{"id":264,"date":"2023-06-21T13:22:50","date_gmt":"2023-06-21T13:22:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/stretch-or-compress-an-exponential-function\/"},"modified":"2023-07-04T04:18:18","modified_gmt":"2023-07-04T04:18:18","slug":"stretch-or-compress-an-exponential-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/stretch-or-compress-an-exponential-function\/","title":{"raw":"\u25aa   Stretching, Compressing, or Reflecting an Exponential Function","rendered":"\u25aa   Stretching, Compressing, or Reflecting an Exponential Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph a stretched or compressed exponential function.<\/li>\r\n \t<li>Graph a reflected exponential function.<\/li>\r\n \t<li>Write the equation of an exponential function that has been transformed.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhile 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] 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].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231151\/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.\" width=\"975\" height=\"445\" \/> (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].[\/caption]\r\n<div class=\"textbox\">\r\n<h3>A General Note: Stretches and Compressions of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)=a{b}^{x}[\/latex]\r\n<ul>\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\u00a0<em>y<\/em>-intercept is [latex]\\left(0,a\\right)[\/latex].<\/li>\r\n \t<li>has a horizontal asymptote of [latex]y=0[\/latex], range of [latex]\\left(0,\\infty \\right)[\/latex], and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are all unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nExponential functions are stretched, compressed or reflected in the same manner you've used to transform other functions. Multipliers or negatives inside the function argument (in the exponent) affect horizontal transformations. Multipliers or negatives outside the function argument affect vertical transformations.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing the Stretch of an Exponential Function<\/h3>\r\nSketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"418729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418729\"]\r\n\r\nBefore graphing, identify the behavior and key points on the graph.\r\n<ul>\r\n \t<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em>x<\/em>-axis as <em>x<\/em>\u00a0increases.<\/li>\r\n \t<li>Since <em>a\u00a0<\/em>= 4, the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched vertically by a factor of 4.<\/li>\r\n \t<li>Create a table of points:\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td>32<\/td>\r\n<td>16<\/td>\r\n<td>8<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>0.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Plot the <em>y-<\/em>intercept, [latex]\\left(0,4\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\r\n \t<li>Draw a smooth curve connecting the points.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231155\/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).\" width=\"487\" height=\"482\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], the horizontal asymptote is y\u00a0= 0.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse an online graphing tool to sketch the graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"796634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796634\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].\u00a0<span id=\"fs-id1165135417835\">\r\n<\/span>\r\n\r\n<img class=\"aligncenter wp-image-3081 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16190801\/CNX_Precalc_Figure_04_02_0122.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"488\" height=\"294\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]129498[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Graphing Reflections<\/h2>\r\nIn addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em>x<\/em>-axis. When we multiply the input by \u20131, we get a reflection about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={2}^{-x}[\/latex], are both shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231158\/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.\" width=\"975\" height=\"628\" \/> (a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]h\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.[\/caption]\r\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><\/figcaption><\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Reflecting the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>x<\/em>-axis.<\/li>\r\n \t<li>has a <em>y<\/em>-intercept of [latex]\\left(0,-1\\right)[\/latex].<\/li>\r\n \t<li>has a range of [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\r\n \t<li>has a horizontal asymptote of [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\nThe function [latex]f\\left(x\\right)={b}^{-x}[\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>y<\/em>-axis.<\/li>\r\n \t<li>has a <em>y<\/em>-intercept of [latex]\\left(0,1\\right)[\/latex], a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing and Graphing the Reflection of an Exponential Function<\/h3>\r\nFind and graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x<\/em>-axis. State its domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"91748\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"91748\"]\r\n\r\nSince we want to reflect the parent function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by \u20131 to get [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\r\n<td>\u201364<\/td>\r\n<td>\u201316<\/td>\r\n<td>\u20134<\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20130.25<\/td>\r\n<td>\u20130.0625<\/td>\r\n<td>\u20130.0156<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the <em>y-<\/em>intercept, [latex]\\left(0,-1\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(1,-0.25\\right)[\/latex].\r\n\r\nDraw a smooth curve connecting the points:\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231202\/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).\" width=\"487\" height=\"407\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,0\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse an online graphing calculator to\u00a0graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={1.25}^{x}[\/latex] about the <em>y<\/em>-axis. State its domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"845922\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"845922\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].\r\n\r\n<img class=\"aligncenter size-full wp-image-3082\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16191705\/CNX_Precalc_Figure_04_02_0152.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" width=\"731\" height=\"482\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summarizing Transformations of the Exponential Function<\/h2>\r\nNow that we have worked with each type of translation for the exponential function, we can summarize them\u00a0to arrive at the general equation for transforming exponential functions.\r\n\r\n1, 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).\"&gt;\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Transformations 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>\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>\r\n \t<li>Stretch if |<em>a<\/em>|&gt;1<\/li>\r\n \t<li>Compression if 0&lt;|<em>a<\/em>|&lt;1<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>x<\/em>-axis<\/td>\r\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflect about the <em>y<\/em>-axis<\/td>\r\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>General equation for all transformations<\/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 class=\"textbox\">\r\n<h3>A General Note: Transformations of Exponential Functions<\/h3>\r\nA transformation of an exponential function has the form\r\n\r\n[latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex], where the parent function, [latex]y={b}^{x}[\/latex], [latex]b&gt;1[\/latex], is\r\n<ul>\r\n \t<li>shifted horizontally <em>c<\/em>\u00a0units to the left.<\/li>\r\n \t<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\r\n \t<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\r\n \t<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\nNote the order of the shifts, transformations, and reflections follow the order of operations.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Function from a Description<\/h3>\r\nWrite the equation for the function described below. Give the horizontal asymptote, domain, and range.\r\n<ul>\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <em>y<\/em>-axis, and then shifted up 4\u00a0units.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"290621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"290621\"]\r\n\r\nWe 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>.\r\n<ul>\r\n \t<li>We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so <em>b\u00a0<\/em>= <em>e<\/em>.<\/li>\r\n \t<li>The function is stretched by a factor of 2, so <em>a\u00a0<\/em>= 2.<\/li>\r\n \t<li>The function is reflected about the <em>y<\/em>-axis. We replace <em>x<\/em>\u00a0with \u2013<em>x<\/em>\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\r\n \t<li>The graph is shifted vertically 4 units, so <em>d\u00a0<\/em>= 4.<\/li>\r\n<\/ul>\r\nSubstituting in the general form, we get:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}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{array}[\/latex]<\/p>\r\nThe 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].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite the equation for the function described below. Give the horizontal asymptote, the domain, and the range.\r\n<ul>\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <em>x<\/em>-axis, and then shifted down 2\u00a0units.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"525289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525289\"]\r\n\r\n[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2[\/latex]; the domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,-2\\right)[\/latex]; the horizontal asymptote is [latex]y=-2[\/latex].[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph a stretched or compressed exponential function.<\/li>\n<li>Graph a reflected exponential function.<\/li>\n<li>Write the equation of an exponential function that has been transformed.<\/li>\n<\/ul>\n<\/div>\n<p>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] 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].<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231151\/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.\" width=\"975\" height=\"445\" \/><\/p>\n<p class=\"wp-caption-text\">(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>\n<div class=\"textbox\">\n<h3>A General Note: Stretches and Compressions of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>The function [latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\n<ul>\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\u00a0<em>y<\/em>-intercept is [latex]\\left(0,a\\right)[\/latex].<\/li>\n<li>has a horizontal asymptote of [latex]y=0[\/latex], range of [latex]\\left(0,\\infty \\right)[\/latex], and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are all unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Exponential functions are stretched, compressed or reflected in the same manner you&#8217;ve used to transform other functions. Multipliers or negatives inside the function argument (in the exponent) affect horizontal transformations. Multipliers or negatives outside the function argument affect vertical transformations.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing the Stretch of an Exponential Function<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q418729\">Show Solution<\/span><\/p>\n<div id=\"q418729\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before graphing, identify the behavior and key points on the graph.<\/p>\n<ul>\n<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em>x<\/em>-axis as <em>x<\/em>\u00a0increases.<\/li>\n<li>Since <em>a\u00a0<\/em>= 4, the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched vertically by a factor of 4.<\/li>\n<li>Create a table of points:<br \/>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td>32<\/td>\n<td>16<\/td>\n<td>8<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>0.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y-<\/em>intercept, [latex]\\left(0,4\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\n<li>Draw a smooth curve connecting the points.<\/li>\n<\/ul>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231155\/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).\" width=\"487\" height=\"482\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], the horizontal asymptote is y\u00a0= 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use an online graphing tool to sketch the graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q796634\">Show Solution<\/span><\/p>\n<div id=\"q796634\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].\u00a0<span id=\"fs-id1165135417835\"><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3081 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16190801\/CNX_Precalc_Figure_04_02_0122.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"488\" height=\"294\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm129498\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129498&theme=oea&iframe_resize_id=ohm129498&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing Reflections<\/h2>\n<p>In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em>x<\/em>-axis. When we multiply the input by \u20131, we get a reflection about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={2}^{-x}[\/latex], are both shown below.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231158\/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.\" width=\"975\" height=\"628\" \/><\/p>\n<p class=\"wp-caption-text\">(a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]h\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.<\/p>\n<\/div>\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><\/figcaption><\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Reflecting the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>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>x<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-intercept of [latex]\\left(0,-1\\right)[\/latex].<\/li>\n<li>has a range of [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\n<li>has a horizontal asymptote of [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>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>y<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-intercept of [latex]\\left(0,1\\right)[\/latex], a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing and Graphing the Reflection of an Exponential Function<\/h3>\n<p>Find and graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q91748\">Show Solution<\/span><\/p>\n<div id=\"q91748\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since we want to reflect the parent function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by \u20131 to get [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\n<td>\u201364<\/td>\n<td>\u201316<\/td>\n<td>\u20134<\/td>\n<td>\u20131<\/td>\n<td>\u20130.25<\/td>\n<td>\u20130.0625<\/td>\n<td>\u20130.0156<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the <em>y-<\/em>intercept, [latex]\\left(0,-1\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(1,-0.25\\right)[\/latex].<\/p>\n<p>Draw a smooth curve connecting the points:<\/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\/wp-content\/uploads\/sites\/896\/2016\/11\/02231202\/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).\" width=\"487\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,0\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use an online graphing calculator to\u00a0graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={1.25}^{x}[\/latex] about the <em>y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q845922\">Show Solution<\/span><\/p>\n<div id=\"q845922\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3082\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16191705\/CNX_Precalc_Figure_04_02_0152.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" width=\"731\" height=\"482\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summarizing Transformations of the Exponential Function<\/h2>\n<p>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 transforming exponential functions.<\/p>\n<p>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).&#8221;&gt;<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Transformations 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>\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>\n<li>Stretch if |<em>a<\/em>|&gt;1<\/li>\n<li>Compression if 0&lt;|<em>a<\/em>|&lt;1<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>x<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>y<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all transformations<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A General Note: Transformations of Exponential Functions<\/h3>\n<p>A transformation of an exponential function has the form<\/p>\n<p>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], where the parent function, [latex]y={b}^{x}[\/latex], [latex]b>1[\/latex], is<\/p>\n<ul>\n<li>shifted horizontally <em>c<\/em>\u00a0units to the left.<\/li>\n<li>stretched vertically by a factor of |<em>a<\/em>| if |<em>a<\/em>| &gt; 0.<\/li>\n<li>compressed vertically by a factor of |<em>a<\/em>| if 0 &lt; |<em>a<\/em>| &lt; 1.<\/li>\n<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\n<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<p>Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Function from a Description<\/h3>\n<p>Write the equation for the function described below. Give the horizontal asymptote, domain, and range.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <em>y<\/em>-axis, and then shifted up 4\u00a0units.<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q290621\">Show Solution<\/span><\/p>\n<div id=\"q290621\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>\n<li>We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so <em>b\u00a0<\/em>= <em>e<\/em>.<\/li>\n<li>The function is stretched by a factor of 2, so <em>a\u00a0<\/em>= 2.<\/li>\n<li>The function is reflected about the <em>y<\/em>-axis. We replace <em>x<\/em>\u00a0with \u2013<em>x<\/em>\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\n<li>The graph is shifted vertically 4 units, so <em>d\u00a0<\/em>= 4.<\/li>\n<\/ul>\n<p>Substituting in the general form, we get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}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{array}[\/latex]<\/p>\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(4,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <em>x<\/em>-axis, and then shifted down 2\u00a0units.<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525289\">Show Solution<\/span><\/p>\n<div id=\"q525289\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2[\/latex]; the domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,-2\\right)[\/latex]; the horizontal asymptote is [latex]y=-2[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-264\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 129498. <strong>Authored by<\/strong>: Day, Alyson. <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>: IMathAS Community LicenseCC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 129498\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community LicenseCC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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