{"id":124,"date":"2021-02-03T21:32:20","date_gmt":"2021-02-03T21:32:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=124"},"modified":"2022-03-11T21:37:50","modified_gmt":"2022-03-11T21:37:50","slug":"transformations-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/transformations-of-functions\/","title":{"raw":"Transformations of Functions","rendered":"Transformations of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170573361979\">We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function [latex]y=x^2[\/latex] to get the function [latex]f(x)=(x-2)^2[\/latex]. This subtraction represents a shift of the function [latex]y=x^2[\/latex] two units to the right. A shift, horizontally or vertically, is a type of <strong>transformation of a function<\/strong>. Other transformations include horizontal and vertical scalings, and reflections about the axes.<\/p>\r\n\r\n<h3>Vertical Shift<\/h3>\r\nA vertical shift of a function occurs if we add or subtract the same constant to each output [latex]y[\/latex]. For [latex]c&gt;0[\/latex], the graph of [latex]f(x)+c[\/latex] is a shift of the graph of [latex]f(x)[\/latex] up [latex]c[\/latex] units, whereas the graph of [latex]f(x)-c[\/latex] is a shift of the graph of [latex]f(x)[\/latex] down [latex]c[\/latex] units. For example, the graph of the function [latex]f(x)=x^3+4[\/latex] is the graph of [latex]y=x^3[\/latex] shifted up 4 units; the graph of the function [latex]f(x)=x^3-4[\/latex] is the graph of [latex]y=x^3[\/latex] shifted down 4 units (Figure 15).\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"708\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202303\/CNX_Calc_Figure_01_02_023.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201ca\u201d and has an x axis that runs from -4 to 4 and a y axis that runs from -1 to 10. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = (x squared) + 4\u201d, which is a parabola that decreases until the point (0, 4) and then increases again after the origin. The two functions are the same in shape, but the second function is shifted up 4 units. The second graph is labeled \u201cb\u201d and has an x axis that runs from -4 to 4 and a y axis that runs from -5 to 6. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = (x squared) - 4\u201d, which is a parabola that decreases until the point (0, -4) and then increases again after the origin. The two functions are the same in shape, but the second function is shifted down 4 units.\" width=\"708\" height=\"505\" \/> Figure 15. (a) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x)+c[\/latex] is a vertical shift up [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex]. (b) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x)-c[\/latex] is a vertical shift down [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<h3>Horizontal Shift<\/h3>\r\n<p id=\"fs-id1170573582792\">A horizontal shift of a function occurs if we add or subtract the same constant to each input [latex]x[\/latex]. For [latex]c&gt;0[\/latex], the graph of [latex]f(x+c)[\/latex] is a shift of the graph of [latex]f(x)[\/latex] to the left [latex]c[\/latex] units; the graph of [latex]f(x-c)[\/latex] is a shift of the graph of [latex]f(x)[\/latex] to the right [latex]c[\/latex] units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let\u2019s look at an example.<\/p>\r\nConsider the function [latex]f(x)=|x+3|[\/latex] and evaluate this function at [latex]x-3.[\/latex] Since [latex]f(x-3)=|x|[\/latex] and [latex]x-3&lt;x[\/latex], the graph of [latex]f(x)=|x+3|[\/latex] is the graph of [latex]y=|x|[\/latex] shifted left 3 units. Similarly, the graph of [latex]f(x)=|x-3|[\/latex] is the graph of [latex]y=|x|[\/latex] shifted right 3 units (Figure 16).\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202307\/CNX_Calc_Figure_01_02_015.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201ca\u201d and has an x axis that runs from -8 to 5 and a y axis that runs from -3 to 5. The graph is of two functions. The first function is \u201cf(x) = absolute value of x\u201d, which decreases in a straight line until the origin and then increases in a straight line again after the origin. The second function is \u201cf(x) = absolute value of (x + 3)\u201d, which decreases in a straight line until the point (-3, 0) and then increases in a straight line again after the point (-3, 0). The two functions are the same in shape, but the second function is shifted left 3 units. The second graph is labeled \u201cb\u201d and has an x axis that runs from -5 to 8 and a y axis that runs from -3 to 5. The graph is of two functions. The first function is \u201cf(x) = absolute value of x\u201d, which decreases in a straight line until the origin and then increases in a straight line again after the origin. The second function is \u201cf(x) = absolute value of (x - 3)\u201d, which decreases in a straight line until the point (3, 0) and then increases in a straight line again after the point (3, 0). The two functions are the same in shape, but the second function is shifted right 3 units.\" width=\"975\" height=\"337\" \/> Figure 16. (a) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x+c)[\/latex] is a horizontal shift left [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex]. (b) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x-c)[\/latex] is a horizontal shift right [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<h3>Vertical Scaling (Stretched\/Compressed)<\/h3>\r\n<p id=\"fs-id1170573582097\">A vertical scaling of a graph occurs if we multiply all outputs [latex]y[\/latex] of a function by the same positive constant. For [latex]c&gt;0[\/latex], the graph of the function [latex]cf(x)[\/latex] is the graph of [latex]f(x)[\/latex] scaled vertically by a factor of [latex]c[\/latex]. If [latex]c&gt;1[\/latex], the values of the outputs for the function [latex]cf(x)[\/latex] are larger than the values of the outputs for the function [latex]f(x)[\/latex]; therefore, the graph has been stretched vertically. If [latex]0&lt;c&lt;1[\/latex], then the outputs of the function [latex]cf(x)[\/latex] are smaller, so the graph has been compressed. For example, the graph of the function [latex]f(x)=3x^2[\/latex] is the graph of [latex]y=x^2[\/latex] stretched vertically by a factor of 3, whereas the graph of [latex]f(x)=\\frac{x^2}{3}[\/latex] is the graph of [latex]y=x^2[\/latex] compressed vertically by a factor of 3 (Figure 17).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"662\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202311\/CNX_Calc_Figure_01_02_024.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201ca\u201d and has an x axis that runs from -3 to 3 and a y axis that runs from -2 to 9. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = 3(x squared)\u201d, which is a parabola that decreases until the origin and then increases again after the origin, but is vertically stretched and thus increases at a quicker rate than the first function. The second graph is labeled \u201cb\u201d and has an x axis that runs from -4 to 4 and a y axis that runs from -2 to 9. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = (1\/3)(x squared)\u201d, which is a parabola that decreases until the origin and then increases again after the origin, but is vertically compressed and thus increases at a slower rate than the first function.\" width=\"662\" height=\"507\" \/> Figure 17. (a) If [latex]c&gt;1[\/latex], the graph of [latex]y=cf(x)[\/latex] is a vertical stretch of the graph of [latex]y=f(x)[\/latex]. (b) If [latex]0&lt;c&lt;1[\/latex], the graph of [latex]y=cf(x)[\/latex] is a vertical compression of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<div class=\"wp-caption-text\"><\/div>\r\n<h3>Horizontal Scaling (Stretched\/Compressed)<\/h3>\r\n<p id=\"fs-id1170573569978\">The horizontal scaling of a function occurs if we multiply the inputs [latex]x[\/latex] by the same positive constant. For [latex]c&gt;0[\/latex], the graph of the function [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] scaled horizontally by a factor of [latex]c[\/latex]. If [latex]c&gt;1[\/latex], the graph of [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] compressed horizontally. If [latex]0&lt;c&lt;1[\/latex], the graph of [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] stretched horizontally. For example, consider the function [latex]f(x)=\\sqrt{2x}[\/latex] and evaluate [latex]f[\/latex] at [latex]\\dfrac{x}{2}.[\/latex] Since [latex]f(\\frac{x}{2})=\\sqrt{x}[\/latex], the graph of [latex]f(x)=\\sqrt{2x}[\/latex] is the graph of [latex]y=\\sqrt{x}[\/latex] compressed horizontally. The graph of [latex]y=\\sqrt{\\frac{x}{2}}[\/latex] is a horizontal stretch of the graph of [latex]y=\\sqrt{x}[\/latex] (Figure 18).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202314\/CNX_Calc_Figure_01_02_017.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -2 to 4 and a y axis that runs from -2 to 5. The first graph is labeled \u201ca\u201d and is of two functions. The first graph is of two functions. The first function is \u201cf(x) = square root of x\u201d, which is a curved function that begins at the origin and increases. The second function is \u201cf(x) = square root of 2x\u201d, which is a curved function that begins at the origin and increases, but increases at a faster rate than the first function. The second graph is labeled \u201cb\u201d and is of two functions. The first function is \u201cf(x) = square root of x\u201d, which is a curved function that begins at the origin and increases. The second function is \u201cf(x) = square root of (x\/2)\u201d, which is a curved function that begins at the origin and increases, but increases at a slower rate than the first function.\" width=\"731\" height=\"351\" \/> Figure 18. (a) If [latex]c&gt;1[\/latex], the graph of [latex]y=f(cx)[\/latex] is a horizontal compression of the graph of [latex]y=f(x)[\/latex]. (b) If [latex]0&lt;c&lt;1[\/latex], the graph of [latex]y=f(cx)[\/latex] is a horizontal stretch of the graph of [latex]y=f(x)[\/latex].[\/caption]\r\n<h3>Reflection<\/h3>\r\n<p id=\"fs-id1170573582412\">We have explored what happens to the graph of a function [latex]f[\/latex] when we multiply [latex]f[\/latex] by a constant [latex]c&gt;0[\/latex] to get a new function [latex]cf(x)[\/latex]. We have also discussed what happens to the graph of a function [latex]f[\/latex] when we multiply the independent variable [latex]x[\/latex] by [latex]c&gt;0[\/latex] to get a new function [latex]f(cx)[\/latex]. However, we have not addressed what happens to the graph of the function if the constant [latex]c[\/latex] is negative. If we have a constant [latex]c&lt;0[\/latex], we can write [latex]c[\/latex] as a positive number multiplied by [latex]-1[\/latex]; but, what kind of transformation do we get when we multiply the function or its argument by [latex]-1[\/latex]? When we multiply all the outputs by [latex]-1[\/latex], we get a reflection about the [latex]x[\/latex]-axis. When we multiply all inputs by [latex]-1[\/latex], we get a reflection about the [latex]y[\/latex]-axis. For example, the graph of [latex]f(x)=\u2212(x^3+1)[\/latex] is the graph of [latex]y=(x^3+1)[\/latex] reflected about the [latex]x[\/latex]-axis. The graph of [latex]f(x)=(\u2212x)^3+1[\/latex] is the graph of [latex]y=x^3+1[\/latex] reflected about the [latex]y[\/latex]-axis (Figure 19).<\/p>\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\/2332\/2018\/01\/11202317\/CNX_Calc_Figure_01_02_018.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -5 to 6. The first graph is labeled \u201ca\u201d and is of two functions. The first graph is of two functions. The first function is \u201cf(x) = x cubed + 1\u201d, which is a curved increasing function that has an x intercept at (-1, 0) and a y intercept at (0, 1). The second function is \u201cf(x) = -(x cubed + 1)\u201d, which is a curved decreasing function that has an x intercept at (-1, 0) and a y intercept at (0, -1). The second graph is labeled \u201cb\u201d and is of two functions. The first function is \u201cf(x) = x cubed + 1\u201d, which is a curved increasing function that has an x intercept at (-1, 0) and a y intercept at (0, 1). The second function is \u201cf(x) = (-x) cubed + 1\u201d, which is a curved decreasing function that has an x intercept at (1, 0) and a y intercept at (0, 1). The first function increases at the same rate the second function decreases for the same values of x.\" width=\"487\" height=\"434\" \/> Figure 19. (a) The graph of [latex]y=\u2212f(x)[\/latex] is the graph of [latex]y=f(x)[\/latex] reflected about the [latex]x[\/latex]-axis. (b) The graph of [latex]y=f(\u2212x)[\/latex] is the graph of [latex]y=f(x)[\/latex] reflected about the [latex]y[\/latex]-axis.[\/caption]\r\n<h3>Multiple Transformations<\/h3>\r\n<p id=\"fs-id1170573580155\">If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function [latex]f(x)[\/latex], the graph of the related function [latex]y=cf(a(x+b))+d[\/latex] can be obtained from the graph of [latex]y=f(x)[\/latex] by performing the transformations in the following order.<\/p>\r\n\r\n<ol id=\"fs-id1170573580235\">\r\n \t<li>Horizontal shift of the graph of [latex]y=f(x)[\/latex]. If [latex]b&gt;0[\/latex], shift left. If [latex]b&lt;0[\/latex], shift right.<\/li>\r\n \t<li>Horizontal scaling of the graph of [latex]y=f(x+b)[\/latex] by a factor of [latex]|a|[\/latex]. If [latex]a&lt;0[\/latex], reflect the graph about the [latex]y[\/latex]-axis.<\/li>\r\n \t<li>Vertical scaling of the graph of [latex]y=f(a(x+b))[\/latex] by a factor of [latex]|c|[\/latex]. If [latex]c&lt;0[\/latex], reflect the graph about the [latex]x[\/latex]-axis.<\/li>\r\n \t<li>Vertical shift of the graph of [latex]y=cf(a(x+b))[\/latex]. If [latex]d&gt;0[\/latex], shift up. If [latex]d&lt;0[\/latex], shift down.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573580474\">We can summarize the different transformations and their related effects on the graph of a function in the following table.<\/p>\r\n\r\n<table id=\"fs-id1170573580486\" summary=\"A table with 8 rows and 2 columns. The first column is labeled \u201cTransformation of f(c &gt; 0)\u201d and has the values \u201cf(x) +c; f(x) -c; f(x + c); f(x - c); cf(x); f(cx); -f(x); f(-x)\u201d. The second column is labeled \u201cEffect on the graph of f\u201d and the values are \u201cVertical shift up c units; Vertical shift down c units; Shift left by c units; Shift right by c units; \u2018Vertical stretch if c &gt; 1, Vertical compression is 0 &lt; c &lt; 1\u2032; \u2018Horizontal stretch if 0 &lt; c &lt; 1, horizontal compression if c &gt; 1\u2032; reflection about the x-axis; reflection about the y-axis\u201d.\"><caption>Transformations of Functions<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><strong>Transformation of [latex]f(c&gt;0)[\/latex]<\/strong><\/th>\r\n<th><strong>Effect on the graph of<\/strong>[latex]f[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]f(x)+c[\/latex]<\/td>\r\n<td>Vertical shift up [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]f(x)-c[\/latex]<\/td>\r\n<td>Vertical shift down [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]f(x+c)[\/latex]<\/td>\r\n<td>Shift left by [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]f(x-c)[\/latex]<\/td>\r\n<td>Shift right by [latex]c[\/latex] units<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]cf(x)[\/latex]<\/td>\r\n<td>Vertical stretch if [latex]c&gt;1[\/latex];\r\nvertical compression if [latex]0&lt;c&lt;1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]f(cx)[\/latex]<\/td>\r\n<td>Horizontal stretch if [latex]0&lt;c&lt;1[\/latex]; horizontal compression if [latex]c&gt;1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u2212f(x)[\/latex]<\/td>\r\n<td>Reflection about the [latex]x[\/latex]-axis<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]f(\u2212x)[\/latex]<\/td>\r\n<td>Reflection about the [latex]y[\/latex]-axis<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1170573580888\" class=\"textbox exercises\">\r\n<h3>Example: Transforming a Function<\/h3>\r\n<p id=\"fs-id1170573580897\">For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function.<\/p>\r\n\r\n<ol id=\"fs-id1170573580902\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(x)=\u2212|x+2|-3[\/latex]<\/li>\r\n \t<li>[latex]f(x)=3\\sqrt{\u2212x}+1[\/latex]<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1170573580983\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573580983\"]\r\n<ol id=\"fs-id1170573580983\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Starting with the graph of [latex]y=|x|[\/latex], shift 2 units to the left, reflect about the [latex]x[\/latex]-axis, and then shift down 3 units.[caption id=\"\" align=\"aligncenter\" width=\"479\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202321\/CNX_Calc_Figure_01_02_019.jpg\" alt=\"An image of a graph. The x axis runs from -7 to 7 and a y axis runs from -7 to 7. The graph contains four functions. The first function is \u201cf(x) = absolute value of x\u201d and is labeled starting function. It decreases in a straight line until the origin and then increases in a straight line again after the origin. The second function is \u201cf(x) = absolute value of (x + 2)\u201d, which decreases in a straight line until the point (-2, 0) and then increases in a straight line again after the point (-2, 0). The second function is the same shape as the first function, but is shifted left 2 units. The third function is \u201cf(x) = -(absolute value of (x + 2))\u201d, which increases in a straight line until the point (-2, 0) and then decreases in a straight line again after the point (-2, 0). The third function is the second function reflected about the x axis. The fourth function is \u201cf(x) = -(absolute value of (x + 2)) - 3\u201d and is labeled \u201ctransformed function\u201d. It increases in a straight line until the point (-2, -3) and then decreases in a straight line again after the point (-2, -3). The fourth function is the third function shifted down 3 units.\" width=\"479\" height=\"489\" \/> Figure 20. The function [latex]f(x)=\u2212|x+2|-3[\/latex] can be viewed as a sequence of three transformations of the function [latex]y=|x|[\/latex].[\/caption]<\/li>\r\n \t<li>Starting with the graph of [latex]y=\\sqrt{x}[\/latex], reflect about the [latex]y[\/latex]-axis, stretch the graph vertically by a factor of 3, and move up 1 unit.[caption id=\"\" align=\"aligncenter\" width=\"479\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202325\/CNX_Calc_Figure_01_02_020.jpg\" alt=\"An image of a graph. The x axis runs from -7 to 7 and a y axis runs from -2 to 10. The graph contains four functions. The first function is \u201cf(x) = square root of x\u201d and is labeled starting function. It is a curved function that begins at the origin and increases. The second function is \u201cf(x) = square root of -x\u201d, which is a curved function that decreases until it reaches the origin, where it stops. The second function is the first function reflected about the y axis. The third function is \u201cf(x) = 3(square root of -x)\u201d, which is a curved function that decreases until it reaches the origin, where it stops. The third function decreases at a quicker rate than the second function. The fourth function is \u201cf(x) = 3(square root of -x) + 1\u201d and is labeled \u201ctransformed function\u201d. Itis a curved function that decreases until it reaches the point (0, 1), where it stops. The fourth function is the third function shifted up 1 unit.\" width=\"479\" height=\"422\" \/> Figure 21. The function [latex]f(x)=3\\sqrt{\u2212x}+1[\/latex] can be viewed as a sequence of three transformations of the function [latex]y=\\sqrt{x}[\/latex].[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Transforming a Function[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/iiBBHtVIk9U?controls=0&amp;start=1635&amp;end=1821&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.2BasicClassesofFunctions1635to1821_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.2 Basic Classes of Functions\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170573581200\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170573581209\">Describe how the function [latex]f(x)=\u2212(x+1)^2-4[\/latex] can be graphed using the graph of [latex]y=x^2[\/latex] and a sequence of transformations.<\/p>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"783667\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"783667\"]\r\n\r\nUse the Transformations of Functions table.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1170573581275\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573581275\"]\r\n<p id=\"fs-id1170573581275\">Shift the graph of [latex]y=x^2[\/latex] to the left 1 unit, reflect about the [latex]x[\/latex]-axis, then shift down 4 units.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]217380[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170573361979\">We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function [latex]y=x^2[\/latex] to get the function [latex]f(x)=(x-2)^2[\/latex]. This subtraction represents a shift of the function [latex]y=x^2[\/latex] two units to the right. A shift, horizontally or vertically, is a type of <strong>transformation of a function<\/strong>. Other transformations include horizontal and vertical scalings, and reflections about the axes.<\/p>\n<h3>Vertical Shift<\/h3>\n<p>A vertical shift of a function occurs if we add or subtract the same constant to each output [latex]y[\/latex]. For [latex]c>0[\/latex], the graph of [latex]f(x)+c[\/latex] is a shift of the graph of [latex]f(x)[\/latex] up [latex]c[\/latex] units, whereas the graph of [latex]f(x)-c[\/latex] is a shift of the graph of [latex]f(x)[\/latex] down [latex]c[\/latex] units. For example, the graph of the function [latex]f(x)=x^3+4[\/latex] is the graph of [latex]y=x^3[\/latex] shifted up 4 units; the graph of the function [latex]f(x)=x^3-4[\/latex] is the graph of [latex]y=x^3[\/latex] shifted down 4 units (Figure 15).<\/p>\n<div style=\"width: 718px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202303\/CNX_Calc_Figure_01_02_023.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201ca\u201d and has an x axis that runs from -4 to 4 and a y axis that runs from -1 to 10. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = (x squared) + 4\u201d, which is a parabola that decreases until the point (0, 4) and then increases again after the origin. The two functions are the same in shape, but the second function is shifted up 4 units. The second graph is labeled \u201cb\u201d and has an x axis that runs from -4 to 4 and a y axis that runs from -5 to 6. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = (x squared) - 4\u201d, which is a parabola that decreases until the point (0, -4) and then increases again after the origin. The two functions are the same in shape, but the second function is shifted down 4 units.\" width=\"708\" height=\"505\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 15. (a) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x)+c[\/latex] is a vertical shift up [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex]. (b) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x)-c[\/latex] is a vertical shift down [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex].<\/p>\n<\/div>\n<h3>Horizontal Shift<\/h3>\n<p id=\"fs-id1170573582792\">A horizontal shift of a function occurs if we add or subtract the same constant to each input [latex]x[\/latex]. For [latex]c>0[\/latex], the graph of [latex]f(x+c)[\/latex] is a shift of the graph of [latex]f(x)[\/latex] to the left [latex]c[\/latex] units; the graph of [latex]f(x-c)[\/latex] is a shift of the graph of [latex]f(x)[\/latex] to the right [latex]c[\/latex] units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let\u2019s look at an example.<\/p>\n<p>Consider the function [latex]f(x)=|x+3|[\/latex] and evaluate this function at [latex]x-3.[\/latex] Since [latex]f(x-3)=|x|[\/latex] and [latex]x-3<x[\/latex], the graph of [latex]f(x)=|x+3|[\/latex] is the graph of [latex]y=|x|[\/latex] shifted left 3 units. Similarly, the graph of [latex]f(x)=|x-3|[\/latex] is the graph of [latex]y=|x|[\/latex] shifted right 3 units (Figure 16).\n\n\n\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202307\/CNX_Calc_Figure_01_02_015.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201ca\u201d and has an x axis that runs from -8 to 5 and a y axis that runs from -3 to 5. The graph is of two functions. The first function is \u201cf(x) = absolute value of x\u201d, which decreases in a straight line until the origin and then increases in a straight line again after the origin. The second function is \u201cf(x) = absolute value of (x + 3)\u201d, which decreases in a straight line until the point (-3, 0) and then increases in a straight line again after the point (-3, 0). The two functions are the same in shape, but the second function is shifted left 3 units. The second graph is labeled \u201cb\u201d and has an x axis that runs from -5 to 8 and a y axis that runs from -3 to 5. The graph is of two functions. The first function is \u201cf(x) = absolute value of x\u201d, which decreases in a straight line until the origin and then increases in a straight line again after the origin. The second function is \u201cf(x) = absolute value of (x - 3)\u201d, which decreases in a straight line until the point (3, 0) and then increases in a straight line again after the point (3, 0). The two functions are the same in shape, but the second function is shifted right 3 units.\" width=\"975\" height=\"337\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 16. (a) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x+c)[\/latex] is a horizontal shift left [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex]. (b) For [latex]c&gt;0[\/latex], the graph of [latex]y=f(x-c)[\/latex] is a horizontal shift right [latex]c[\/latex] units of the graph of [latex]y=f(x)[\/latex].<\/p>\n<\/div>\n<h3>Vertical Scaling (Stretched\/Compressed)<\/h3>\n<p id=\"fs-id1170573582097\">A vertical scaling of a graph occurs if we multiply all outputs [latex]y[\/latex] of a function by the same positive constant. For [latex]c>0[\/latex], the graph of the function [latex]cf(x)[\/latex] is the graph of [latex]f(x)[\/latex] scaled vertically by a factor of [latex]c[\/latex]. If [latex]c>1[\/latex], the values of the outputs for the function [latex]cf(x)[\/latex] are larger than the values of the outputs for the function [latex]f(x)[\/latex]; therefore, the graph has been stretched vertically. If [latex]0<c<1[\/latex], then the outputs of the function [latex]cf(x)[\/latex] are smaller, so the graph has been compressed. For example, the graph of the function [latex]f(x)=3x^2[\/latex] is the graph of [latex]y=x^2[\/latex] stretched vertically by a factor of 3, whereas the graph of [latex]f(x)=\\frac{x^2}{3}[\/latex] is the graph of [latex]y=x^2[\/latex] compressed vertically by a factor of 3 (Figure 17).<\/p>\n<div style=\"width: 672px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202311\/CNX_Calc_Figure_01_02_024.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201ca\u201d and has an x axis that runs from -3 to 3 and a y axis that runs from -2 to 9. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = 3(x squared)\u201d, which is a parabola that decreases until the origin and then increases again after the origin, but is vertically stretched and thus increases at a quicker rate than the first function. The second graph is labeled \u201cb\u201d and has an x axis that runs from -4 to 4 and a y axis that runs from -2 to 9. The graph is of two functions. The first function is \u201cf(x) = x squared\u201d, which is a parabola that decreases until the origin and then increases again after the origin. The second function is \u201cf(x) = (1\/3)(x squared)\u201d, which is a parabola that decreases until the origin and then increases again after the origin, but is vertically compressed and thus increases at a slower rate than the first function.\" width=\"662\" height=\"507\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 17. (a) If [latex]c&gt;1[\/latex], the graph of [latex]y=cf(x)[\/latex] is a vertical stretch of the graph of [latex]y=f(x)[\/latex]. (b) If [latex]0&lt;c&lt;1[\/latex], the graph of [latex]y=cf(x)[\/latex] is a vertical compression of the graph of [latex]y=f(x)[\/latex].<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<h3>Horizontal Scaling (Stretched\/Compressed)<\/h3>\n<p id=\"fs-id1170573569978\">The horizontal scaling of a function occurs if we multiply the inputs [latex]x[\/latex] by the same positive constant. For [latex]c>0[\/latex], the graph of the function [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] scaled horizontally by a factor of [latex]c[\/latex]. If [latex]c>1[\/latex], the graph of [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] compressed horizontally. If [latex]0<c<1[\/latex], the graph of [latex]f(cx)[\/latex] is the graph of [latex]f(x)[\/latex] stretched horizontally. For example, consider the function [latex]f(x)=\\sqrt{2x}[\/latex] and evaluate [latex]f[\/latex] at [latex]\\dfrac{x}{2}.[\/latex] Since [latex]f(\\frac{x}{2})=\\sqrt{x}[\/latex], the graph of [latex]f(x)=\\sqrt{2x}[\/latex] is the graph of [latex]y=\\sqrt{x}[\/latex] compressed horizontally. The graph of [latex]y=\\sqrt{\\frac{x}{2}}[\/latex] is a horizontal stretch of the graph of [latex]y=\\sqrt{x}[\/latex] (Figure 18).<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202314\/CNX_Calc_Figure_01_02_017.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -2 to 4 and a y axis that runs from -2 to 5. The first graph is labeled \u201ca\u201d and is of two functions. The first graph is of two functions. The first function is \u201cf(x) = square root of x\u201d, which is a curved function that begins at the origin and increases. The second function is \u201cf(x) = square root of 2x\u201d, which is a curved function that begins at the origin and increases, but increases at a faster rate than the first function. The second graph is labeled \u201cb\u201d and is of two functions. The first function is \u201cf(x) = square root of x\u201d, which is a curved function that begins at the origin and increases. The second function is \u201cf(x) = square root of (x\/2)\u201d, which is a curved function that begins at the origin and increases, but increases at a slower rate than the first function.\" width=\"731\" height=\"351\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 18. (a) If [latex]c&gt;1[\/latex], the graph of [latex]y=f(cx)[\/latex] is a horizontal compression of the graph of [latex]y=f(x)[\/latex]. (b) If [latex]0&lt;c&lt;1[\/latex], the graph of [latex]y=f(cx)[\/latex] is a horizontal stretch of the graph of [latex]y=f(x)[\/latex].<\/p>\n<\/div>\n<h3>Reflection<\/h3>\n<p id=\"fs-id1170573582412\">We have explored what happens to the graph of a function [latex]f[\/latex] when we multiply [latex]f[\/latex] by a constant [latex]c>0[\/latex] to get a new function [latex]cf(x)[\/latex]. We have also discussed what happens to the graph of a function [latex]f[\/latex] when we multiply the independent variable [latex]x[\/latex] by [latex]c>0[\/latex] to get a new function [latex]f(cx)[\/latex]. However, we have not addressed what happens to the graph of the function if the constant [latex]c[\/latex] is negative. If we have a constant [latex]c<0[\/latex], we can write [latex]c[\/latex] as a positive number multiplied by [latex]-1[\/latex]; but, what kind of transformation do we get when we multiply the function or its argument by [latex]-1[\/latex]? When we multiply all the outputs by [latex]-1[\/latex], we get a reflection about the [latex]x[\/latex]-axis. When we multiply all inputs by [latex]-1[\/latex], we get a reflection about the [latex]y[\/latex]-axis. For example, the graph of [latex]f(x)=\u2212(x^3+1)[\/latex] is the graph of [latex]y=(x^3+1)[\/latex] reflected about the [latex]x[\/latex]-axis. The graph of [latex]f(x)=(\u2212x)^3+1[\/latex] is the graph of [latex]y=x^3+1[\/latex] reflected about the [latex]y[\/latex]-axis (Figure 19).<\/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\/2332\/2018\/01\/11202317\/CNX_Calc_Figure_01_02_018.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -5 to 6. The first graph is labeled \u201ca\u201d and is of two functions. The first graph is of two functions. The first function is \u201cf(x) = x cubed + 1\u201d, which is a curved increasing function that has an x intercept at (-1, 0) and a y intercept at (0, 1). The second function is \u201cf(x) = -(x cubed + 1)\u201d, which is a curved decreasing function that has an x intercept at (-1, 0) and a y intercept at (0, -1). The second graph is labeled \u201cb\u201d and is of two functions. The first function is \u201cf(x) = x cubed + 1\u201d, which is a curved increasing function that has an x intercept at (-1, 0) and a y intercept at (0, 1). The second function is \u201cf(x) = (-x) cubed + 1\u201d, which is a curved decreasing function that has an x intercept at (1, 0) and a y intercept at (0, 1). The first function increases at the same rate the second function decreases for the same values of x.\" width=\"487\" height=\"434\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 19. (a) The graph of [latex]y=\u2212f(x)[\/latex] is the graph of [latex]y=f(x)[\/latex] reflected about the [latex]x[\/latex]-axis. (b) The graph of [latex]y=f(\u2212x)[\/latex] is the graph of [latex]y=f(x)[\/latex] reflected about the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<h3>Multiple Transformations<\/h3>\n<p id=\"fs-id1170573580155\">If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function [latex]f(x)[\/latex], the graph of the related function [latex]y=cf(a(x+b))+d[\/latex] can be obtained from the graph of [latex]y=f(x)[\/latex] by performing the transformations in the following order.<\/p>\n<ol id=\"fs-id1170573580235\">\n<li>Horizontal shift of the graph of [latex]y=f(x)[\/latex]. If [latex]b>0[\/latex], shift left. If [latex]b<0[\/latex], shift right.<\/li>\n<li>Horizontal scaling of the graph of [latex]y=f(x+b)[\/latex] by a factor of [latex]|a|[\/latex]. If [latex]a<0[\/latex], reflect the graph about the [latex]y[\/latex]-axis.<\/li>\n<li>Vertical scaling of the graph of [latex]y=f(a(x+b))[\/latex] by a factor of [latex]|c|[\/latex]. If [latex]c<0[\/latex], reflect the graph about the [latex]x[\/latex]-axis.<\/li>\n<li>Vertical shift of the graph of [latex]y=cf(a(x+b))[\/latex]. If [latex]d>0[\/latex], shift up. If [latex]d<0[\/latex], shift down.<\/li>\n<\/ol>\n<p id=\"fs-id1170573580474\">We can summarize the different transformations and their related effects on the graph of a function in the following table.<\/p>\n<table id=\"fs-id1170573580486\" summary=\"A table with 8 rows and 2 columns. The first column is labeled \u201cTransformation of f(c &gt; 0)\u201d and has the values \u201cf(x) +c; f(x) -c; f(x + c); f(x - c); cf(x); f(cx); -f(x); f(-x)\u201d. The second column is labeled \u201cEffect on the graph of f\u201d and the values are \u201cVertical shift up c units; Vertical shift down c units; Shift left by c units; Shift right by c units; \u2018Vertical stretch if c &gt; 1, Vertical compression is 0 &lt; c &lt; 1\u2032; \u2018Horizontal stretch if 0 &lt; c &lt; 1, horizontal compression if c &gt; 1\u2032; reflection about the x-axis; reflection about the y-axis\u201d.\">\n<caption>Transformations of Functions<\/caption>\n<thead>\n<tr valign=\"top\">\n<th><strong>Transformation of [latex]f(c>0)[\/latex]<\/strong><\/th>\n<th><strong>Effect on the graph of<\/strong>[latex]f[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]f(x)+c[\/latex]<\/td>\n<td>Vertical shift up [latex]c[\/latex] units<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]f(x)-c[\/latex]<\/td>\n<td>Vertical shift down [latex]c[\/latex] units<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]f(x+c)[\/latex]<\/td>\n<td>Shift left by [latex]c[\/latex] units<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]f(x-c)[\/latex]<\/td>\n<td>Shift right by [latex]c[\/latex] units<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]cf(x)[\/latex]<\/td>\n<td>Vertical stretch if [latex]c>1[\/latex];<br \/>\nvertical compression if [latex]0<c<1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]f(cx)[\/latex]<\/td>\n<td>Horizontal stretch if [latex]0<c<1[\/latex]; horizontal compression if [latex]c>1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u2212f(x)[\/latex]<\/td>\n<td>Reflection about the [latex]x[\/latex]-axis<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]f(\u2212x)[\/latex]<\/td>\n<td>Reflection about the [latex]y[\/latex]-axis<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1170573580888\" class=\"textbox exercises\">\n<h3>Example: Transforming a Function<\/h3>\n<p id=\"fs-id1170573580897\">For each of the following functions, a. and b., sketch a graph by using a sequence of transformations of a well-known function.<\/p>\n<ol id=\"fs-id1170573580902\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x)=\u2212|x+2|-3[\/latex]<\/li>\n<li>[latex]f(x)=3\\sqrt{\u2212x}+1[\/latex]<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170573580983\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170573580983\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170573580983\" style=\"list-style-type: lower-alpha;\">\n<li>Starting with the graph of [latex]y=|x|[\/latex], shift 2 units to the left, reflect about the [latex]x[\/latex]-axis, and then shift down 3 units.\n<div style=\"width: 489px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202321\/CNX_Calc_Figure_01_02_019.jpg\" alt=\"An image of a graph. The x axis runs from -7 to 7 and a y axis runs from -7 to 7. The graph contains four functions. The first function is \u201cf(x) = absolute value of x\u201d and is labeled starting function. It decreases in a straight line until the origin and then increases in a straight line again after the origin. The second function is \u201cf(x) = absolute value of (x + 2)\u201d, which decreases in a straight line until the point (-2, 0) and then increases in a straight line again after the point (-2, 0). The second function is the same shape as the first function, but is shifted left 2 units. The third function is \u201cf(x) = -(absolute value of (x + 2))\u201d, which increases in a straight line until the point (-2, 0) and then decreases in a straight line again after the point (-2, 0). The third function is the second function reflected about the x axis. The fourth function is \u201cf(x) = -(absolute value of (x + 2)) - 3\u201d and is labeled \u201ctransformed function\u201d. It increases in a straight line until the point (-2, -3) and then decreases in a straight line again after the point (-2, -3). The fourth function is the third function shifted down 3 units.\" width=\"479\" height=\"489\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 20. The function [latex]f(x)=\u2212|x+2|-3[\/latex] can be viewed as a sequence of three transformations of the function [latex]y=|x|[\/latex].<\/p>\n<\/div>\n<\/li>\n<li>Starting with the graph of [latex]y=\\sqrt{x}[\/latex], reflect about the [latex]y[\/latex]-axis, stretch the graph vertically by a factor of 3, and move up 1 unit.\n<div style=\"width: 489px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202325\/CNX_Calc_Figure_01_02_020.jpg\" alt=\"An image of a graph. The x axis runs from -7 to 7 and a y axis runs from -2 to 10. The graph contains four functions. The first function is \u201cf(x) = square root of x\u201d and is labeled starting function. It is a curved function that begins at the origin and increases. The second function is \u201cf(x) = square root of -x\u201d, which is a curved function that decreases until it reaches the origin, where it stops. The second function is the first function reflected about the y axis. The third function is \u201cf(x) = 3(square root of -x)\u201d, which is a curved function that decreases until it reaches the origin, where it stops. The third function decreases at a quicker rate than the second function. The fourth function is \u201cf(x) = 3(square root of -x) + 1\u201d and is labeled \u201ctransformed function\u201d. Itis a curved function that decreases until it reaches the point (0, 1), where it stops. The fourth function is the third function shifted up 1 unit.\" width=\"479\" height=\"422\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 21. The function [latex]f(x)=3\\sqrt{\u2212x}+1[\/latex] can be viewed as a sequence of three transformations of the function [latex]y=\\sqrt{x}[\/latex].<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Transforming a Function<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/iiBBHtVIk9U?controls=0&amp;start=1635&amp;end=1821&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.2BasicClassesofFunctions1635to1821_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.2 Basic Classes of Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170573581200\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170573581209\">Describe how the function [latex]f(x)=\u2212(x+1)^2-4[\/latex] can be graphed using the graph of [latex]y=x^2[\/latex] and a sequence of transformations.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783667\">Hint<\/span><\/p>\n<div id=\"q783667\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Transformations of Functions table.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170573581275\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170573581275\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573581275\">Shift the graph of [latex]y=x^2[\/latex] to the left 1 unit, reflect about the [latex]x[\/latex]-axis, then shift down 4 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm217380\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=217380&theme=oea&iframe_resize_id=ohm217380&show_question_numbers\" width=\"100%\" height=\"150\"><\/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-124\">\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>1.2 Basic Classes of Functions. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.2 Basic Classes of Functions\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-124","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/124","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":26,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/124\/revisions"}],"predecessor-version":[{"id":4746,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/124\/revisions\/4746"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/124\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=124"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=124"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=124"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}