/p>\n<div id=\"fs-id1165134312214\" class=\"bc-section section\">\n<h4>Graphing a Vertical Shift<\/h4>\n<p id=\"fs-id1165137911544\">The first transformation occurs when we add a constant [latex]d[\/latex] to the exponential function [latex]f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">vertical shift\u00a0<\/span>[latex]d[\/latex] units in the same direction as the sign.\u00a0For example, if we begin by graphing the function, [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph two vertical shifts alongside it, using [latex]d=3[\/latex] and [latex]d=-3:[\/latex] the upward shift, [latex]g\\left(x\\right)=f\\left(x\\right)+3={2}^{x}+3[\/latex] and the downward shift, [latex]h\\left(x\\right)=f\\left(x\\right)-3={2}^{x}-3.[\/latex] Both vertical shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_006\">Figure 5<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_006\" class=\"small\">\n<div style=\"width: 326px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210315\/CNX_Precalc_Figure_04_02_006.jpg\" alt=\"Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions\u2019 transformations are described in the text.\" width=\"316\" height=\"408\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137464499\">Observe the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically:<\/p>\n<ul id=\"fs-id1165135203774\">\n<li>The domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex] remains unchanged.<\/li>\n<li>When the function is shifted up [latex]3[\/latex] units to [latex]g\\left(x\\right)={2}^{x}+3:[\/latex]\n<ul id=\"fs-id1165137601587\">\n<li>The <em>y-<\/em>intercept shifts up [latex]3[\/latex] units to [latex]\\left(0,4\\right).[\/latex]<\/li>\n<li>The horizontal asymptote shifts up [latex]3[\/latex] units to [latex]y=3[\/latex] on the left side.<\/li>\n<li>The range becomes [latex]\\left(3,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>When the function is shifted down [latex]3[\/latex] units to [latex]h\\left(x\\right)={2}^{x}-3:[\/latex]\n<ul id=\"fs-id1165137784817\">\n<li>The <em>y-<\/em>intercept shifts down [latex]3[\/latex] units to [latex]\\left(0,-2\\right).[\/latex]<\/li>\n<li>The horizontal asymptote also shifts down [latex]3[\/latex] units to [latex]y=-3[\/latex] on the left side.<\/li>\n<li>The range becomes [latex]\\left(-3,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137566517\" class=\"bc-section section\">\n<h4>Graphing a Horizontal Shift<\/h4>\n<p id=\"fs-id1165137748336\">The next transformation occurs when we subtract a constant [latex]c[\/latex] from the input of the exponential function [latex]f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">horizontal shift\u00a0<\/span>[latex]c[\/latex] units in the direction of the sign of [latex]c[\/latex]. The equation is given by [latex]f\\left(x-c\\right)=b^{x-c}.[\/latex] For example, if we begin by graphing the function [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph two horizontal shifts alongside it, using<\/p>\n<p style=\"padding-left: 30px;\">[latex]c=-3:[\/latex] the shift left, [latex]g\\left(x\\right)=f\\left(x+3\\right)={2}^{x+3},[\/latex] and using<\/p>\n<p style=\"padding-left: 30px;\">[latex]c=3:[\/latex] the shift right, [latex]h\\left(x\\right)=f\\left(x-3\\right)={2}^{x-3}.[\/latex]<\/p>\n<p>Both horizontal shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_007\">Figure 6<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_007\" class=\"medium\">\n<div style=\"width: 460px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210318\/CNX_Precalc_Figure_04_02_007.jpg\" alt=\"Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions\u2019 asymptotes are at y=0Note that each functions\u2019 transformations are described in the text.\" width=\"450\" height=\"294\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137411256\">Observe the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] horizontally:<\/p>\n<ul id=\"fs-id1165135187815\">\n<li>The domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] remains unchanged.<\/li>\n<li>The horizontal asymptote, [latex]y=0,[\/latex] remains unchanged.<\/li>\n<li>The vertical intercept shifts such that:\n<ul id=\"fs-id1165137482879\">\n<li>When the function is shifted left [latex]3[\/latex] units to [latex]g\\left(x\\right)={2}^{x+3},[\/latex] the vertical intercept becomes [latex]\\left(0,8\\right).[\/latex] This is because [latex]{2}^{x+3}={2}^{x}{2}^{3}=\\left(8\\right){2}^{x}[\/latex] using the rules of exponents, so the initial value of the function is [latex]8.[\/latex]<\/li>\n<li>When the function is shifted right [latex]3[\/latex] units to [latex]h\\left(x\\right)={2}^{x-3},[\/latex] the vertical intercept becomes [latex]\\left(0,\\frac{1}{8}\\right).[\/latex] Again, see that [latex]{2}^{x-3}={2}^{x}{2}^{-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 id=\"fs-id1165134042183\">\n<h3>Shifts of the Function y\u00a0= <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165134037589\">For any constants [latex]c[\/latex] and [latex]d,[\/latex] the function [latex]f\\left(x\\right)={b}^{x-c}+d[\/latex] shifts the exponential function [latex]y={b}^{x}[\/latex]<\/p>\n<ul>\n<li>vertically [latex]d[\/latex] units, in the direction of the sign of [latex]d,[\/latex] and<\/li>\n<li>horizontally [latex]c[\/latex] units, in the direction of the sign of [latex]c.[\/latex]<\/li>\n<li>The vertical intercept becomes [latex]\\left(0,{b}^{-c}+d\\right).[\/latex]<\/li>\n<li>The horizontal asymptote becomes [latex]y=d[\/latex] on the same side.<\/li>\n<li>The range becomes [latex]\\left(d,\\infty \\right).[\/latex]<\/li>\n<li>The domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] remains unchanged.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135500732\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135500706\"><strong>Given an exponential function with the form [latex]f\\left(x\\right)={b}^{x-c}+d,[\/latex] graph the translation.<\/strong><\/p>\n<ol id=\"fs-id1165137767676\" type=\"1\">\n<li>Draw the horizontal asymptote [latex]y=d.[\/latex]<\/li>\n<li>Identify [latex]c[\/latex] and [latex]d[\/latex].\u00a0 Shift the graph of [latex]y={b}^{x}[\/latex] right [latex]c[\/latex] units if [latex]c[\/latex] is positive, and left [latex]c[\/latex] units if [latex]c[\/latex] is negative.<\/li>\n<li>Shift the graph of [latex]y={b}^{x}[\/latex] up [latex]d[\/latex] units if [latex]d[\/latex] is positive, and down [latex]d[\/latex] units if [latex]d[\/latex] is negative.<\/li>\n<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right),[\/latex] the range, [latex]\\left(d,\\infty \\right),[\/latex] and the horizontal asymptote [latex]y=d.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_02_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137834201\">\n<div id=\"fs-id1165137416701\">\n<h3>Example 2:\u00a0 Graphing a Shift of an Exponential Function<\/h3>\n<p id=\"fs-id1165137563667\">Graph [latex]f\\left(x\\right)={2}^{x+1}-3.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135175234\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135175234\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135175234\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137923482\">We have an exponential equation of the form [latex]f\\left(x\\right)={b}^{x-c}+d,[\/latex] with [latex]b=2,[\/latex] [latex]c=-1,[\/latex] and [latex]d=-3.[\/latex]<\/p>\n<p id=\"fs-id1165137469681\">Draw the horizontal asymptote [latex]y=d[\/latex], so draw [latex]y=-3.[\/latex]\u00a0 Shift the graph of [latex]y={2}^{x}[\/latex] left 1 unit and down 3 units.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_008\" class=\"small\">\n<div style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210320\/CNX_Precalc_Figure_04_02_008.jpg\" alt=\"Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).\" width=\"278\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134199602\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(-3,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=-3[\/latex] on the left side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241073\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_04_02_02\">\n<div id=\"fs-id1165137805939\">\n<p id=\"fs-id1165137805941\">Graph [latex]f\\left(x\\right)={2}^{x-1}+3.[\/latex] State domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137731918\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137731918\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137731918\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135513714\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(3,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=3[\/latex] on the left side.<\/p>\n<p><span id=\"fs-id1165137628194\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210323\/CNX_Precalc_Figure_04_02_009.jpg\" alt=\"Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).\" width=\"319\" height=\"321\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137431154\" class=\"bc-section section\">\n<h4>Graphing a Vertical Stretch or Compression<\/h4>\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <span class=\"no-emphasis\">stretch<\/span> or <span class=\"no-emphasis\">compression<\/span> occurs when we multiply the exponential function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|>0.[\/latex] For example, if we begin by graphing the function [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph the vertical stretch, using [latex]a=3,[\/latex] to get [latex]g\\left(x\\right)=3f\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] as shown on the left in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">Figure 8a<\/a>, and the vertical compression, using [latex]a=\\frac{1}{3},[\/latex] to get [latex]h\\left(x\\right)=\\frac{1}{3}f\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] as shown on the right in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">Figure 8b<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_010\" class=\"wp-caption aligncenter\" style=\"width: 957px;\">\n<div style=\"width: 681px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210327\/CNX_Precalc_Figure_04_02_010.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" width=\"671\" height=\"306\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. (a) [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] stretches the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]3.[\/latex] (b) [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] compresses the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]\\frac{1}{3}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627908\">\n<h3>Stretches and Compressions of the Function <i>y<\/i>\u00a0= <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137696285\">For any factor [latex]a>0,[\/latex] the function [latex]f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex]<\/p>\n<ul id=\"fs-id1165137476370\">\n<li>stretches [latex]y={b}^{x}[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|>1.[\/latex]<\/li>\n<li>compresses [latex]y={b}^{x}[\/latex] vertically by a factor of [latex]a[\/latex] if [latex]|a|<1.[\/latex]<\/li>\n<li>has a vertical intercept of [latex]\\left(0,a\\right).[\/latex]<\/li>\n<li>has a horizontal asymptote at [latex]y=0,[\/latex] a range of [latex]\\left(0,\\infty \\right),[\/latex] and a domain of [latex]\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from [latex]y={b}^{x}.[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135528997\">\n<div id=\"fs-id1165135656098\">\n<h3 id=\"fs-id1165135656100\">Example 3:\u00a0 Graphing the Stretch of an Exponential Function<\/h3>\n<p id=\"fs-id1165135656104\">Sketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137657436\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137657436\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137657436\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>Before graphing, identify the behavior and key points on the graph.<\/p>\n<ul id=\"fs-id1165137657441\">\n<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left end behavior of the graph will increase without bound as [latex]x[\/latex] decreases without bound, and the right end behavior will approach the <em>x<\/em>-axis as [latex]x[\/latex] increases without bound.<\/li>\n<li>Since [latex]a=4,[\/latex] the graph of [latex]y={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched by a factor of [latex]4.[\/latex]<\/li>\n<li>Create a table of points as shown in <a class=\"autogenerated-content\" href=\"#Table_04_02_04\">Table 8<\/a>.<br \/>\n<table id=\"Table_04_02_04\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=4(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 32), (-2, 16), (-1, 8), (0, 4), (1, 2), (2, 1), and (3, 0.5).\">\n<caption>Table 8<\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 163.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 163.656px; text-align: center;\">[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]32[\/latex]<\/td>\n<td class=\"border\" style=\"width: 77.6563px; text-align: center;\">[latex]16[\/latex]<\/td>\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]8[\/latex]<\/td>\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 71.6563px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]0.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the vertical intercept, [latex]\\left(0,4\\right),[\/latex] along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135319502\">Draw a smooth curve connecting the points, as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_011\">Figure 9<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_011\" class=\"small\">\n<div style=\"width: 306px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210330\/CNX_Precalc_Figure_04_02_011.jpg\" alt=\"Graph of the function, f(x) = 4(1\/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).\" width=\"296\" height=\"293\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137442037\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541809\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_04_02_04\">\n<div>\n<p id=\"fs-id1165137452032\">Sketch the graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}.[\/latex] State the domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137694067\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137694067\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137694067\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137653325\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the left side.<\/p>\n<p><span id=\"fs-id1165135417835\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210333\/CNX_Precalc_Figure_04_02_012.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"407\" height=\"245\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135433028\" class=\"bc-section section\">\n<h4>Graphing Reflections<\/h4>\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect an exponential function about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the exponential function [latex]f\\left(x\\right)={b}^{x}[\/latex] by [latex]-1,[\/latex] we get a vertical reflection about the <em>x<\/em>-axis. When we multiply the input by [latex]-1,[\/latex] we get a <span class=\"no-emphasis\">reflection<\/span> about the <em>y<\/em>-axis. For example, if we begin by graphing the function [latex]f\\left(x\\right)={2}^{x},[\/latex] we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x},[\/latex] is shown on the left side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">Figure 10a<\/a>, and the reflection about the <em>y<\/em>-axis [latex]h\\left(x\\right)={2}^{-x},[\/latex] is shown on the right side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">Figure 10b<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_013\" class=\"wp-caption aligncenter\" style=\"width: 966px;\">\n<div style=\"width: 622px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210337\/CNX_Precalc_Figure_04_02_013.jpg\" alt=\"Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.\" width=\"612\" height=\"394\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10 (a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]g\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135477501\">\n<h3>Reflections of the\u00a0 Function <i>y<\/i>\u00a0= f(x) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137455888\">The function [latex]g\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\n<ul>\n<li>reflects the function [latex]y={b}^{x}[\/latex] over the <em>x<\/em>-axis.<\/li>\n<li>has a vertical intercept of [latex]\\left(0,-1\\right).[\/latex]<\/li>\n<li>has a range of [latex]\\left(-\\infty ,0\\right).[\/latex]<\/li>\n<li>has a horizontal asymptote at [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from the function [latex]y={b}^{x}.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137742185\">The function [latex]h\\left(x\\right)={b}^{-x}[\/latex]<\/p>\n<ul id=\"fs-id1165137551240\">\n<li>reflects the\u00a0 function [latex]y={b}^{x}[\/latex] over the <em>y<\/em>-axis.<\/li>\n<li>has a vertical intercept of [latex]\\left(0,1\\right),[\/latex] a horizontal asymptote at [latex]y=0,[\/latex] a range of [latex]\\left(0,\\infty \\right),[\/latex] and a domain of [latex]\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from the function [latex]y={b}^{x}.[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137406134\">\n<div id=\"fs-id1165137406136\">\n<h3>Example 4:\u00a0 Writing and Graphing the Reflection of an Exponential Function<\/h3>\n<p id=\"fs-id1165137896193\">Find and graph the equation for a function, [latex]g\\left(x\\right),[\/latex] that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] over the <em>x<\/em>-axis. State its domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137937537\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137937537\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137937537\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137937539\">Since we want to reflect the function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by [latex]-1[\/latex] to get, [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}.[\/latex] Next we create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_005\">Table 9<\/a>.<\/p>\n<table id=\"Table_04_02_005\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=-(1\/4)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, -64), (-2, -16), (-1, -4), (0, -1), (1, -0.25), (2, -0.0625), and (3, -0.0156).\">\n<caption>Table 9<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 105.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"width: 91.6563px; text-align: center;\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 105.656px; text-align: center;\"><strong>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-64[\/latex]<\/td>\n<td class=\"border\" style=\"width: 81.6563px; text-align: center;\">[latex]-16[\/latex]<\/td>\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-4[\/latex]<\/td>\n<td class=\"border\" style=\"width: 75.6563px; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td class=\"border\" style=\"width: 91.6563px; text-align: center;\">[latex]-0.25[\/latex]<\/td>\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]-0.0625[\/latex]<\/td>\n<td class=\"border\" style=\"width: 104.656px; text-align: center;\">[latex]-0.0156[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"eip-id1167546794019\">Plot the <em>y-<\/em>intercept, [latex]\\left(0,-1\\right),[\/latex] along with two other points. We can use [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(1,-0.25\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135369275\">Draw a smooth curve connecting the points:<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_014\" class=\"small\">\n<div style=\"width: 318px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210340\/CNX_Precalc_Figure_04_02_014.jpg\" alt=\"Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).\" width=\"308\" height=\"257\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 11.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137828154\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(-\\infty ,0\\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205992\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_04_02_05\">\n<div id=\"fs-id1165135254653\">\n<p id=\"fs-id1165135254655\">Find and graph the equation for a function, [latex]g\\left(x\\right),[\/latex] that reflects [latex]f\\left(x\\right)={1.25}^{x}[\/latex] over the <em>y<\/em>-axis. State its domain, range, and horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135368458\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135368458\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135368458\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135368461\">The function is [latex]g\\left(x\\right)={1.25}^{-x}={0.8}^{x}.[\/latex] The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(0,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=0[\/latex] on the right side.<\/p>\n<p><span id=\"fs-id1165137828034\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08210343\/CNX_Precalc_Figure_04_02_015.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" width=\"452\" height=\"298\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501015\" class=\"bc-section section\">\n<h4>Summarizing Translations of the Exponential Function<\/h4>\n<p>Now that we have worked with each type of translation for the exponential function, we can summarize them in <a class=\"autogenerated-content\" href=\"#Table_04_02_006\">Table 10<\/a>\u00a0to arrive at the general equation for translating exponential functions.<\/p>\n<table id=\"Table_04_02_006\">\n<caption>Table 10<\/caption>\n<thead>\n<tr>\n<th colspan=\"2\">Translations of the Function [latex]y={b}^{x}[\/latex]<\/th>\n<\/tr>\n<tr>\n<th class=\"border\">Translation<\/th>\n<th class=\"border\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\">Shift<\/p>\n<ul id=\"fs-id1165137640731\">\n<li>Horizontally [latex]c[\/latex] units to the left or right<\/li>\n<li>Vertically [latex]d[\/latex] units up or down<\/li>\n<\/ul>\n<\/td>\n<td class=\"border\">[latex]f\\left(x\\right)={b}^{x-c}+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Vertical Stretch and Compression<\/p>\n<ul id=\"fs-id1165134074993\">\n<li>Stretch if [latex]|a|>1[\/latex]<\/li>\n<li>Compression if [latex]0<|a|<1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td class=\"border\">[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Reflect about the <em>x<\/em>-axis<\/td>\n<td class=\"border\">[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Reflect about the <em>y<\/em>-axis<\/td>\n<td class=\"border\">[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">General equation for all translations<\/td>\n<td class=\"border\">[latex]f\\left(x\\right)=a{b}^{x-c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div>\n<div class=\"textbox key-takeaways\">\n<h3>Q&amp;A<\/h3>\n<p><strong>Why isn&#8217;t there a discussion on horizontal stretches and compressions?<\/strong><\/p>\n<p><em>Recall the exponential rule [latex]b^{mn}={\\left(b^m\\right)}^n.[\/latex] Essentially a horizontal compression would be a change in the base of the function.\u00a0 For example,\u00a0[latex]b^{3x}={\\left(b^3\\right)}^x.[\/latex] The original base is [latex]b[\/latex] with a horizontal compression by a factor of [latex]\\frac{1}{3}[\/latex], but we can also simply consider this as a function with the new base [latex]b^3[\/latex].\u00a0\u00a0<\/em><\/p>\n<p><em>Think of [latex]f\\left(x\\right)=2^{3x}.[\/latex] We can think of this as a horizontal compression by a factor of [latex]\\frac{1}{3}[\/latex] of the function [latex]y=2^{x}.[\/latex] The point (1,2) will be compressed to the point [latex]\\left(\\frac{1}{3},2\\right).[\/latex] Notice that if we used the function [latex]g\\left(x\\right)=\\left(2^{3}\\right)^{x}\\text{ or }g\\left(x\\right)=8^{x},[\/latex] we would also see the point\u00a0[latex]\\left(\\frac{1}{3},2\\right).[\/latex] This helps us see that we can achieve the same results as horizontal compressions by rewriting the function with a different base.<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"Example_04_02_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137937623\">\n<div id=\"fs-id1165135250578\">\n<h3 id=\"fs-id1165135250580\">Example 5:\u00a0 Writing a Function from a Description<\/h3>\n<p id=\"fs-id1165135250584\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul id=\"fs-id1165137724821\">\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of [latex]2[\/latex], reflected across the <em>y<\/em>-axis, and then shifted up [latex]4[\/latex] units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135532412\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135532412\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135532412\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135532414\">We want to find an equation of the general form [latex]g\\left(x\\right)=a{b}^{x-c}+d.[\/latex] We use the description provided to find [latex]a,[\/latex] [latex]b,[\/latex] [latex]c,[\/latex] and [latex]d.[\/latex]<\/p>\n<ul id=\"fs-id1165137807102\">\n<li>We are given the function [latex]f\\left(x\\right)={e}^{x},[\/latex] so [latex]b=e.[\/latex]<\/li>\n<li>The function is stretched by a factor of [latex]2[\/latex], so [latex]a=2.[\/latex]<\/li>\n<li>The function is reflected about the <em>y<\/em>-axis. We replace [latex]x[\/latex] 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 id=\"fs-id1165137634849\">Substituting in the general form we get,<\/p>\n<div id=\"eip-id1165137832492\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*} g\\left(x\\right)&=a{b}^{x-c}+d\\hfill\\\\ \\hfill&=2{e}^{-x-0}+4\\hfill\\\\ \\hfill&=2{e}^{-x}+4\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137665666\">The domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(4,\\infty \\right);[\/latex] the horizontal asymptote is [latex]y=4[\/latex] on the right side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137553895\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"ti_04_02_06\">\n<div id=\"fs-id1165137724079\">\n<p id=\"fs-id1165137724081\">Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul id=\"fs-id1165137539693\">\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3},[\/latex] reflected across the <em>x<\/em>-axis and then shifted down [latex]2[\/latex] units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137724110\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137724110\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137724110\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137724112\">[latex]g\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2;[\/latex] the domain is [latex]\\left(-\\infty ,\\infty \\right);[\/latex] the range is [latex]\\left(-\\infty ,-2\\right);[\/latex] the horizontal asymptote is [latex]y=-2[\/latex] on the right side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560786\" class=\"precalculus media\">\n<div id=\"fs-id1165137566517\" class=\"bc-section section\">\n<h3>Approximating Solutions to an Exponential Equation with the Calculator<\/h3>\n<p>Sometimes we want to find out when an exponential function will have a particular output value.\u00a0 The next sections will focus on being able to do this algebraically.\u00a0 Currently, we can use technology to determine what input will give a particular output for the transformed exponential function.<\/p>\n<div id=\"fs-id1165137639988\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137756810\"><strong>Given an equation of the form [latex]y=a{b}^{x-c}+d,[\/latex] use a graphing calculator to approximate the solution.<\/strong><\/p>\n<ul id=\"fs-id1165137842461\">\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]y[\/latex] in the line headed \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em>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]y.[\/latex]<\/li>\n<li>To find the value of [latex]x,[\/latex] 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 [latex]x[\/latex] for the indicated output value of the function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137618985\">\n<div id=\"fs-id1165137618987\">\n<h3>Example 6:\u00a0 Approximating the Solution of an Exponential Equation<\/h3>\n<p id=\"fs-id1165135449598\">Solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165137653309\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137653309\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137653309\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137737383\">Press <strong>[Y=]<\/strong> and enter [latex]1.2{\\left(5\\right)}^{x}+2.8[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong>Y2=<\/strong>. For a window, use the values \u20133 to 3 for [latex]x[\/latex] and \u20135 to 55 for [latex]y.[\/latex] Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near [latex]x=2.[\/latex]<\/p>\n<p id=\"fs-id1165137460953\">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>x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be slightly 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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545893\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"ti_04_02_03\">\n<div id=\"fs-id1165137838712\">\n<p id=\"fs-id1165137838714\">Solve [latex]4=7.85{\\left(1.15\\right)}^{x}-2.27[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165137854192\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137854192\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137854192\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137854194\">[latex]x\\approx -1.608[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137785000\">Access this online resource for additional instruction and practice with graphing exponential functions.<\/p>\n<ul id=\"fs-id1165137785004\">\n<li><a href=\"http:\/\/openstax.org\/l\/graphexpfunc\">Graph Exponential Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661989\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id2055298\" summary=\"...\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 414px;\">General Form for the Translation of the Function [latex]y={b}^{x}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 225px;\">[latex]f\\left(x\\right)=a{b}^{x-c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137447701\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137447708\">\n<li>The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a <em>y-<\/em>intercept at [latex]\\left(0, 1\\right),[\/latex] domain [latex]\\left(-\\infty , \\infty \\right),[\/latex] range[latex]\\left(0, \\infty \\right),[\/latex] and horizontal asymptote [latex]y=0.[\/latex]<\/li>\n<li>End behavior describes what happens to the output if you go very far to the left or right.\n<ul>\n<li>If [latex]b>1,[\/latex] the function is increasing. The left end behavior of the graph will approach the horizontal asymptote [latex]y=0,[\/latex] and the right end behavior will increase without bound.<\/li>\n<li>If [latex]0 \\lt b \\lt 1,[\/latex] the function is decreasing. The left end behavior of the graph will increase without bound, and the right end behavior will approach the horizontal asymptote [latex]y=0.[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the exponential function [latex]y={b}^{x}.[\/latex]<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x-c}[\/latex] represents a horizontal shift of the exponential function [latex]y={b}^{x}.[\/latex]<\/li>\n<li>The equation [latex]f\\left(x\\right)=a{b}^{x},[\/latex] where [latex]a>0,[\/latex] represents a vertical stretch if [latex]|a|>1[\/latex] or compression if [latex]0<|a|<1[\/latex] of the exponential function [latex]y={b}^{x}.[\/latex]<\/li>\n<li>When the exponential function [latex]y={b}^{x}[\/latex] is multiplied by [latex]-1,[\/latex]\u00a0 the result, [latex]g\\left(x\\right)=-{b}^{x},[\/latex] is a reflection about the <em>x<\/em>-axis. When the input is multiplied by [latex]-1,[\/latex] the result, [latex]h\\left(x\\right)={b}^{-x},[\/latex] is a reflection about the <em>y<\/em>-axis.<\/li>\n<li>All translations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x-c}+d.[\/latex]<\/li>\n<li>Using the general equation [latex]f\\left(x\\right)=a{b}^{x-c}+d,[\/latex] we can write the equation of a function given its description.<\/li>\n<li>Approximate solutions of the equation [latex]y={b}^{x-c}+d[\/latex] can be found using a graphing calculator.<\/li>\n<\/ul>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-295\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Graphs of Exponential Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:c8aEyW2u@16\/Graphs-of-Exponential-Functions\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:c8aEyW2u@16\/Graphs-of-Exponential-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Graphs of Exponential Functions\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:c8aEyW2u@16\/Graphs-of-Exponential-Functions\",\"project\":\"Essential Precalcus, Part 1\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-295","chapter","type-chapter","status-publish","hentry"],"part":223,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/295","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":32,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/295\/revisions"}],"predecessor-version":[{"id":3287,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/295\/revisions\/3287"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/223"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/295\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=295"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=295"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=295"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}