{"id":2631,"date":"2022-06-10T21:36:02","date_gmt":"2022-06-10T21:36:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2631"},"modified":"2026-01-22T19:45:48","modified_gmt":"2026-01-22T19:45:48","slug":"5-2-2-transformations-of-the-exponential-function-stretches-and-reflections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/5-2-2-transformations-of-the-exponential-function-stretches-and-reflections\/","title":{"raw":"5.2.2: Transformations of the Exponential Function\u2013\u2013Stretches, Compressions and Reflections","rendered":"5.2.2: Transformations of the Exponential Function\u2013\u2013Stretches, Compressions and Reflections"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h1>Learning Outcomes<\/h1>\r\nFor the exponential function [latex]f(x)=r^x[\/latex],\r\n<ul>\r\n \t<li>Perform vertical compressions and\u00a0stretches<\/li>\r\n \t<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\r\n \t<li>Perform reflections across the [latex]y[\/latex]-axis<\/li>\r\n \t<li>Determine the function given specific transformations<\/li>\r\n \t<li>Determine the transformations of the exponential function\u00a0[latex]f(x)=ar^{(x-h)}+k[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Stretching Up and Compressing Down<\/h2>\r\nIf we vertically stretch the graph of the function [latex]f(x)=2^x[\/latex] by a factor of two, all of the [latex]y[\/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[\/latex]-coordinates remain the same. The equation of the function after the graph is stretched by a factor of 2 is [latex]f(x)=2\\left(2^x\\right)[\/latex]. The reason for multiplying\u00a0 [latex]2^x[\/latex] by 2 is that each [latex]y[\/latex]-coordinate is doubled, and since [latex]y=2^x[\/latex], [latex]2^x[\/latex] is doubled. Table 1 shows this change and the graph is shown in figure 1. Any point [latex](x, y)[\/latex] on the graph of [latex]f(x)=2^x[\/latex] moves to [latex](x, 2y)[\/latex] when the graph is stretched by a factor of 2.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 248px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 33.3333%; height: 19px;\">[latex]2^x[\/latex]<\/th>\r\n<th style=\"width: 16.6667%; height: 19px;\">[latex]2\\left(2^x\\right)[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 248px;\" rowspan=\"9\">\r\n\r\n[caption id=\"attachment_2744\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/17022638\/desmos-graph-2022-06-16T202557.534-300x300.png\" alt=\"Stretching the graph\" width=\"380\" height=\"380\" \/> Figure 1. Stretching the graph vertically by a factor of 2[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 96px;\">\r\n<td style=\"width: 83.3333%; height: 96px;\" colspan=\"3\">Table 1.\u00a0Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2\\left(2^x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn the other hand, if we vertically compress the graph of the function [latex]f(x)=2^x[\/latex] by half, all of the [latex]y[\/latex]-coordinates of the points on the graph are halved, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are divided by 2, or multiplied by [latex]\\frac{1}{2}[\/latex]. The equation of the function after being compressed by half is [latex]f(x)=\\dfrac{1}{2}\\left(2^x\\right)[\/latex]. The reason for multiplying [latex]2^x[\/latex] by [latex]\\frac{1}{2}[\/latex] is that each [latex]y[\/latex]-coordinate becomes half of the original value when it is divided by 2. Table 2 shows this change and the graph is shown in figure 2.\u00a0Any point [latex](x, y)[\/latex] on the graph of [latex]f(x)=2^x[\/latex] moves to [latex]\\left(x, \\frac{1}{2}y\\right)[\/latex] when the graph is compressed.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 20.9915%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 27.9536%; height: 19px;\">[latex]2^x[\/latex]<\/th>\r\n<th style=\"width: 34.3882%; height: 19px;\">[latex]\\dfrac{1}{2}\\left(2^x\\right)[\/latex]<\/th>\r\n<td style=\"width: 16.6667%;\" rowspan=\"9\">\r\n\r\n[caption id=\"attachment_2745\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2745\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/17023728\/desmos-graph-2022-06-16T203704.058-300x300.png\" alt=\"Compressing the graph\" width=\"380\" height=\"380\" \/> Figure 2. Compressing the graph.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">[latex]\\dfrac{1}{16}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 83.3333%;\" colspan=\"3\">Table 2.\u00a0Compressing the graph vertically by half transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=\\dfrac{1}{2}\\left(2^x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nStretching and compression can, of course, be applied to any exponential function [latex]f(x)=r^x[\/latex], with [latex]r&gt;0[\/latex] and [latex]r\\neq1[\/latex].\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>vertical stretching and compressing<\/h3>\r\n<p id=\"fs-id1165137770279\">A stretch or compression of the graph of [latex]f(x)=r^x[\/latex] can be represented by\u00a0multiplying the function by a constant\u00a0 [latex]a&gt;0[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=a\\cdot r^x[\/latex]<\/p>\r\nThe magnitude of [latex]a[\/latex]\u00a0indicates the stretch or compression of the graph. If [latex]a&gt;1[\/latex], the graph is stretched upwards by a factor of [latex]a.[\/latex] If [latex]0&lt;a&lt;1[\/latex], the graph is compressed down to [latex]a[\/latex] times its original height.\r\n\r\nIn addition, [latex]f(0)=a[\/latex], so the\u00a0<em><strong>initial value<\/strong><\/em> of the function is transformed from [latex]1[\/latex] to [latex]a[\/latex].\r\n\r\n<\/div>\r\nWe can change the value of [latex]a[\/latex] in figure 3 by moving the purple dot to see what happens to the graph of [latex]f(x)=r^x[\/latex]. Moving the red dot changes the value of [latex]r[\/latex].\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/yvwnta9ypu?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 3. Manipulation of [latex]f(x)=ar^x[\/latex]<\/p>\r\nNotice that the value of [latex]a[\/latex] becomes the\u00a0<strong><em>initial value\u00a0<\/em><\/strong>of [latex]f(x)[\/latex]. That is [latex]f(0)=a[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nManipulate the animation in figure 3 to answer the following questions:\r\n<ol>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[\/latex] when it is transformed to [latex]f(x)=4\\left(2^x\\right)[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[\/latex] when it is transformed to [latex]f(x)=4\\left(3^x\\right)[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=\\left(\\dfrac{1}{2}\\right)^x[\/latex] when it is transformed to [latex]f(x)=4\\left(\\dfrac{1}{2}\\right)^x[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=4r^x[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=ar^x[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>With [latex]r=2[\/latex] and [latex]a=4[\/latex], the point (0, 1) moves to (0, 4).<\/li>\r\n \t<li>With [latex]r=3[\/latex] and [latex]a=4[\/latex], the point (0, 1) moves to (0, 4).<\/li>\r\n \t<li>With [latex]r=\\dfrac{1}{2}[\/latex] and [latex]a=4[\/latex], the point (0, 1) moves to (0, 4).<\/li>\r\n \t<li>With [latex]a=4[\/latex], the point (0, 1) moves to [latex](0, 4)[\/latex].<\/li>\r\n \t<li>The point (0, 1) moves to [latex](0, a)[\/latex].<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nManipulate the animation in figure 3 to answer the following questions:\r\n<ol>\r\n \t<li>What happens to the point (1, 2) on the graph of [latex]f(x)=2^x[\/latex] when it is transformed to [latex]f(x)=4\\left(2^x\\right)[\/latex]?<\/li>\r\n \t<li>What happens to the point (1, 3) on the graph of [latex]f(x)=3^x[\/latex] when it is transformed to [latex]f(x)=4\\left(3^x\\right)[\/latex]?<\/li>\r\n \t<li>What happens to the point [latex]\\left(1, \\dfrac{1}{2}\\right)[\/latex] on the graph of [latex]f(x)=\\left(\\dfrac{1}{2}\\right)^x[\/latex] when it is transformed to [latex]f(x)=4\\left(\\dfrac{1}{2}\\right)^x[\/latex]?<\/li>\r\n \t<li>What happens to the point [latex](1, r)[\/latex] on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=4\\left(r^x\\right)[\/latex]?<\/li>\r\n \t<li>What happens to the point [latex](1, r)[\/latex] on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=a\\left(r^x\\right)[\/latex]?<\/li>\r\n \t<li>What happens to the asymptote [latex]y=0[\/latex] when [latex]f(x)[\/latex] is stretched or compressed?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm531\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm531\"]\r\n<ol>\r\n \t<li>With [latex]r=2[\/latex] and [latex]a=4[\/latex], the point (1, 2) moves to (1, 8).<\/li>\r\n \t<li>With [latex]r=3[\/latex] and [latex]a=4[\/latex], the point (1, 3) moves to (1, 12).<\/li>\r\n \t<li>With [latex]r=\\dfrac{1}{2}[\/latex] and [latex]a=4[\/latex], the point [latex]\\left(1, \\dfrac{1}{2}\\right)[\/latex] moves to (1, 2).<\/li>\r\n \t<li>With [latex]a=4[\/latex], the point [latex](1, r)[\/latex] moves to [latex](1, 4r)[\/latex].<\/li>\r\n \t<li>The point [latex](1, r)[\/latex] moves to [latex](1, ar)[\/latex].<\/li>\r\n \t<li>The asymptote stays the same.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nDetermine what happens to the point (2, 9) when the function [latex]f(x)=3^x[\/latex] is:\r\n<ol>\r\n \t<li>stretched by a factor of 5<\/li>\r\n \t<li>compressed to [latex]\\dfrac{1}{3}[\/latex] its height<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>When a graph is stretched the [latex]x[\/latex]-coordinate stays the same, while the [latex]y[\/latex]-coordinate is multiplied by the stretch factor, 5. Hence, (2, 9) becomes (2, 45).<\/li>\r\n \t<li>When a graph is compressed the [latex]x[\/latex]-coordinate stays the same, while the [latex]y[\/latex]-coordinate is multiplied by[latex]\\dfrac{1}{3}[\/latex]. Hence, (2, 9) becomes (2, 3).<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine what happens to the point (\u20132, 9) when the function [latex]f(x)=\\left(\\dfrac{1}{3}\\right)^x[\/latex] is:\r\n<ol>\r\n \t<li>stretched by a factor of 4<\/li>\r\n \t<li>compressed to [latex]\\dfrac{1}{6}[\/latex] its height<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm160\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm160\"]\r\n<ol>\r\n \t<li>(\u20132, 9) becomes (\u20132, 36).<\/li>\r\n \t<li>(\u20132, 9) becomes [latex]\\left(\u20132, \\dfrac{3}{2}\\right)[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nThe function [latex]f(x)=4^x[\/latex] is transformed. Determine the equation of the\u00a0transformed function when:\r\n<ol>\r\n \t<li>[latex]f(x)[\/latex] is stretched by a factor of 2.<\/li>\r\n \t<li>[latex]f(x)[\/latex] is stretched by a factor of 5.<\/li>\r\n \t<li>[latex]f(x)[\/latex] is compressed to [latex]\\dfrac{1}{3}[\/latex] its height.<\/li>\r\n \t<li>[latex]f(x)[\/latex] is compressed to [latex]\\dfrac{1}{4}[\/latex] its height.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>Since [latex]a=2[\/latex] the transformed function is [latex]f(x)=2\\left(4^x\\right)[\/latex].<\/li>\r\n \t<li>Since [latex]a=5[\/latex] the transformed function is [latex]f(x)=5\\left(4^x\\right)[\/latex].<\/li>\r\n \t<li>Since [latex]a=\\dfrac{1}{3}[\/latex] the transformed function is [latex]f(x)=\\dfrac{1}{3}\\left(4^x\\right)[\/latex].<\/li>\r\n \t<li>Since [latex]a=\\dfrac{1}{4}[\/latex] the transformed function is [latex]f(x)=\\dfrac{1}{4}\\left(4^x\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nDetermine the transformation made to the parent function [latex]g(x)=5^x[\/latex].\r\n<ol>\r\n \t<li>[latex]g(x)=4\\left(5^x\\right)[\/latex]<\/li>\r\n \t<li>[latex]g(x)=\\dfrac{1}{4}\\left(5^x\\right)[\/latex]<\/li>\r\n \t<li>[latex]g(x)=7\\left(5^x\\right)[\/latex]<\/li>\r\n \t<li>[latex]g(x)=\\dfrac{1}{3}\\left(5^x\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm669\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm669\"]\r\n<ol>\r\n \t<li>[latex]g(x)=5^x[\/latex] is stretched by a factor of 4.<\/li>\r\n \t<li>[latex]g(x)=5^x[\/latex] is compressed to [latex]\\dfrac{1}{4}[\/latex] its height.<\/li>\r\n \t<li>[latex]g(x)=5^x[\/latex] is stretched by a factor of 7.<\/li>\r\n \t<li>[latex]g(x)=5^x[\/latex] is compressed to [latex]\\dfrac{1}{3}[\/latex] its height.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflection across the [latex]x[\/latex]-axis<\/h2>\r\nWhen the graph of the function [latex]f(x)=2^x[\/latex] is reflected across the\u00a0[latex]x[\/latex]-axis, the\u00a0[latex]y[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values, while the [latex]x[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=2^x[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-2^x[\/latex]. The graph changes from increasing upwards to decreasing downwards. Table 3 shows the effect of such a reflection on the function values and the graph is shown in figure 4. While the [latex]x[\/latex]-coordinate stays the same, the [latex]y[\/latex]-coordinate becomes [latex]-y[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%; height: 213px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 33.3333%; height: 19px;\">[latex]y=2^x[\/latex]<\/th>\r\n<th style=\"width: 16.6667%; height: 19px;\">[latex]-y=2^x[\/latex] or [latex]y=-2^x[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 213px; vertical-align: top;\" rowspan=\"9\">[caption id=\"attachment_2645\" align=\"aligncenter\" width=\"395\"]<img class=\"wp-image-2645 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/11213556\/desmos-graph-2022-06-11T153524.687-300x300.png\" alt=\"reflection across x-axis\" width=\"395\" height=\"395\" \/> Figure 4. Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]x[\/latex]-axis.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-\\dfrac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 154px;\">\r\n<td style=\"width: 83.3333%; height: 61px;\" colspan=\"3\">Table 3.\u00a0Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=-2^x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nThe function [latex]f(x)=4^x[\/latex] is reflected across the [latex]x[\/latex]-axis. What happens to the point (2, 16) that lies on the parent function after the transformation?\r\n<h4>Solution<\/h4>\r\nWhen a function is reflected across the [latex]x[\/latex]-axis, the [latex]x[\/latex]-coordinate stays the same while the [latex]y[\/latex]-coordinate changes sign. So (2, 16) is transformed to (2, \u201316).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nThe function [latex]f(x)=\\left(\\dfrac{2}{3}\\right)^x[\/latex] is reflected across the [latex]x[\/latex]-axis. What happens to the point [latex]\\left(-1, \\dfrac{3}{2}\\right)[\/latex] that lies on the parent function after the transformation?\r\n\r\n[reveal-answer q=\"hjm231\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm231\"]\r\n\r\n[latex]\\left(-1, \\dfrac{3}{2}\\right)[\/latex] is transformed to\u00a0[latex]\\left(-1, -\\dfrac{3}{2}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflection across the [latex]y[\/latex]-axis<\/h2>\r\nWhen the graph of the function [latex]f(x)=2^x[\/latex] is reflected across the\u00a0[latex]y[\/latex]-axis, the\u00a0[latex]x[\/latex]-coordinates of all of the points on the graph change their signs while the [latex]y[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=2^x[\/latex] is reflected across the [latex]y[\/latex]-axis is [latex]f(x)=2^{-x}[\/latex]. The graph changes from increasing from the left to decreasing from the left. Table 4 shows the effect of such a reflection on the functions values and the graph is shown in figure 5.\u00a0While the [latex]y[\/latex]-coordinate stays the same, the [latex]x[\/latex]-coordinate becomes [latex]-x[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%; height: 211px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 28.9793%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 29.677%; height: 19px;\">[latex]2^x[\/latex]<\/th>\r\n<th style=\"width: 1.29199%; height: 19px;\">[latex]2^{-x}[\/latex]<\/th>\r\n<td style=\"width: 40.05174625322997%; height: 211px; vertical-align: top;\" rowspan=\"9\">\r\n<div class=\"mceTemp\"><\/div>\r\n[caption id=\"attachment_3235\" align=\"aligncenter\" width=\"400\"]<img class=\"wp-image-3235\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/08154633\/5-2-2-ReflectAcrossYNew-300x300.png\" alt=\"Reflection across the y-axis\" width=\"400\" height=\"400\" \/> Figure 5. Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]y[\/latex]-axis.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 154px;\">\r\n<td style=\"width: 59.94825374677003%; height: 59px;\" colspan=\"3\">Table 4.\u00a0Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]y[\/latex]-axis transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2^{-x}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nThe function [latex]f(x)=4^x[\/latex] is reflected across the [latex]y[\/latex]-axis. What happens to the point (2, 16) that lies on the parent function, after the transformation?\r\n<h4>Solution<\/h4>\r\nWhen a function is reflected across the [latex]y[\/latex]-axis, the [latex]y[\/latex]-coordinate stays the same while the [latex]x[\/latex]-coordinate changes sign. So (2, 16) is transformed to (\u20132, 16).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nThe function [latex]f(x)=7^x[\/latex] is reflected across the [latex]y[\/latex]-axis. What happens to the point [latex]\\left(-1, \\dfrac{1}{7}\\right)[\/latex] that lies on the parent function, after the transformation?\r\n\r\n[reveal-answer q=\"hjm429\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm429\"]\r\n\r\n[latex]\\left(-1, \\dfrac{1}{7}\\right)[\/latex] is transformed to\u00a0[latex]\\left(1, \\dfrac{1}{7}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Determine the transformations of the exponential function\u00a0[latex]f(x)=ar^{x-h}+k[\/latex]<\/h2>\r\nAll of the transformations\u00a0<span style=\"font-size: 1em;\">we have applied to<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0the parent function [latex]f(x)=r^x[\/latex] can be combined. The result is a general exponential function [latex]f(x)=ar^{x-h}+k[\/latex]. Given any function in the form\u00a0[latex]f(x)=ar^{x-h}+k[\/latex], we can determine from the values of [latex]a, h,[\/latex] and [latex]k[\/latex] the transformations that were performed on the parent function [latex]f(x)=r^x[\/latex]. Likewise, if we know the transformations, we can write the equation of the transformed function.<\/span>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nWhat transformations were performed on the parent function [latex]f(x)=2^x[\/latex] to get the function [latex]f(x)=-3\\left(2^{x+4}\\right)-6[\/latex]?\r\n<h4>Solution<\/h4>\r\nFirst we identify\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]a, h,[\/latex] and [latex]k[\/latex]:<\/span>\r\n\r\n[latex]a=-3,\\;h=-4,\\;k=-6[\/latex]\r\n\r\nA negative value of [latex]a[\/latex] means the function\u00a0[latex]f(x)=2^x[\/latex] has been reflected across the [latex]x[\/latex]-axis. [latex]a=-3[\/latex] means it has also been stretched by a factor of 3.\r\n\r\n[latex]h=-4[\/latex] means the parent function has been shifted horizontally 4 units left, while [latex]k=-6[\/latex] means it has been shifted vertically down by 6 units.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nWhat transformations were performed on the parent function [latex]f(x)=2^x[\/latex] to get the function [latex]f(x)=\\dfrac{1}{4}\\left(2^{x-6}\\right)+3[\/latex]?\r\n\r\n[reveal-answer q=\"hjm581\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm581\"]\r\n\r\nThe parent function has been compressed to [latex]\\dfrac{1}{4}[\/latex] its height, shifted horizontally 6 units right, and shifted vertically 3 units up.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nIf the parent function [latex]g(x)=3^x[\/latex] is stretched by a factor of 7, reflected across the [latex]y[\/latex]-axis, and shifted vertically 3 units down, what is the equation of the transformed function?\r\n<h4>Solution<\/h4>\r\nStretched by a factor of 7 means [latex]a=7[\/latex].\r\n\r\nReflected across the [latex]y[\/latex]-axis means [latex]x[\/latex] becomes [latex]-x[\/latex].\r\n\r\nShifted vertically down by 3 units means [latex]k=-3[\/latex].\r\n\r\nSo, the transformed function is [latex]f(x)=7\\left(3^{-x}\\right)-3[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nIf the parent function [latex]g(x)=5^x[\/latex] is compressed to [latex]\\dfrac{1}{4}[\/latex] its height, reflected across the [latex]x[\/latex]-axis, shifted horizontally left by 9 units, and shifted vertically up by 4 units, what is the equation of the transformed function?\r\n\r\n[reveal-answer q=\"hjm766\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm766\"]\r\n\r\n[latex]f(x)=-\\dfrac{1}{4}\\left(5^{x+9}\\right)+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nMove the dots in figure 6, to see what happens to the graph of the parent function [latex]f(x)=r^x[\/latex] when it is transformed by changing the values of the constants [latex]a, h, k[\/latex] and by making [latex]x[\/latex] negative to reflect across the [latex]y-axis[\/latex].\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/dhnb3kpap2?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 6. Transformation of [latex]f(x)=r^x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h1>Learning Outcomes<\/h1>\n<p>For the exponential function [latex]f(x)=r^x[\/latex],<\/p>\n<ul>\n<li>Perform vertical compressions and\u00a0stretches<\/li>\n<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\n<li>Perform reflections across the [latex]y[\/latex]-axis<\/li>\n<li>Determine the function given specific transformations<\/li>\n<li>Determine the transformations of the exponential function\u00a0[latex]f(x)=ar^{(x-h)}+k[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Stretching Up and Compressing Down<\/h2>\n<p>If we vertically stretch the graph of the function [latex]f(x)=2^x[\/latex] by a factor of two, all of the [latex]y[\/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[\/latex]-coordinates remain the same. The equation of the function after the graph is stretched by a factor of 2 is [latex]f(x)=2\\left(2^x\\right)[\/latex]. The reason for multiplying\u00a0 [latex]2^x[\/latex] by 2 is that each [latex]y[\/latex]-coordinate is doubled, and since [latex]y=2^x[\/latex], [latex]2^x[\/latex] is doubled. Table 1 shows this change and the graph is shown in figure 1. Any point [latex](x, y)[\/latex] on the graph of [latex]f(x)=2^x[\/latex] moves to [latex](x, 2y)[\/latex] when the graph is stretched by a factor of 2.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 248px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 33.3333%; height: 19px;\">[latex]2^x[\/latex]<\/th>\n<th style=\"width: 16.6667%; height: 19px;\">[latex]2\\left(2^x\\right)[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 248px;\" rowspan=\"9\">\n<div id=\"attachment_2744\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2744\" class=\"wp-image-2744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/17022638\/desmos-graph-2022-06-16T202557.534-300x300.png\" alt=\"Stretching the graph\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2744\" class=\"wp-caption-text\">Figure 1. Stretching the graph vertically by a factor of 2<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 96px;\">\n<td style=\"width: 83.3333%; height: 96px;\" colspan=\"3\">Table 1.\u00a0Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2\\left(2^x\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>On the other hand, if we vertically compress the graph of the function [latex]f(x)=2^x[\/latex] by half, all of the [latex]y[\/latex]-coordinates of the points on the graph are halved, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are divided by 2, or multiplied by [latex]\\frac{1}{2}[\/latex]. The equation of the function after being compressed by half is [latex]f(x)=\\dfrac{1}{2}\\left(2^x\\right)[\/latex]. The reason for multiplying [latex]2^x[\/latex] by [latex]\\frac{1}{2}[\/latex] is that each [latex]y[\/latex]-coordinate becomes half of the original value when it is divided by 2. Table 2 shows this change and the graph is shown in figure 2.\u00a0Any point [latex](x, y)[\/latex] on the graph of [latex]f(x)=2^x[\/latex] moves to [latex]\\left(x, \\frac{1}{2}y\\right)[\/latex] when the graph is compressed.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 20.9915%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 27.9536%; height: 19px;\">[latex]2^x[\/latex]<\/th>\n<th style=\"width: 34.3882%; height: 19px;\">[latex]\\dfrac{1}{2}\\left(2^x\\right)[\/latex]<\/th>\n<td style=\"width: 16.6667%;\" rowspan=\"9\">\n<div id=\"attachment_2745\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2745\" class=\"wp-image-2745\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/17023728\/desmos-graph-2022-06-16T203704.058-300x300.png\" alt=\"Compressing the graph\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2745\" class=\"wp-caption-text\">Figure 2. Compressing the graph.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">[latex]\\dfrac{1}{16}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 83.3333%;\" colspan=\"3\">Table 2.\u00a0Compressing the graph vertically by half transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=\\dfrac{1}{2}\\left(2^x\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Stretching and compression can, of course, be applied to any exponential function [latex]f(x)=r^x[\/latex], with [latex]r>0[\/latex] and [latex]r\\neq1[\/latex].<\/p>\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>vertical stretching and compressing<\/h3>\n<p id=\"fs-id1165137770279\">A stretch or compression of the graph of [latex]f(x)=r^x[\/latex] can be represented by\u00a0multiplying the function by a constant\u00a0 [latex]a>0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a\\cdot r^x[\/latex]<\/p>\n<p>The magnitude of [latex]a[\/latex]\u00a0indicates the stretch or compression of the graph. If [latex]a>1[\/latex], the graph is stretched upwards by a factor of [latex]a.[\/latex] If [latex]0<a<1[\/latex], the graph is compressed down to [latex]a[\/latex] times its original height.\n\nIn addition, [latex]f(0)=a[\/latex], so the\u00a0<em><strong>initial value<\/strong><\/em> of the function is transformed from [latex]1[\/latex] to [latex]a[\/latex].<\/p>\n<\/div>\n<p>We can change the value of [latex]a[\/latex] in figure 3 by moving the purple dot to see what happens to the graph of [latex]f(x)=r^x[\/latex]. Moving the red dot changes the value of [latex]r[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/yvwnta9ypu?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 3. Manipulation of [latex]f(x)=ar^x[\/latex]<\/p>\n<p>Notice that the value of [latex]a[\/latex] becomes the\u00a0<strong><em>initial value\u00a0<\/em><\/strong>of [latex]f(x)[\/latex]. That is [latex]f(0)=a[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Manipulate the animation in figure 3 to answer the following questions:<\/p>\n<ol>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[\/latex] when it is transformed to [latex]f(x)=4\\left(2^x\\right)[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[\/latex] when it is transformed to [latex]f(x)=4\\left(3^x\\right)[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=\\left(\\dfrac{1}{2}\\right)^x[\/latex] when it is transformed to [latex]f(x)=4\\left(\\dfrac{1}{2}\\right)^x[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=4r^x[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=ar^x[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>With [latex]r=2[\/latex] and [latex]a=4[\/latex], the point (0, 1) moves to (0, 4).<\/li>\n<li>With [latex]r=3[\/latex] and [latex]a=4[\/latex], the point (0, 1) moves to (0, 4).<\/li>\n<li>With [latex]r=\\dfrac{1}{2}[\/latex] and [latex]a=4[\/latex], the point (0, 1) moves to (0, 4).<\/li>\n<li>With [latex]a=4[\/latex], the point (0, 1) moves to [latex](0, 4)[\/latex].<\/li>\n<li>The point (0, 1) moves to [latex](0, a)[\/latex].<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Manipulate the animation in figure 3 to answer the following questions:<\/p>\n<ol>\n<li>What happens to the point (1, 2) on the graph of [latex]f(x)=2^x[\/latex] when it is transformed to [latex]f(x)=4\\left(2^x\\right)[\/latex]?<\/li>\n<li>What happens to the point (1, 3) on the graph of [latex]f(x)=3^x[\/latex] when it is transformed to [latex]f(x)=4\\left(3^x\\right)[\/latex]?<\/li>\n<li>What happens to the point [latex]\\left(1, \\dfrac{1}{2}\\right)[\/latex] on the graph of [latex]f(x)=\\left(\\dfrac{1}{2}\\right)^x[\/latex] when it is transformed to [latex]f(x)=4\\left(\\dfrac{1}{2}\\right)^x[\/latex]?<\/li>\n<li>What happens to the point [latex](1, r)[\/latex] on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=4\\left(r^x\\right)[\/latex]?<\/li>\n<li>What happens to the point [latex](1, r)[\/latex] on the graph of [latex]f(x)=r^x[\/latex] when it is transformed to [latex]f(x)=a\\left(r^x\\right)[\/latex]?<\/li>\n<li>What happens to the asymptote [latex]y=0[\/latex] when [latex]f(x)[\/latex] is stretched or compressed?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm531\">Show Answer<\/span><\/p>\n<div id=\"qhjm531\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>With [latex]r=2[\/latex] and [latex]a=4[\/latex], the point (1, 2) moves to (1, 8).<\/li>\n<li>With [latex]r=3[\/latex] and [latex]a=4[\/latex], the point (1, 3) moves to (1, 12).<\/li>\n<li>With [latex]r=\\dfrac{1}{2}[\/latex] and [latex]a=4[\/latex], the point [latex]\\left(1, \\dfrac{1}{2}\\right)[\/latex] moves to (1, 2).<\/li>\n<li>With [latex]a=4[\/latex], the point [latex](1, r)[\/latex] moves to [latex](1, 4r)[\/latex].<\/li>\n<li>The point [latex](1, r)[\/latex] moves to [latex](1, ar)[\/latex].<\/li>\n<li>The asymptote stays the same.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Determine what happens to the point (2, 9) when the function [latex]f(x)=3^x[\/latex] is:<\/p>\n<ol>\n<li>stretched by a factor of 5<\/li>\n<li>compressed to [latex]\\dfrac{1}{3}[\/latex] its height<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>When a graph is stretched the [latex]x[\/latex]-coordinate stays the same, while the [latex]y[\/latex]-coordinate is multiplied by the stretch factor, 5. Hence, (2, 9) becomes (2, 45).<\/li>\n<li>When a graph is compressed the [latex]x[\/latex]-coordinate stays the same, while the [latex]y[\/latex]-coordinate is multiplied by[latex]\\dfrac{1}{3}[\/latex]. Hence, (2, 9) becomes (2, 3).<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine what happens to the point (\u20132, 9) when the function [latex]f(x)=\\left(\\dfrac{1}{3}\\right)^x[\/latex] is:<\/p>\n<ol>\n<li>stretched by a factor of 4<\/li>\n<li>compressed to [latex]\\dfrac{1}{6}[\/latex] its height<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm160\">Show Answer<\/span><\/p>\n<div id=\"qhjm160\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>(\u20132, 9) becomes (\u20132, 36).<\/li>\n<li>(\u20132, 9) becomes [latex]\\left(\u20132, \\dfrac{3}{2}\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>The function [latex]f(x)=4^x[\/latex] is transformed. Determine the equation of the\u00a0transformed function when:<\/p>\n<ol>\n<li>[latex]f(x)[\/latex] is stretched by a factor of 2.<\/li>\n<li>[latex]f(x)[\/latex] is stretched by a factor of 5.<\/li>\n<li>[latex]f(x)[\/latex] is compressed to [latex]\\dfrac{1}{3}[\/latex] its height.<\/li>\n<li>[latex]f(x)[\/latex] is compressed to [latex]\\dfrac{1}{4}[\/latex] its height.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>Since [latex]a=2[\/latex] the transformed function is [latex]f(x)=2\\left(4^x\\right)[\/latex].<\/li>\n<li>Since [latex]a=5[\/latex] the transformed function is [latex]f(x)=5\\left(4^x\\right)[\/latex].<\/li>\n<li>Since [latex]a=\\dfrac{1}{3}[\/latex] the transformed function is [latex]f(x)=\\dfrac{1}{3}\\left(4^x\\right)[\/latex].<\/li>\n<li>Since [latex]a=\\dfrac{1}{4}[\/latex] the transformed function is [latex]f(x)=\\dfrac{1}{4}\\left(4^x\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Determine the transformation made to the parent function [latex]g(x)=5^x[\/latex].<\/p>\n<ol>\n<li>[latex]g(x)=4\\left(5^x\\right)[\/latex]<\/li>\n<li>[latex]g(x)=\\dfrac{1}{4}\\left(5^x\\right)[\/latex]<\/li>\n<li>[latex]g(x)=7\\left(5^x\\right)[\/latex]<\/li>\n<li>[latex]g(x)=\\dfrac{1}{3}\\left(5^x\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm669\">Show Answer<\/span><\/p>\n<div id=\"qhjm669\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]g(x)=5^x[\/latex] is stretched by a factor of 4.<\/li>\n<li>[latex]g(x)=5^x[\/latex] is compressed to [latex]\\dfrac{1}{4}[\/latex] its height.<\/li>\n<li>[latex]g(x)=5^x[\/latex] is stretched by a factor of 7.<\/li>\n<li>[latex]g(x)=5^x[\/latex] is compressed to [latex]\\dfrac{1}{3}[\/latex] its height.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflection across the [latex]x[\/latex]-axis<\/h2>\n<p>When the graph of the function [latex]f(x)=2^x[\/latex] is reflected across the\u00a0[latex]x[\/latex]-axis, the\u00a0[latex]y[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values, while the [latex]x[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=2^x[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-2^x[\/latex]. The graph changes from increasing upwards to decreasing downwards. Table 3 shows the effect of such a reflection on the function values and the graph is shown in figure 4. While the [latex]x[\/latex]-coordinate stays the same, the [latex]y[\/latex]-coordinate becomes [latex]-y[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 213px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 33.3333%; height: 19px;\">[latex]y=2^x[\/latex]<\/th>\n<th style=\"width: 16.6667%; height: 19px;\">[latex]-y=2^x[\/latex] or [latex]y=-2^x[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 213px; vertical-align: top;\" rowspan=\"9\">\n<div id=\"attachment_2645\" style=\"width: 405px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2645\" class=\"wp-image-2645\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/11213556\/desmos-graph-2022-06-11T153524.687-300x300.png\" alt=\"reflection across x-axis\" width=\"395\" height=\"395\" \/><\/p>\n<p id=\"caption-attachment-2645\" class=\"wp-caption-text\">Figure 4. Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]x[\/latex]-axis.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-\\dfrac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]-8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 154px;\">\n<td style=\"width: 83.3333%; height: 61px;\" colspan=\"3\">Table 3.\u00a0Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=-2^x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>The function [latex]f(x)=4^x[\/latex] is reflected across the [latex]x[\/latex]-axis. What happens to the point (2, 16) that lies on the parent function after the transformation?<\/p>\n<h4>Solution<\/h4>\n<p>When a function is reflected across the [latex]x[\/latex]-axis, the [latex]x[\/latex]-coordinate stays the same while the [latex]y[\/latex]-coordinate changes sign. So (2, 16) is transformed to (2, \u201316).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>The function [latex]f(x)=\\left(\\dfrac{2}{3}\\right)^x[\/latex] is reflected across the [latex]x[\/latex]-axis. What happens to the point [latex]\\left(-1, \\dfrac{3}{2}\\right)[\/latex] that lies on the parent function after the transformation?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm231\">Show Answer<\/span><\/p>\n<div id=\"qhjm231\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-1, \\dfrac{3}{2}\\right)[\/latex] is transformed to\u00a0[latex]\\left(-1, -\\dfrac{3}{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflection across the [latex]y[\/latex]-axis<\/h2>\n<p>When the graph of the function [latex]f(x)=2^x[\/latex] is reflected across the\u00a0[latex]y[\/latex]-axis, the\u00a0[latex]x[\/latex]-coordinates of all of the points on the graph change their signs while the [latex]y[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=2^x[\/latex] is reflected across the [latex]y[\/latex]-axis is [latex]f(x)=2^{-x}[\/latex]. The graph changes from increasing from the left to decreasing from the left. Table 4 shows the effect of such a reflection on the functions values and the graph is shown in figure 5.\u00a0While the [latex]y[\/latex]-coordinate stays the same, the [latex]x[\/latex]-coordinate becomes [latex]-x[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 211px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 28.9793%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 29.677%; height: 19px;\">[latex]2^x[\/latex]<\/th>\n<th style=\"width: 1.29199%; height: 19px;\">[latex]2^{-x}[\/latex]<\/th>\n<td style=\"width: 40.05174625322997%; height: 211px; vertical-align: top;\" rowspan=\"9\">\n<div class=\"mceTemp\"><\/div>\n<div id=\"attachment_3235\" style=\"width: 410px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3235\" class=\"wp-image-3235\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/08154633\/5-2-2-ReflectAcrossYNew-300x300.png\" alt=\"Reflection across the y-axis\" width=\"400\" height=\"400\" \/><\/p>\n<p id=\"caption-attachment-3235\" class=\"wp-caption-text\">Figure 5. Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 28.979294832041347%; height: 19px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 29.676969250645996%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 1.2919896640826867%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 154px;\">\n<td style=\"width: 59.94825374677003%; height: 59px;\" colspan=\"3\">Table 4.\u00a0Reflecting the graph of [latex]f(x)=2^x[\/latex] across the [latex]y[\/latex]-axis transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2^{-x}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>The function [latex]f(x)=4^x[\/latex] is reflected across the [latex]y[\/latex]-axis. What happens to the point (2, 16) that lies on the parent function, after the transformation?<\/p>\n<h4>Solution<\/h4>\n<p>When a function is reflected across the [latex]y[\/latex]-axis, the [latex]y[\/latex]-coordinate stays the same while the [latex]x[\/latex]-coordinate changes sign. So (2, 16) is transformed to (\u20132, 16).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>The function [latex]f(x)=7^x[\/latex] is reflected across the [latex]y[\/latex]-axis. What happens to the point [latex]\\left(-1, \\dfrac{1}{7}\\right)[\/latex] that lies on the parent function, after the transformation?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm429\">Show Answer<\/span><\/p>\n<div id=\"qhjm429\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-1, \\dfrac{1}{7}\\right)[\/latex] is transformed to\u00a0[latex]\\left(1, \\dfrac{1}{7}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Determine the transformations of the exponential function\u00a0[latex]f(x)=ar^{x-h}+k[\/latex]<\/h2>\n<p>All of the transformations\u00a0<span style=\"font-size: 1em;\">we have applied to<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0the parent function [latex]f(x)=r^x[\/latex] can be combined. The result is a general exponential function [latex]f(x)=ar^{x-h}+k[\/latex]. Given any function in the form\u00a0[latex]f(x)=ar^{x-h}+k[\/latex], we can determine from the values of [latex]a, h,[\/latex] and [latex]k[\/latex] the transformations that were performed on the parent function [latex]f(x)=r^x[\/latex]. Likewise, if we know the transformations, we can write the equation of the transformed function.<\/span><\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>What transformations were performed on the parent function [latex]f(x)=2^x[\/latex] to get the function [latex]f(x)=-3\\left(2^{x+4}\\right)-6[\/latex]?<\/p>\n<h4>Solution<\/h4>\n<p>First we identify\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]a, h,[\/latex] and [latex]k[\/latex]:<\/span><\/p>\n<p>[latex]a=-3,\\;h=-4,\\;k=-6[\/latex]<\/p>\n<p>A negative value of [latex]a[\/latex] means the function\u00a0[latex]f(x)=2^x[\/latex] has been reflected across the [latex]x[\/latex]-axis. [latex]a=-3[\/latex] means it has also been stretched by a factor of 3.<\/p>\n<p>[latex]h=-4[\/latex] means the parent function has been shifted horizontally 4 units left, while [latex]k=-6[\/latex] means it has been shifted vertically down by 6 units.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>What transformations were performed on the parent function [latex]f(x)=2^x[\/latex] to get the function [latex]f(x)=\\dfrac{1}{4}\\left(2^{x-6}\\right)+3[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm581\">Show Answer<\/span><\/p>\n<div id=\"qhjm581\" class=\"hidden-answer\" style=\"display: none\">\n<p>The parent function has been compressed to [latex]\\dfrac{1}{4}[\/latex] its height, shifted horizontally 6 units right, and shifted vertically 3 units up.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>If the parent function [latex]g(x)=3^x[\/latex] is stretched by a factor of 7, reflected across the [latex]y[\/latex]-axis, and shifted vertically 3 units down, what is the equation of the transformed function?<\/p>\n<h4>Solution<\/h4>\n<p>Stretched by a factor of 7 means [latex]a=7[\/latex].<\/p>\n<p>Reflected across the [latex]y[\/latex]-axis means [latex]x[\/latex] becomes [latex]-x[\/latex].<\/p>\n<p>Shifted vertically down by 3 units means [latex]k=-3[\/latex].<\/p>\n<p>So, the transformed function is [latex]f(x)=7\\left(3^{-x}\\right)-3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>If the parent function [latex]g(x)=5^x[\/latex] is compressed to [latex]\\dfrac{1}{4}[\/latex] its height, reflected across the [latex]x[\/latex]-axis, shifted horizontally left by 9 units, and shifted vertically up by 4 units, what is the equation of the transformed function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm766\">Show Answer<\/span><\/p>\n<div id=\"qhjm766\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)=-\\dfrac{1}{4}\\left(5^{x+9}\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Move the dots in figure 6, to see what happens to the graph of the parent function [latex]f(x)=r^x[\/latex] when it is transformed by changing the values of the constants [latex]a, h, k[\/latex] and by making [latex]x[\/latex] negative to reflect across the [latex]y-axis[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/dhnb3kpap2?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 6. Transformation of [latex]f(x)=r^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-2631\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>5.2.2: Transformations of the Exponential Functionu2013u2013Stretches, Compressions and Reflections. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All examples and Try its: hjm766; hjm581; hjm429; hjm231; hjm669; hjm160; hjm531. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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":370291,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"5.2.2: Transformations of the Exponential Functionu2013u2013Stretches, Compressions and Reflections\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All examples and Try its: hjm766; hjm581; hjm429; hjm231; hjm669; hjm160; hjm531\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"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-2631","chapter","type-chapter","status-publish","hentry"],"part":2116,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2631","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":29,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2631\/revisions"}],"predecessor-version":[{"id":4856,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2631\/revisions\/4856"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2116"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2631\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2631"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2631"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2631"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}