{"id":153,"date":"2023-06-21T13:22:38","date_gmt":"2023-06-21T13:22:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/sequences-of-transformations\/"},"modified":"2023-08-24T06:07:53","modified_gmt":"2023-08-24T06:07:53","slug":"sequences-of-transformations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/sequences-of-transformations\/","title":{"raw":"\u25aa   Sequences of Transformations","rendered":"\u25aa   Sequences of Transformations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Combine vertical and horizontal shifts.<\/li>\r\n \t<li>Follow a pattern when combining shifts and stretches.<\/li>\r\n<\/ul>\r\n<\/div>\r\nNow that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output ( [latex]y\\text{-}[\/latex] ) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( [latex]x\\text{-}[\/latex] ) axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down <i>and<\/i> right or left.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function and both a vertical and a horizontal shift, sketch the graph.<\/h3>\r\n<ol>\r\n \t<li>Identify the vertical and horizontal shifts from the formula.<\/li>\r\n \t<li>The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.<\/li>\r\n \t<li>The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.<\/li>\r\n \t<li>Apply the shifts to the graph in either order.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing Combined Vertical and Horizontal Shifts<\/h3>\r\nGiven [latex]f\\left(x\\right)=|x|[\/latex], sketch a graph of [latex]h\\left(x\\right)=f\\left(x+1\\right)-3[\/latex].\r\n\r\nThe function [latex]f[\/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of [latex]h[\/latex] has transformed [latex]f[\/latex] in two ways: [latex]f\\left(x+1\\right)[\/latex] is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in [latex]f\\left(x+1\\right)-3[\/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated\u00a0below.\r\n\r\nLet us follow one point of the graph of [latex]f\\left(x\\right)=|x|[\/latex].\r\n<ul>\r\n \t<li>The point [latex]\\left(0,0\\right)[\/latex] is transformed first by shifting left 1 unit: [latex]\\left(0,0\\right)\\to \\left(-1,0\\right)[\/latex]<\/li>\r\n \t<li>The point [latex]\\left(-1,0\\right)[\/latex] is transformed next by shifting down 3 units: [latex]\\left(-1,0\\right)\\to \\left(-1,-3\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203632\/CNX_Precalc_Figure_01_05_009a2.jpg\" alt=\"Graph of an absolute function, y=|x|, and how it was transformed to y=|x+1|-3.\" width=\"487\" height=\"401\" \/>\r\n\r\nBelow is\u00a0the graph of [latex]h[\/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\/18203634\/CNX_Precalc_Figure_01_05_009b2.jpg\" alt=\"The final function y=|x+1|-3.\" width=\"487\" height=\"401\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]f\\left(x\\right)=|x|[\/latex], sketch a graph of [latex]h\\left(x\\right)=f\\left(x - 2\\right)+4[\/latex].\r\nCheck your work with an online graphing calculator.\r\n\r\n[reveal-answer q=\"807890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807890\"]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\"><img class=\"size-full wp-image-2752 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\" alt=\"cnx_precalc_figure_01_05_010\" width=\"487\" height=\"402\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74730&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Combined Vertical and Horizontal Shifts<\/h3>\r\nWrite a formula for the graph shown below, which is a transformation of the toolkit square root function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203636\/CNX_Precalc_Figure_01_05_0112.jpg\" alt=\"Graph of a square root function transposed right one unit and up 2.\" width=\"487\" height=\"292\" \/>\r\n[reveal-answer q=\"639112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"639112\"]\r\n\r\nThe graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as\r\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)=f\\left(x - 1\\right)+2[\/latex]<\/p>\r\nUsing the formula for the square root function, we can write\r\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)=\\sqrt{x - 1}+2[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that this transformation has changed the domain and range of the function. This new graph has domain [latex]\\left[1,\\infty \\right)[\/latex] and range [latex]\\left[2,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite a formula for a transformation of the toolkit reciprocal function [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] that shifts the function\u2019s graph three units to the left and one unit down.\r\n\r\n[reveal-answer q=\"126023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"126023\"]\r\n\r\n[latex]g(x)=\\dfrac{1}{x+3}-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying a Learning Model Equation<\/h3>\r\nA common model for learning has an equation similar to [latex]k\\left(t\\right)=-{2}^{-t}+1[\/latex], where [latex]k[\/latex] is the percentage of mastery that can be achieved after [latex]t[\/latex] practice sessions. This is a transformation of the function [latex]f\\left(t\\right)={2}^{t}[\/latex] shown below. Sketch a graph of [latex]k\\left(t\\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\/18203639\/CNX_Precalc_Figure_01_05_0162.jpg\" alt=\"Graph of k(t)\" width=\"487\" height=\"442\" \/>\r\n[reveal-answer q=\"533018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"533018\"]\r\n\r\nThis equation combines three transformations into one equation.\r\n<ul>\r\n \t<li>A horizontal reflection: [latex]f\\left(-t\\right)={2}^{-t}[\/latex]<\/li>\r\n \t<li>A vertical reflection: [latex]-f\\left(-t\\right)=-{2}^{-t}[\/latex]<\/li>\r\n \t<li>A vertical shift: [latex]-f\\left(-t\\right)+1=-{2}^{-t}+1[\/latex]<\/li>\r\n<\/ul>\r\nWe can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points [latex](0, 1)[\/latex] and [latex](1, 2)[\/latex].\r\n<ol>\r\n \t<li>First, we apply a horizontal reflection: [latex](0, 1) (\u20131, 2)[\/latex].<\/li>\r\n \t<li>Then, we apply a vertical reflection: [latex](0, \u22121) (1, \u20132)[\/latex].<\/li>\r\n \t<li>Finally, we apply a vertical shift: [latex](0, 0) (1, 1)[\/latex].<\/li>\r\n<\/ol>\r\nThis means that the original points, [latex](0,1)[\/latex] and [latex](1,2)[\/latex] become [latex](0,0)[\/latex] and [latex](1,1)[\/latex] after we apply the transformations.\r\n\r\nIn the graphs below, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203642\/CNX_Precalc_Figure_01_05_017abc2.jpg\" alt=\"Graphs of all the transformations.\" width=\"975\" height=\"413\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nAs a model for learning, this function would be limited to a domain of [latex]t\\ge 0[\/latex], with corresponding range [latex]\\left[0,1\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the toolkit function [latex]f\\left(x\\right)={x}^{2}[\/latex], graph [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]. Take note of any surprising behavior for these functions.\r\n\r\n[reveal-answer q=\"386010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"386010\"]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\"><img class=\"aligncenter size-full wp-image-2755\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\" alt=\"cnx_precalc_figure_01_05_020\" width=\"487\" height=\"438\" \/><\/a>\r\n\r\nNotice: [latex]g(x)=f(\u2212x)[\/latex]\u2009looks the same as [latex]f(x)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Combine Shifts and Stretches<\/h2>\r\nWhen combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.\r\n\r\nWhen we see an expression such as [latex]2f\\left(x\\right)+3[\/latex], which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of [latex]f\\left(x\\right)[\/latex], we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.\r\n\r\nHorizontal transformations are a little trickier to think about. When we write [latex]g\\left(x\\right)=f\\left(2x+3\\right)[\/latex], for example, we have to think about how the inputs to the function [latex]g[\/latex] relate to the inputs to the function [latex]f[\/latex]. Suppose we know [latex]f\\left(7\\right)=12[\/latex]. What input to [latex]g[\/latex] would produce that output? In other words, what value of [latex]x[\/latex] will allow [latex]g\\left(x\\right)=f\\left(2x+3\\right)=12[\/latex]? We would need [latex]2x+3=7[\/latex]. To solve for [latex]x[\/latex], we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.\r\n\r\nThis format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.\r\n<p style=\"text-align: center;\">[latex]f\\left(bx+p\\right)=f\\left(b\\left(x+\\frac{p}{b}\\right)\\right)[\/latex]<\/p>\r\nLet\u2019s work through an example.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2x+4\\right)}^{2}[\/latex]<\/p>\r\nWe can factor out a 2.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2\\left(x+2\\right)\\right)}^{2}[\/latex]<\/p>\r\nNow we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Combining Transformations<\/h3>\r\nWhen combining vertical transformations written in the form [latex]af\\left(x\\right)+k[\/latex], first vertically stretch by [latex]a[\/latex] and then vertically shift by [latex]k[\/latex].\r\n\r\nWhen combining horizontal transformations written in the form [latex]f\\left(bx-h\\right)[\/latex], first horizontally shift by [latex]\\frac{h}{b}[\/latex] and then horizontally stretch by [latex]\\frac{1}{b}[\/latex].\r\n\r\nWhen combining horizontal transformations written in the form [latex]f\\left(b\\left(x-h\\right)\\right)[\/latex], first horizontally stretch by [latex]\\frac{1}{b}[\/latex] and then horizontally shift by [latex]h[\/latex].\r\n\r\nHorizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Triple Transformation of a Tabular Function<\/h3>\r\nGiven the table below\u00a0for the function [latex]f\\left(x\\right)[\/latex], create a table of values for the function [latex]g\\left(x\\right)=2f\\left(3x\\right)+1[\/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>6<\/td>\r\n<td>12<\/td>\r\n<td>18<\/td>\r\n<td>24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>10<\/td>\r\n<td>14<\/td>\r\n<td>15<\/td>\r\n<td>17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"669282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"669282\"]\r\n\r\nThere are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, [latex]f\\left(3x\\right)[\/latex] is a horizontal compression by [latex]\\frac{1}{3}[\/latex], which means we multiply each [latex]x\\text{-}[\/latex] value by [latex]\\frac{1}{3}[\/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(3x\\right)[\/latex] <\/strong><\/td>\r\n<td>10<\/td>\r\n<td>14<\/td>\r\n<td>15<\/td>\r\n<td>17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLooking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation.\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]2f\\left(3x\\right)[\/latex] <\/strong><\/td>\r\n<td>20<\/td>\r\n<td>28<\/td>\r\n<td>30<\/td>\r\n<td>34<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFinally, we can apply the vertical shift, which will add 1 to all the output values.\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]g\\left(x\\right)=2f\\left(3x\\right)+1[\/latex]<\/strong><\/td>\r\n<td>21<\/td>\r\n<td>29<\/td>\r\n<td>31<\/td>\r\n<td>35<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=113225&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: Finding a Triple Transformation of a Graph<\/h3>\r\nUse the graph of [latex]f\\left(x\\right)[\/latex]\u00a0to sketch a graph of [latex]k\\left(x\\right)=f\\left(\\frac{1}{2}x+1\\right)-3[\/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\/18203644\/CNX_Precalc_Figure_01_05_034.jpg\" alt=\"Graph of a half-circle.\" width=\"487\" height=\"442\" \/>\r\n\r\n[reveal-answer q=\"697686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"697686\"]\r\n\r\nTo simplify, let\u2019s start by factoring out the inside of the function.\r\n<p style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}x+1\\right)-3=f\\left(\\frac{1}{2}\\left(x+2\\right)\\right)-3[\/latex]<\/p>\r\nBy factoring the inside, we can first horizontally stretch by 2, as indicated by the [latex]\\frac{1}{2}[\/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point [latex](0,2)[\/latex] remains at [latex](0,2)[\/latex] while the point [latex](2,0)[\/latex] will stretch to [latex](4,0)[\/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\/18203647\/CNX_Precalc_Figure_01_05_035.jpg\" alt=\"Graph of a vertically stretch half-circle.\" width=\"487\" height=\"442\" \/>\r\n\r\nNext, we horizontally shift left by 2 units, as indicated by [latex]x+2[\/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\/18203649\/CNX_Precalc_Figure_01_05_036.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"487\" height=\"442\" \/>\r\n\r\nLast, we vertically shift down by 3 to complete our sketch, as indicated by the [latex]-3[\/latex] on the outside of the function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203651\/CNX_Precalc_Figure_01_05_037.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"487\" height=\"442\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Combine vertical and horizontal shifts.<\/li>\n<li>Follow a pattern when combining shifts and stretches.<\/li>\n<\/ul>\n<\/div>\n<p>Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output ( [latex]y\\text{-}[\/latex] ) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( [latex]x\\text{-}[\/latex] ) axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down <i>and<\/i> right or left.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function and both a vertical and a horizontal shift, sketch the graph.<\/h3>\n<ol>\n<li>Identify the vertical and horizontal shifts from the formula.<\/li>\n<li>The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.<\/li>\n<li>The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.<\/li>\n<li>Apply the shifts to the graph in either order.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing Combined Vertical and Horizontal Shifts<\/h3>\n<p>Given [latex]f\\left(x\\right)=|x|[\/latex], sketch a graph of [latex]h\\left(x\\right)=f\\left(x+1\\right)-3[\/latex].<\/p>\n<p>The function [latex]f[\/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of [latex]h[\/latex] has transformed [latex]f[\/latex] in two ways: [latex]f\\left(x+1\\right)[\/latex] is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in [latex]f\\left(x+1\\right)-3[\/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated\u00a0below.<\/p>\n<p>Let us follow one point of the graph of [latex]f\\left(x\\right)=|x|[\/latex].<\/p>\n<ul>\n<li>The point [latex]\\left(0,0\\right)[\/latex] is transformed first by shifting left 1 unit: [latex]\\left(0,0\\right)\\to \\left(-1,0\\right)[\/latex]<\/li>\n<li>The point [latex]\\left(-1,0\\right)[\/latex] is transformed next by shifting down 3 units: [latex]\\left(-1,0\\right)\\to \\left(-1,-3\\right)[\/latex]<\/li>\n<\/ul>\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\/18203632\/CNX_Precalc_Figure_01_05_009a2.jpg\" alt=\"Graph of an absolute function, y=|x|, and how it was transformed to y=|x+1|-3.\" width=\"487\" height=\"401\" \/><\/p>\n<p>Below is\u00a0the graph of [latex]h[\/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\/18203634\/CNX_Precalc_Figure_01_05_009b2.jpg\" alt=\"The final function y=|x+1|-3.\" width=\"487\" height=\"401\" \/><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]f\\left(x\\right)=|x|[\/latex], sketch a graph of [latex]h\\left(x\\right)=f\\left(x - 2\\right)+4[\/latex].<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=\"q807890\">Show Solution<\/span><\/p>\n<div id=\"q807890\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2752 alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\" alt=\"cnx_precalc_figure_01_05_010\" width=\"487\" height=\"402\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74730&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Combined Vertical and Horizontal Shifts<\/h3>\n<p>Write a formula for the graph shown below, which is a transformation of the toolkit square root function.<\/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\/18203636\/CNX_Precalc_Figure_01_05_0112.jpg\" alt=\"Graph of a square root function transposed right one unit and up 2.\" width=\"487\" height=\"292\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q639112\">Show Solution<\/span><\/p>\n<div id=\"q639112\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)=f\\left(x - 1\\right)+2[\/latex]<\/p>\n<p>Using the formula for the square root function, we can write<\/p>\n<p style=\"text-align: center;\">[latex]h\\left(x\\right)=\\sqrt{x - 1}+2[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that this transformation has changed the domain and range of the function. This new graph has domain [latex]\\left[1,\\infty \\right)[\/latex] and range [latex]\\left[2,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write a formula for a transformation of the toolkit reciprocal function [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] that shifts the function\u2019s graph three units to the left and one unit down.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q126023\">Show Solution<\/span><\/p>\n<div id=\"q126023\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)=\\dfrac{1}{x+3}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying a Learning Model Equation<\/h3>\n<p>A common model for learning has an equation similar to [latex]k\\left(t\\right)=-{2}^{-t}+1[\/latex], where [latex]k[\/latex] is the percentage of mastery that can be achieved after [latex]t[\/latex] practice sessions. This is a transformation of the function [latex]f\\left(t\\right)={2}^{t}[\/latex] shown below. Sketch a graph of [latex]k\\left(t\\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\/18203639\/CNX_Precalc_Figure_01_05_0162.jpg\" alt=\"Graph of k(t)\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q533018\">Show Solution<\/span><\/p>\n<div id=\"q533018\" class=\"hidden-answer\" style=\"display: none\">\n<p>This equation combines three transformations into one equation.<\/p>\n<ul>\n<li>A horizontal reflection: [latex]f\\left(-t\\right)={2}^{-t}[\/latex]<\/li>\n<li>A vertical reflection: [latex]-f\\left(-t\\right)=-{2}^{-t}[\/latex]<\/li>\n<li>A vertical shift: [latex]-f\\left(-t\\right)+1=-{2}^{-t}+1[\/latex]<\/li>\n<\/ul>\n<p>We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points [latex](0, 1)[\/latex] and [latex](1, 2)[\/latex].<\/p>\n<ol>\n<li>First, we apply a horizontal reflection: [latex](0, 1) (\u20131, 2)[\/latex].<\/li>\n<li>Then, we apply a vertical reflection: [latex](0, \u22121) (1, \u20132)[\/latex].<\/li>\n<li>Finally, we apply a vertical shift: [latex](0, 0) (1, 1)[\/latex].<\/li>\n<\/ol>\n<p>This means that the original points, [latex](0,1)[\/latex] and [latex](1,2)[\/latex] become [latex](0,0)[\/latex] and [latex](1,1)[\/latex] after we apply the transformations.<\/p>\n<p>In the graphs below, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.<\/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\/18203642\/CNX_Precalc_Figure_01_05_017abc2.jpg\" alt=\"Graphs of all the transformations.\" width=\"975\" height=\"413\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>As a model for learning, this function would be limited to a domain of [latex]t\\ge 0[\/latex], with corresponding range [latex]\\left[0,1\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the toolkit function [latex]f\\left(x\\right)={x}^{2}[\/latex], graph [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]. Take note of any surprising behavior for these functions.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q386010\">Show Solution<\/span><\/p>\n<div id=\"q386010\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2755\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\" alt=\"cnx_precalc_figure_01_05_020\" width=\"487\" height=\"438\" \/><\/a><\/p>\n<p>Notice: [latex]g(x)=f(\u2212x)[\/latex]\u2009looks the same as [latex]f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Combine Shifts and Stretches<\/h2>\n<p>When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.<\/p>\n<p>When we see an expression such as [latex]2f\\left(x\\right)+3[\/latex], which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of [latex]f\\left(x\\right)[\/latex], we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.<\/p>\n<p>Horizontal transformations are a little trickier to think about. When we write [latex]g\\left(x\\right)=f\\left(2x+3\\right)[\/latex], for example, we have to think about how the inputs to the function [latex]g[\/latex] relate to the inputs to the function [latex]f[\/latex]. Suppose we know [latex]f\\left(7\\right)=12[\/latex]. What input to [latex]g[\/latex] would produce that output? In other words, what value of [latex]x[\/latex] will allow [latex]g\\left(x\\right)=f\\left(2x+3\\right)=12[\/latex]? We would need [latex]2x+3=7[\/latex]. To solve for [latex]x[\/latex], we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.<\/p>\n<p>This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(bx+p\\right)=f\\left(b\\left(x+\\frac{p}{b}\\right)\\right)[\/latex]<\/p>\n<p>Let\u2019s work through an example.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2x+4\\right)}^{2}[\/latex]<\/p>\n<p>We can factor out a 2.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2\\left(x+2\\right)\\right)}^{2}[\/latex]<\/p>\n<p>Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Combining Transformations<\/h3>\n<p>When combining vertical transformations written in the form [latex]af\\left(x\\right)+k[\/latex], first vertically stretch by [latex]a[\/latex] and then vertically shift by [latex]k[\/latex].<\/p>\n<p>When combining horizontal transformations written in the form [latex]f\\left(bx-h\\right)[\/latex], first horizontally shift by [latex]\\frac{h}{b}[\/latex] and then horizontally stretch by [latex]\\frac{1}{b}[\/latex].<\/p>\n<p>When combining horizontal transformations written in the form [latex]f\\left(b\\left(x-h\\right)\\right)[\/latex], first horizontally stretch by [latex]\\frac{1}{b}[\/latex] and then horizontally shift by [latex]h[\/latex].<\/p>\n<p>Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Triple Transformation of a Tabular Function<\/h3>\n<p>Given the table below\u00a0for the function [latex]f\\left(x\\right)[\/latex], create a table of values for the function [latex]g\\left(x\\right)=2f\\left(3x\\right)+1[\/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>6<\/td>\n<td>12<\/td>\n<td>18<\/td>\n<td>24<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>10<\/td>\n<td>14<\/td>\n<td>15<\/td>\n<td>17<\/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=\"q669282\">Show Solution<\/span><\/p>\n<div id=\"q669282\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, [latex]f\\left(3x\\right)[\/latex] is a horizontal compression by [latex]\\frac{1}{3}[\/latex], which means we multiply each [latex]x\\text{-}[\/latex] value by [latex]\\frac{1}{3}[\/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(3x\\right)[\/latex] <\/strong><\/td>\n<td>10<\/td>\n<td>14<\/td>\n<td>15<\/td>\n<td>17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation.<\/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]2f\\left(3x\\right)[\/latex] <\/strong><\/td>\n<td>20<\/td>\n<td>28<\/td>\n<td>30<\/td>\n<td>34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Finally, we can apply the vertical shift, which will add 1 to all the output values.<\/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]g\\left(x\\right)=2f\\left(3x\\right)+1[\/latex]<\/strong><\/td>\n<td>21<\/td>\n<td>29<\/td>\n<td>31<\/td>\n<td>35<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=113225&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Triple Transformation of a Graph<\/h3>\n<p>Use the graph of [latex]f\\left(x\\right)[\/latex]\u00a0to sketch a graph of [latex]k\\left(x\\right)=f\\left(\\frac{1}{2}x+1\\right)-3[\/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\/18203644\/CNX_Precalc_Figure_01_05_034.jpg\" alt=\"Graph of a half-circle.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q697686\">Show Solution<\/span><\/p>\n<div id=\"q697686\" class=\"hidden-answer\" style=\"display: none\">\n<p>To simplify, let\u2019s start by factoring out the inside of the function.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}x+1\\right)-3=f\\left(\\frac{1}{2}\\left(x+2\\right)\\right)-3[\/latex]<\/p>\n<p>By factoring the inside, we can first horizontally stretch by 2, as indicated by the [latex]\\frac{1}{2}[\/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point [latex](0,2)[\/latex] remains at [latex](0,2)[\/latex] while the point [latex](2,0)[\/latex] will stretch to [latex](4,0)[\/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\/18203647\/CNX_Precalc_Figure_01_05_035.jpg\" alt=\"Graph of a vertically stretch half-circle.\" width=\"487\" height=\"442\" \/><\/p>\n<p>Next, we horizontally shift left by 2 units, as indicated by [latex]x+2[\/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\/18203649\/CNX_Precalc_Figure_01_05_036.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"487\" height=\"442\" \/><\/p>\n<p>Last, we vertically shift down by 3 to complete our sketch, as indicated by the [latex]-3[\/latex] on the outside of the function.<\/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\/18203651\/CNX_Precalc_Figure_01_05_037.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"487\" height=\"442\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-153\">\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 113225. <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 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