{"id":1532,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1532"},"modified":"2020-03-12T17:26:43","modified_gmt":"2020-03-12T17:26:43","slug":"graph-exponential-functions-using-transformations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/graph-exponential-functions-using-transformations\/","title":{"raw":"Graph exponential functions using transformations","rendered":"Graph exponential functions using transformations"},"content":{"raw":"<section data-depth=\"1\">\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\" data-depth=\"2\">\r\n<h2 data-type=\"title\">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\/924\/2015\/11\/25201806\/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\" data-media-type=\"image\/jpg\" \/> <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\" data-bullet-style=\"open-circle\">\r\n \t<li>The <em data-effect=\"italics\">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\" data-bullet-style=\"open-circle\">\r\n \t<li>The <em data-effect=\"italics\">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\" data-depth=\"2\">\r\n<h2 data-type=\"title\">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 data-effect=\"italics\">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\/924\/2015\/11\/25201808\/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\" data-media-type=\"image\/jpg\" \/> <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 data-effect=\"italics\">y-<\/em>intercept shifts such that:\r\n<ul id=\"fs-id1165137482879\" data-bullet-style=\"open-circle\">\r\n \t<li>When the function is shifted left 3\u00a0units to [latex]g\\left(x\\right)={2}^{x+3}[\/latex], the <em data-effect=\"italics\">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 data-effect=\"italics\">y<\/em>-intercept becomes [latex]\\left(0,\\frac{1}{8}\\right)[\/latex]. Again, see that [latex]{2}^{x - 3}=\\left(\\frac{1}{8}\\right){2}^{x}[\/latex], so the initial value of the function is [latex]\\frac{1}{8}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1165134042183\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"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>\r\n \t<li>vertically <em>d<\/em>\u00a0units, in the <em data-effect=\"italics\">same<\/em> direction of the sign of <em>d<\/em>.<\/li>\r\n \t<li>horizontally <em>c<\/em>\u00a0units, in the <em data-effect=\"italics\">opposite<\/em> direction of the sign of <em>c<\/em>.<\/li>\r\n \t<li>The <em data-effect=\"italics\">y<\/em>-intercept becomes [latex]\\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\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\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\" data-number-style=\"arabic\">\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\" data-type=\"example\">\r\n<div id=\"fs-id1165137834201\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137416701\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 1: 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\r\n<\/div>\r\n<div id=\"fs-id1165135175234\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\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\" data-type=\"media\" data-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).\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201809\/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\" data-media-type=\"image\/jpg\" \/><\/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\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 2<\/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<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137639988\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\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 data-effect=\"bold\">[Y=]<\/strong>. Enter the given exponential equation in the line headed \"<strong data-effect=\"bold\">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 data-effect=\"bold\">Y<sub>2<\/sub>=<\/strong>.\"<\/li>\r\n \t<li>Press <strong data-effect=\"bold\">[WINDOW]<\/strong>. Adjust the <em data-effect=\"italics\">y<\/em>-axis so that it includes the value entered for \"<strong data-effect=\"bold\">Y<sub>2<\/sub>=<\/strong>.\"<\/li>\r\n \t<li>Press <strong data-effect=\"bold\">[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 data-effect=\"bold\">[2ND] <\/strong>then <strong data-effect=\"bold\">[CALC]<\/strong>. Select \"intersect\" and press <strong data-effect=\"bold\">[ENTER]<\/strong> three times. The point of intersection gives the value of <em data-effect=\"italics\">x <\/em>for the indicated value of the function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_02_03\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165137618985\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137618987\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 2: Approximating the Solution of an Exponential Equation<\/h3>\r\n<p id=\"fs-id1165135449598\">Solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically. Round to the nearest thousandth.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137653309\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137737383\">Press <strong data-effect=\"bold\">[Y=]<\/strong> and enter [latex]1.2{\\left(5\\right)}^{x}+2.8[\/latex] next to <strong data-effect=\"bold\">Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong data-effect=\"bold\">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 data-effect=\"bold\">[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 data-effect=\"bold\">[2ND] <\/strong>then <strong data-effect=\"bold\">[CALC]<\/strong>. Select <strong data-effect=\"bold\">[5: intersect]<\/strong> and press <strong data-effect=\"bold\">[ENTER]<\/strong> three times. The <em data-effect=\"italics\">x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for <strong data-effect=\"bold\">Guess?<\/strong>) To the nearest thousandth, [latex]x\\approx 2.166[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 3<\/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<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137431154\" data-depth=\"2\">\r\n<h2 data-type=\"title\">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\/924\/2015\/11\/25201810\/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.\" data-media-type=\"image\/jpg\" \/>\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\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: Stretches and Compressions of the Parent Function <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">b<\/em><sup><em data-effect=\"italics\">x<\/em><\/sup><\/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 data-effect=\"italics\">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\" data-type=\"example\">\r\n<div id=\"fs-id1165135528997\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135656098\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 id=\"fs-id1165135656100\"><span data-type=\"title\">Example 3: Graphing the Stretch of an Exponential Function<\/span><\/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\r\n<\/div>\r\n<div id=\"fs-id1165137657436\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\nBefore graphing, identify the behavior and key points on the graph.\r\n<ul id=\"fs-id1165137657441\">\r\n \t<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em data-effect=\"italics\">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 data-effect=\"italics\">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\" data-type=\"media\" data-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).\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201811\/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).\" data-media-type=\"image\/jpg\" \/><\/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\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 4<\/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<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135433028\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Graphing Reflections<\/h2>\r\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em data-effect=\"italics\">x<\/em>-axis or the <em data-effect=\"italics\">y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When we multiply the input by \u20131, we get a <strong>reflection<\/strong> about the <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], is shown on the left side, and the reflection about the <em data-effect=\"italics\">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\/924\/2015\/11\/25201813\/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.\" data-media-type=\"image\/jpg\" \/>\r\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><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 data-effect=\"italics\">y<\/em>-axis.<\/figcaption><\/figure>\r\n<div id=\"fs-id1165135477501\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: Reflections of the Parent Function <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">b<\/em><sup><em data-effect=\"italics\">x<\/em><\/sup><\/h3>\r\n<p id=\"fs-id1165137455888\">The function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\r\n \t<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex]\\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 data-effect=\"italics\">y<\/em>-axis.<\/li>\r\n \t<li>has a <em data-effect=\"italics\">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\" data-type=\"example\">\r\n<div id=\"fs-id1165137406134\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137406136\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 4: 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 data-effect=\"italics\">x<\/em>-axis. State its domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137937537\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\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 data-effect=\"italics\">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 data-width=\"75\" \/><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 data-effect=\"italics\">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\" data-type=\"media\" data-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).\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201814\/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).\" data-media-type=\"image\/jpg\" \/><\/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\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 5<\/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 data-effect=\"italics\">y<\/em>-axis. State its domain, range, and asymptote.<\/p>\r\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135501015\" data-depth=\"2\">\r\n<h2 data-type=\"title\">Summarizing Translations of the Exponential Function<\/h2>\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 data-effect=\"italics\">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 data-effect=\"italics\">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\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"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\" data-type=\"equation\">[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 [latex]|a|[\/latex] if [latex]|a| &gt; 1[\/latex].<\/li>\r\n \t<li>compressed vertically by a factor of [latex]|a|[\/latex]\u00a0if [latex]0 &lt; |a| &lt; 1[\/latex].<\/li>\r\n \t<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>reflected about the <em data-effect=\"italics\">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\" data-type=\"example\">\r\n<div id=\"fs-id1165137937623\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135250578\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 id=\"fs-id1165135250580\"><span data-type=\"title\">Example 5: Writing a Function from a Description<\/span><\/h3>\r\n<p id=\"fs-id1165135250584\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\r\n\r\n<ul id=\"fs-id1165137724821\">\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <em data-effect=\"italics\">y<\/em>-axis, and then shifted up 4\u00a0units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135532412\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\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 data-effect=\"italics\">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\r\n<div id=\"eip-id1165137832492\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases} f\\left(x\\right)\\hfill &amp; =a{b}^{x+c}+d\\hfill \\\\ \\hfill &amp; =2{e}^{-x+0}+4\\hfill \\\\ \\hfill &amp; =2{e}^{-x}+4\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137665666\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(4,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=4[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 6<\/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\r\n<ul id=\"fs-id1165137539693\">\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <em data-effect=\"italics\">x<\/em>-axis and then shifted down 2\u00a0units.<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section>","rendered":"<section data-depth=\"1\">\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\" data-depth=\"2\">\n<h2 data-type=\"title\">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\/924\/2015\/11\/25201806\/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\" data-media-type=\"image\/jpg\" \/><\/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\" data-bullet-style=\"open-circle\">\n<li>The <em data-effect=\"italics\">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\" data-bullet-style=\"open-circle\">\n<li>The <em data-effect=\"italics\">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\" data-depth=\"2\">\n<h2 data-type=\"title\">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 data-effect=\"italics\">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\/924\/2015\/11\/25201808\/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\" data-media-type=\"image\/jpg\" \/><\/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 data-effect=\"italics\">y-<\/em>intercept shifts such that:\n<ul id=\"fs-id1165137482879\" data-bullet-style=\"open-circle\">\n<li>When the function is shifted left 3\u00a0units to [latex]g\\left(x\\right)={2}^{x+3}[\/latex], the <em data-effect=\"italics\">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 data-effect=\"italics\">y<\/em>-intercept becomes [latex]\\left(0,\\frac{1}{8}\\right)[\/latex]. Again, see that [latex]{2}^{x - 3}=\\left(\\frac{1}{8}\\right){2}^{x}[\/latex], so the initial value of the function is [latex]\\frac{1}{8}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div id=\"fs-id1165134042183\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"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>\n<li>vertically <em>d<\/em>\u00a0units, in the <em data-effect=\"italics\">same<\/em> direction of the sign of <em>d<\/em>.<\/li>\n<li>horizontally <em>c<\/em>\u00a0units, in the <em data-effect=\"italics\">opposite<\/em> direction of the sign of <em>c<\/em>.<\/li>\n<li>The <em data-effect=\"italics\">y<\/em>-intercept becomes [latex]\\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\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\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\" data-number-style=\"arabic\">\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\" data-type=\"example\">\n<div id=\"fs-id1165137834201\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137416701\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 1: 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>\n<div id=\"fs-id1165135175234\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\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\" data-type=\"media\" data-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).\"><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\/924\/2015\/11\/25201809\/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\" data-media-type=\"image\/jpg\" \/><\/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 class=\"bcc-box bcc-success\">\n<h3>Try It 2<\/h3>\n<p id=\"fs-id1165137805941\">Graph [latex]f\\left(x\\right)={2}^{x - 1}+3[\/latex]. State domain, range, and asymptote.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<div id=\"fs-id1165137639988\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\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 data-effect=\"bold\">[Y=]<\/strong>. Enter the given exponential equation in the line headed &#8220;<strong data-effect=\"bold\">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 data-effect=\"bold\">Y<sub>2<\/sub>=<\/strong>.&#8221;<\/li>\n<li>Press <strong data-effect=\"bold\">[WINDOW]<\/strong>. Adjust the <em data-effect=\"italics\">y<\/em>-axis so that it includes the value entered for &#8220;<strong data-effect=\"bold\">Y<sub>2<\/sub>=<\/strong>.&#8221;<\/li>\n<li>Press <strong data-effect=\"bold\">[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 data-effect=\"bold\">[2ND] <\/strong>then <strong data-effect=\"bold\">[CALC]<\/strong>. Select &#8220;intersect&#8221; and press <strong data-effect=\"bold\">[ENTER]<\/strong> three times. The point of intersection gives the value of <em data-effect=\"italics\">x <\/em>for the indicated value of the function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_03\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137618985\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137618987\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 2: Approximating the Solution of an Exponential Equation<\/h3>\n<p id=\"fs-id1165135449598\">Solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165137653309\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137737383\">Press <strong data-effect=\"bold\">[Y=]<\/strong> and enter [latex]1.2{\\left(5\\right)}^{x}+2.8[\/latex] next to <strong data-effect=\"bold\">Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong data-effect=\"bold\">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 data-effect=\"bold\">[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 data-effect=\"bold\">[2ND] <\/strong>then <strong data-effect=\"bold\">[CALC]<\/strong>. Select <strong data-effect=\"bold\">[5: intersect]<\/strong> and press <strong data-effect=\"bold\">[ENTER]<\/strong> three times. The <em data-effect=\"italics\">x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for <strong data-effect=\"bold\">Guess?<\/strong>) To the nearest thousandth, [latex]x\\approx 2.166[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 3<\/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<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137431154\" data-depth=\"2\">\n<h2 data-type=\"title\">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\/924\/2015\/11\/25201810\/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.\" data-media-type=\"image\/jpg\" \/><\/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\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Stretches and Compressions of the Parent Function <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">b<\/em><sup><em data-effect=\"italics\">x<\/em><\/sup><\/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 data-effect=\"italics\">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\" data-type=\"example\">\n<div id=\"fs-id1165135528997\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135656098\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 id=\"fs-id1165135656100\"><span data-type=\"title\">Example 3: Graphing the Stretch of an Exponential Function<\/span><\/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>\n<div id=\"fs-id1165137657436\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p>Before graphing, identify the behavior and key points on the graph.<\/p>\n<ul id=\"fs-id1165137657441\">\n<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em data-effect=\"italics\">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 data-effect=\"italics\">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\" data-type=\"media\" data-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).\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201811\/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).\" data-media-type=\"image\/jpg\" \/><\/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 class=\"bcc-box bcc-success\">\n<h3>Try It 4<\/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<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135433028\" data-depth=\"2\">\n<h2 data-type=\"title\">Graphing Reflections<\/h2>\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em data-effect=\"italics\">x<\/em>-axis or the <em data-effect=\"italics\">y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When we multiply the input by \u20131, we get a <strong>reflection<\/strong> about the <em data-effect=\"italics\">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 data-effect=\"italics\">x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], is shown on the left side, and the reflection about the <em data-effect=\"italics\">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\/924\/2015\/11\/25201813\/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.\" data-media-type=\"image\/jpg\" \/><\/p>\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><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 data-effect=\"italics\">y<\/em>-axis.<\/figcaption><\/figure>\n<div id=\"fs-id1165135477501\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Reflections of the Parent Function <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">b<\/em><sup><em data-effect=\"italics\">x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137455888\">The function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n<li>has a <em data-effect=\"italics\">y<\/em>-intercept of [latex]\\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 data-effect=\"italics\">y<\/em>-axis.<\/li>\n<li>has a <em data-effect=\"italics\">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\" data-type=\"example\">\n<div id=\"fs-id1165137406134\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137406136\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 4: 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 data-effect=\"italics\">x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137937537\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\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 data-effect=\"italics\">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 data-width=\"75\" \/>\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 data-effect=\"italics\">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\" data-type=\"media\" data-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).\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201814\/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).\" data-media-type=\"image\/jpg\" \/><\/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 class=\"bcc-box bcc-success\">\n<h3>Try It 5<\/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 data-effect=\"italics\">y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135501015\" data-depth=\"2\">\n<h2 data-type=\"title\">Summarizing Translations of the Exponential Function<\/h2>\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 data-effect=\"italics\">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 data-effect=\"italics\">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\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"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\" data-type=\"equation\">[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 [latex]|a|[\/latex] if [latex]|a| > 1[\/latex].<\/li>\n<li>compressed vertically by a factor of [latex]|a|[\/latex]\u00a0if [latex]0 < |a| < 1[\/latex].<\/li>\n<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\n<li>reflected about the <em data-effect=\"italics\">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\" data-type=\"example\">\n<div id=\"fs-id1165137937623\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135250578\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 id=\"fs-id1165135250580\"><span data-type=\"title\">Example 5: Writing a Function from a Description<\/span><\/h3>\n<p id=\"fs-id1165135250584\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul id=\"fs-id1165137724821\">\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <em data-effect=\"italics\">y<\/em>-axis, and then shifted up 4\u00a0units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135532412\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\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 data-effect=\"italics\">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<div id=\"eip-id1165137832492\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases} f\\left(x\\right)\\hfill & =a{b}^{x+c}+d\\hfill \\\\ \\hfill & =2{e}^{-x+0}+4\\hfill \\\\ \\hfill & =2{e}^{-x}+4\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137665666\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(4,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=4[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 6<\/h3>\n<p id=\"fs-id1165137724081\">Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul id=\"fs-id1165137539693\">\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <em data-effect=\"italics\">x<\/em>-axis and then shifted down 2\u00a0units.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-26\/\" target=\"_blank\" rel=\"noopener\">Solution<\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\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-1532\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1532","chapter","type-chapter","status-publish","hentry"],"part":1518,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1532","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1532\/revisions"}],"predecessor-version":[{"id":3181,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1532\/revisions\/3181"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1518"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1532\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1532"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1532"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1532"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1532"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}