{"id":152,"date":"2023-06-21T13:22:38","date_gmt":"2023-06-21T13:22:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/compressions-and-stretches\/"},"modified":"2023-08-24T06:01:30","modified_gmt":"2023-08-24T06:01:30","slug":"compressions-and-stretches","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/compressions-and-stretches\/","title":{"raw":"\u25aa   Compressions and Stretches","rendered":"\u25aa   Compressions and Stretches"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph Functions Using Compressions and Stretches.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAdding 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.\r\n\r\nWe 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.\r\n<h2>Vertical Stretches and Compressions<\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/18203611\/CNX_Precalc_Figure_01_05_0242.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"487\" height=\"326\" \/> Vertical stretch and compression[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Vertical Stretches and Compressions<\/h3>\r\nGiven 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 <strong>vertical stretch<\/strong> or <strong>vertical compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].\r\n<ul>\r\n \t<li>If [latex]a&gt;1[\/latex], then the graph will be stretched.<\/li>\r\n \t<li>If [latex]0 &lt; a &lt; 1[\/latex], 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 class=\"textbox\">\r\n<h3>How To: Given a function, graph its vertical stretch.<\/h3>\r\n<ol>\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>If [latex]a&gt;1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].\r\nIf [latex]{ 0 }&lt;{ a }&lt;{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].\r\nIf [latex]a&lt;0[\/latex], the graph is either stretched or compressed and also reflected about the [latex]x[\/latex]-axis.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Vertical Stretch<\/h3>\r\nA function [latex]P\\left(t\\right)[\/latex] models the number\u00a0of fruit flies in a population over time, and is graphed below.\r\n\r\nA 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.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203613\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"487\" height=\"367\" \/>\r\n[reveal-answer q=\"951851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951851\"]\r\n\r\nBecause the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values.\r\n\r\nIf we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.\r\n\r\nThe following shows where the new points for the new graph will be located.\r\n\r\n[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]\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\/10\/18203615\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"487\" height=\"367\" \/> <b>Figure 16<\/b>[\/caption]\r\n\r\nSymbolically, the relationship is written as\r\n\r\n[latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]\r\n\r\nThis 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.\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\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74700&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><span style=\"width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Finding a Vertical Compression of a Tabular Function<\/h3>\r\nA 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].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\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[reveal-answer q=\"798923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"798923\"]\r\n\r\nThe 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\r\n\r\n[latex]g\\left(4\\right)=\\frac{1}{2}\\cdot{f}(4) =\\frac{1}{2}\\cdot\\left(3\\right)=\\frac{3}{2}[\/latex]\r\n\r\nWe do the same for the other values to produce this table.\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{7}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Analysis of the Solution<\/h4>\r\nThe 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.\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\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<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<td>12<\/td>\r\n<td>16<\/td>\r\n<td>20<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"805921\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"805921\"]\r\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled, \"><colgroup> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\r\n<td>9<\/td>\r\n<td>12<\/td>\r\n<td>15<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=112707&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Recognizing a Vertical Stretch<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203618\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"487\" height=\"442\" \/>\r\n\r\nThe 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].\r\n\r\n[reveal-answer q=\"289067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"289067\"]\r\n\r\nWhen 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].\r\n\r\nWe can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f[\/latex].\r\n<p style=\"text-align: center;\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/p>\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 formula for the function that we get when we vertically stretch (or scale) the identity toolkit function by a factor of 3, and then shift it down by 2 units.\r\nCheck your work with an online graphing calculator.\r\n\r\n[reveal-answer q=\"473017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473017\"]\r\n\r\n[latex]g(x)=3x-2[\/latex]\r\n\r\n<img class=\"wp-image-6746 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08201018\/Screen-Shot-2019-07-08-at-1.09.59-PM.png\" alt=\"Graph of f(x)=x and f(x)=3x-2\" width=\"452\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=112726&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Horizontal Stretches and Compressions<\/h2>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203621\/CNX_Precalc_Figure_01_05_028.jpg\" alt=\"Graph of the vertical stretch and compression of x^2.\" width=\"487\" height=\"514\" \/>\r\n\r\nNow 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.\r\n\r\nGiven 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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal Stretches and Compressions<\/h3>\r\nGiven 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 <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].\r\n<ul>\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 class=\"textbox\">\r\n<h3>How To: Given a description of a function, sketch a horizontal compression or stretch.<strong>\r\n<\/strong><\/h3>\r\n<ol>\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 class=\"textbox exercises\">\r\n<h3>Example: Graphing a Horizontal Compression<\/h3>\r\nSuppose 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.\r\n\r\n[reveal-answer q=\"855794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"855794\"]\r\n\r\nSymbolically, we could write\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;R\\left(1\\right)=P\\left(2\\right), \\\\ &amp;R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,} \\\\ &amp;R\\left(t\\right)=P\\left(2t\\right). \\end{align}[\/latex]<\/p>\r\nSee below\u00a0for a graphical comparison of the original population and the compressed population.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203623\/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\" \/> (a) Original population graph (b) Compressed population graph[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Horizontal Stretch for a Tabular Function<\/h3>\r\nA 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].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\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[reveal-answer q=\"261935\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261935\"]\r\n\r\nThe 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]\r\n<p style=\"text-align: center;\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/p>\r\nWe do the same for the other values to produce the table below.\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203626\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"975\" height=\"333\" \/>\r\n\r\nThis figure shows the graphs of both of these sets of points.\r\n<h4>Analysis of the Solution<\/h4>\r\nBecause 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: 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].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203628\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"487\" height=\"291\" \/>\r\n[reveal-answer q=\"396995\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"396995\"]\r\n\r\nThe 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].\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice 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.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite a formula for the toolkit square root function horizontally stretched by a factor of 3.\r\nUse an online graphing calculator to check your work.\r\n\r\n[reveal-answer q=\"35233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35233\"]\r\n\r\n[latex]g\\left(x\\right)=\\sqrt{\\frac{1}{3}x}[\/latex]\r\n\r\n<img class=\"wp-image-6743 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08200821\/Screen-Shot-2019-07-08-at-1.08.04-PM.png\" alt=\"Graph of f(x)=sqrt(x) and f(x)=sqrt(1\/3 x)\" width=\"419\" height=\"274\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=60791&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom11\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=60790&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph Functions Using Compressions and Stretches.<\/li>\n<\/ul>\n<\/div>\n<p>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>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<h2>Vertical Stretches and Compressions<\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/18203611\/CNX_Precalc_Figure_01_05_0242.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"487\" height=\"326\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical stretch and compression<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Stretches and Compressions<\/h3>\n<p>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 <strong>vertical stretch<\/strong> or <strong>vertical compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\n<ul>\n<li>If [latex]a>1[\/latex], then the graph will be stretched.<\/li>\n<li>If [latex]0 < a < 1[\/latex], 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 class=\"textbox\">\n<h3>How To: Given a function, graph its vertical stretch.<\/h3>\n<ol>\n<li>Identify the value of [latex]a[\/latex].<\/li>\n<li>Multiply all range values by [latex]a[\/latex].<\/li>\n<li>If [latex]a>1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].<br \/>\nIf [latex]{ 0 }<{ a }<{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].\nIf [latex]a<0[\/latex], the graph is either stretched or compressed and also reflected about the [latex]x[\/latex]-axis.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Vertical Stretch<\/h3>\n<p>A function [latex]P\\left(t\\right)[\/latex] models the number\u00a0of fruit flies in a population over time, and is graphed below.<\/p>\n<p>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<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203613\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"487\" height=\"367\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951851\">Show Solution<\/span><\/p>\n<div id=\"q951851\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>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>The following shows where the new points for the new graph will be located.<\/p>\n<p>[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]<\/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\/10\/18203615\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<p>Symbolically, the relationship is written as<\/p>\n<p>[latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]<\/p>\n<p>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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74700&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><span style=\"width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Finding a Vertical Compression of a Tabular Function<\/h3>\n<p>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 summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\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>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798923\">Show Solution<\/span><\/p>\n<div id=\"q798923\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p>[latex]g\\left(4\\right)=\\frac{1}{2}\\cdot{f}(4) =\\frac{1}{2}\\cdot\\left(3\\right)=\\frac{3}{2}[\/latex]<\/p>\n<p>We do the same for the other values to produce this table.<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\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>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\frac{7}{2}[\/latex]<\/td>\n<td>[latex]\\frac{11}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Analysis of the Solution<\/h4>\n<p>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=\"textbox key-takeaways\">\n<h3>Try It<\/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<table summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<td>20<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q805921\">Show Solution<\/span><\/p>\n<div id=\"q805921\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup> <\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>9<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=112707&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Recognizing a Vertical Stretch<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203618\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"487\" height=\"442\" \/><\/p>\n<p>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].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q289067\">Show Solution<\/span><\/p>\n<div id=\"q289067\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>We can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the formula for the function that we get when we vertically stretch (or scale) the identity toolkit function by a factor of 3, and then shift it down by 2 units.<br \/>\nCheck your work with an online graphing calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473017\">Show Solution<\/span><\/p>\n<div id=\"q473017\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)=3x-2[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6746 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08201018\/Screen-Shot-2019-07-08-at-1.09.59-PM.png\" alt=\"Graph of f(x)=x and f(x)=3x-2\" width=\"452\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=112726&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Horizontal Stretches and Compressions<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203621\/CNX_Precalc_Figure_01_05_028.jpg\" alt=\"Graph of the vertical stretch and compression of x^2.\" width=\"487\" height=\"514\" \/><\/p>\n<p>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.<\/p>\n<p>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 class=\"textbox\">\n<h3>A General Note: Horizontal Stretches and Compressions<\/h3>\n<p>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 <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\n<ul>\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 class=\"textbox\">\n<h3>How To: Given a description of a function, sketch a horizontal compression or stretch.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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 class=\"textbox exercises\">\n<h3>Example: Graphing a Horizontal Compression<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q855794\">Show Solution<\/span><\/p>\n<div id=\"q855794\" class=\"hidden-answer\" style=\"display: none\">\n<p>Symbolically, we could write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&R\\left(1\\right)=P\\left(2\\right), \\\\ &R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,} \\\\ &R\\left(t\\right)=P\\left(2t\\right). \\end{align}[\/latex]<\/p>\n<p>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\/wp-content\/uploads\/sites\/896\/2016\/10\/18203623\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is &#091;0,7&#093; and range is &#091;0,3&#093;. 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 &#091;0, 3.5&#093; and a range of &#091;0,3&#093;. The maximum occurs at (1.5, 3).\" width=\"976\" height=\"401\" \/><\/p>\n<p class=\"wp-caption-text\">(a) Original population graph (b) Compressed population graph<\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>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 class=\"textbox exercises\">\n<h3>Example: Finding a Horizontal Stretch for a Tabular Function<\/h3>\n<p>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 summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261935\">Show Solution<\/span><\/p>\n<div id=\"q261935\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p style=\"text-align: center;\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/p>\n<p>We do the same for the other values to produce the table below.<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203626\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"975\" height=\"333\" \/><\/p>\n<p>This figure shows the graphs of both of these sets of points.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>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 class=\"textbox exercises\">\n<h3>Example: 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].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203628\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"487\" height=\"291\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396995\">Show Solution<\/span><\/p>\n<div id=\"q396995\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<h4>Analysis of the Solution<\/h4>\n<p>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><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<br \/>\nUse an online graphing calculator to check your work.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q35233\">Show Solution<\/span><\/p>\n<div id=\"q35233\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g\\left(x\\right)=\\sqrt{\\frac{1}{3}x}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6743 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08200821\/Screen-Shot-2019-07-08-at-1.08.04-PM.png\" alt=\"Graph of f(x)=sqrt(x) and f(x)=sqrt(1\/3 x)\" width=\"419\" height=\"274\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=60791&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom11\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=60790&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"250\"><\/iframe><\/p>\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-152\">\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><li>Question ID 112707, 112726. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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><li>Question ID 74696. <strong>Authored by<\/strong>: Meacham,William. <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 License CC-BY + GPL<\/li><li>Question ID 60791, 60790. <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 License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and 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