{"id":98,"date":"2021-06-04T18:08:56","date_gmt":"2021-06-04T18:08:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/graphs-of-exponential-functions\/"},"modified":"2022-11-28T14:35:59","modified_gmt":"2022-11-28T14:35:59","slug":"graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/graphs-of-exponential-functions\/","title":{"raw":"5.2 Graphs of Exponential Functions","rendered":"5.2 Graphs of Exponential Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph exponential functions.<\/li>\r\n \t<li>Graph exponential functions using transformations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<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<h2>Graph exponential functions<\/h2>\r\n<section id=\"fs-id1165135407520\">\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 the table below\u00a0change as the input increases by 1.<\/p>\r\n\r\n<table id=\"Table_04_02_01\" summary=\"Two rows and eight columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/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, 2. We call the base 2 the <em>constant ratio<\/em>. In fact, for any exponential function with the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex], <em>b<\/em>\u00a0is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.<\/p>\r\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\r\n\r\n<ul id=\"fs-id1165137658509\">\r\n \t<li>the output values are positive for all values of <em>x<\/em>;<\/li>\r\n \t<li>as <em>x<\/em>\u00a0increases, the output values increase without bound; and<\/li>\r\n \t<li>as <em>x<\/em>\u00a0decreases, the output values grow smaller, approaching zero.<\/li>\r\n<\/ul>\r\nFigure 1\u00a0shows the exponential growth function [latex]f\\left(x\\right)={2}^{x}[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010811\/CNX_Precalc_Figure_04_02_0012.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=\"487\" height=\"520\" \/> <b>Figure 1.<\/b> Notice that the graph gets close to the x-axis, but never touches it.[\/caption]\r\n<p id=\"fs-id1165137459614\">The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\r\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <strong>exponential decay<\/strong>, 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 the table below\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, \">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td>8<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135347846\">Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio [latex]\\frac{1}{2}[\/latex].<\/p>\r\n<p id=\"fs-id1165137452063\">Notice from the table that<\/p>\r\n\r\n<ul id=\"fs-id1165135499992\">\r\n \t<li>the output values are positive for all values of <em>x<\/em>;<\/li>\r\n \t<li>as <em>x<\/em>\u00a0increases, the output values grow smaller, approaching zero; and<\/li>\r\n \t<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137405421\">The graph shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex].<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_0022.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=\"487\" height=\"520\" \/>\r\n<p id=\"fs-id1165137723586\" style=\"text-align: center;\"><strong>Figure 2.\u00a0<\/strong>The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165135571835\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Characteristics of the Graph of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\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 id=\"fs-id1165135186684\">\r\n \t<li><strong>one-to-one<\/strong> function<\/li>\r\n \t<li>horizontal asymptote: [latex]y=0[\/latex]<\/li>\r\n \t<li>domain: [latex]\\left(-\\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<\/ul>\r\n<p id=\"fs-id1165137471878\">Compare the graphs of <strong>exponential growth<\/strong> and decay functions.<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_003new2.jpg\" alt=\"&quot;Graph\" \/>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134195243\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135194093\">How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function.<\/h3>\r\n<ol id=\"fs-id1165137435782\">\r\n \t<li>Create a table of points.<\/li>\r\n \t<li>Plot at least 3\u00a0point from the table, including the <em>y<\/em>-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 the horizontal asymptote, [latex]y=0[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_04_02_01\" class=\"example\">\r\n<div id=\"fs-id1165135208984\" class=\"exercise\">\r\n<div id=\"fs-id1165137453336\" class=\"problem textbox shaded\">\r\n<h3>Example 1: 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 asymptote.<\/p>\r\n[reveal-answer q=\"170706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"170706\"]\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 id=\"fs-id1165137566570\">\r\n \t<li>Since <em>b\u00a0<\/em>= 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote <em>y\u00a0<\/em>= 0.<\/li>\r\n \t<li>Create a table of points.\r\n<table id=\"Table_04_02_03\" summary=\"Two rows and eight columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\r\n<td>64<\/td>\r\n<td>16<\/td>\r\n<td>4<\/td>\r\n<td>1<\/td>\r\n<td>0.25<\/td>\r\n<td>0.0625<\/td>\r\n<td>0.015625<\/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.<span id=\"fs-id1165137940681\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_0042.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).\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137548870\" style=\"text-align: center;\"><strong>Figure 4.\u00a0<\/strong>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].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137548853\">Sketch the graph of [latex]f\\left(x\\right)={4}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"680272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"680272\"]\r\n\r\nThe 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].<span id=\"fs-id1165137437648\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_0052.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>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"fs-id1165137731723\" class=\"solution\">Graph exponential functions using transformations<\/h2>\r\n<section id=\"fs-id1165137694074\">\r\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function [latex]f\\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\r\n<section id=\"fs-id1165134312214\">\r\n<h2>Graphing a Vertical Shift<\/h2>\r\nThe first transformation occurs when we add a constant <em>d<\/em>\u00a0to the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex], giving us a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two vertical shifts alongside it, using [latex]d=3[\/latex]: the upward shift, [latex]g\\left(x\\right)={2}^{x}+3[\/latex] and the downward shift, [latex]h\\left(x\\right)={2}^{x}-3[\/latex]. Both vertical shifts are shown in Figure 5.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0062.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=\"487\" height=\"628\" \/> <b>Figure 5<\/b>[\/caption]\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 3\u00a0units 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 3\u00a0units to [latex]\\left(0,4\\right)[\/latex].<\/li>\r\n \t<li>The asymptote shifts up 3\u00a0units to [latex]y=3[\/latex].<\/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 3\u00a0units 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 3\u00a0units to [latex]\\left(0,-2\\right)[\/latex].<\/li>\r\n \t<li>The asymptote also shifts down 3\u00a0units to [latex]y=-3[\/latex].<\/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<\/section><section id=\"fs-id1165137566517\">\r\n<h2>Graphing a Horizontal Shift<\/h2>\r\n<p id=\"fs-id1165137748336\">The next transformation occurs when we add a constant <em>c<\/em>\u00a0to the input of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex], giving us a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em>opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two horizontal shifts alongside it, using [latex]c=3[\/latex]: the shift left, [latex]g\\left(x\\right)={2}^{x+3}[\/latex], and the shift right, [latex]h\\left(x\\right)={2}^{x - 3}[\/latex]. Both horizontal shifts are shown in Figure 6.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0072.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=\"731\" height=\"478\" \/> <b>Figure 6<\/b>[\/caption]\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 asymptote, [latex]y=0[\/latex], remains unchanged.<\/li>\r\n \t<li>The <em>y-<\/em>intercept shifts such that:\r\n<ul id=\"fs-id1165137482879\">\r\n \t<li>When the function is shifted left 3\u00a0units to [latex]g\\left(x\\right)={2}^{x+3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,8\\right)[\/latex]. This is because [latex]{2}^{x+3}=\\left(8\\right){2}^{x}[\/latex], so the initial value of the function is 8.<\/li>\r\n \t<li>When the function is shifted right 3\u00a0units to [latex]h\\left(x\\right)={2}^{x - 3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,\\frac{1}{8}\\right)[\/latex]. Again, see that [latex]{2}^{x - 3}=\\left(\\frac{1}{8}\\right){2}^{x}[\/latex], so the initial value of the function is [latex]\\frac{1}{8}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1165134042183\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Shifts of the Parent Function\u00a0[latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\n<p id=\"fs-id1165134037589\">For any constants <em>c<\/em>\u00a0and <em>d<\/em>, the function [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137638569\">\r\n \t<li>vertically <em>d<\/em>\u00a0units, in the <em>same<\/em> direction of the sign of <em>d<\/em>.<\/li>\r\n \t<li>horizontally <em>c<\/em>\u00a0units, in the <em>opposite<\/em> direction of the sign of <em>c<\/em>.<\/li>\r\n \t<li>The <em>y<\/em>-intercept becomes [latex]\\left(0,{b}^{c}+d\\right)[\/latex].<\/li>\r\n \t<li>The horizontal asymptote becomes <em>y<\/em> =\u00a0<em>d<\/em>.<\/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=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135500706\">How To: Given an exponential function with the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex], graph the translation.<\/h3>\r\n<ol id=\"fs-id1165137767676\">\r\n \t<li>Draw the horizontal asymptote <em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\r\n \t<li>Identify the shift as [latex]\\left(-c,d\\right)[\/latex]. Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left <em>c<\/em>\u00a0units if <em>c<\/em>\u00a0is positive, and right [latex]c[\/latex] units if <em>c<\/em>\u00a0is negative.<\/li>\r\n \t<li>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] up <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is positive, and down <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is 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=\"example\">\r\n<div id=\"fs-id1165137834201\" class=\"exercise\">\r\n<div id=\"fs-id1165137416701\" class=\"problem textbox shaded\">\r\n<h3>Example 2: 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 asymptote.<\/p>\r\n[reveal-answer q=\"918334\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"918334\"]\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].<\/p>\r\n<p id=\"fs-id1165137661814\">Identify the shift as [latex]\\left(-c,d\\right)[\/latex], so the shift is [latex]\\left(-1,-3\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165137693953\">Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left 1 units and down 3 units.<span id=\"fs-id1165137591826\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0082.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=\"487\" height=\"519\" \/><\/span><\/p>\r\n<p id=\"fs-id1165134199602\" style=\"text-align: center;\"><strong>Figure 7.\u00a0<\/strong>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].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137805941\">Graph [latex]f\\left(x\\right)={2}^{x - 1}+3[\/latex]. State domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"309365\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"309365\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(3,\\infty \\right)[\/latex]; the horizontal asymptote is <em>y\u00a0<\/em>= 3.<span id=\"fs-id1165137628194\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0092.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).\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174256[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137639988\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137756810\">How To: Given an equation of the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] for [latex]x[\/latex], use a graphing calculator to approximate the solution.<\/h3>\r\n<ul id=\"fs-id1165137842461\">\r\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given exponential equation in the line headed \"<strong>Y<sub>1<\/sub>=<\/strong>.\"<\/li>\r\n \t<li>Enter the given value for [latex]f\\left(x\\right)[\/latex] in the line headed \"<strong>Y<sub>2<\/sub>=<\/strong>.\"<\/li>\r\n \t<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em>y<\/em>-axis so that it includes the value entered for \"<strong>Y<sub>2<\/sub>=<\/strong>.\"<\/li>\r\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graph of the exponential function along with the line for the specified value of [latex]f\\left(x\\right)[\/latex].<\/li>\r\n \t<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \"intersect\" and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x <\/em>for the indicated value of the function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_03\" class=\"example\">\r\n<div id=\"fs-id1165137618985\" class=\"exercise\">\r\n<div id=\"fs-id1165137618987\" class=\"problem textbox shaded\">\r\n<h3>Example 3: 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[reveal-answer q=\"423771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"423771\"]\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 <em>x<\/em>\u00a0and \u20135 to 55 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near <em>x\u00a0<\/em>= 2.<\/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 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 class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\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[reveal-answer q=\"772725\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"772725\"]\r\n\r\n[latex]x\\approx -1.608[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137431154\">\r\n<h2>Graphing a Stretch or Compression<\/h2>\r\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <strong>stretch<\/strong> or <strong>compression<\/strong> occurs when we multiply the parent 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 parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the stretch, using [latex]a=3[\/latex], to get [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] as shown on the left in Figure 8, and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] as shown on the right in\u00a0Figure 8.<\/p>\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010814\/CNX_Precalc_Figure_04_02_0102.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" \/>\r\n<p style=\"text-align: center;\"><strong>Figure 8.\u00a0<\/strong>(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 3. (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>\r\n\r\n<div id=\"fs-id1165137627908\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Stretches and Compressions of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165137696285\">For any factor <em>a<\/em> &gt; 0, 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>is stretched vertically by a factor of <em>a\u00a0<\/em>if [latex]|a|&gt;1[\/latex].<\/li>\r\n \t<li>is compressed vertically by a factor of <em>a<\/em>\u00a0if [latex]|a|&lt;1[\/latex].<\/li>\r\n \t<li>has a <em>y<\/em>-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 the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_04\" class=\"example\">\r\n<div id=\"fs-id1165135528997\" class=\"exercise\">\r\n<div id=\"fs-id1165135656098\" class=\"problem textbox shaded\">\r\n<h3 id=\"fs-id1165135656100\">Example 4: 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 asymptote.<\/p>\r\n[reveal-answer q=\"44474\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44474\"]\r\n<p id=\"fs-id1165137657438\">Before graphing, identify the behavior and key points on the graph.<\/p>\r\n\r\n<ul id=\"fs-id1165137657441\">\r\n \t<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em>x<\/em>-axis as <em>x<\/em>\u00a0increases.<\/li>\r\n \t<li>Since <em>a\u00a0<\/em>= 4, the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched by a factor of 4.<\/li>\r\n \t<li>Create a table of points.\r\n<table id=\"Table_04_02_04\" summary=\"Two rows and eight columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td>32<\/td>\r\n<td>16<\/td>\r\n<td>8<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>0.5<\/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(0,4\\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.<span id=\"fs-id1165135453156\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010814\/CNX_Precalc_Figure_04_02_0112.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).\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137442037\" style=\"text-align: center;\"><strong>Figure 9.\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is <em>y<\/em>\u00a0= 0.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\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 asymptote.<\/p>\r\n[reveal-answer q=\"697611\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"697611\"]\r\n\r\nThe 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].\u00a0<span id=\"fs-id1165135417835\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010814\/CNX_Precalc_Figure_04_02_0122.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).\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]34500[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Reflections<\/span>\r\n\r\n<\/section><section id=\"fs-id1165135433028\">\r\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em>x<\/em>-axis. When we multiply the input by \u20131, we get a <strong>reflection<\/strong> about the <em>y<\/em>-axis. For example, if we begin by graphing the parent 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, and the reflection about the <em>y<\/em>-axis [latex]h\\left(x\\right)={2}^{-x}[\/latex], is shown on the right side.<\/p>\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010816\/CNX_Precalc_Figure_04_02_0132.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.\" \/>\r\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption>\r\n<div style=\"text-align: center;\"><strong>Figure 10.<\/strong>\r\n(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.\r\n(b) [latex]g\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the <em>y<\/em>-axis.<\/div>\r\n<\/figcaption><\/figure>\r\n<div id=\"fs-id1165135477501\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Reflections of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165137455888\">The function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137838801\">\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>x<\/em>-axis.<\/li>\r\n \t<li>has a <em>y<\/em>-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 parent function.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137742185\">The function [latex]f\\left(x\\right)={b}^{-x}[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165137551240\">\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>y<\/em>-axis.<\/li>\r\n \t<li>has a <em>y<\/em>-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 parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_05\" class=\"example\">\r\n<div id=\"fs-id1165137406134\" class=\"exercise\">\r\n<div id=\"fs-id1165137406136\" class=\"problem textbox shaded\">\r\n<h3>Example 5: 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] about the <em>x<\/em>-axis. State its domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"992175\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"992175\"]\r\n<p id=\"fs-id1165137937539\">Since we want to reflect the parent 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 \u20131 to get, [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.<\/p>\r\n\r\n<table id=\"Table_04_02_005\" summary=\"Two rows and eight columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\r\n<td>\u201364<\/td>\r\n<td>\u201316<\/td>\r\n<td>\u20134<\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20130.25<\/td>\r\n<td>\u20130.0625<\/td>\r\n<td>\u20130.0156<\/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:<span id=\"fs-id1165137736449\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010817\/CNX_Precalc_Figure_04_02_0142.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).\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137828154\" style=\"text-align: center;\"><strong>Figure 11.\u00a0<\/strong>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].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\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] about the <em>y<\/em>-axis. State its domain, range, and asymptote.<\/p>\r\n[reveal-answer q=\"123124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"123124\"]\r\n\r\nThe 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].<span id=\"fs-id1165137828034\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010817\/CNX_Precalc_Figure_04_02_0152.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).\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]34535[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Summarizing Translations of the Exponential Function<\/span>\r\n\r\n<\/section><section id=\"fs-id1165135501015\">\r\n<p id=\"fs-id1165135501021\">Now that we have worked with each type of translation for the exponential function, we can summarize them\u00a0to arrive at the general equation for translating exponential functions.<\/p>\r\n\r\n<table id=\"Table_04_02_006\" style=\"border: 1px dashed #bbbbbb;\" summary=\"Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center;\">Translation<\/th>\r\n<th style=\"text-align: center;\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Shift\r\n<ul id=\"fs-id1165137640731\">\r\n \t<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\r\n \t<li>Vertically <em>d<\/em>\u00a0units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Stretch and Compress\r\n<ul id=\"fs-id1165134074993\">\r\n \t<li>Stretch if |<em>a<\/em>|&gt;1<\/li>\r\n \t<li>Compression if 0&lt;|<em>a<\/em>|&lt;1<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>x<\/em>-axis<\/td>\r\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>y<\/em>-axis<\/td>\r\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>General equation for all translations<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165137635134\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Translations of Exponential Functions<\/h3>\r\n<p id=\"fs-id1165137806521\">A translation of an exponential function has the form<\/p>\r\n\r\n<div id=\"fs-id1165137806525\" class=\"equation unnumered\">[latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/div>\r\n<p id=\"fs-id1165137805520\">Where the parent function, [latex]y={b}^{x}[\/latex], [latex]b&gt;1[\/latex], is<\/p>\r\n\r\n<ul id=\"fs-id1165137678290\">\r\n \t<li>shifted horizontally <em>c<\/em>\u00a0units to the left.<\/li>\r\n \t<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\r\n \t<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\r\n \t<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137937613\">Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_04_02_06\" class=\"example\">\r\n<div id=\"fs-id1165137937623\" class=\"exercise\">\r\n<div id=\"fs-id1165135250578\" class=\"problem textbox shaded\">\r\n<h3 id=\"fs-id1165135250580\">Example 6: 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<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\">[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <\/span><em style=\"font-size: 0.9em;\">y<\/em><span style=\"font-size: 0.9em;\">-axis, and then shifted up 4\u00a0units.<\/span><\/p>\r\n[reveal-answer q=\"251290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"251290\"]\r\n<p id=\"fs-id1165135532414\">We want to find an equation of the general form [latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex]. We use the description provided to find <em>a<\/em>, <em>b<\/em>, <em>c<\/em>, and <em>d<\/em>.<\/p>\r\n\r\n<ul id=\"fs-id1165137807102\">\r\n \t<li>We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so <em>b\u00a0<\/em>= <em>e<\/em>.<\/li>\r\n \t<li>The function is stretched by a factor of 2, so <em>a\u00a0<\/em>= 2.<\/li>\r\n \t<li>The function is reflected about the <em>y<\/em>-axis. We replace <em>x<\/em>\u00a0with \u2013<em>x<\/em>\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\r\n \t<li>The graph is shifted vertically 4 units, so <em>d\u00a0<\/em>= 4.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137634849\">Substituting in the general form we get,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(x\\right)&amp; =a{b}^{x+c}+d \\\\ &amp; =2{e}^{-x+0}+4\\\\ &amp; =2{e}^{-x}+4 \\end{align}[\/latex]<\/p>\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].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\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<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\">[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <\/span><em style=\"font-size: 0.9em;\">x<\/em><span style=\"font-size: 0.9em;\">-axis and then shifted down 2\u00a0units.<\/span><\/p>\r\n[reveal-answer q=\"474538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474538\"]\r\n\r\n<span id=\"fs-id1165137828034\">[latex]f\\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].<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><\/section><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span>\r\n\r\n<section id=\"fs-id1165137661989\" class=\"key-equations\">\r\n<table id=\"fs-id2055298\" summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td>General Form for the Translation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137447701\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\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>If [latex]b&gt;1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\r\n \t<li>If 0 &lt;\u00a0<em>b<\/em> &lt; 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={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 parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n \t<li>Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/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 parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n \t<li>When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the <em>x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\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<\/ul>\r\n<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph 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<h2>Graph exponential functions<\/h2>\n<section id=\"fs-id1165135407520\">\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 the table below\u00a0change as the input increases by 1.<\/p>\n<table id=\"Table_04_02_01\" summary=\"Two rows and eight columns. The first row is labeled,\">\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><em><strong>x<\/strong><\/em><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>8<\/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, 2. We call the base 2 the <em>constant ratio<\/em>. In fact, for any exponential function with the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex], <em>b<\/em>\u00a0is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.<\/p>\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\n<ul id=\"fs-id1165137658509\">\n<li>the output values are positive for all values of <em>x<\/em>;<\/li>\n<li>as <em>x<\/em>\u00a0increases, the output values increase without bound; and<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow smaller, approaching zero.<\/li>\n<\/ul>\n<p>Figure 1\u00a0shows the exponential growth function [latex]f\\left(x\\right)={2}^{x}[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010811\/CNX_Precalc_Figure_04_02_0012.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=\"487\" height=\"520\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> Notice that the graph gets close to the x-axis, but never touches it.<\/p>\n<\/div>\n<p id=\"fs-id1165137459614\">The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <strong>exponential decay<\/strong>, 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 the table below\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,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135347846\">Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio [latex]\\frac{1}{2}[\/latex].<\/p>\n<p id=\"fs-id1165137452063\">Notice from the table that<\/p>\n<ul id=\"fs-id1165135499992\">\n<li>the output values are positive for all values of <em>x<\/em>;<\/li>\n<li>as <em>x<\/em>\u00a0increases, the output values grow smaller, approaching zero; and<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p id=\"fs-id1165137405421\">The graph shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_0022.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=\"487\" height=\"520\" \/><\/p>\n<p id=\"fs-id1165137723586\" style=\"text-align: center;\"><strong>Figure 2.\u00a0<\/strong>The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<div id=\"fs-id1165135571835\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Characteristics of the Graph of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\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 id=\"fs-id1165135186684\">\n<li><strong>one-to-one<\/strong> function<\/li>\n<li>horizontal asymptote: [latex]y=0[\/latex]<\/li>\n<li>domain: [latex]\\left(-\\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<\/ul>\n<p id=\"fs-id1165137471878\">Compare the graphs of <strong>exponential growth<\/strong> and decay functions.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_003new2.jpg\" alt=\"&quot;Graph\" \/><\/p>\n<\/div>\n<div id=\"fs-id1165134195243\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135194093\">How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function.<\/h3>\n<ol id=\"fs-id1165137435782\">\n<li>Create a table of points.<\/li>\n<li>Plot at least 3\u00a0point from the table, including the <em>y<\/em>-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 the horizontal asymptote, [latex]y=0[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_04_02_01\" class=\"example\">\n<div id=\"fs-id1165135208984\" class=\"exercise\">\n<div id=\"fs-id1165137453336\" class=\"problem textbox shaded\">\n<h3>Example 1: 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 asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q170706\">Show Solution<\/span><\/p>\n<div id=\"q170706\" 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 id=\"fs-id1165137566570\">\n<li>Since <em>b\u00a0<\/em>= 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote <em>y\u00a0<\/em>= 0.<\/li>\n<li>Create a table of points.<br \/>\n<table id=\"Table_04_02_03\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td>64<\/td>\n<td>16<\/td>\n<td>4<\/td>\n<td>1<\/td>\n<td>0.25<\/td>\n<td>0.0625<\/td>\n<td>0.015625<\/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.<span id=\"fs-id1165137940681\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_0042.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).\" \/><\/span><\/p>\n<p id=\"fs-id1165137548870\" style=\"text-align: center;\"><strong>Figure 4.\u00a0<\/strong>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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137548853\">Sketch the graph of [latex]f\\left(x\\right)={4}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680272\">Show Solution<\/span><\/p>\n<div id=\"q680272\" class=\"hidden-answer\" style=\"display: none\">\n<p>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].<span id=\"fs-id1165137437648\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010812\/CNX_Precalc_Figure_04_02_0052.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><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 id=\"fs-id1165137731723\" class=\"solution\">Graph exponential functions using transformations<\/h2>\n<section id=\"fs-id1165137694074\">\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function [latex]f\\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<section id=\"fs-id1165134312214\">\n<h2>Graphing a Vertical Shift<\/h2>\n<p>The first transformation occurs when we add a constant <em>d<\/em>\u00a0to the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex], giving us a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two vertical shifts alongside it, using [latex]d=3[\/latex]: the upward shift, [latex]g\\left(x\\right)={2}^{x}+3[\/latex] and the downward shift, [latex]h\\left(x\\right)={2}^{x}-3[\/latex]. Both vertical shifts are shown in Figure 5.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0062.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=\"487\" height=\"628\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\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 3\u00a0units to [latex]g\\left(x\\right)={2}^{x}+3[\/latex]:\n<ul id=\"fs-id1165137601587\">\n<li>The <em>y-<\/em>intercept shifts up 3\u00a0units to [latex]\\left(0,4\\right)[\/latex].<\/li>\n<li>The asymptote shifts up 3\u00a0units to [latex]y=3[\/latex].<\/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 3\u00a0units to [latex]h\\left(x\\right)={2}^{x}-3[\/latex]:\n<ul id=\"fs-id1165137784817\">\n<li>The <em>y-<\/em>intercept shifts down 3\u00a0units to [latex]\\left(0,-2\\right)[\/latex].<\/li>\n<li>The asymptote also shifts down 3\u00a0units to [latex]y=-3[\/latex].<\/li>\n<li>The range becomes [latex]\\left(-3,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1165137566517\">\n<h2>Graphing a Horizontal Shift<\/h2>\n<p id=\"fs-id1165137748336\">The next transformation occurs when we add a constant <em>c<\/em>\u00a0to the input of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex], giving us a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em>opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two horizontal shifts alongside it, using [latex]c=3[\/latex]: the shift left, [latex]g\\left(x\\right)={2}^{x+3}[\/latex], and the shift right, [latex]h\\left(x\\right)={2}^{x - 3}[\/latex]. Both horizontal shifts are shown in Figure 6.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0072.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=\"731\" height=\"478\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\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 asymptote, [latex]y=0[\/latex], remains unchanged.<\/li>\n<li>The <em>y-<\/em>intercept shifts such that:\n<ul id=\"fs-id1165137482879\">\n<li>When the function is shifted left 3\u00a0units to [latex]g\\left(x\\right)={2}^{x+3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,8\\right)[\/latex]. This is because [latex]{2}^{x+3}=\\left(8\\right){2}^{x}[\/latex], so the initial value of the function is 8.<\/li>\n<li>When the function is shifted right 3\u00a0units to [latex]h\\left(x\\right)={2}^{x - 3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,\\frac{1}{8}\\right)[\/latex]. Again, see that [latex]{2}^{x - 3}=\\left(\\frac{1}{8}\\right){2}^{x}[\/latex], so the initial value of the function is [latex]\\frac{1}{8}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div id=\"fs-id1165134042183\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Shifts of the Parent Function\u00a0[latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p id=\"fs-id1165134037589\">For any constants <em>c<\/em>\u00a0and <em>d<\/em>, the function [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/p>\n<ul id=\"fs-id1165137638569\">\n<li>vertically <em>d<\/em>\u00a0units, in the <em>same<\/em> direction of the sign of <em>d<\/em>.<\/li>\n<li>horizontally <em>c<\/em>\u00a0units, in the <em>opposite<\/em> direction of the sign of <em>c<\/em>.<\/li>\n<li>The <em>y<\/em>-intercept becomes [latex]\\left(0,{b}^{c}+d\\right)[\/latex].<\/li>\n<li>The horizontal asymptote becomes <em>y<\/em> =\u00a0<em>d<\/em>.<\/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=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135500706\">How To: Given an exponential function with the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex], graph the translation.<\/h3>\n<ol id=\"fs-id1165137767676\">\n<li>Draw the horizontal asymptote <em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\n<li>Identify the shift as [latex]\\left(-c,d\\right)[\/latex]. Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left <em>c<\/em>\u00a0units if <em>c<\/em>\u00a0is positive, and right [latex]c[\/latex] units if <em>c<\/em>\u00a0is negative.<\/li>\n<li>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] up <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is positive, and down <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is 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=\"example\">\n<div id=\"fs-id1165137834201\" class=\"exercise\">\n<div id=\"fs-id1165137416701\" class=\"problem textbox shaded\">\n<h3>Example 2: 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 asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q918334\">Show Solution<\/span><\/p>\n<div id=\"q918334\" 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].<\/p>\n<p id=\"fs-id1165137661814\">Identify the shift as [latex]\\left(-c,d\\right)[\/latex], so the shift is [latex]\\left(-1,-3\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137693953\">Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left 1 units and down 3 units.<span id=\"fs-id1165137591826\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0082.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=\"487\" height=\"519\" \/><\/span><\/p>\n<p id=\"fs-id1165134199602\" style=\"text-align: center;\"><strong>Figure 7.\u00a0<\/strong>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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137805941\">Graph [latex]f\\left(x\\right)={2}^{x - 1}+3[\/latex]. State domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q309365\">Show Solution<\/span><\/p>\n<div id=\"q309365\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(3,\\infty \\right)[\/latex]; the horizontal asymptote is <em>y\u00a0<\/em>= 3.<span id=\"fs-id1165137628194\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010813\/CNX_Precalc_Figure_04_02_0092.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).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174256\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174256&theme=oea&iframe_resize_id=ohm174256\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165137639988\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137756810\">How To: Given an equation of the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] for [latex]x[\/latex], use a graphing calculator to approximate the solution.<\/h3>\n<ul id=\"fs-id1165137842461\">\n<li>Press <strong>[Y=]<\/strong>. Enter the given exponential equation in the line headed &#8220;<strong>Y<sub>1<\/sub>=<\/strong>.&#8221;<\/li>\n<li>Enter the given value for [latex]f\\left(x\\right)[\/latex] in the line headed &#8220;<strong>Y<sub>2<\/sub>=<\/strong>.&#8221;<\/li>\n<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em>y<\/em>-axis so that it includes the value entered for &#8220;<strong>Y<sub>2<\/sub>=<\/strong>.&#8221;<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graph of the exponential function along with the line for the specified value of [latex]f\\left(x\\right)[\/latex].<\/li>\n<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select &#8220;intersect&#8221; and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x <\/em>for the indicated value of the function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_03\" class=\"example\">\n<div id=\"fs-id1165137618985\" class=\"exercise\">\n<div id=\"fs-id1165137618987\" class=\"problem textbox shaded\">\n<h3>Example 3: 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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q423771\">Show Solution<\/span><\/p>\n<div id=\"q423771\" 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 <em>x<\/em>\u00a0and \u20135 to 55 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near <em>x\u00a0<\/em>= 2.<\/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 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 class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q772725\">Show Solution<\/span><\/p>\n<div id=\"q772725\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x\\approx -1.608[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137431154\">\n<h2>Graphing a Stretch or Compression<\/h2>\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <strong>stretch<\/strong> or <strong>compression<\/strong> occurs when we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|>0[\/latex]. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the stretch, using [latex]a=3[\/latex], to get [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] as shown on the left in Figure 8, and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] as shown on the right in\u00a0Figure 8.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010814\/CNX_Precalc_Figure_04_02_0102.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 8.\u00a0<\/strong>(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 3. (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 id=\"fs-id1165137627908\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Stretches and Compressions of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137696285\">For any factor <em>a<\/em> &gt; 0, the function [latex]f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex]<\/p>\n<ul id=\"fs-id1165137476370\">\n<li>is stretched vertically by a factor of <em>a\u00a0<\/em>if [latex]|a|>1[\/latex].<\/li>\n<li>is compressed vertically by a factor of <em>a<\/em>\u00a0if [latex]|a|<1[\/latex].<\/li>\n<li>has a <em>y<\/em>-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 the parent function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_04\" class=\"example\">\n<div id=\"fs-id1165135528997\" class=\"exercise\">\n<div id=\"fs-id1165135656098\" class=\"problem textbox shaded\">\n<h3 id=\"fs-id1165135656100\">Example 4: 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 asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44474\">Show Solution<\/span><\/p>\n<div id=\"q44474\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137657438\">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 tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em>x<\/em>-axis as <em>x<\/em>\u00a0increases.<\/li>\n<li>Since <em>a\u00a0<\/em>= 4, the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched by a factor of 4.<\/li>\n<li>Create a table of points.<br \/>\n<table id=\"Table_04_02_04\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td>32<\/td>\n<td>16<\/td>\n<td>8<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>0.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y-<\/em>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.<span id=\"fs-id1165135453156\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010814\/CNX_Precalc_Figure_04_02_0112.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).\" \/><\/span><\/p>\n<p id=\"fs-id1165137442037\" style=\"text-align: center;\"><strong>Figure 9.\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is <em>y<\/em>\u00a0= 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\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 asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q697611\">Show Solution<\/span><\/p>\n<div id=\"q697611\" class=\"hidden-answer\" style=\"display: none\">\n<p>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].\u00a0<span id=\"fs-id1165135417835\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010814\/CNX_Precalc_Figure_04_02_0122.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).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm34500\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34500&theme=oea&iframe_resize_id=ohm34500\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Graphing Reflections<\/span><\/p>\n<\/section>\n<section id=\"fs-id1165135433028\">\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em>x<\/em>-axis. When we multiply the input by \u20131, we get a <strong>reflection<\/strong> about the <em>y<\/em>-axis. For example, if we begin by graphing the parent 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, and the reflection about the <em>y<\/em>-axis [latex]h\\left(x\\right)={2}^{-x}[\/latex], is shown on the right side.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010816\/CNX_Precalc_Figure_04_02_0132.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.\" \/><\/p>\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption>\n<strong>Figure 10.<\/strong><br \/>\n(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.<br \/>\n(b) [latex]g\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the <em>y<\/em>-axis.<br \/>\n<\/figcaption><\/figure>\n<div id=\"fs-id1165135477501\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Reflections of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137455888\">The function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\n<ul id=\"fs-id1165137838801\">\n<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>x<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-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 parent function.<\/li>\n<\/ul>\n<p id=\"fs-id1165137742185\">The function [latex]f\\left(x\\right)={b}^{-x}[\/latex]<\/p>\n<ul id=\"fs-id1165137551240\">\n<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>y<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-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 parent function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_05\" class=\"example\">\n<div id=\"fs-id1165137406134\" class=\"exercise\">\n<div id=\"fs-id1165137406136\" class=\"problem textbox shaded\">\n<h3>Example 5: 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] about the <em>x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992175\">Show Solution<\/span><\/p>\n<div id=\"q992175\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137937539\">Since we want to reflect the parent 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 \u20131 to get, [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.<\/p>\n<table id=\"Table_04_02_005\" summary=\"Two rows and eight columns. The first row is labeled,\">\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><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\n<td>\u201364<\/td>\n<td>\u201316<\/td>\n<td>\u20134<\/td>\n<td>\u20131<\/td>\n<td>\u20130.25<\/td>\n<td>\u20130.0625<\/td>\n<td>\u20130.0156<\/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:<span id=\"fs-id1165137736449\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010817\/CNX_Precalc_Figure_04_02_0142.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).\" \/><\/span><\/p>\n<p id=\"fs-id1165137828154\" style=\"text-align: center;\"><strong>Figure 11.\u00a0<\/strong>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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\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] about the <em>y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q123124\">Show Solution<\/span><\/p>\n<div id=\"q123124\" class=\"hidden-answer\" style=\"display: none\">\n<p>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].<span id=\"fs-id1165137828034\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010817\/CNX_Precalc_Figure_04_02_0152.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).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm34535\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34535&theme=oea&iframe_resize_id=ohm34535\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Summarizing Translations of the Exponential Function<\/span><\/p>\n<\/section>\n<section id=\"fs-id1165135501015\">\n<p id=\"fs-id1165135501021\">Now that we have worked with each type of translation for the exponential function, we can summarize them\u00a0to arrive at the general equation for translating exponential functions.<\/p>\n<table id=\"Table_04_02_006\" style=\"border: 1px dashed #bbbbbb;\" summary=\"Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Translation<\/th>\n<th style=\"text-align: center;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul id=\"fs-id1165137640731\">\n<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\n<li>Vertically <em>d<\/em>\u00a0units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compress<\/p>\n<ul id=\"fs-id1165134074993\">\n<li>Stretch if |<em>a<\/em>|&gt;1<\/li>\n<li>Compression if 0&lt;|<em>a<\/em>|&lt;1<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>x<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>y<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137635134\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Translations of Exponential Functions<\/h3>\n<p id=\"fs-id1165137806521\">A translation of an exponential function has the form<\/p>\n<div id=\"fs-id1165137806525\" class=\"equation unnumered\">[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/div>\n<p id=\"fs-id1165137805520\">Where the parent function, [latex]y={b}^{x}[\/latex], [latex]b>1[\/latex], is<\/p>\n<ul id=\"fs-id1165137678290\">\n<li>shifted horizontally <em>c<\/em>\u00a0units to the left.<\/li>\n<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\n<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\n<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\n<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<p id=\"fs-id1165137937613\">Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\n<\/div>\n<div id=\"Example_04_02_06\" class=\"example\">\n<div id=\"fs-id1165137937623\" class=\"exercise\">\n<div id=\"fs-id1165135250578\" class=\"problem textbox shaded\">\n<h3 id=\"fs-id1165135250580\">Example 6: 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<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\">[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <\/span><em style=\"font-size: 0.9em;\">y<\/em><span style=\"font-size: 0.9em;\">-axis, and then shifted up 4\u00a0units.<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q251290\">Show Solution<\/span><\/p>\n<div id=\"q251290\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135532414\">We want to find an equation of the general form [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]. We use the description provided to find <em>a<\/em>, <em>b<\/em>, <em>c<\/em>, and <em>d<\/em>.<\/p>\n<ul id=\"fs-id1165137807102\">\n<li>We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so <em>b\u00a0<\/em>= <em>e<\/em>.<\/li>\n<li>The function is stretched by a factor of 2, so <em>a\u00a0<\/em>= 2.<\/li>\n<li>The function is reflected about the <em>y<\/em>-axis. We replace <em>x<\/em>\u00a0with \u2013<em>x<\/em>\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\n<li>The graph is shifted vertically 4 units, so <em>d\u00a0<\/em>= 4.<\/li>\n<\/ul>\n<p id=\"fs-id1165137634849\">Substituting in the general form we get,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(x\\right)& =a{b}^{x+c}+d \\\\ & =2{e}^{-x+0}+4\\\\ & =2{e}^{-x}+4 \\end{align}[\/latex]<\/p>\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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137724081\">Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 0.9em;\">[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <\/span><em style=\"font-size: 0.9em;\">x<\/em><span style=\"font-size: 0.9em;\">-axis and then shifted down 2\u00a0units.<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q474538\">Show Solution<\/span><\/p>\n<div id=\"q474538\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1165137828034\">[latex]f\\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].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/section>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span><\/p>\n<section id=\"fs-id1165137661989\" class=\"key-equations\">\n<table id=\"fs-id2055298\" summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Translation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137447701\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\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>If [latex]b>1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\n<li>If 0 &lt;\u00a0<em>b<\/em> &lt; 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/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 parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the <em>x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\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<\/ul>\n<\/section>\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-98\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <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":23485,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"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-98","chapter","type-chapter","status-publish","hentry"],"part":96,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/98","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/98\/revisions"}],"predecessor-version":[{"id":774,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/98\/revisions\/774"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/parts\/96"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/98\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/media?parent=98"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=98"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/contributor?post=98"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/license?post=98"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}