{"id":16023,"date":"2019-09-26T17:24:02","date_gmt":"2019-09-26T17:24:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-transform-linear-functions\/"},"modified":"2024-05-02T15:54:15","modified_gmt":"2024-05-02T15:54:15","slug":"read-transform-linear-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-transform-linear-functions\/","title":{"raw":"Transforming Linear Functions","rendered":"Transforming Linear Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use simple transformations to graph linear functions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing a Linear Function Using Transformations<\/h2>\r\nAnother option for graphing linear functions is to use <strong>transformations<\/strong> of the identity function [latex]f\\left(x\\right)=x[\/latex] . A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.\r\n<h3>Vertical Stretch or Compression<\/h3>\r\nIn the equation [latex]f\\left(x\\right)=mx[\/latex], the <em>m<\/em>\u00a0is acting as the <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the identity function. When <em>m<\/em>\u00a0is negative, there is also a vertical reflection of the graph. Notice in the figure below\u00a0that multiplying the equation of [latex]f\\left(x\\right)=x[\/latex] by <em>m<\/em>\u00a0vertically stretches the graph of <i>f<\/i>\u00a0by a factor of <em>m<\/em>\u00a0units if [latex]m&gt;1[\/latex] and vertically compresses the graph of <em>f<\/em>\u00a0by a factor of <em>m<\/em>\u00a0units if [latex]0&lt;m&lt;1[\/latex]. This means the larger the absolute value of <em>m<\/em>, the steeper the slope. Below you can see a variety of vertical stretches, compressions, and\/or reflections on the function [latex]f\\left(x\\right)=x[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201051\/CNX_Precalc_Figure_02_02_0052.jpg\" alt=\"Graph with several linear functions including y = 3x, y = 2x, y = x, y = (1\/2)x, y = (1\/3)x, y = (-1\/2)x, y = -x, and y = -2x\" width=\"900\" height=\"759\" \/>\r\n<h3>Vertical Shift<\/h3>\r\nIn [latex]f\\left(x\\right)=mx+b[\/latex], the <em>b<\/em>\u00a0acts as the <strong>vertical shift<\/strong>, moving the graph up and down without affecting the slope of the line. Notice in the figure below\u00a0that adding a value of <em>b<\/em>\u00a0to the equation of [latex]f\\left(x\\right)=x[\/latex] shifts the graph of\u00a0<em>f<\/em>\u00a0a total of <em>b<\/em>\u00a0units up if <em>b<\/em>\u00a0is positive and [latex]|b|[\/latex] units down if <em>b<\/em>\u00a0is negative.\u00a0The graph below illustrates vertical shifts of the function [latex]f\\left(x\\right)=x[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201052\/CNX_Precalc_Figure_02_02_0062.jpg\" alt=\"graph showing y = x , y = x+2, y = x+4, y = x-2, y = x-4\" width=\"900\" height=\"759\" \/>\r\n\r\nUsing vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.\r\n<div class=\"textbox\">\r\n<h3>How To: Given the equation of a linear function, use transformations to graph A linear function OF the form [latex]f\\left(x\\right)=mx+b[\/latex]<\/h3>\r\n<ol>\r\n \t<li>Graph [latex]f\\left(x\\right)=x[\/latex].<\/li>\r\n \t<li>Vertically stretch or compress the graph by a factor of |<em>m|<\/em>.<\/li>\r\n \t<li>Shift the graph up or down <em>b<\/em>\u00a0units.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the first example, we will see how a vertical compression changes the graph of the identity function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDescribe the\u00a0transformations to the function [latex]f(x)=\\dfrac{2}{3}x[\/latex] and\u00a0draw a graph.\r\n[reveal-answer q=\"955903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"955903\"]\r\n\r\nIn this case, [latex]m=\\dfrac{2}{3}[\/latex], so this is a vertical compression since [latex]0&lt;m&lt;1[\/latex].\r\n\r\nThe graph of\u00a0[latex]f(x)=\\dfrac{2}{3}x[\/latex]\u00a0is plotted below with the identity:\r\n\r\n<img class=\"size-medium wp-image-2245 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07171247\/Screen-Shot-2016-07-07-at-10.12.10-AM-300x289.png\" alt=\"Screen Shot 2016-07-07 at 10.12.10 AM\" width=\"300\" height=\"289\" \/>\r\n\r\nNote how the identity is compressed because the rate of change is \"slowed\" due to the vertical compression of [latex]\\dfrac{2}{3}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example we will vertically stretch the identity by a factor of\u00a0[latex]2[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDescribe the\u00a0transformations to the function [latex]f(x)=2x[\/latex] and\u00a0draw a graph.\r\n[reveal-answer q=\"50534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"50534\"]\r\n\r\nIn this case, [latex]m=2[\/latex], so this is a vertical stretch\u00a0since [latex]m&gt;1[\/latex].\r\n\r\nThe graph of\u00a0[latex]f(x)=2x[\/latex] is plotted below with the identity:\r\n\r\n[caption id=\"attachment_2246\" align=\"aligncenter\" width=\"368\"]<img class=\"wp-image-2246\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07172241\/Screen-Shot-2016-07-07-at-10.22.15-AM-300x292.png\" alt=\"Graph of the lines y equals 2 x and y equals x. The two lines intersect at the origin. y equals 2 x goes through the points (2, 4) and (-2, -4). y equals x goes through the points (2, 2) and (-2, -2).\" width=\"368\" height=\"358\" \/> y=2x and y=x[\/caption]\r\n\r\nNote how the identity is\u00a0more steep\u00a0because the rate of change is \"faster\" due to the vertical stretch\u00a0of [latex]2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nOur next example shows how making the slope negative reflects the identity across the <em>y<\/em>-axis.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDescribe the\u00a0transformations to the function [latex]f(x)=-2x[\/latex] and\u00a0draw a graph.\r\n[reveal-answer q=\"54449\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"54449\"]\r\n\r\nIn this case, [latex]m=-2[\/latex], so this is a vertical stretch since [latex]|m|&gt;1[\/latex]. The negative sign reflects the graph across the y-axis.\r\n\r\nThe graph of\u00a0[latex]f(x)=-2x[\/latex] is plotted below with the identity:\r\n\r\n[caption id=\"attachment_2248\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-2248\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07173037\/Screen-Shot-2016-07-07-at-10.30.09-AM-300x291.png\" alt=\"y = x and y = -2x\" width=\"300\" height=\"291\" \/> y = x and y = -2x[\/caption]\r\n\r\nNote how the steepness of the graph of \u00a0[latex]f(x)=-2x[\/latex] is similar to[latex]f(x)=2x[\/latex] but it points in the opposite direction because of the negative.[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last example, we will combine a vertical compression and a vertical shift to transform [latex]f(x)=x[\/latex] into\u00a0[latex]f\\left(x\\right)=\\dfrac{1}{2}x - 3[\/latex] and draw the graph.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph [latex]f\\left(x\\right)=\\dfrac{1}{2}x - 3[\/latex] using transformations.\r\n\r\n[reveal-answer q=\"208708\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"208708\"]\r\n\r\nThe equation for the function shows that [latex]m=\\dfrac{1}{2}[\/latex], so the identity function is vertically compressed by [latex]\\dfrac{1}{2}[\/latex]. The equation for the function also shows that [latex]b=\u20133[\/latex], so the identity function is vertically shifted down\u00a0[latex]3[\/latex] units. First, graph the identity function and show the vertical compression.\u00a0Below is the function [latex]y=x[\/latex], compressed by a factor of [latex]\\dfrac{1}{2}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201054\/CNX_Precalc_Figure_02_02_0072.jpg\" alt=\"graph showing the lines y = x and y = (1\/2)x\" width=\"487\" height=\"378\" \/>\r\n\r\nNow show the vertical shift.\u00a0The function [latex]y=\\dfrac{1}{2}x[\/latex] shifted down\u00a0[latex]3[\/latex] units is shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201055\/CNX_Precalc_Figure_02_02_0082.jpg\" alt=\"Graph showing the lines y = (1\/2)x, and y = (1\/2) + 3\" width=\"487\" height=\"377\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video example describes another linear transformation of the identity and its corresponding graph.\r\n\r\nhttps:\/\/youtu.be\/h9zn_ODlgbM\r\n<div class=\"textbox\">\r\n<h2>Q &amp; A<\/h2>\r\nIn the example above, could we have sketched the graph by reversing the order of the transformations?\u00a0No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first.\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>Vertical compressions of the identity happen when the slope is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex].<\/li>\r\n \t<li>Vertical stretches of the identity happen when the slope is greater than\u00a0[latex]1[\/latex].<\/li>\r\n \t<li>Reflections happen when the slope is negative.<\/li>\r\n \t<li>Vertical shifts happen when the intercept is not equal to\u00a0[latex]0[\/latex].<\/li>\r\n \t<li>Multiple transformations can be made to a function.<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use simple transformations to graph linear functions<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing a Linear Function Using Transformations<\/h2>\n<p>Another option for graphing linear functions is to use <strong>transformations<\/strong> of the identity function [latex]f\\left(x\\right)=x[\/latex] . A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.<\/p>\n<h3>Vertical Stretch or Compression<\/h3>\n<p>In the equation [latex]f\\left(x\\right)=mx[\/latex], the <em>m<\/em>\u00a0is acting as the <strong>vertical stretch<\/strong> or <strong>compression<\/strong> of the identity function. When <em>m<\/em>\u00a0is negative, there is also a vertical reflection of the graph. Notice in the figure below\u00a0that multiplying the equation of [latex]f\\left(x\\right)=x[\/latex] by <em>m<\/em>\u00a0vertically stretches the graph of <i>f<\/i>\u00a0by a factor of <em>m<\/em>\u00a0units if [latex]m>1[\/latex] and vertically compresses the graph of <em>f<\/em>\u00a0by a factor of <em>m<\/em>\u00a0units if [latex]0<m<1[\/latex]. This means the larger the absolute value of <em>m<\/em>, the steeper the slope. Below you can see a variety of vertical stretches, compressions, and\/or reflections on the function [latex]f\\left(x\\right)=x[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201051\/CNX_Precalc_Figure_02_02_0052.jpg\" alt=\"Graph with several linear functions including y = 3x, y = 2x, y = x, y = (1\/2)x, y = (1\/3)x, y = (-1\/2)x, y = -x, and y = -2x\" width=\"900\" height=\"759\" \/><\/p>\n<h3>Vertical Shift<\/h3>\n<p>In [latex]f\\left(x\\right)=mx+b[\/latex], the <em>b<\/em>\u00a0acts as the <strong>vertical shift<\/strong>, moving the graph up and down without affecting the slope of the line. Notice in the figure below\u00a0that adding a value of <em>b<\/em>\u00a0to the equation of [latex]f\\left(x\\right)=x[\/latex] shifts the graph of\u00a0<em>f<\/em>\u00a0a total of <em>b<\/em>\u00a0units up if <em>b<\/em>\u00a0is positive and [latex]|b|[\/latex] units down if <em>b<\/em>\u00a0is negative.\u00a0The graph below illustrates vertical shifts of the function [latex]f\\left(x\\right)=x[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201052\/CNX_Precalc_Figure_02_02_0062.jpg\" alt=\"graph showing y = x , y = x+2, y = x+4, y = x-2, y = x-4\" width=\"900\" height=\"759\" \/><\/p>\n<p>Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the equation of a linear function, use transformations to graph A linear function OF the form [latex]f\\left(x\\right)=mx+b[\/latex]<\/h3>\n<ol>\n<li>Graph [latex]f\\left(x\\right)=x[\/latex].<\/li>\n<li>Vertically stretch or compress the graph by a factor of |<em>m|<\/em>.<\/li>\n<li>Shift the graph up or down <em>b<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/div>\n<p>In the first example, we will see how a vertical compression changes the graph of the identity function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Describe the\u00a0transformations to the function [latex]f(x)=\\dfrac{2}{3}x[\/latex] and\u00a0draw a graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q955903\">Show Solution<\/span><\/p>\n<div id=\"q955903\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this case, [latex]m=\\dfrac{2}{3}[\/latex], so this is a vertical compression since [latex]0<m<1[\/latex].\n\nThe graph of\u00a0[latex]f(x)=\\dfrac{2}{3}x[\/latex]\u00a0is plotted below with the identity:\n\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2245 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07171247\/Screen-Shot-2016-07-07-at-10.12.10-AM-300x289.png\" alt=\"Screen Shot 2016-07-07 at 10.12.10 AM\" width=\"300\" height=\"289\" \/><\/p>\n<p>Note how the identity is compressed because the rate of change is &#8220;slowed&#8221; due to the vertical compression of [latex]\\dfrac{2}{3}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the next example we will vertically stretch the identity by a factor of\u00a0[latex]2[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Describe the\u00a0transformations to the function [latex]f(x)=2x[\/latex] and\u00a0draw a graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q50534\">Show Solution<\/span><\/p>\n<div id=\"q50534\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this case, [latex]m=2[\/latex], so this is a vertical stretch\u00a0since [latex]m>1[\/latex].<\/p>\n<p>The graph of\u00a0[latex]f(x)=2x[\/latex] is plotted below with the identity:<\/p>\n<div id=\"attachment_2246\" style=\"width: 378px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2246\" class=\"wp-image-2246\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07172241\/Screen-Shot-2016-07-07-at-10.22.15-AM-300x292.png\" alt=\"Graph of the lines y equals 2 x and y equals x. The two lines intersect at the origin. y equals 2 x goes through the points (2, 4) and (-2, -4). y equals x goes through the points (2, 2) and (-2, -2).\" width=\"368\" height=\"358\" \/><\/p>\n<p id=\"caption-attachment-2246\" class=\"wp-caption-text\">y=2x and y=x<\/p>\n<\/div>\n<p>Note how the identity is\u00a0more steep\u00a0because the rate of change is &#8220;faster&#8221; due to the vertical stretch\u00a0of [latex]2[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>Our next example shows how making the slope negative reflects the identity across the <em>y<\/em>-axis.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Describe the\u00a0transformations to the function [latex]f(x)=-2x[\/latex] and\u00a0draw a graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q54449\">Show Solution<\/span><\/p>\n<div id=\"q54449\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this case, [latex]m=-2[\/latex], so this is a vertical stretch since [latex]|m|>1[\/latex]. The negative sign reflects the graph across the y-axis.<\/p>\n<p>The graph of\u00a0[latex]f(x)=-2x[\/latex] is plotted below with the identity:<\/p>\n<div id=\"attachment_2248\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2248\" class=\"size-medium wp-image-2248\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07173037\/Screen-Shot-2016-07-07-at-10.30.09-AM-300x291.png\" alt=\"y = x and y = -2x\" width=\"300\" height=\"291\" \/><\/p>\n<p id=\"caption-attachment-2248\" class=\"wp-caption-text\">y = x and y = -2x<\/p>\n<\/div>\n<p>Note how the steepness of the graph of \u00a0[latex]f(x)=-2x[\/latex] is similar to[latex]f(x)=2x[\/latex] but it points in the opposite direction because of the negative.<\/p><\/div>\n<\/div>\n<\/div>\n<p>In our last example, we will combine a vertical compression and a vertical shift to transform [latex]f(x)=x[\/latex] into\u00a0[latex]f\\left(x\\right)=\\dfrac{1}{2}x - 3[\/latex] and draw the graph.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph [latex]f\\left(x\\right)=\\dfrac{1}{2}x - 3[\/latex] using transformations.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q208708\">Show Solution<\/span><\/p>\n<div id=\"q208708\" class=\"hidden-answer\" style=\"display: none\">\n<p>The equation for the function shows that [latex]m=\\dfrac{1}{2}[\/latex], so the identity function is vertically compressed by [latex]\\dfrac{1}{2}[\/latex]. The equation for the function also shows that [latex]b=\u20133[\/latex], so the identity function is vertically shifted down\u00a0[latex]3[\/latex] units. First, graph the identity function and show the vertical compression.\u00a0Below is the function [latex]y=x[\/latex], compressed by a factor of [latex]\\dfrac{1}{2}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201054\/CNX_Precalc_Figure_02_02_0072.jpg\" alt=\"graph showing the lines y = x and y = (1\/2)x\" width=\"487\" height=\"378\" \/><\/p>\n<p>Now show the vertical shift.\u00a0The function [latex]y=\\dfrac{1}{2}x[\/latex] shifted down\u00a0[latex]3[\/latex] units is shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201055\/CNX_Precalc_Figure_02_02_0082.jpg\" alt=\"Graph showing the lines y = (1\/2)x, and y = (1\/2) + 3\" width=\"487\" height=\"377\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video example describes another linear transformation of the identity and its corresponding graph.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Graph a Linear Function as a Transformation of f(x)=x\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox\">\n<h2>Q &amp; A<\/h2>\n<p>In the example above, could we have sketched the graph by reversing the order of the transformations?\u00a0No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first.<\/p>\n<\/div>\n<h2>Summary<\/h2>\n<ul>\n<li>Vertical compressions of the identity happen when the slope is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex].<\/li>\n<li>Vertical stretches of the identity happen when the slope is greater than\u00a0[latex]1[\/latex].<\/li>\n<li>Reflections happen when the slope is negative.<\/li>\n<li>Vertical shifts happen when the intercept is not equal to\u00a0[latex]0[\/latex].<\/li>\n<li>Multiple transformations can be made to a function.<\/li>\n<\/ul>\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-16023\">\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>Graph a Linear Function as a Transformation of f(x)=x. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/h9zn_ODlgbM\">https:\/\/youtu.be\/h9zn_ODlgbM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/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":169554,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"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.\"},{\"type\":\"original\",\"description\":\"Graph a Linear Function as a Transformation of f(x)=x\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/h9zn_ODlgbM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"59bc4e998d814b60802c43c965682ffc, 4d48a7a26d5f415ba56358b55d0042d9 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