{"id":4664,"date":"2017-12-26T16:54:20","date_gmt":"2017-12-26T16:54:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/transformations-of-functions\/"},"modified":"2018-05-17T00:51:14","modified_gmt":"2018-05-17T00:51:14","slug":"transformations-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/transformations-of-functions\/","title":{"raw":"Transformations of Functions","rendered":"Transformations of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li class=\"p1\"><span class=\"s1\">Transformations of functions<\/span>\r\n<ul>\r\n \t<li class=\"p1\"><span class=\"s1\">Graph functions using vertical and horizontal shifts<\/span><\/li>\r\n \t<li class=\"p1\"><span class=\"s1\">Graph functions using reflections about the\u00a0<\/span><em><span class=\"s6\">x<\/span><\/em><span class=\"s1\">-axis and the <\/span><span class=\"s4\"><i>y<\/i><\/span><span class=\"s1\">-axis<\/span><\/li>\r\n \t<li class=\"p1\"><span class=\"s1\">Graph functions using compressions and stretches<\/span><\/li>\r\n \t<li class=\"p1\"><span class=\"s1\">Combine transformations<\/span><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Transformations of quadratic functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165307\/CNX_Precalc_Figure_01_05_038n2.jpg\" alt=\"Figure_01_05_038\" width=\"488\" height=\"325\" \/> <strong>Figure 1.\u00a0<\/strong>(credit: \"Misko\"\/Flickr)[\/caption]\r\n<h2>Shifts<\/h2>\r\nOne simple kind of <strong>transformation<\/strong> involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a <strong>vertical shift<\/strong>, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function [latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex], the function [latex]f\\left(x\\right)[\/latex] is shifted vertically [latex]k[\/latex] units.\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\/2862\/2017\/12\/26165309\/CNX_Precalc_Figure_01_05_0012.jpg\" alt=\"Graph of f of x equals the cubed root of x shifted upward one unit, the resulting graph passes through the point (0,1) instead of (0,0), (1, 2) instead of (1,1) and (-1, 0) instead of (-1, -1)\" width=\"487\" height=\"292\" \/> Vertical shift by [latex]k=1[\/latex] of the cube root function [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex].[\/caption]To help you visualize the concept of a vertical shift, consider that [latex]y=f\\left(x\\right)[\/latex]. Therefore, [latex]f\\left(x\\right)+k[\/latex] is equivalent to [latex]y+k[\/latex]. Every unit of [latex]y[\/latex] is replaced by [latex]y+k[\/latex], so the [latex]y\\text{-}[\/latex] value increases or decreases depending on the value of [latex]k[\/latex]. The result is a shift upward or downward.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Vertical Shift<\/h3>\r\nGiven a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex], where [latex]k[\/latex] is a constant, is a <strong>vertical shift<\/strong> of the function [latex]f\\left(x\\right)[\/latex]. All the output values change by [latex]k[\/latex] units. If [latex]k[\/latex] is positive, the graph will shift up. If [latex]k[\/latex] is negative, the graph will shift down.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Adding a Constant to a Function<\/h3>\r\nTo regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 2\u00a0shows the area of open vents [latex]V[\/latex] (in square feet) throughout the day in hours after midnight, [latex]t[\/latex]. During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165311\/CNX_Precalc_Figure_01_05_0022.jpg\" alt=\"image\" width=\"487\" height=\"326\" \/>\r\n[reveal-answer q=\"716887\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"716887\"]\r\nWe can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165313\/CNX_Precalc_Figure_01_05_003a2.jpg\" alt=\"image\" width=\"487\" height=\"329\" \/>\r\n\r\nNotice that\u00a0for each input value, the output value has increased by 20, so if we call the new function [latex]S\\left(t\\right)[\/latex], we could write\r\n<p style=\"text-align: center\">[latex]S\\left(t\\right)=V\\left(t\\right)+20[\/latex]<\/p>\r\nThis notation tells us that, for any value of [latex]t,S\\left(t\\right)[\/latex] can be found by evaluating the function [latex]V[\/latex] at the same input and then adding 20 to the result. This defines [latex]S[\/latex] as a transformation of the function [latex]V[\/latex], in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change.\r\n<table summary=\"Three rows and seven columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>8<\/td>\r\n<td>10<\/td>\r\n<td>17<\/td>\r\n<td>19<\/td>\r\n<td>24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]V\\left(t\\right)[\/latex] <\/strong><\/td>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td>220<\/td>\r\n<td>220<\/td>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]S\\left(t\\right)[\/latex] <\/strong><\/td>\r\n<td>20<\/td>\r\n<td>20<\/td>\r\n<td>240<\/td>\r\n<td>240<\/td>\r\n<td>20<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a tabular function, create a new row to represent a vertical shift.<\/h3>\r\n<ol>\r\n \t<li>Identify the output row or column.<\/li>\r\n \t<li>Determine the <strong>magnitude<\/strong> of the shift.<\/li>\r\n \t<li>Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Shifting a Tabular Function Vertically<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"603330\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"603330\"]\r\nThe formula [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex] tells us that we can find the output values of [latex]g[\/latex] by subtracting 3 from the output values of [latex]f[\/latex]. For example:\r\n<p style=\"text-align: center\">[latex]\\begin{cases}f\\left(2\\right)=1\\hfill &amp; \\text{Given}\\hfill \\\\ g\\left(x\\right)=f\\left(x\\right)-3\\hfill &amp; \\text{Given transformation}\\hfill \\\\ g\\left(2\\right)=f\\left(2\\right)-3\\hfill &amp; \\hfill \\\\ =1 - 3\\hfill &amp; \\hfill \\\\ =-2\\hfill &amp; \\hfill \\end{cases}[\/latex]<\/p>\r\nSubtracting 3 from each [latex]f\\left(x\\right)[\/latex] value, we can complete a table of values for [latex]g\\left(x\\right)[\/latex].\r\n<table summary=\"Three rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>0<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Analysis of the Solution<\/h4>\r\nAs with the earlier vertical shift, notice the input values stay the same and only the output values change.\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\nThe function [latex]h\\left(t\\right)=-4.9{t}^{2}+30t[\/latex] gives the height [latex]h[\/latex] of a ball (in meters) thrown upward from the ground after [latex]t[\/latex] seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function [latex]b\\left(t\\right)[\/latex] to [latex]h\\left(t\\right)[\/latex], and then find a formula for [latex]b\\left(t\\right)[\/latex].\r\n\r\n[reveal-answer q=\"901752\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"901752\"][latex]b\\left(t\\right)=h\\left(t\\right)+10=-4.9{t}^{2}+30t+10[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Identifying Horizontal Shifts<\/h3>\r\nWe just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a <strong>horizontal shift<\/strong>.\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\/2862\/2017\/12\/26165316\/CNX_Precalc_Figure_01_05_0042.jpg\" alt=\"Graph of f of x equals the cubed root of x shifted left one unit, the resulting graph passes through the point (0,-1) instead of (0,0), (0, 1) instead of (1,1) and (-2, -1) instead of (-1, -1)\" width=\"487\" height=\"288\" \/> Horizontal shift of the function [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]. Note that [latex]h=+1[\/latex] shifts the graph to the left, that is, towards negative values of [latex]x[\/latex].[\/caption]&nbsp;\r\n<h3><\/h3>\r\nFor example, if [latex]f\\left(x\\right)={x}^{2}[\/latex], then [latex]g\\left(x\\right)={\\left(x - 2\\right)}^{2}[\/latex] is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in [latex]f[\/latex].\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal Shift<\/h3>\r\nGiven a function [latex]f[\/latex], a new function [latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex], where [latex]h[\/latex] is a constant, is a <strong>horizontal shift<\/strong> of the function [latex]f[\/latex]. If [latex]h[\/latex] is positive, the graph will shift right. If [latex]h[\/latex] is negative, the graph will shift left.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Adding a Constant to an Input<\/h3>\r\nReturning to our building airflow example from Example 2, suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.\r\n\r\n[reveal-answer q=\"839866\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"839866\"]\r\n\r\nWe can set [latex]V\\left(t\\right)[\/latex] to be the original program and [latex]F\\left(t\\right)[\/latex] to be the revised program.\r\n<p style=\"text-align: center\">[latex]\\begin{cases}{c}V\\left(t\\right)=\\text{ the original venting plan}\\\\ \\text{F}\\left(t\\right)=\\text{starting 2 hrs sooner}\\end{cases}[\/latex]<\/p>\r\nIn the new graph, at each time, the airflow is the same as the original function [latex]V[\/latex] was 2 hours later. For example, in the original function [latex]V[\/latex], the airflow starts to change at 8 a.m., whereas for the function [latex]F[\/latex], the airflow starts to change at 6 a.m. The comparable function values are [latex]V\\left(8\\right)=F\\left(6\\right)[\/latex].\u00a0Notice also that the vents first opened to [latex]220{\\text{ ft}}^{2}[\/latex] at 10 a.m. under the original plan, while under the new plan the vents reach [latex]220{\\text{ ft}}^{\\text{2}}[\/latex] at 8 a.m., so [latex]V\\left(10\\right)=F\\left(8\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165318\/CNX_Precalc_Figure_01_05_005a2.jpg\" alt=\"image\" width=\"487\" height=\"329\" \/>\r\n\r\nIn both cases, we see that, because [latex]F\\left(t\\right)[\/latex] starts 2 hours sooner, [latex]h=-2[\/latex]. That means that the same output values are reached when [latex]F\\left(t\\right)=V\\left(t-\\left(-2\\right)\\right)=V\\left(t+2\\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that [latex]V\\left(t+2\\right)[\/latex] has the effect of shifting the graph to the <em>left<\/em>.\r\n\r\nHorizontal changes or \"inside changes\" affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function [latex]F\\left(t\\right)[\/latex] uses the same outputs as [latex]V\\left(t\\right)[\/latex], but matches those outputs to inputs 2 hours earlier than those of [latex]V\\left(t\\right)[\/latex]. Said another way, we must add 2 hours to the input of [latex]V[\/latex] to find the corresponding output for [latex]F:F\\left(t\\right)=V\\left(t+2\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a tabular function, create a new row to represent a horizontal shift.<\/h3>\r\n<ol>\r\n \t<li>Identify the input row or column.<\/li>\r\n \t<li>Determine the magnitude of the shift.<\/li>\r\n \t<li>Add the shift to the value in each input cell.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Shifting a Tabular Function Horizontally<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(x - 3\\right)[\/latex].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"719880\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"719880\"]\r\nThe formula [latex]g\\left(x\\right)=f\\left(x - 3\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are the same as the output value of [latex]f[\/latex] when the input value is 3 less than the original value. For example, we know that [latex]f\\left(2\\right)=1[\/latex]. To get the same output from the function [latex]g[\/latex], we will need an input value that is 3 <em>larger<\/em>. We input a value that is 3 larger for [latex]g\\left(x\\right)[\/latex] because the function takes 3 away before evaluating the function [latex]f[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{cases}g\\left(5\\right)=f\\left(5 - 3\\right)\\hfill \\\\ =f\\left(2\\right)\\hfill \\\\ =1\\hfill \\end{cases}[\/latex]<\/p>\r\nWe continue with the other values to create this table.\r\n<table summary=\"Three 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>5<\/td>\r\n<td>7<\/td>\r\n<td>9<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]x - 3[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe result is that the function [latex]g\\left(x\\right)[\/latex] has been shifted to the right by 3. Notice the output values for [latex]g\\left(x\\right)[\/latex] remain the same as the output values for [latex]f\\left(x\\right)[\/latex], but the corresponding input values, [latex]x[\/latex], have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph below represents both of the functions. We can see the horizontal shift in each point.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165320\/CNX_Precalc_Figure_01_05_0062.jpg\" alt=\"Graph of the points from the previous table for f(x) and g(x)=f(x-3).\" width=\"487\" height=\"549\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying a Horizontal Shift of a Toolkit Function<\/h3>\r\nThis graph represents a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{2}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165321\/CNX_Precalc_Figure_01_05_0072.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"328\" \/>\r\n[reveal-answer q=\"937293\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"937293\"]\r\nNotice that the graph is identical in shape to the [latex]f\\left(x\\right)={x}^{2}[\/latex] function, but the <em>x-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so\r\n<p style=\"text-align: center\">[latex]g\\left(x\\right)=f\\left(x - 2\\right)[\/latex]<\/p>\r\nNotice how we must input the value [latex]x=2[\/latex] to get the output value [latex]y=0[\/latex]; the<em> x<\/em>-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the [latex]f\\left(x\\right)[\/latex] function to write a formula for [latex]g\\left(x\\right)[\/latex] by evaluating [latex]f\\left(x - 2\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{cases}f\\left(x\\right)={x}^{2}\\hfill \\\\ g\\left(x\\right)=f\\left(x - 2\\right)\\hfill \\\\ g\\left(x\\right)=f\\left(x - 2\\right)={\\left(x - 2\\right)}^{2}\\hfill \\end{cases}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nTo determine whether the shift is [latex]+2[\/latex] or [latex]-2[\/latex] , consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function, [latex]f\\left(0\\right)=0[\/latex]. In our shifted function, [latex]g\\left(2\\right)=0[\/latex]. To obtain the output value of 0 from the function [latex]f[\/latex], we need to decide whether a plus or a minus sign will work to satisfy [latex]g\\left(2\\right)=f\\left(x - 2\\right)=f\\left(0\\right)=0[\/latex]. For this to work, we will need to <em>subtract<\/em> 2 units from our input values.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Interpreting Horizontal versus Vertical Shifts<\/h3>\r\nThe function [latex]G\\left(m\\right)[\/latex] gives the number of gallons of gas required to drive [latex]m[\/latex] miles. Interpret [latex]G\\left(m\\right)+10[\/latex] and [latex]G\\left(m+10\\right)[\/latex].\r\n\r\n[reveal-answer q=\"792859\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"792859\"]\r\n[latex]G\\left(m\\right)+10[\/latex] can be interpreted as adding 10 to the output, gallons. This is the gas required to drive [latex]m[\/latex] miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.\r\n\r\n[latex]G\\left(m+10\\right)[\/latex] can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than [latex]m[\/latex] miles. The graph would indicate a horizontal shift.\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 function [latex]f\\left(x\\right)=\\sqrt{x}[\/latex], graph the original function [latex]f\\left(x\\right)[\/latex] and the transformation [latex]g\\left(x\\right)=f\\left(x+2\\right)[\/latex] on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?\r\n\r\n[reveal-answer q=\"193388\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"193388\"]\r\n\r\nA horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflections<\/h2>\r\nAnother transformation that can be applied to a function is a reflection over the <em>x<\/em>- or <em>y<\/em>-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the <em>x<\/em>-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the <em>y<\/em>-axis. The reflections are shown in Figure 9.\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\/2862\/2017\/12\/26165323\/CNX_Precalc_Figure_01_05_0122.jpg\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"487\" height=\"442\" \/> Vertical and horizontal reflections of a function.[\/caption]\r\n\r\nNotice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the <em>x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the <em>y<\/em>-axis.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Reflections<\/h3>\r\nGiven a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] is a <strong>vertical reflection<\/strong> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about (or over, or through) the <em>x<\/em>-axis.\r\n\r\nGiven a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] is a <strong>horizontal reflection<\/strong> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about the <em>y<\/em>-axis.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function, reflect the graph both vertically and horizontally.<\/h3>\r\n<ol>\r\n \t<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the <em>x<\/em>-axis.<\/li>\r\n \t<li>Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the <em>y<\/em>-axis.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reflecting a Graph Horizontally and Vertically<\/h3>\r\nReflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex] (a) vertically and (b) horizontally.\r\n\r\n[reveal-answer q=\"211400\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"211400\"]\r\na. Reflecting the graph vertically means that each output value will be reflected over the horizontal <em>t-<\/em>axis as shown below.\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\/2862\/2017\/12\/26165327\/CNX_Precalc_Figure_01_05_0132.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"975\" height=\"442\" \/> Vertical reflection of the square root function[\/caption]\r\n\r\nBecause each output value is the opposite of the original output value, we can write\r\n<p style=\"text-align: center\">[latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt{t}[\/latex]<\/p>\r\nNotice that this is an outside change, or vertical shift, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.\r\n\r\nb. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.\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\/2862\/2017\/12\/26165329\/CNX_Precalc_Figure_01_05_0142.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"975\" height=\"442\" \/> Horizontal reflection of the square root function[\/caption]\r\n\r\nBecause each input value is the opposite of the original input value, we can write\r\n<p style=\"text-align: center\">[latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/p>\r\nNotice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.\r\n\r\nNote that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\\left[0,\\infty \\right)[\/latex] and range [latex]\\left[0,\\infty \\right)[\/latex], the vertical reflection gives the [latex]V\\left(t\\right)[\/latex] function the range [latex]\\left(-\\infty ,0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,0\\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\nReflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.\r\n\r\n[reveal-answer q=\"362828\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"362828\"]\r\n\r\na)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165331\/CNX_Precalc_Figure_01_05_015a2.jpg\" alt=\"Graph of a vertically reflected absolute function.\" width=\"487\" height=\"213\" \/>\r\n\r\nb)\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165333\/CNX_Precalc_Figure_01_05_015b2.jpg\" alt=\"Graph of an absolute function translated one unit left.\" width=\"487\" height=\"251\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reflecting a Tabular Function Horizontally and Vertically<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.\r\n<ol>\r\n \t<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\r\n \t<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<table id=\"Table_01_05_05\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"608272\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"608272\"]\r\n<ol>\r\n \t<li>For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the <em>x<\/em>-values stay the same and each output value will be the opposite of the original output value.\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20137<\/td>\r\n<td>\u201311<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>For [latex]h\\left(x\\right)[\/latex], the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\\left(x\\right)[\/latex] values stay the same as the [latex]f\\left(x\\right)[\/latex] values.\r\n<table id=\"Table_01_05_07\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22124<\/td>\r\n<td>\u22126<\/td>\r\n<td>\u22128<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<table id=\"Table_01_05_08\" 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>\u22122<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>5<\/td>\r\n<td>10<\/td>\r\n<td>15<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing the function [latex]f\\left(x\\right)[\/latex] given in the table above, create a table for the functions below.\r\n\r\na. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]\r\n\r\nb. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]\r\n\r\n[reveal-answer q=\"230301\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"230301\"]\r\n<ol>\r\n \t<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>-2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]-10[\/latex]<\/td>\r\n<td>[latex]-15[\/latex]<\/td>\r\n<td>[latex]-20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>-2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]h\\left(x\\right)[\/latex]<\/td>\r\n<td>15<\/td>\r\n<td>10<\/td>\r\n<td>5<\/td>\r\n<td>unknown<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Compressions and Stretches<\/h2>\r\nAdding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.\r\n\r\nWe can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.\r\n<h3>Vertical Stretches and Compressions<\/h3>\r\nWhen we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a<strong> vertical compression<\/strong>. The graph below\u00a0shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165339\/CNX_Precalc_Figure_01_05_0242.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"487\" height=\"326\" \/> Vertical stretch and compression[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Vertical Stretches and Compressions<\/h3>\r\nGiven a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=af\\left(x\\right)[\/latex], where [latex]a[\/latex] is a constant, is a <strong>vertical stretch<\/strong> or <strong>vertical compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].\r\n<ul>\r\n \t<li>If [latex]a&gt;1[\/latex], then the graph will be stretched.<\/li>\r\n \t<li>If 0 &lt; a &lt; 1, then the graph will be compressed.<\/li>\r\n \t<li>If [latex]a&lt;0[\/latex], then there will be combination of a vertical stretch or compression with a vertical reflection.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function, graph its vertical stretch.<\/h3>\r\n<ol>\r\n \t<li>Identify the value of [latex]a[\/latex].<\/li>\r\n \t<li>Multiply all range values by [latex]a[\/latex].<\/li>\r\n \t<li>If [latex]a&gt;1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].\r\nIf [latex]{ 0 }&lt;{ a }&lt;{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].\r\nIf [latex]a&lt;0[\/latex], the graph is either stretched or compressed and also reflected about the <em>x<\/em>-axis.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Vertical Stretch<\/h3>\r\nA function [latex]P\\left(t\\right)[\/latex] models the number\u00a0of fruit flies in a population over time, and is graphed below.\r\n\r\nA scientist is comparing this population to another population, [latex]Q[\/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165342\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"487\" height=\"367\" \/>\r\n[reveal-answer q=\"951851\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"951851\"]\r\n\r\nBecause the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values.\r\n\r\nIf we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.\r\n\r\nThe following shows where the new points for the new graph will be located.\r\n\r\n[latex]\\begin{cases}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{cases}[\/latex]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165344\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"487\" height=\"367\" \/> <b>Figure 16<\/b>[\/caption]\r\n\r\nSymbolically, the relationship is written as\r\n\r\n[latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]\r\n\r\nThis means that for any input [latex]t[\/latex], the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P[\/latex]. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t[\/latex], stay the same while the output values are twice as large as before.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<\/h3>\r\n<ol>\r\n \t<li>Determine the value of [latex]a[\/latex].<\/li>\r\n \t<li>Multiply all of the output values by [latex]a[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Vertical Compression of a Tabular Function<\/h3>\r\nA function [latex]f[\/latex] is given in the table below. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"798923\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"798923\"]\r\nThe formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3[\/latex]. Then\r\n\r\n[latex]g\\left(4\\right)=\\frac{1}{2}f\\left(4\\right)=\\frac{1}{2}\\left(3\\right)=\\frac{3}{2}[\/latex]\r\nWe do the same for the other values to produce this table.\r\n<table summary=\"Two rows and five columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{7}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Analysis of the Solution<\/h4>\r\nThe result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by [latex]\\frac{1}{2}[\/latex]. Each output value is divided in half, so the graph is half the original height.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA function [latex]f[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right)[\/latex].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<td>12<\/td>\r\n<td>16<\/td>\r\n<td>20<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"805921\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"805921\"]\r\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled,\"><colgroup><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x\\\\[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)\\\\[\/latex]<\/td>\r\n<td>9<\/td>\r\n<td>12<\/td>\r\n<td>15<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Recognizing a Vertical Stretch<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165346\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"487\" height=\"442\" \/>\r\n\r\nThe graph\u00a0is a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].\r\n\r\n[reveal-answer q=\"289067\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"289067\"]\r\nWhen trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that [latex]g\\left(2\\right)=2[\/latex]. With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8[\/latex]. Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]g\\left(2\\right)=\\frac{1}{4}f\\left(2\\right)[\/latex]. From this we can fairly safely conclude that [latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)[\/latex].\r\n\r\nWe can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f[\/latex].\r\n<p style=\"text-align: center\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It 6<\/h3>\r\nWrite the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.\r\n\r\n[reveal-answer q=\"473017\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"473017\"]\r\n\r\n[latex]g(x)=3x-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Horizontal Stretches and Compressions<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165348\/CNX_Precalc_Figure_01_05_028.jpg\" alt=\"Graph of the vertical stretch and compression of x^2.\" width=\"487\" height=\"514\" \/>\r\n\r\nNow we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.\r\n\r\nGiven a function [latex]y=f\\left(x\\right)[\/latex], the form [latex]y=f\\left(bx\\right)[\/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[\/latex].\u00a0The graph of [latex]y={\\left(0.5x\\right)}^{2}[\/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2. The graph of [latex]y={\\left(2x\\right)}^{2}[\/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal Stretches and Compressions<\/h3>\r\nGiven a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex], where [latex]b[\/latex] is a constant, is a <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].\r\n<ul>\r\n \t<li>If [latex]b&gt;1[\/latex], then the graph will be compressed by [latex]\\frac{1}{b}[\/latex].<\/li>\r\n \t<li>If [latex]0&lt;b&lt;1[\/latex], then the graph will be stretched by [latex]\\frac{1}{b}[\/latex].<\/li>\r\n \t<li>If [latex]b&lt;0[\/latex], then there will be combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a description of a function, sketch a horizontal compression or stretch.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Write a formula to represent the function.<\/li>\r\n \t<li>Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b&gt;1[\/latex] for a compression or [latex]0&lt;b&lt;1[\/latex]\r\nfor a stretch.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Horizontal Compression<\/h3>\r\nSuppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, [latex]R[\/latex], will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.\r\n\r\n[reveal-answer q=\"855794\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"855794\"]\r\nSymbolically, we could write\r\n<p style=\"text-align: center\">[latex]\\begin{cases}R\\left(1\\right)=P\\left(2\\right),\\hfill \\\\ R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,}\\hfill \\\\ R\\left(t\\right)=P\\left(2t\\right).\\hfill \\end{cases}[\/latex]<\/p>\r\nSee below\u00a0for a graphical comparison of the original population and the compressed population.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"976\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165351\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).\" width=\"976\" height=\"401\" \/> (a) Original population graph (b) Compressed population graph[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding a Horizontal Stretch for a Tabular Function<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"261935\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"261935\"]\r\nThe formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right)[\/latex], and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]\r\n<p style=\"text-align: center\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/p>\r\nWe do the same for the other values to produce the table below.\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<td>12<\/td>\r\n<td>16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165353\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"975\" height=\"333\" \/>\r\n\r\nThis figure shows the graphs of both of these sets of points.\r\n<h4>Analysis of the Solution<\/h4>\r\nBecause each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Recognizing a Horizontal Compression on a Graph<\/h3>\r\nRelate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165355\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"487\" height=\"291\" \/>\r\n[reveal-answer q=\"396995\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"396995\"]\r\nThe graph of [latex]g\\left(x\\right)[\/latex] looks like the graph of [latex]f\\left(x\\right)[\/latex] horizontally compressed. Because [latex]f\\left(x\\right)[\/latex] ends at [latex]\\left(6,4\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] ends at [latex]\\left(2,4\\right)[\/latex], we can see that the [latex]x\\text{-}[\/latex] values have been compressed by [latex]\\frac{1}{3}[\/latex], because [latex]6\\left(\\frac{1}{3}\\right)=2[\/latex]. We might also notice that [latex]g\\left(2\\right)=f\\left(6\\right)[\/latex] and [latex]g\\left(1\\right)=f\\left(3\\right)[\/latex]. Either way, we can describe this relationship as [latex]g\\left(x\\right)=f\\left(3x\\right)[\/latex]. This is a horizontal compression by [latex]\\frac{1}{3}[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\\frac{1}{4}[\/latex] in our function: [latex]f\\left(\\frac{1}{4}x\\right)[\/latex]. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite a formula for the toolkit square root function horizontally stretched by a factor of 3.\r\n\r\n[reveal-answer q=\"35233\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"35233\"][latex]g\\left(x\\right)=|x - 1|-3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Sequences of Transformations<\/h2>\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\/2862\/2017\/12\/26165357\/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\/2862\/2017\/12\/26165359\/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\n\r\n[reveal-answer q=\"807890\"]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\/2862\/2017\/12\/26165401\/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\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\/2862\/2017\/12\/26165404\/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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"639112\"]\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)=\\frac{1}{x}[\/latex] that shifts the function\u2019s graph one unit to the right and one unit up.\r\n\r\n[reveal-answer q=\"126023\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"126023\"]\r\n\r\n[latex]g\\left(x\\right)=\\frac{1}{{\\left(x+4\\right)}^{2}}+2[\/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\/2862\/2017\/12\/26165406\/CNX_Precalc_Figure_01_05_0162.jpg\" alt=\"Graph of k(t)\" width=\"487\" height=\"442\" \/>\r\n[reveal-answer q=\"533018\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"533018\"]\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 (0, 1) and (1, 2).\r\n<ol>\r\n \t<li>First, we apply a horizontal reflection: (0, 1) (\u20131, 2).<\/li>\r\n \t<li>Then, we apply a vertical reflection: (0, \u22121) (1, \u20132).<\/li>\r\n \t<li>Finally, we apply a vertical shift: (0, 0) (1, 1).<\/li>\r\n<\/ol>\r\nThis means that the original points, (0,1) and (1,2) become (0,0) and (1,1) 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\/2862\/2017\/12\/26165408\/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\"]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\/2862\/2017\/12\/26165410\/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<h3>Combine Shifts and Stretches<\/h3>\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]h[\/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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"669282\"]\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 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\/2862\/2017\/12\/26165413\/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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"697686\"]\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 (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0).\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165415\/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\/2862\/2017\/12\/26165417\/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\/2862\/2017\/12\/26165419\/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>\r\n<h2>Transformations of Quadratic Functions<\/h2>\r\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n\r\n<div id=\"fs-id1165135320100\" class=\"equation\" style=\"text-align: center\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\r\n<p id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\r\nThe standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. The figure below\u00a0is the graph of this basic function.\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\/25201255\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/>\r\n\r\n&nbsp;\r\n<h2>Shift Up and Down by Changing the value of k<\/h2>\r\n<p id=\"fs-id1165137770279\">You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, k.<\/p>\r\n<p style=\"text-align: center\">[latex]f(x)=x^2 + k[\/latex]<\/p>\r\n\u00a0If [latex]k&gt;0[\/latex], the graph shifts upward, whereas if [latex]k&lt;0[\/latex], the graph shifts downward.\r\n\r\n<strong><span style=\"text-decoration: underline\">Instructions:<\/span><\/strong>\r\n<ol>\r\n \t<li>Use the slider in the interactive below to shift the graph of [latex]f(x)=x^2[\/latex] down 4 units, then up 4 units.<\/li>\r\n \t<li>Use the textbox below the graph to write both transformed equations.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/fpatj6tbcn\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Make Note<\/h3>\r\nWrite the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted up 4 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow check yourself!\r\n[reveal-answer q=\"725488\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725488\"]The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units is\r\n\r\n[latex]f(x)=x^2+4[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units is\r\n\r\n[latex]f(x)=x^2-4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Shift left and right by changing the value of h.<\/h3>\r\n<p id=\"fs-id1165137770279\">You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, h, to the variable x, before squaring.<\/p>\r\n<p style=\"text-align: center\">[latex]f(x)=(x-h)^2 [\/latex]<\/p>\r\nIf [latex]h&gt;0[\/latex], the graph shifts toward the right and if [latex]h&lt;0[\/latex], the graph shifts to the left.\r\n\r\n<strong><span style=\"text-decoration: underline\">Instructions:<\/span><\/strong>\r\n<ol>\r\n \t<li>Use the interactive graph below to shift the graph of [latex]f(x)=x^2[\/latex] 2 units to the right, then 2 units to the left.<\/li>\r\n \t<li>Use the textbox below the graph to write both transformed equations<\/li>\r\n<\/ol>\r\nhttps:\/\/www.desmos.com\/calculator\/5g3xfhkklq\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Make Note<\/h3>\r\nWrite the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted right 2 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted left 2 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow check yourself!\r\n[reveal-answer q=\"725588\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725588\"]The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is\r\n\r\n[latex]f(x)=(x-2)^2[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is\r\n\r\n[latex]f(x)=(x+2)^2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Stretch or compress by changing the value of a.<\/h3>\r\n<p id=\"fs-id1165137770279\">You can represent a stretch or compression (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, a.<\/p>\r\n<p style=\"text-align: center\">[latex]f(x)=ax^2 [\/latex]<\/p>\r\nThe magnitude of <em>a<\/em>\u00a0indicates the stretch of the graph. If [latex]|a|&gt;1[\/latex], the point associated with a particular <em>x<\/em>-value shifts farther from the <em>x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|&lt;1[\/latex], the point associated with a particular <em>x<\/em>-value shifts closer to the <em>x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression.\r\n\r\n<strong><span style=\"text-decoration: underline\">Instructions:<\/span><\/strong>\r\n<ol>\r\n \t<li>Use the interactive graph below to make a graph of the function\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex],<\/li>\r\n \t<li>And another that has been vertically stretched by a factor of 3.<\/li>\r\n \t<li>Use the textbox below the graph to write your equations.<\/li>\r\n<\/ol>\r\nhttps:\/\/www.desmos.com\/calculator\/ha6gh59rq7\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Make Note<\/h3>\r\nWrite the equation for the graph of [latex]f(x)=x^2[\/latex] that has been has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nThen, \u00a0write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow check yourself!\r\n[reveal-answer q=\"725489\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725489\"]The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex]\r\n\r\n[latex]f(x)=\\frac{1}{2}x^2[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.is\r\n\r\n[latex]f(x)=3x^2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135353112\">The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\r\n\r\n<div id=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}a{\\left(x-h\\right)}^{2}+k=a{x}^{2}+bx+c\\hfill \\\\ a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)=a{x}^{2}+bx+c\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137409211\">For the linear terms to be equal, the coefficients must be equal.<\/p>\r\n\r\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]-2ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex].<\/div>\r\n<p id=\"fs-id1165134118295\">This is the <strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal:<\/p>\r\n\r\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}a{h}^{2}+k=c\\hfill \\\\ \\text{ }k=c-a{h}^{2}\\hfill \\\\ \\text{ }=c-a-{\\left(\\frac{b}{2a}\\right)}^{2}\\hfill \\\\ \\text{ }=c-\\frac{{b}^{2}}{4a}\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em>k<\/em> is the output value of the function when the input is <em>h<\/em>, so [latex]f\\left(h\\right)=k[\/latex].<\/p>\r\nNow you try it.\r\n\r\nUse the interactive graph below to define two quadratic functions whose axis of symmetry is x = -3, and whose vertex is (-3, 2). Use the sliders for a, h, k below to help you.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/pimelalx4i\r\n\r\nNow answer the following questions about the graphs you made.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Take Note<\/h3>\r\nHow many potential values are there for h in this scenario?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\nHow about k?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\nHow about a?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n[reveal-answer q=\"349748\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"349748\"]\r\n\r\nThere is only one [latex](h,k)[\/latex] pair that will satisfy these conditions,\u00a0[latex](-3,2)[\/latex]. \u00a0The value of a does not affect the line of symmetry or the vertex of a quadratic graph, so a can be an infinite number of values.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>\u00a0Summary of Transformations<\/h2>\r\n<section id=\"fs-id1165135499979\" class=\"key-equations\">\r\n<table id=\"eip-id1165134474082\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>Vertical shift<\/td>\r\n<td>[latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex] (up for [latex]k&gt;0[\/latex] )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Horizontal shift<\/td>\r\n<td>[latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex] (right for [latex]h&gt;0[\/latex] )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Vertical reflection<\/td>\r\n<td>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Horizontal reflection<\/td>\r\n<td>[latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Vertical stretch<\/td>\r\n<td>[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] ( [latex]a&gt;0[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Vertical compression<\/td>\r\n<td>[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] [latex]\\left(0&lt;a&lt;1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Horizontal stretch<\/td>\r\n<td>[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] [latex]\\left(0&lt;b&lt;1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Horizontal compression<\/td>\r\n<td>[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] ( [latex]b&gt;1[\/latex] )<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165135264626\" class=\"key-concepts\">\r\n<div><\/div>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li class=\"p1\"><span class=\"s1\">Transformations of functions<\/span>\n<ul>\n<li class=\"p1\"><span class=\"s1\">Graph functions using vertical and horizontal shifts<\/span><\/li>\n<li class=\"p1\"><span class=\"s1\">Graph functions using reflections about the\u00a0<\/span><em><span class=\"s6\">x<\/span><\/em><span class=\"s1\">-axis and the <\/span><span class=\"s4\"><i>y<\/i><\/span><span class=\"s1\">-axis<\/span><\/li>\n<li class=\"p1\"><span class=\"s1\">Graph functions using compressions and stretches<\/span><\/li>\n<li class=\"p1\"><span class=\"s1\">Combine transformations<\/span><\/li>\n<\/ul>\n<\/li>\n<li>Transformations of quadratic functions<\/li>\n<\/ul>\n<\/div>\n<p>We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165307\/CNX_Precalc_Figure_01_05_038n2.jpg\" alt=\"Figure_01_05_038\" width=\"488\" height=\"325\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.\u00a0<\/strong>(credit: &#8220;Misko&#8221;\/Flickr)<\/p>\n<\/div>\n<h2>Shifts<\/h2>\n<p>One simple kind of <strong>transformation<\/strong> involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a <strong>vertical shift<\/strong>, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function [latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex], the function [latex]f\\left(x\\right)[\/latex] is shifted vertically [latex]k[\/latex] units.<\/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\/2862\/2017\/12\/26165309\/CNX_Precalc_Figure_01_05_0012.jpg\" alt=\"Graph of f of x equals the cubed root of x shifted upward one unit, the resulting graph passes through the point (0,1) instead of (0,0), (1, 2) instead of (1,1) and (-1, 0) instead of (-1, -1)\" width=\"487\" height=\"292\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical shift by [latex]k=1[\/latex] of the cube root function [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex].<\/p>\n<\/div>\n<p>To help you visualize the concept of a vertical shift, consider that [latex]y=f\\left(x\\right)[\/latex]. Therefore, [latex]f\\left(x\\right)+k[\/latex] is equivalent to [latex]y+k[\/latex]. Every unit of [latex]y[\/latex] is replaced by [latex]y+k[\/latex], so the [latex]y\\text{-}[\/latex] value increases or decreases depending on the value of [latex]k[\/latex]. The result is a shift upward or downward.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Shift<\/h3>\n<p>Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex], where [latex]k[\/latex] is a constant, is a <strong>vertical shift<\/strong> of the function [latex]f\\left(x\\right)[\/latex]. All the output values change by [latex]k[\/latex] units. If [latex]k[\/latex] is positive, the graph will shift up. If [latex]k[\/latex] is negative, the graph will shift down.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding a Constant to a Function<\/h3>\n<p>To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 2\u00a0shows the area of open vents [latex]V[\/latex] (in square feet) throughout the day in hours after midnight, [latex]t[\/latex]. During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new 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\/2862\/2017\/12\/26165311\/CNX_Precalc_Figure_01_05_0022.jpg\" alt=\"image\" width=\"487\" height=\"326\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716887\">Solution<\/span><\/p>\n<div id=\"q716887\" class=\"hidden-answer\" style=\"display: none\">\nWe can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165313\/CNX_Precalc_Figure_01_05_003a2.jpg\" alt=\"image\" width=\"487\" height=\"329\" \/><\/p>\n<p>Notice that\u00a0for each input value, the output value has increased by 20, so if we call the new function [latex]S\\left(t\\right)[\/latex], we could write<\/p>\n<p style=\"text-align: center\">[latex]S\\left(t\\right)=V\\left(t\\right)+20[\/latex]<\/p>\n<p>This notation tells us that, for any value of [latex]t,S\\left(t\\right)[\/latex] can be found by evaluating the function [latex]V[\/latex] at the same input and then adding 20 to the result. This defines [latex]S[\/latex] as a transformation of the function [latex]V[\/latex], in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change.<\/p>\n<table summary=\"Three rows and seven columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]t[\/latex]<\/strong><\/td>\n<td>0<\/td>\n<td>8<\/td>\n<td>10<\/td>\n<td>17<\/td>\n<td>19<\/td>\n<td>24<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]V\\left(t\\right)[\/latex] <\/strong><\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td>220<\/td>\n<td>220<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]S\\left(t\\right)[\/latex] <\/strong><\/td>\n<td>20<\/td>\n<td>20<\/td>\n<td>240<\/td>\n<td>240<\/td>\n<td>20<\/td>\n<td>20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a tabular function, create a new row to represent a vertical shift.<\/h3>\n<ol>\n<li>Identify the output row or column.<\/li>\n<li>Determine the <strong>magnitude<\/strong> of the shift.<\/li>\n<li>Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Shifting a Tabular Function Vertically<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex].<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q603330\">Solution<\/span><\/p>\n<div id=\"q603330\" class=\"hidden-answer\" style=\"display: none\">\nThe formula [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex] tells us that we can find the output values of [latex]g[\/latex] by subtracting 3 from the output values of [latex]f[\/latex]. For example:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{cases}f\\left(2\\right)=1\\hfill & \\text{Given}\\hfill \\\\ g\\left(x\\right)=f\\left(x\\right)-3\\hfill & \\text{Given transformation}\\hfill \\\\ g\\left(2\\right)=f\\left(2\\right)-3\\hfill & \\hfill \\\\ =1 - 3\\hfill & \\hfill \\\\ =-2\\hfill & \\hfill \\end{cases}[\/latex]<\/p>\n<p>Subtracting 3 from each [latex]f\\left(x\\right)[\/latex] value, we can complete a table of values for [latex]g\\left(x\\right)[\/latex].<\/p>\n<table summary=\"Three rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u22122<\/td>\n<td>0<\/td>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Analysis of the Solution<\/h4>\n<p>As with the earlier vertical shift, notice the input values stay the same and only the output values change.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The function [latex]h\\left(t\\right)=-4.9{t}^{2}+30t[\/latex] gives the height [latex]h[\/latex] of a ball (in meters) thrown upward from the ground after [latex]t[\/latex] seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function [latex]b\\left(t\\right)[\/latex] to [latex]h\\left(t\\right)[\/latex], and then find a formula for [latex]b\\left(t\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q901752\">Solution<\/span><\/p>\n<div id=\"q901752\" class=\"hidden-answer\" style=\"display: none\">[latex]b\\left(t\\right)=h\\left(t\\right)+10=-4.9{t}^{2}+30t+10[\/latex]<\/div>\n<\/div>\n<\/div>\n<h3>Identifying Horizontal Shifts<\/h3>\n<p>We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a <strong>horizontal shift<\/strong>.<\/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\/2862\/2017\/12\/26165316\/CNX_Precalc_Figure_01_05_0042.jpg\" alt=\"Graph of f of x equals the cubed root of x shifted left one unit, the resulting graph passes through the point (0,-1) instead of (0,0), (0, 1) instead of (1,1) and (-2, -1) instead of (-1, -1)\" width=\"487\" height=\"288\" \/><\/p>\n<p class=\"wp-caption-text\">Horizontal shift of the function [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]. Note that [latex]h=+1[\/latex] shifts the graph to the left, that is, towards negative values of [latex]x[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3><\/h3>\n<p>For example, if [latex]f\\left(x\\right)={x}^{2}[\/latex], then [latex]g\\left(x\\right)={\\left(x - 2\\right)}^{2}[\/latex] is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in [latex]f[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Horizontal Shift<\/h3>\n<p>Given a function [latex]f[\/latex], a new function [latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex], where [latex]h[\/latex] is a constant, is a <strong>horizontal shift<\/strong> of the function [latex]f[\/latex]. If [latex]h[\/latex] is positive, the graph will shift right. If [latex]h[\/latex] is negative, the graph will shift left.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding a Constant to an Input<\/h3>\n<p>Returning to our building airflow example from Example 2, suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q839866\">Solution<\/span><\/p>\n<div id=\"q839866\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can set [latex]V\\left(t\\right)[\/latex] to be the original program and [latex]F\\left(t\\right)[\/latex] to be the revised program.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{cases}{c}V\\left(t\\right)=\\text{ the original venting plan}\\\\ \\text{F}\\left(t\\right)=\\text{starting 2 hrs sooner}\\end{cases}[\/latex]<\/p>\n<p>In the new graph, at each time, the airflow is the same as the original function [latex]V[\/latex] was 2 hours later. For example, in the original function [latex]V[\/latex], the airflow starts to change at 8 a.m., whereas for the function [latex]F[\/latex], the airflow starts to change at 6 a.m. The comparable function values are [latex]V\\left(8\\right)=F\\left(6\\right)[\/latex].\u00a0Notice also that the vents first opened to [latex]220{\\text{ ft}}^{2}[\/latex] at 10 a.m. under the original plan, while under the new plan the vents reach [latex]220{\\text{ ft}}^{\\text{2}}[\/latex] at 8 a.m., so [latex]V\\left(10\\right)=F\\left(8\\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\/2862\/2017\/12\/26165318\/CNX_Precalc_Figure_01_05_005a2.jpg\" alt=\"image\" width=\"487\" height=\"329\" \/><\/p>\n<p>In both cases, we see that, because [latex]F\\left(t\\right)[\/latex] starts 2 hours sooner, [latex]h=-2[\/latex]. That means that the same output values are reached when [latex]F\\left(t\\right)=V\\left(t-\\left(-2\\right)\\right)=V\\left(t+2\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that [latex]V\\left(t+2\\right)[\/latex] has the effect of shifting the graph to the <em>left<\/em>.<\/p>\n<p>Horizontal changes or &#8220;inside changes&#8221; affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function [latex]F\\left(t\\right)[\/latex] uses the same outputs as [latex]V\\left(t\\right)[\/latex], but matches those outputs to inputs 2 hours earlier than those of [latex]V\\left(t\\right)[\/latex]. Said another way, we must add 2 hours to the input of [latex]V[\/latex] to find the corresponding output for [latex]F:F\\left(t\\right)=V\\left(t+2\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a tabular function, create a new row to represent a horizontal shift.<\/h3>\n<ol>\n<li>Identify the input row or column.<\/li>\n<li>Determine the magnitude of the shift.<\/li>\n<li>Add the shift to the value in each input cell.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Shifting a Tabular Function Horizontally<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(x - 3\\right)[\/latex].<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719880\">Solution<\/span><\/p>\n<div id=\"q719880\" class=\"hidden-answer\" style=\"display: none\">\nThe formula [latex]g\\left(x\\right)=f\\left(x - 3\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are the same as the output value of [latex]f[\/latex] when the input value is 3 less than the original value. For example, we know that [latex]f\\left(2\\right)=1[\/latex]. To get the same output from the function [latex]g[\/latex], we will need an input value that is 3 <em>larger<\/em>. We input a value that is 3 larger for [latex]g\\left(x\\right)[\/latex] because the function takes 3 away before evaluating the function [latex]f[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{cases}g\\left(5\\right)=f\\left(5 - 3\\right)\\hfill \\\\ =f\\left(2\\right)\\hfill \\\\ =1\\hfill \\end{cases}[\/latex]<\/p>\n<p>We continue with the other values to create this table.<\/p>\n<table summary=\"Three rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>5<\/td>\n<td>7<\/td>\n<td>9<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]x - 3[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The result is that the function [latex]g\\left(x\\right)[\/latex] has been shifted to the right by 3. Notice the output values for [latex]g\\left(x\\right)[\/latex] remain the same as the output values for [latex]f\\left(x\\right)[\/latex], but the corresponding input values, [latex]x[\/latex], have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph below represents both of the functions. We can see the horizontal shift in each point.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165320\/CNX_Precalc_Figure_01_05_0062.jpg\" alt=\"Graph of the points from the previous table for f(x) and g(x)=f(x-3).\" width=\"487\" height=\"549\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying a Horizontal Shift of a Toolkit Function<\/h3>\n<p>This graph represents a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{2}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165321\/CNX_Precalc_Figure_01_05_0072.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"328\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q937293\">Solution<\/span><\/p>\n<div id=\"q937293\" class=\"hidden-answer\" style=\"display: none\">\nNotice that the graph is identical in shape to the [latex]f\\left(x\\right)={x}^{2}[\/latex] function, but the <em>x-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so<\/p>\n<p style=\"text-align: center\">[latex]g\\left(x\\right)=f\\left(x - 2\\right)[\/latex]<\/p>\n<p>Notice how we must input the value [latex]x=2[\/latex] to get the output value [latex]y=0[\/latex]; the<em> x<\/em>-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the [latex]f\\left(x\\right)[\/latex] function to write a formula for [latex]g\\left(x\\right)[\/latex] by evaluating [latex]f\\left(x - 2\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{cases}f\\left(x\\right)={x}^{2}\\hfill \\\\ g\\left(x\\right)=f\\left(x - 2\\right)\\hfill \\\\ g\\left(x\\right)=f\\left(x - 2\\right)={\\left(x - 2\\right)}^{2}\\hfill \\end{cases}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>To determine whether the shift is [latex]+2[\/latex] or [latex]-2[\/latex] , consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function, [latex]f\\left(0\\right)=0[\/latex]. In our shifted function, [latex]g\\left(2\\right)=0[\/latex]. To obtain the output value of 0 from the function [latex]f[\/latex], we need to decide whether a plus or a minus sign will work to satisfy [latex]g\\left(2\\right)=f\\left(x - 2\\right)=f\\left(0\\right)=0[\/latex]. For this to work, we will need to <em>subtract<\/em> 2 units from our input values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Interpreting Horizontal versus Vertical Shifts<\/h3>\n<p>The function [latex]G\\left(m\\right)[\/latex] gives the number of gallons of gas required to drive [latex]m[\/latex] miles. Interpret [latex]G\\left(m\\right)+10[\/latex] and [latex]G\\left(m+10\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792859\">Solution<\/span><\/p>\n<div id=\"q792859\" class=\"hidden-answer\" style=\"display: none\">\n[latex]G\\left(m\\right)+10[\/latex] can be interpreted as adding 10 to the output, gallons. This is the gas required to drive [latex]m[\/latex] miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.<\/p>\n<p>[latex]G\\left(m+10\\right)[\/latex] can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than [latex]m[\/latex] miles. The graph would indicate a horizontal shift.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=\\sqrt{x}[\/latex], graph the original function [latex]f\\left(x\\right)[\/latex] and the transformation [latex]g\\left(x\\right)=f\\left(x+2\\right)[\/latex] on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q193388\">Solution<\/span><\/p>\n<div id=\"q193388\" class=\"hidden-answer\" style=\"display: none\">\n<p>A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflections<\/h2>\n<p>Another transformation that can be applied to a function is a reflection over the <em>x<\/em>&#8211; or <em>y<\/em>-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the <em>x<\/em>-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the <em>y<\/em>-axis. The reflections are shown in Figure 9.<\/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\/2862\/2017\/12\/26165323\/CNX_Precalc_Figure_01_05_0122.jpg\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical and horizontal reflections of a function.<\/p>\n<\/div>\n<p>Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the <em>x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the <em>y<\/em>-axis.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Reflections<\/h3>\n<p>Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] is a <strong>vertical reflection<\/strong> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about (or over, or through) the <em>x<\/em>-axis.<\/p>\n<p>Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] is a <strong>horizontal reflection<\/strong> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about the <em>y<\/em>-axis.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function, reflect the graph both vertically and horizontally.<\/h3>\n<ol>\n<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the <em>x<\/em>-axis.<\/li>\n<li>Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the <em>y<\/em>-axis.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Reflecting a Graph Horizontally and Vertically<\/h3>\n<p>Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex] (a) vertically and (b) horizontally.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211400\">Solution<\/span><\/p>\n<div id=\"q211400\" class=\"hidden-answer\" style=\"display: none\">\na. Reflecting the graph vertically means that each output value will be reflected over the horizontal <em>t-<\/em>axis as shown below.<\/p>\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\/2862\/2017\/12\/26165327\/CNX_Precalc_Figure_01_05_0132.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"975\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical reflection of the square root function<\/p>\n<\/div>\n<p>Because each output value is the opposite of the original output value, we can write<\/p>\n<p style=\"text-align: center\">[latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt{t}[\/latex]<\/p>\n<p>Notice that this is an outside change, or vertical shift, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.<\/p>\n<p>b. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.<\/p>\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\/2862\/2017\/12\/26165329\/CNX_Precalc_Figure_01_05_0142.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"975\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Horizontal reflection of the square root function<\/p>\n<\/div>\n<p>Because each input value is the opposite of the original input value, we can write<\/p>\n<p style=\"text-align: center\">[latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/p>\n<p>Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.<\/p>\n<p>Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\\left[0,\\infty \\right)[\/latex] and range [latex]\\left[0,\\infty \\right)[\/latex], the vertical reflection gives the [latex]V\\left(t\\right)[\/latex] function the range [latex]\\left(-\\infty ,0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,0\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Reflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q362828\">Solution<\/span><\/p>\n<div id=\"q362828\" class=\"hidden-answer\" style=\"display: none\">\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165331\/CNX_Precalc_Figure_01_05_015a2.jpg\" alt=\"Graph of a vertically reflected absolute function.\" width=\"487\" height=\"213\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165333\/CNX_Precalc_Figure_01_05_015b2.jpg\" alt=\"Graph of an absolute function translated one unit left.\" width=\"487\" height=\"251\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Reflecting a Tabular Function Horizontally and Vertically<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.<\/p>\n<ol>\n<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\n<\/ol>\n<table id=\"Table_01_05_05\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q608272\">Solution<\/span><\/p>\n<div id=\"q608272\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the <em>x<\/em>-values stay the same and each output value will be the opposite of the original output value.<br \/>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u20131<\/td>\n<td>\u20133<\/td>\n<td>\u20137<\/td>\n<td>\u201311<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>For [latex]h\\left(x\\right)[\/latex], the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\\left(x\\right)[\/latex] values stay the same as the [latex]f\\left(x\\right)[\/latex] values.<br \/>\n<table id=\"Table_01_05_07\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22124<\/td>\n<td>\u22126<\/td>\n<td>\u22128<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<table id=\"Table_01_05_08\" 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>\u22122<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>5<\/td>\n<td>10<\/td>\n<td>15<\/td>\n<td>20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using the function [latex]f\\left(x\\right)[\/latex] given in the table above, create a table for the functions below.<\/p>\n<p>a. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<p>b. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q230301\">Solution<\/span><\/p>\n<div id=\"q230301\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<br \/>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>-2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]-10[\/latex]<\/td>\n<td>[latex]-15[\/latex]<\/td>\n<td>[latex]-20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<br \/>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>-2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]h\\left(x\\right)[\/latex]<\/td>\n<td>15<\/td>\n<td>10<\/td>\n<td>5<\/td>\n<td>unknown<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Compressions and Stretches<\/h2>\n<p>Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\n<p>We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\n<h3>Vertical Stretches and Compressions<\/h3>\n<p>When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a<strong> vertical compression<\/strong>. The graph below\u00a0shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165339\/CNX_Precalc_Figure_01_05_0242.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"487\" height=\"326\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical stretch and compression<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Stretches and Compressions<\/h3>\n<p>Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=af\\left(x\\right)[\/latex], where [latex]a[\/latex] is a constant, is a <strong>vertical stretch<\/strong> or <strong>vertical compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\n<ul>\n<li>If [latex]a>1[\/latex], then the graph will be stretched.<\/li>\n<li>If 0 &lt; a &lt; 1, then the graph will be compressed.<\/li>\n<li>If [latex]a<0[\/latex], then there will be combination of a vertical stretch or compression with a vertical reflection.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function, graph its vertical stretch.<\/h3>\n<ol>\n<li>Identify the value of [latex]a[\/latex].<\/li>\n<li>Multiply all range values by [latex]a[\/latex].<\/li>\n<li>If [latex]a>1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].<br \/>\nIf [latex]{ 0 }<{ a }<{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].\nIf [latex]a<0[\/latex], the graph is either stretched or compressed and also reflected about the <em>x<\/em>-axis.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Vertical Stretch<\/h3>\n<p>A function [latex]P\\left(t\\right)[\/latex] models the number\u00a0of fruit flies in a population over time, and is graphed below.<\/p>\n<p>A scientist is comparing this population to another population, [latex]Q[\/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165342\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"487\" height=\"367\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951851\">Show Answer<\/span><\/p>\n<div id=\"q951851\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values.<\/p>\n<p>If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.<\/p>\n<p>The following shows where the new points for the new graph will be located.<\/p>\n<p>[latex]\\begin{cases}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{cases}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165344\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<p>Symbolically, the relationship is written as<\/p>\n<p>[latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]<\/p>\n<p>This means that for any input [latex]t[\/latex], the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P[\/latex]. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t[\/latex], stay the same while the output values are twice as large as before.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.<\/h3>\n<ol>\n<li>Determine the value of [latex]a[\/latex].<\/li>\n<li>Multiply all of the output values by [latex]a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Vertical Compression of a Tabular Function<\/h3>\n<p>A function [latex]f[\/latex] is given in the table below. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex].<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798923\">Solution<\/span><\/p>\n<div id=\"q798923\" class=\"hidden-answer\" style=\"display: none\">\nThe formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3[\/latex]. Then<\/p>\n<p>[latex]g\\left(4\\right)=\\frac{1}{2}f\\left(4\\right)=\\frac{1}{2}\\left(3\\right)=\\frac{3}{2}[\/latex]<br \/>\nWe do the same for the other values to produce this table.<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\frac{7}{2}[\/latex]<\/td>\n<td>[latex]\\frac{11}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Analysis of the Solution<\/h4>\n<p>The result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by [latex]\\frac{1}{2}[\/latex]. Each output value is divided in half, so the graph is half the original height.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A function [latex]f[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right)[\/latex].<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<td>20<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q805921\">Solution<\/span><\/p>\n<div id=\"q805921\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x\\\\[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)\\\\[\/latex]<\/td>\n<td>9<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Recognizing a Vertical Stretch<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165346\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"487\" height=\"442\" \/><\/p>\n<p>The graph\u00a0is a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q289067\">Solution<\/span><\/p>\n<div id=\"q289067\" class=\"hidden-answer\" style=\"display: none\">\nWhen trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that [latex]g\\left(2\\right)=2[\/latex]. With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8[\/latex]. Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]g\\left(2\\right)=\\frac{1}{4}f\\left(2\\right)[\/latex]. From this we can fairly safely conclude that [latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)[\/latex].<\/p>\n<p>We can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It 6<\/h3>\n<p>Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473017\">Solution<\/span><\/p>\n<div id=\"q473017\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)=3x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Horizontal Stretches and Compressions<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165348\/CNX_Precalc_Figure_01_05_028.jpg\" alt=\"Graph of the vertical stretch and compression of x^2.\" width=\"487\" height=\"514\" \/><\/p>\n<p>Now we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.<\/p>\n<p>Given a function [latex]y=f\\left(x\\right)[\/latex], the form [latex]y=f\\left(bx\\right)[\/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[\/latex].\u00a0The graph of [latex]y={\\left(0.5x\\right)}^{2}[\/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2. The graph of [latex]y={\\left(2x\\right)}^{2}[\/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Horizontal Stretches and Compressions<\/h3>\n<p>Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex], where [latex]b[\/latex] is a constant, is a <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f\\left(x\\right)[\/latex].<\/p>\n<ul>\n<li>If [latex]b>1[\/latex], then the graph will be compressed by [latex]\\frac{1}{b}[\/latex].<\/li>\n<li>If [latex]0<b<1[\/latex], then the graph will be stretched by [latex]\\frac{1}{b}[\/latex].<\/li>\n<li>If [latex]b<0[\/latex], then there will be combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of a function, sketch a horizontal compression or stretch.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Write a formula to represent the function.<\/li>\n<li>Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b>1[\/latex] for a compression or [latex]0<b<1[\/latex]\nfor a stretch.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Horizontal Compression<\/h3>\n<p>Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, [latex]R[\/latex], will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q855794\">Solution<\/span><\/p>\n<div id=\"q855794\" class=\"hidden-answer\" style=\"display: none\">\nSymbolically, we could write<\/p>\n<p style=\"text-align: center\">[latex]\\begin{cases}R\\left(1\\right)=P\\left(2\\right),\\hfill \\\\ R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,}\\hfill \\\\ R\\left(t\\right)=P\\left(2t\\right).\\hfill \\end{cases}[\/latex]<\/p>\n<p>See below\u00a0for a graphical comparison of the original population and the compressed population.<\/p>\n<div style=\"width: 986px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165351\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is &#091;0,7&#093; and range is &#091;0,3&#093;. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of &#091;0, 3.5&#093; and a range of &#091;0,3&#093;. The maximum occurs at (1.5, 3).\" width=\"976\" height=\"401\" \/><\/p>\n<p class=\"wp-caption-text\">(a) Original population graph (b) Compressed population graph<\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Horizontal Stretch for a Tabular Function<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex].<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261935\">Solution<\/span><\/p>\n<div id=\"q261935\" class=\"hidden-answer\" style=\"display: none\">\nThe formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right)[\/latex], and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/p>\n<p>We do the same for the other values to produce the table below.<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>4<\/td>\n<td>8<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165353\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"975\" height=\"333\" \/><\/p>\n<p>This figure shows the graphs of both of these sets of points.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Because each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Recognizing a Horizontal Compression on a Graph<\/h3>\n<p>Relate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165355\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"487\" height=\"291\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396995\">Solution<\/span><\/p>\n<div id=\"q396995\" class=\"hidden-answer\" style=\"display: none\">\nThe graph of [latex]g\\left(x\\right)[\/latex] looks like the graph of [latex]f\\left(x\\right)[\/latex] horizontally compressed. Because [latex]f\\left(x\\right)[\/latex] ends at [latex]\\left(6,4\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] ends at [latex]\\left(2,4\\right)[\/latex], we can see that the [latex]x\\text{-}[\/latex] values have been compressed by [latex]\\frac{1}{3}[\/latex], because [latex]6\\left(\\frac{1}{3}\\right)=2[\/latex]. We might also notice that [latex]g\\left(2\\right)=f\\left(6\\right)[\/latex] and [latex]g\\left(1\\right)=f\\left(3\\right)[\/latex]. Either way, we can describe this relationship as [latex]g\\left(x\\right)=f\\left(3x\\right)[\/latex]. This is a horizontal compression by [latex]\\frac{1}{3}[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\\frac{1}{4}[\/latex] in our function: [latex]f\\left(\\frac{1}{4}x\\right)[\/latex]. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q35233\">Solution<\/span><\/p>\n<div id=\"q35233\" class=\"hidden-answer\" style=\"display: none\">[latex]g\\left(x\\right)=|x - 1|-3[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Sequences of Transformations<\/h2>\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\/2862\/2017\/12\/26165357\/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\/2862\/2017\/12\/26165359\/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].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q807890\">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\/2862\/2017\/12\/26165401\/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<\/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\/2862\/2017\/12\/26165404\/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\">Solution<\/span><\/p>\n<div id=\"q639112\" class=\"hidden-answer\" style=\"display: none\">\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<\/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)=\\frac{1}{x}[\/latex] that shifts the function\u2019s graph one unit to the right and one unit up.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q126023\">Solution<\/span><\/p>\n<div id=\"q126023\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g\\left(x\\right)=\\frac{1}{{\\left(x+4\\right)}^{2}}+2[\/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\/2862\/2017\/12\/26165406\/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\">Solution<\/span><\/p>\n<div id=\"q533018\" class=\"hidden-answer\" style=\"display: none\">\nThis 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 (0, 1) and (1, 2).<\/p>\n<ol>\n<li>First, we apply a horizontal reflection: (0, 1) (\u20131, 2).<\/li>\n<li>Then, we apply a vertical reflection: (0, \u22121) (1, \u20132).<\/li>\n<li>Finally, we apply a vertical shift: (0, 0) (1, 1).<\/li>\n<\/ol>\n<p>This means that the original points, (0,1) and (1,2) become (0,0) and (1,1) 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\/2862\/2017\/12\/26165408\/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\">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\/2862\/2017\/12\/26165410\/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<h3>Combine Shifts and Stretches<\/h3>\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]h[\/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\">Solution<\/span><\/p>\n<div id=\"q669282\" class=\"hidden-answer\" style=\"display: none\">\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].<\/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 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\/2862\/2017\/12\/26165413\/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\">Solution<\/span><\/p>\n<div id=\"q697686\" class=\"hidden-answer\" style=\"display: none\">\nTo 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 (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165415\/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\/2862\/2017\/12\/26165417\/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\/2862\/2017\/12\/26165419\/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<h2>Transformations of Quadratic Functions<\/h2>\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<div id=\"fs-id1165135320100\" class=\"equation\" style=\"text-align: center\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\n<p id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\n<p>The standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. The figure below\u00a0is the graph of this basic function.<\/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\/25201255\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/><\/p>\n<p>&nbsp;<\/p>\n<h2>Shift Up and Down by Changing the value of k<\/h2>\n<p id=\"fs-id1165137770279\">You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, k.<\/p>\n<p style=\"text-align: center\">[latex]f(x)=x^2 + k[\/latex]<\/p>\n<p>\u00a0If [latex]k>0[\/latex], the graph shifts upward, whereas if [latex]k<0[\/latex], the graph shifts downward.\n\n<strong><span style=\"text-decoration: underline\">Instructions:<\/span><\/strong><\/p>\n<ol>\n<li>Use the slider in the interactive below to shift the graph of [latex]f(x)=x^2[\/latex] down 4 units, then up 4 units.<\/li>\n<li>Use the textbox below the graph to write both transformed equations.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/fpatj6tbcn<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Make Note<\/h3>\n<p>Write the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted up 4 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now check yourself!<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725488\">Show Answer<\/span><\/p>\n<div id=\"q725488\" class=\"hidden-answer\" style=\"display: none\">The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units is<\/p>\n<p>[latex]f(x)=x^2+4[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units is<\/p>\n<p>[latex]f(x)=x^2-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Shift left and right by changing the value of h.<\/h3>\n<p id=\"fs-id1165137770279\">You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, h, to the variable x, before squaring.<\/p>\n<p style=\"text-align: center\">[latex]f(x)=(x-h)^2[\/latex]<\/p>\n<p>If [latex]h>0[\/latex], the graph shifts toward the right and if [latex]h<0[\/latex], the graph shifts to the left.\n\n<strong><span style=\"text-decoration: underline\">Instructions:<\/span><\/strong><\/p>\n<ol>\n<li>Use the interactive graph below to shift the graph of [latex]f(x)=x^2[\/latex] 2 units to the right, then 2 units to the left.<\/li>\n<li>Use the textbox below the graph to write both transformed equations<\/li>\n<\/ol>\n<p>https:\/\/www.desmos.com\/calculator\/5g3xfhkklq<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Make Note<\/h3>\n<p>Write the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted right 2 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted left 2 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now check yourself!<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725588\">Show Answer<\/span><\/p>\n<div id=\"q725588\" class=\"hidden-answer\" style=\"display: none\">The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is<\/p>\n<p>[latex]f(x)=(x-2)^2[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is<\/p>\n<p>[latex]f(x)=(x+2)^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Stretch or compress by changing the value of a.<\/h3>\n<p id=\"fs-id1165137770279\">You can represent a stretch or compression (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, a.<\/p>\n<p style=\"text-align: center\">[latex]f(x)=ax^2[\/latex]<\/p>\n<p>The magnitude of <em>a<\/em>\u00a0indicates the stretch of the graph. If [latex]|a|>1[\/latex], the point associated with a particular <em>x<\/em>-value shifts farther from the <em>x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|<1[\/latex], the point associated with a particular <em>x<\/em>-value shifts closer to the <em>x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression.<\/p>\n<p><strong><span style=\"text-decoration: underline\">Instructions:<\/span><\/strong><\/p>\n<ol>\n<li>Use the interactive graph below to make a graph of the function\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex],<\/li>\n<li>And another that has been vertically stretched by a factor of 3.<\/li>\n<li>Use the textbox below the graph to write your equations.<\/li>\n<\/ol>\n<p>https:\/\/www.desmos.com\/calculator\/ha6gh59rq7<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Make Note<\/h3>\n<p>Write the equation for the graph of [latex]f(x)=x^2[\/latex] that has been has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Then, \u00a0write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now check yourself!<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725489\">Show Answer<\/span><\/p>\n<div id=\"q725489\" class=\"hidden-answer\" style=\"display: none\">The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex]<\/p>\n<p>[latex]f(x)=\\frac{1}{2}x^2[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.is<\/p>\n<p>[latex]f(x)=3x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135353112\">The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\n<div id=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}a{\\left(x-h\\right)}^{2}+k=a{x}^{2}+bx+c\\hfill \\\\ a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)=a{x}^{2}+bx+c\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137409211\">For the linear terms to be equal, the coefficients must be equal.<\/p>\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]-2ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex].<\/div>\n<p id=\"fs-id1165134118295\">This is the <strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal:<\/p>\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{cases}a{h}^{2}+k=c\\hfill \\\\ \\text{ }k=c-a{h}^{2}\\hfill \\\\ \\text{ }=c-a-{\\left(\\frac{b}{2a}\\right)}^{2}\\hfill \\\\ \\text{ }=c-\\frac{{b}^{2}}{4a}\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em>k<\/em> is the output value of the function when the input is <em>h<\/em>, so [latex]f\\left(h\\right)=k[\/latex].<\/p>\n<p>Now you try it.<\/p>\n<p>Use the interactive graph below to define two quadratic functions whose axis of symmetry is x = -3, and whose vertex is (-3, 2). Use the sliders for a, h, k below to help you.<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/pimelalx4i<\/p>\n<p>Now answer the following questions about the graphs you made.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Take Note<\/h3>\n<p>How many potential values are there for h in this scenario?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>How about k?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>How about a?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q349748\">Show Answer<\/span><\/p>\n<div id=\"q349748\" class=\"hidden-answer\" style=\"display: none\">\n<p>There is only one [latex](h,k)[\/latex] pair that will satisfy these conditions,\u00a0[latex](-3,2)[\/latex]. \u00a0The value of a does not affect the line of symmetry or the vertex of a quadratic graph, so a can be an infinite number of values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>\u00a0Summary of Transformations<\/h2>\n<section id=\"fs-id1165135499979\" class=\"key-equations\">\n<table id=\"eip-id1165134474082\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>Vertical shift<\/td>\n<td>[latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex] (up for [latex]k>0[\/latex] )<\/td>\n<\/tr>\n<tr>\n<td>Horizontal shift<\/td>\n<td>[latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex] (right for [latex]h>0[\/latex] )<\/td>\n<\/tr>\n<tr>\n<td>Vertical reflection<\/td>\n<td>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Horizontal reflection<\/td>\n<td>[latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Vertical stretch<\/td>\n<td>[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] ( [latex]a>0[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Vertical compression<\/td>\n<td>[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] [latex]\\left(0<a<1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Horizontal stretch<\/td>\n<td>[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] [latex]\\left(0<b<1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Horizontal compression<\/td>\n<td>[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] ( [latex]b>1[\/latex] )<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135264626\" class=\"key-concepts\">\n<div><\/div>\n<\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4664\">\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 113437, 60789, 112701, 60650, 113454, 112703, 112707, 112726, 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><li>Interactive: Transform Quadratic 1. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/fpatj6tbcn\">https:\/\/www.desmos.com\/calculator\/fpatj6tbcn<\/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>Interactive: Transform Quadratic 2. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 3. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/ha6gh59rq7\">https:\/\/www.desmos.com\/calculator\/ha6gh59rq7<\/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>Interactive: Transform Quadratic 4. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/pimelalx4i\">https:\/\/www.desmos.com\/calculator\/pimelalx4i<\/a>. <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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 75586, 75929, 75931. <strong>Authored by<\/strong>: Shahbazian, Roy. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 33066. <strong>Authored by<\/strong>: Smart, Jim. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 74696, 74730. <strong>Authored by<\/strong>: Meacham, William. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 60791, 60790. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":23485,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 75586, 75929, 75931\",\"author\":\"Shahbazian, Roy\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Question ID 113437, 60789, 112701, 60650, 113454, 112703, 112707, 112726, 113225\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 33066\",\"author\":\"Smart, Jim\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 74696, 74730\",\"author\":\"Meacham, William\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 60791, 60790\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 1\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/fpatj6tbcn\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 2\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 3\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/ha6gh59rq7\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 4\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/pimelalx4i\",\"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-4664","chapter","type-chapter","status-publish","hentry"],"part":1897,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4664","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4664\/revisions"}],"predecessor-version":[{"id":5109,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4664\/revisions\/5109"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1897"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4664\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=4664"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4664"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=4664"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=4664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}