{"id":2004,"date":"2016-11-02T23:19:05","date_gmt":"2016-11-02T23:19:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2004"},"modified":"2017-04-21T00:40:26","modified_gmt":"2017-04-21T00:40:26","slug":"stretch-or-compress-an-exponential-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/stretch-or-compress-an-exponential-function\/","title":{"raw":"Stretch, Compress, or Reflect an Exponential Function","rendered":"Stretch, Compress, or Reflect an Exponential Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/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] as shown on the left in Figure 8, and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] as shown on the right in the figure 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\/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\" data-media-type=\"image\/jpg\" \/> (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 <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\r\nFor any factor <em>a<\/em> &gt; 0, the function [latex]f\\left(x\\right)=a{\\left(b\\right)}^{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 <em>y<\/em>-intercept of [latex]\\left(0,a\\right)[\/latex].<\/li>\r\n \t<li>has a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex], which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div 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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"418729\"]\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 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\" data-media-type=\"image\/jpg\" \/> 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 the sliders in the graph below 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\nhttps:\/\/www.desmos.com\/calculator\/u8kysdu1wl\r\n\r\n[reveal-answer q=\"796634\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"796634\"]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\" data-type=\"media\" data-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).\" data-display=\"block\">\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<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=129498&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\">\r\n<\/iframe>\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 <strong>reflection<\/strong> about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], is shown on the left side, and the reflection about the <em>y<\/em>-axis [latex]h\\left(x\\right)={2}^{-x}[\/latex], is shown on the right side.\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\" data-media-type=\"image\/jpg\" \/> (a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]g\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.[\/caption]\r\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><\/figcaption><\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Reflections of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/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 at [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex], which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\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\"]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 data-width=\"75\" \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\r\n<td>\u201364<\/td>\r\n<td>\u201316<\/td>\r\n<td>\u20134<\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20130.25<\/td>\r\n<td>\u20130.0625<\/td>\r\n<td>\u20130.0156<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\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\" data-media-type=\"image\/jpg\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,0\\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse Desmos 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<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\n\r\n[reveal-answer q=\"845922\"]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].<span id=\"fs-id1165137828034\" data-type=\"media\" data-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).\">\r\n<\/span>\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 Translations 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 translating exponential functions.\r\n<table summary=\"Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center;\">Translation<\/th>\r\n<th style=\"text-align: center;\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Shift\r\n<ul>\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 translations<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Translations of Exponential Functions<\/h3>\r\nA translation of an exponential function has the form\r\n\r\n[latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex]\r\n\r\nWhere 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, the domain, and the 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\"]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} 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 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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"525289\"][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 Objectives<\/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] as shown on the left in Figure 8, and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] as shown on the right in the figure 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\/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\" data-media-type=\"image\/jpg\" \/><\/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 <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p>For any factor <em>a<\/em> &gt; 0, the function [latex]f\\left(x\\right)=a{\\left(b\\right)}^{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 <em>y<\/em>-intercept of [latex]\\left(0,a\\right)[\/latex].<\/li>\n<li>has a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex], which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div 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\">Solution<\/span><\/p>\n<div id=\"q418729\" class=\"hidden-answer\" style=\"display: none\">\nBefore 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 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\" data-media-type=\"image\/jpg\" \/><\/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 the sliders in the graph below 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<p>https:\/\/www.desmos.com\/calculator\/u8kysdu1wl<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q796634\">Solution<\/span><\/p>\n<div id=\"q796634\" class=\"hidden-answer\" style=\"display: none\">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\" data-type=\"media\" data-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).\" data-display=\"block\"><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=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=129498&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\"><br \/>\n<\/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 <strong>reflection<\/strong> about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], is shown on the left side, and the reflection about the <em>y<\/em>-axis [latex]h\\left(x\\right)={2}^{-x}[\/latex], is shown on the right side.<\/p>\n<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\" data-media-type=\"image\/jpg\" \/><\/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]g\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.<\/p>\n<\/div>\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><\/figcaption><\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Reflections of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p>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 at [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex], which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>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\">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 data-width=\"75\" \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\n<td>\u201364<\/td>\n<td>\u201316<\/td>\n<td>\u20134<\/td>\n<td>\u20131<\/td>\n<td>\u20130.25<\/td>\n<td>\u20130.0625<\/td>\n<td>\u20130.0156<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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\" data-media-type=\"image\/jpg\" \/><\/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]; 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 Desmos 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<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q845922\">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].<span id=\"fs-id1165137828034\" data-type=\"media\" data-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).\"><br \/>\n<\/span><\/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 Translations 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 translating exponential functions.<\/p>\n<table summary=\"Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Translations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Translation<\/th>\n<th style=\"text-align: center;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul>\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 translations<\/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: Translations of Exponential Functions<\/h3>\n<p>A translation of an exponential function has the form<\/p>\n<p>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/p>\n<p>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, the domain, and the 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\">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} 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 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\">Solution<\/span><\/p>\n<div id=\"q525289\" class=\"hidden-answer\" style=\"display: none\">[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].<\/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-2004\">\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":21,"menu_order":9,"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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