{"id":947,"date":"2015-11-12T18:37:58","date_gmt":"2015-11-12T18:37:58","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=947"},"modified":"2017-03-31T21:01:41","modified_gmt":"2017-03-31T21:01:41","slug":"graph-functions-using-compressions-and-stretches","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/graph-functions-using-compressions-and-stretches\/","title":{"raw":"Graph functions using compressions and stretches","rendered":"Graph functions using compressions and stretches"},"content":{"raw":"<section id=\"fs-id1165137654768\" data-depth=\"1\">\r\n<p id=\"fs-id1165137654773\">Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\r\n<p id=\"fs-id1165137675403\">We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\r\n\r\n<section id=\"fs-id1165137793506\" data-depth=\"2\">\r\n<h2 style=\"text-align: center;\" data-type=\"title\"><\/h2>\r\n<h2 style=\"text-align: center;\" data-type=\"title\"><\/h2>\r\n<h2 style=\"text-align: center;\" data-type=\"title\"><span style=\"text-decoration: underline;\">Vertical Stretches and Compressions<\/span><\/h2>\r\nWhen we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a<strong> vertical compression<\/strong>. The graph below\u00a0shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200813\/CNX_Precalc_Figure_01_05_0242.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"487\" height=\"326\" data-media-type=\"image\/jpg\" \/> <b>Figure 14.<\/b> Vertical stretch and compression[\/caption]\r\n\r\n<div id=\"fs-id1165137472530\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: Vertical Stretches and Compressions<\/h3>\r\n<p id=\"fs-id1165137444261\">Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=af\\left(x\\right)[\/latex], where [latex]a[\/latex] is a constant, is a <span data-type=\"term\">vertical stretch<\/span> or <span data-type=\"term\">vertical compression<\/span> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<ul id=\"fs-id1165135553621\">\r\n \t<li>If [latex]a&gt;1[\/latex], then the graph will be stretched.<\/li>\r\n \t<li>If 0 &lt; a &lt; 1, then the graph will be compressed.<\/li>\r\n \t<li>If [latex]a&lt;0[\/latex], then there will be combination of a vertical stretch or compression with a vertical reflection.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135173107\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 id=\"fs-id1165132939216\">How To: Given a function, graph its vertical stretch.<\/h3>\r\n<ol id=\"fs-id1165134190735\" data-number-style=\"arabic\">\r\n \t<li>Identify the value of [latex]a[\/latex].<\/li>\r\n \t<li>Multiply all range values by [latex]a[\/latex].<\/li>\r\n \t<li>\r\n<p id=\"eip-id1165133107002\">If [latex]a&gt;1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].<\/p>\r\n<p id=\"eip-id1165135191979\">If [latex]{ 0 }&lt;{ a }&lt;{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].<\/p>\r\n<p id=\"eip-id1165134061939\">If [latex]a&lt;0[\/latex], the graph is either stretched or compressed and also reflected about the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_13\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165137619296\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137619298\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 10: Graphing a Vertical Stretch<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200814\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"487\" height=\"367\" data-media-type=\"image\/jpg\" \/> <b>Figure 15<\/b>[\/caption]\r\n<p id=\"fs-id1165137552971\">A function [latex]P\\left(t\\right)[\/latex] models the population of fruit flies.<span id=\"fs-id1165137501364\" data-type=\"media\" data-alt=\"Graph to represent the growth of the population of fruit flies.\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135349865\">A scientist is comparing this population to another population, [latex]Q[\/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137482306\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137482308\">Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values.<\/p>\r\n<p id=\"fs-id1165137482312\">If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.<\/p>\r\n<p id=\"fs-id1165137705889\">The following shows where the new points for the new graph will be located.<\/p>\r\n\r\n<div id=\"fs-id1165133281393\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{cases}[\/latex]<span data-type=\"media\" data-alt=\"Graph of the population function doubled.\"><span data-type=\"media\" data-alt=\"Graph of the population function doubled.\">\r\n<\/span><\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200816\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"487\" height=\"367\" data-media-type=\"image\/jpg\" \/> <b>Figure 16<\/b>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165132939230\">Symbolically, the relationship is written as<\/p>\r\n\r\n<div id=\"fs-id1165135524747\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165135305830\">This means that for any input [latex]t[\/latex], the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P[\/latex]. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t[\/latex], stay the same while the output values are twice as large as before.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137647092\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 id=\"fs-id1165137442797\">How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137812146\" data-number-style=\"arabic\">\r\n \t<li>Determine the value of [latex]a[\/latex].<\/li>\r\n \t<li>Multiply all of the output values by [latex]a[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_14\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165135436655\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135436657\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 11: Finding a Vertical Compression of a Tabular Function<\/h3>\r\n<p id=\"fs-id1165134237296\">A function [latex]f[\/latex] is given in the table below. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<table id=\"Table_01_05_09\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup><col \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"left\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td data-align=\"left\">2<\/td>\r\n<td data-align=\"left\">4<\/td>\r\n<td data-align=\"left\">6<\/td>\r\n<td data-align=\"left\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\"><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td data-align=\"left\">1<\/td>\r\n<td data-align=\"left\">3<\/td>\r\n<td data-align=\"left\">7<\/td>\r\n<td data-align=\"left\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137780720\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137889748\">The formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3[\/latex]. Then<\/p>\r\n\r\n<div id=\"fs-id1165134350257\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]g\\left(4\\right)=\\frac{1}{2}f\\left(4\\right)=\\frac{1}{2}\\left(3\\right)=\\frac{3}{2}[\/latex]<\/div>\r\nWe do the same for the other values to produce this table.\r\n<table id=\"Table_01_05_10\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup><col \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td data-align=\"center\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\frac{3}{2}[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\frac{7}{2}[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\frac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165133316477\" class=\"commentary\" data-type=\"commentary\">\r\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165135419787\">The result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by [latex]\\frac{1}{2}[\/latex]. Each output value is divided in half, so the graph is half the original height.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 5<\/h3>\r\nA function [latex]f[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right)[\/latex].\r\n<div class=\"bcc-box bcc-success\">\r\n<table id=\"Table_01_05_011\" summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td data-align=\"left\">[latex]x[\/latex]<\/td>\r\n<td data-align=\"left\">2<\/td>\r\n<td data-align=\"left\">4<\/td>\r\n<td data-align=\"left\">6<\/td>\r\n<td data-align=\"left\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"left\">[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">12<\/td>\r\n<td data-align=\"left\">16<\/td>\r\n<td data-align=\"left\">20<\/td>\r\n<td data-align=\"left\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-5\/\" target=\"_blank\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_15\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165135530673\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165135530675\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 12: Recognizing a Vertical Stretch<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200818\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"487\" height=\"442\" data-media-type=\"image\/jpg\" \/> <b>Figure 17<\/b>[\/caption]\r\n<p id=\"fs-id1165135519281\">The graph\u00a0is a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].<span id=\"fs-id1165137431034\" data-type=\"media\" data-alt=\"Graph of a transformation of f(x)=x^3.\">\r\n<\/span><\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135173356\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137424163\">When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that [latex]g\\left(2\\right)=2[\/latex]. With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8[\/latex]. Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]g\\left(2\\right)=\\frac{1}{4}f\\left(2\\right)[\/latex]. From this we can fairly safely conclude that [latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135154389\">We can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137634248\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 6<\/h3>\r\n<p id=\"fs-id1165137643555\">Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.<\/p>\r\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-5\/\" target=\"_blank\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135344103\" data-depth=\"2\">\r\n<h2 data-type=\"title\"><\/h2>\r\n<h2 data-type=\"title\"><\/h2>\r\n<h2 style=\"text-align: center;\" data-type=\"title\"><span style=\"text-decoration: underline;\">Horizontal Stretches and Compressions<\/span><\/h2>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200819\/CNX_Precalc_Figure_01_05_028.jpg\" alt=\"Graph of the vertical stretch and compression of x^2.\" width=\"487\" height=\"514\" data-media-type=\"image\/jpg\" \/> <b>Figure 18<\/b>[\/caption]\r\n<p id=\"fs-id1165133167751\">Now we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.<span id=\"fs-id1165137659669\" data-type=\"media\" data-alt=\"Graph of the vertical stretch and compression of x^2.\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165133366207\">Given a function [latex]y=f\\left(x\\right)[\/latex], the form [latex]y=f\\left(bx\\right)[\/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[\/latex].\u00a0The graph of [latex]y={\\left(0.5x\\right)}^{2}[\/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2. The graph of [latex]y={\\left(2x\\right)}^{2}[\/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2.<\/p>\r\n\r\n<div id=\"fs-id1165137732896\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: Horizontal Stretches and Compressions<\/h3>\r\n<p id=\"fs-id1165134573810\">Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex], where [latex]b[\/latex] is a constant, is a <span data-type=\"term\">horizontal stretch<\/span> or <span data-type=\"term\">horizontal compression<\/span> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<ul id=\"eip-456\">\r\n \t<li>If [latex]b&gt;1[\/latex], then the graph will be compressed by [latex]\\frac{1}{b}[\/latex].<\/li>\r\n \t<li>If [latex]0&lt;b&lt;1[\/latex], then the graph will be stretched by [latex]\\frac{1}{b}[\/latex].<\/li>\r\n \t<li>If [latex]b&lt;0[\/latex], then there will be combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137832347\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 id=\"fs-id1165137784900\">How To: Given a description of a function, sketch a horizontal compression or stretch.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137784904\" data-number-style=\"arabic\">\r\n \t<li>Write a formula to represent the function.<\/li>\r\n \t<li>Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b&gt;1[\/latex] for a compression or [latex]0&lt;b&lt;1[\/latex]\r\nfor a stretch.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_16\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165137455653\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137470240\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 13: Graphing a Horizontal Compression<\/h3>\r\n<p id=\"fs-id1165137470245\">Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, [latex]R[\/latex], will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137837246\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137837248\">Symbolically, we could write<\/p>\r\n\r\n<div id=\"fs-id1165137767577\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}R\\left(1\\right)=P\\left(2\\right),\\hfill \\\\ R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,}\\hfill \\\\ R\\left(t\\right)=P\\left(2t\\right).\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165134380331\">See below\u00a0for a graphical comparison of the original population and the compressed population.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200820\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).\" width=\"976\" height=\"401\" data-media-type=\"image\/jpg\" \/> <b>Figure 19.<\/b> (a) Original population graph (b) Compressed population graph[\/caption]<\/div>\r\n<div id=\"fs-id1165134297638\" class=\"commentary\" data-type=\"commentary\">\r\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165134297643\">Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_17\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165134071619\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134071621\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 14: Finding a Horizontal Stretch for a Tabular Function<\/h3>\r\n<p id=\"fs-id1165134071627\">A function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex].<\/p>\r\n\r\n<table id=\"Table_01_05_12\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup><col \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">7<\/td>\r\n<td data-align=\"center\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137401572\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137401575\">The formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right)[\/latex], and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135531534\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/div>\r\n<p id=\"fs-id1165137827502\">We do the same for the other values to produce the table below.<\/p>\r\n\r\n<table id=\"Table_01_05_13\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup><col \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<td>12<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200822\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"975\" height=\"333\" data-media-type=\"image\/jpg\" \/> <b>Figure 20<\/b>[\/caption]\r\n<p id=\"fs-id1165135190052\">This figure shows the graphs of both of these sets of points.<span id=\"fs-id1165137899007\" data-type=\"media\" data-alt=\"Graph of the previous table.\">\r\n<\/span><\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135389013\" class=\"commentary\" data-type=\"commentary\">\r\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165133364838\">Because each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_18\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165137783980\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165137783983\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 15: Recognizing a Horizontal Compression on a Graph<\/h3>\r\nRelate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex] in Figure 21.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200823\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"487\" height=\"291\" data-media-type=\"image\/jpg\" \/> <b>Figure 21<\/b>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137737597\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137895242\">The graph of [latex]g\\left(x\\right)[\/latex] looks like the graph of [latex]f\\left(x\\right)[\/latex] horizontally compressed. Because [latex]f\\left(x\\right)[\/latex] ends at [latex]\\left(6,4\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] ends at [latex]\\left(2,4\\right)[\/latex], we can see that the [latex]x\\text{-}[\/latex] values have been compressed by [latex]\\frac{1}{3}[\/latex], because [latex]6\\left(\\frac{1}{3}\\right)=2[\/latex]. We might also notice that [latex]g\\left(2\\right)=f\\left(6\\right)[\/latex] and [latex]g\\left(1\\right)=f\\left(3\\right)[\/latex]. Either way, we can describe this relationship as [latex]g\\left(x\\right)=f\\left(3x\\right)[\/latex]. This is a horizontal compression by [latex]\\frac{1}{3}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137793556\" class=\"commentary\" data-type=\"commentary\">\r\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165137404580\">Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\\frac{1}{4}[\/latex] in our function: [latex]f\\left(\\frac{1}{4}x\\right)[\/latex]. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 7<\/h3>\r\n<p id=\"fs-id1165137451793\">Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<\/p>\r\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-5\/\" target=\"_blank\">Solution<\/a>\r\n\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id1165137676302\" data-depth=\"1\"><\/section>","rendered":"<section id=\"fs-id1165137654768\" data-depth=\"1\">\n<p id=\"fs-id1165137654773\">Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\n<p id=\"fs-id1165137675403\">We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\n<section id=\"fs-id1165137793506\" data-depth=\"2\">\n<h2 style=\"text-align: center;\" data-type=\"title\"><\/h2>\n<h2 style=\"text-align: center;\" data-type=\"title\"><\/h2>\n<h2 style=\"text-align: center;\" data-type=\"title\"><span style=\"text-decoration: underline;\">Vertical Stretches and Compressions<\/span><\/h2>\n<p>When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a<strong> vertical compression<\/strong>. The graph below\u00a0shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200813\/CNX_Precalc_Figure_01_05_0242.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"487\" height=\"326\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14.<\/b> Vertical stretch and compression<\/p>\n<\/div>\n<div id=\"fs-id1165137472530\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Vertical Stretches and Compressions<\/h3>\n<p id=\"fs-id1165137444261\">Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=af\\left(x\\right)[\/latex], where [latex]a[\/latex] is a constant, is a <span data-type=\"term\">vertical stretch<\/span> or <span data-type=\"term\">vertical compression<\/span> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\n<ul id=\"fs-id1165135553621\">\n<li>If [latex]a>1[\/latex], then the graph will be stretched.<\/li>\n<li>If 0 &lt; a &lt; 1, then the graph will be compressed.<\/li>\n<li>If [latex]a<0[\/latex], then there will be combination of a vertical stretch or compression with a vertical reflection.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135173107\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165132939216\">How To: Given a function, graph its vertical stretch.<\/h3>\n<ol id=\"fs-id1165134190735\" data-number-style=\"arabic\">\n<li>Identify the value of [latex]a[\/latex].<\/li>\n<li>Multiply all range values by [latex]a[\/latex].<\/li>\n<li>\n<p id=\"eip-id1165133107002\">If [latex]a>1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].<\/p>\n<p id=\"eip-id1165135191979\">If [latex]{ 0 }<{ a }<{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].<\/p>\n<p id=\"eip-id1165134061939\">If [latex]a<0[\/latex], the graph is either stretched or compressed and also reflected about the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_13\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137619296\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137619298\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 10: Graphing a Vertical Stretch<\/h3>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200814\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"487\" height=\"367\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137552971\">A function [latex]P\\left(t\\right)[\/latex] models the population of fruit flies.<span id=\"fs-id1165137501364\" data-type=\"media\" data-alt=\"Graph to represent the growth of the population of fruit flies.\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135349865\">A scientist is comparing this population to another population, [latex]Q[\/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.<\/p>\n<\/div>\n<div id=\"fs-id1165137482306\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137482308\">Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values.<\/p>\n<p id=\"fs-id1165137482312\">If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.<\/p>\n<p id=\"fs-id1165137705889\">The following shows where the new points for the new graph will be located.<\/p>\n<div id=\"fs-id1165133281393\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{cases}[\/latex]<span data-type=\"media\" data-alt=\"Graph of the population function doubled.\"><span data-type=\"media\" data-alt=\"Graph of the population function doubled.\"><br \/>\n<\/span><\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200816\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"487\" height=\"367\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132939230\">Symbolically, the relationship is written as<\/p>\n<div id=\"fs-id1165135524747\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135305830\">This means that for any input [latex]t[\/latex], the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P[\/latex]. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t[\/latex], stay the same while the output values are twice as large as before.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137647092\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137442797\">How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137812146\" data-number-style=\"arabic\">\n<li>Determine the value of [latex]a[\/latex].<\/li>\n<li>Multiply all of the output values by [latex]a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_14\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135436655\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135436657\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 11: Finding a Vertical Compression of a Tabular Function<\/h3>\n<p id=\"fs-id1165134237296\">A function [latex]f[\/latex] is given in the table below. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex].<\/p>\n<table id=\"Table_01_05_09\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"left\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td data-align=\"left\">2<\/td>\n<td data-align=\"left\">4<\/td>\n<td data-align=\"left\">6<\/td>\n<td data-align=\"left\">8<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td data-align=\"left\">1<\/td>\n<td data-align=\"left\">3<\/td>\n<td data-align=\"left\">7<\/td>\n<td data-align=\"left\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137780720\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137889748\">The formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3[\/latex]. Then<\/p>\n<div id=\"fs-id1165134350257\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]g\\left(4\\right)=\\frac{1}{2}f\\left(4\\right)=\\frac{1}{2}\\left(3\\right)=\\frac{3}{2}[\/latex]<\/div>\n<p>We do the same for the other values to produce this table.<\/p>\n<table id=\"Table_01_05_10\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-align=\"center\">[latex]6[\/latex]<\/td>\n<td data-align=\"center\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<td data-align=\"center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{3}{2}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{7}{2}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{11}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165133316477\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165135419787\">The result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by [latex]\\frac{1}{2}[\/latex]. Each output value is divided in half, so the graph is half the original height.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 5<\/h3>\n<p>A function [latex]f[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right)[\/latex].<\/p>\n<div class=\"bcc-box bcc-success\">\n<table id=\"Table_01_05_011\" summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td data-align=\"left\">[latex]x[\/latex]<\/td>\n<td data-align=\"left\">2<\/td>\n<td data-align=\"left\">4<\/td>\n<td data-align=\"left\">6<\/td>\n<td data-align=\"left\">8<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">[latex]f\\left(x\\right)[\/latex]<\/td>\n<td data-align=\"left\">12<\/td>\n<td data-align=\"left\">16<\/td>\n<td data-align=\"left\">20<\/td>\n<td data-align=\"left\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-5\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_15\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165135530673\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165135530675\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 12: Recognizing a Vertical Stretch<\/h3>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200818\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"487\" height=\"442\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 17<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135519281\">The graph\u00a0is a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].<span id=\"fs-id1165137431034\" data-type=\"media\" data-alt=\"Graph of a transformation of f(x)=x^3.\"><br \/>\n<\/span><\/p>\n<\/div>\n<div id=\"fs-id1165135173356\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137424163\">When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that [latex]g\\left(2\\right)=2[\/latex]. With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8[\/latex]. Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]g\\left(2\\right)=\\frac{1}{4}f\\left(2\\right)[\/latex]. From this we can fairly safely conclude that [latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135154389\">We can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f[\/latex].<\/p>\n<div id=\"fs-id1165137634248\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 6<\/h3>\n<p id=\"fs-id1165137643555\">Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-5\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135344103\" data-depth=\"2\">\n<h2 data-type=\"title\"><\/h2>\n<h2 data-type=\"title\"><\/h2>\n<h2 style=\"text-align: center;\" data-type=\"title\"><span style=\"text-decoration: underline;\">Horizontal Stretches and Compressions<\/span><\/h2>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200819\/CNX_Precalc_Figure_01_05_028.jpg\" alt=\"Graph of the vertical stretch and compression of x^2.\" width=\"487\" height=\"514\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 18<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165133167751\">Now we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.<span id=\"fs-id1165137659669\" data-type=\"media\" data-alt=\"Graph of the vertical stretch and compression of x^2.\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165133366207\">Given a function [latex]y=f\\left(x\\right)[\/latex], the form [latex]y=f\\left(bx\\right)[\/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[\/latex].\u00a0The graph of [latex]y={\\left(0.5x\\right)}^{2}[\/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2. The graph of [latex]y={\\left(2x\\right)}^{2}[\/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2.<\/p>\n<div id=\"fs-id1165137732896\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Horizontal Stretches and Compressions<\/h3>\n<p id=\"fs-id1165134573810\">Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex], where [latex]b[\/latex] is a constant, is a <span data-type=\"term\">horizontal stretch<\/span> or <span data-type=\"term\">horizontal compression<\/span> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\n<ul id=\"eip-456\">\n<li>If [latex]b>1[\/latex], then the graph will be compressed by [latex]\\frac{1}{b}[\/latex].<\/li>\n<li>If [latex]0<b<1[\/latex], then the graph will be stretched by [latex]\\frac{1}{b}[\/latex].<\/li>\n<li>If [latex]b<0[\/latex], then there will be combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137832347\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165137784900\">How To: Given a description of a function, sketch a horizontal compression or stretch.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137784904\" data-number-style=\"arabic\">\n<li>Write a formula to represent the function.<\/li>\n<li>Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b>1[\/latex] for a compression or [latex]0<b<1[\/latex]\nfor a stretch.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_16\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137455653\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137470240\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 13: Graphing a Horizontal Compression<\/h3>\n<p id=\"fs-id1165137470245\">Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, [latex]R[\/latex], will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.<\/p>\n<\/div>\n<div id=\"fs-id1165137837246\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137837248\">Symbolically, we could write<\/p>\n<div id=\"fs-id1165137767577\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}R\\left(1\\right)=P\\left(2\\right),\\hfill \\\\ R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,}\\hfill \\\\ R\\left(t\\right)=P\\left(2t\\right).\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165134380331\">See below\u00a0for a graphical comparison of the original population and the compressed population.<\/p>\n<div style=\"width: 986px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200820\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).\" width=\"976\" height=\"401\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 19.<\/b> (a) Original population graph (b) Compressed population graph<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134297638\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165134297643\">Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_17\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165134071619\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134071621\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 14: Finding a Horizontal Stretch for a Tabular Function<\/h3>\n<p id=\"fs-id1165134071627\">A function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex].<\/p>\n<table id=\"Table_01_05_12\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">7<\/td>\n<td data-align=\"center\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137401572\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137401575\">The formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right)[\/latex], and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]<\/p>\n<div id=\"fs-id1165135531534\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/div>\n<p id=\"fs-id1165137827502\">We do the same for the other values to produce the table below.<\/p>\n<table id=\"Table_01_05_13\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>4<\/td>\n<td>8<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200822\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"975\" height=\"333\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135190052\">This figure shows the graphs of both of these sets of points.<span id=\"fs-id1165137899007\" data-type=\"media\" data-alt=\"Graph of the previous table.\"><br \/>\n<\/span><\/p>\n<\/div>\n<div id=\"fs-id1165135389013\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165133364838\">Because each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_18\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165137783980\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165137783983\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 15: Recognizing a Horizontal Compression on a Graph<\/h3>\n<p>Relate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex] in Figure 21.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200823\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"487\" height=\"291\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 21<\/b><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737597\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137895242\">The graph of [latex]g\\left(x\\right)[\/latex] looks like the graph of [latex]f\\left(x\\right)[\/latex] horizontally compressed. Because [latex]f\\left(x\\right)[\/latex] ends at [latex]\\left(6,4\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] ends at [latex]\\left(2,4\\right)[\/latex], we can see that the [latex]x\\text{-}[\/latex] values have been compressed by [latex]\\frac{1}{3}[\/latex], because [latex]6\\left(\\frac{1}{3}\\right)=2[\/latex]. We might also notice that [latex]g\\left(2\\right)=f\\left(6\\right)[\/latex] and [latex]g\\left(1\\right)=f\\left(3\\right)[\/latex]. Either way, we can describe this relationship as [latex]g\\left(x\\right)=f\\left(3x\\right)[\/latex]. This is a horizontal compression by [latex]\\frac{1}{3}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137793556\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165137404580\">Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\\frac{1}{4}[\/latex] in our function: [latex]f\\left(\\frac{1}{4}x\\right)[\/latex]. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 7<\/h3>\n<p id=\"fs-id1165137451793\">Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-5\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1165137676302\" data-depth=\"1\"><\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-947\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-947","chapter","type-chapter","status-publish","hentry"],"part":921,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/947","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/947\/revisions"}],"predecessor-version":[{"id":2817,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/947\/revisions\/2817"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/921"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/947\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=947"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=947"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=947"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}