{"id":44,"date":"2025-02-13T22:43:07","date_gmt":"2025-02-13T22:43:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/transformation-of-functions\/"},"modified":"2026-01-08T18:17:55","modified_gmt":"2026-01-08T18:17:55","slug":"transformation-of-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/transformation-of-functions\/","title":{"raw":"Transformation of Functions: Learn It 1","rendered":"Transformation of Functions: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Graph functions using a single transformation.<\/li>\r\n \t<li style=\"font-weight: 400;\">Graph functions using a combination of transformations.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine whether a function is even, odd, or neither from its graph.<\/li>\r\n \t<li style=\"font-weight: 400;\">Describe transformations based on a function formula.<\/li>\r\n \t<li style=\"font-weight: 400;\">Give the formula of a function based on its transformations.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<figure id=\"Figure_01_05_001\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010515\/CNX_Precalc_Figure_01_05_038n2.jpg\" alt=\"Figure_01_05_038\" width=\"488\" height=\"325\" \/> (credit: \"Misko\"\/Flickr)[\/caption]<\/figure>\r\n<p id=\"fs-id1165137742090\">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>\r\n\r\n<h2>Graphing Functions Using Reflections about the Axes<\/h2>\r\nAnother transformation that can be applied to a function is a reflection over the [latex]x[\/latex]- or [latex]y[\/latex]-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis.\r\n\r\n<img class=\"wp-image-4920 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_-294x300.png\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"487\" height=\"496\" \/>\r\n\r\n&nbsp;\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>reflections<\/h3>\r\nA <strong>vertical reflection<\/strong> <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">reflects a graph vertically across the [latex]x[\/latex]-axis. <\/span>This transformation changes the sign of the output values of [latex]f(x)[\/latex].\r\n<ul>\r\n \t<li>If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]x[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]g(x) = -f(x)[\/latex]<\/p>\r\n&nbsp;\r\n\r\nA <strong>horizontal reflection<\/strong> <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">reflects a graph horizontally across the [latex]y[\/latex]-axis. <\/span>This transformation changes the sign of the input values of [latex]f(x)[\/latex].\r\n<ul>\r\n \t<li>If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]y[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]g(x) = f(-x)[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a function, reflect the graph both vertically and horizontally.<\/strong>\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 [latex]x[\/latex]-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 [latex]y[\/latex]-axis.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>vertically<\/li>\r\n \t<li>horizontally<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"211400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211400\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Reflecting the graph vertically means that each output value will be reflected over the horizontal [latex]t[\/latex]<em>-<\/em>axis as shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"777\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203559\/CNX_Precalc_Figure_01_05_0132.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"777\" height=\"352\" \/> Vertical reflection of the square root function[\/caption]\r\n\r\n<strong>Table of values<\/strong>\r\n<table style=\"border-collapse: collapse; width: 89.9698%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 18.8123%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 30.0399%;\">[latex]s(t) = \\sqrt{t}[\/latex]<\/td>\r\n<td style=\"width: 41.1178%;\">Reflected Function [latex]V(t)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.8123%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 30.0399%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 41.1178%;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.8123%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 30.0399%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 41.1178%;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.8123%;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 30.0399%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 41.1178%;\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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.<\/li>\r\n \t<li>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=\"777\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203602\/CNX_Precalc_Figure_01_05_0142.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"777\" height=\"352\" \/> Horizontal reflection of the square root function[\/caption]\r\n\r\n<strong>Table for [latex]s(t) = \\sqrt{t}[\/latex]<\/strong>\r\n<table style=\"border-collapse: collapse; width: 77.5426%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17.3652%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 17.6647%;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]s(t) = \\sqrt{t}[\/latex]<\/td>\r\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17.3652%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 17.6647%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Table for reflected function\u00a0<\/strong>\r\n<table style=\"border-collapse: collapse; width: 77.5426%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\r\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17.3652%;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 17.6647%;\">[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">[latex]H(t)[\/latex]<\/td>\r\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17.3652%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 17.6647%;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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.<\/li>\r\n<\/ol>\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].[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A 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>[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]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"608272\"]Show 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 [latex]x[\/latex]-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>[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]\u20131[\/latex]<\/td>\r\n<td>[latex]\u20133[\/latex]<\/td>\r\n<td>[latex]\u20137[\/latex]<\/td>\r\n<td>[latex]\u201311[\/latex]<\/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>[latex]\u22122[\/latex]<\/td>\r\n<td>[latex]\u22124[\/latex]<\/td>\r\n<td>[latex]\u22126[\/latex]<\/td>\r\n<td>[latex]\u22128[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317914[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317915[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317916[\/ohm_question]<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Graph functions using a single transformation.<\/li>\n<li style=\"font-weight: 400;\">Graph functions using a combination of transformations.<\/li>\n<li style=\"font-weight: 400;\">Determine whether a function is even, odd, or neither from its graph.<\/li>\n<li style=\"font-weight: 400;\">Describe transformations based on a function formula.<\/li>\n<li style=\"font-weight: 400;\">Give the formula of a function based on its transformations.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<figure id=\"Figure_01_05_001\" class=\"medium\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010515\/CNX_Precalc_Figure_01_05_038n2.jpg\" alt=\"Figure_01_05_038\" width=\"488\" height=\"325\" \/><figcaption class=\"wp-caption-text\">(credit: &#8220;Misko&#8221;\/Flickr)<\/figcaption><\/figure>\n<\/figure>\n<p id=\"fs-id1165137742090\">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<h2>Graphing Functions Using Reflections about the Axes<\/h2>\n<p>Another transformation that can be applied to a function is a reflection over the [latex]x[\/latex]&#8211; or [latex]y[\/latex]-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4920 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_-294x300.png\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"487\" height=\"496\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_-294x300.png 294w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_-65x66.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_-225x230.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_-350x357.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02212751\/2.2.L1.Graph_.png 588w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/p>\n<p>&nbsp;<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>reflections<\/h3>\n<p>A <strong>vertical reflection<\/strong> <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">reflects a graph vertically across the [latex]x[\/latex]-axis. <\/span>This transformation changes the sign of the output values of [latex]f(x)[\/latex].<\/p>\n<ul>\n<li>If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]x[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]g(x) = -f(x)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>A <strong>horizontal reflection<\/strong> <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">reflects a graph horizontally across the [latex]y[\/latex]-axis. <\/span>This transformation changes the sign of the input values of [latex]f(x)[\/latex].<\/p>\n<ul>\n<li>If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]y[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]g(x) = f(-x)[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a function, reflect the graph both vertically and horizontally.<\/strong><\/p>\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 [latex]x[\/latex]-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 [latex]y[\/latex]-axis.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex]<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>vertically<\/li>\n<li>horizontally<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q211400\">Show Solution<\/button><\/p>\n<div id=\"q211400\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Reflecting the graph vertically means that each output value will be reflected over the horizontal [latex]t[\/latex]<em>&#8211;<\/em>axis as shown below.<br \/>\n<figure style=\"width: 777px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203559\/CNX_Precalc_Figure_01_05_0132.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"777\" height=\"352\" \/><figcaption class=\"wp-caption-text\">Vertical reflection of the square root function<\/figcaption><\/figure>\n<p><strong>Table of values<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 89.9698%;\">\n<tbody>\n<tr>\n<td style=\"width: 18.8123%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 30.0399%;\">[latex]s(t) = \\sqrt{t}[\/latex]<\/td>\n<td style=\"width: 41.1178%;\">Reflected Function [latex]V(t)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.8123%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 30.0399%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 41.1178%;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.8123%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 30.0399%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 41.1178%;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.8123%;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 30.0399%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 41.1178%;\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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.<\/li>\n<li>Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.<br \/>\n<figure style=\"width: 777px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203602\/CNX_Precalc_Figure_01_05_0142.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"777\" height=\"352\" \/><figcaption class=\"wp-caption-text\">Horizontal reflection of the square root function<\/figcaption><\/figure>\n<p><strong>Table for [latex]s(t) = \\sqrt{t}[\/latex]<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 77.5426%;\">\n<tbody>\n<tr>\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17.3652%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 17.6647%;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]s(t) = \\sqrt{t}[\/latex]<\/td>\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17.3652%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 17.6647%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Table for reflected function\u00a0<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 77.5426%;\">\n<tbody>\n<tr>\n<td style=\"width: 25%;\">[latex]t[\/latex]<\/td>\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17.3652%;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 17.6647%;\">[latex]-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">[latex]H(t)[\/latex]<\/td>\n<td style=\"width: 17.515%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17.3652%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 17.6647%;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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.<\/li>\n<\/ol>\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><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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>[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]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q608272\">Show Solution<\/button><\/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 [latex]x[\/latex]-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>[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]\u20131[\/latex]<\/td>\n<td>[latex]\u20133[\/latex]<\/td>\n<td>[latex]\u20137[\/latex]<\/td>\n<td>[latex]\u201311[\/latex]<\/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. 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