{"id":154,"date":"2023-06-21T13:22:38","date_gmt":"2023-06-21T13:22:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-transformations-of-functions\/"},"modified":"2023-07-13T21:25:45","modified_gmt":"2023-07-13T21:25:45","slug":"summary-transformations-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-transformations-of-functions\/","title":{"raw":"Summary: Transformations of Functions","rendered":"Summary: Transformations of Functions"},"content":{"raw":"<h2>\u00a0Key Equations<\/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;1[\/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<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135264630\">\r\n \t<li>A function can be shifted vertically by adding a constant to the output.<\/li>\r\n \t<li>A function can be shifted horizontally by adding a constant to the input.<\/li>\r\n \t<li>Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.<\/li>\r\n \t<li>Vertical and horizontal shifts are often combined.<\/li>\r\n \t<li>A vertical reflection reflects a graph about the [latex]x\\text{-}[\/latex] axis. A graph can be reflected vertically by multiplying the output by \u20131.<\/li>\r\n \t<li>A horizontal reflection reflects a graph about the [latex]y\\text{-}[\/latex] axis. A graph can be reflected horizontally by multiplying the input by \u20131.<\/li>\r\n \t<li>A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.<\/li>\r\n \t<li>A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.<\/li>\r\n \t<li>A function presented as an equation can be reflected by applying transformations one at a time.<\/li>\r\n \t<li>Even functions are symmetric about the [latex]y\\text{-}[\/latex] axis, whereas odd functions are symmetric about the origin.<\/li>\r\n \t<li>Even functions satisfy the condition [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].<\/li>\r\n \t<li>Odd functions satisfy the condition [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex].<\/li>\r\n \t<li>A function can be odd, even, or neither.<\/li>\r\n \t<li>A function can be compressed or stretched vertically by multiplying the output by a constant.<\/li>\r\n \t<li>A function can be compressed or stretched horizontally by multiplying the input by a constant.<\/li>\r\n \t<li>The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137448239\" class=\"definition\">\r\n \t<dt>even function<\/dt>\r\n \t<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex], and is symmetric about the [latex]y\\text{-}[\/latex] axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133242964\" class=\"definition\">\r\n \t<dt>horizontal compression<\/dt>\r\n \t<dd id=\"fs-id1165137833874\">a transformation that compresses a function\u2019s graph horizontally, by multiplying the input by a constant [latex]b&gt;1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135440170\" class=\"definition\">\r\n \t<dt>horizontal reflection<\/dt>\r\n \t<dd id=\"fs-id1165137602051\">a transformation that reflects a function\u2019s graph across the <em>y<\/em>-axis by multiplying the input by [latex]-1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137922367\" class=\"definition\">\r\n \t<dt>horizontal shift<\/dt>\r\n \t<dd id=\"fs-id1165137922373\">a transformation that shifts a function\u2019s graph left or right by adding a positive or negative constant to the input<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137922379\" class=\"definition\">\r\n \t<dt>horizontal stretch<\/dt>\r\n \t<dd id=\"fs-id1165135675238\">a transformation that stretches a function\u2019s graph horizontally by multiplying the input by a constant [latex]0&lt;b&lt;1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134259240\" class=\"definition\">\r\n \t<dt>odd function<\/dt>\r\n \t<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex], and is symmetric about the origin<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137676545\" class=\"definition\">\r\n \t<dt>vertical compression<\/dt>\r\n \t<dd id=\"fs-id1165137676551\">a function transformation that compresses the function\u2019s graph vertically by multiplying the output by a constant [latex]0&lt;a&lt;1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137662611\" class=\"definition\">\r\n \t<dt>vertical reflection<\/dt>\r\n \t<dd id=\"fs-id1165137834403\">a transformation that reflects a function\u2019s graph across the <em>x<\/em>-axis by multiplying the output by [latex]-1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135580354\" class=\"definition\">\r\n \t<dt>vertical shift<\/dt>\r\n \t<dd id=\"fs-id1165137862443\">a transformation that shifts a function\u2019s graph up or down by adding a positive or negative constant to the output<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137862450\" class=\"definition\">\r\n \t<dt>vertical stretch<\/dt>\r\n \t<dd id=\"fs-id1165132971698\">a transformation that stretches a function\u2019s graph vertically by multiplying the output by a constant [latex]a&gt;1[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<h2>\u00a0Key Equations<\/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>1[\/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<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135264630\">\n<li>A function can be shifted vertically by adding a constant to the output.<\/li>\n<li>A function can be shifted horizontally by adding a constant to the input.<\/li>\n<li>Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.<\/li>\n<li>Vertical and horizontal shifts are often combined.<\/li>\n<li>A vertical reflection reflects a graph about the [latex]x\\text{-}[\/latex] axis. A graph can be reflected vertically by multiplying the output by \u20131.<\/li>\n<li>A horizontal reflection reflects a graph about the [latex]y\\text{-}[\/latex] axis. A graph can be reflected horizontally by multiplying the input by \u20131.<\/li>\n<li>A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.<\/li>\n<li>A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.<\/li>\n<li>A function presented as an equation can be reflected by applying transformations one at a time.<\/li>\n<li>Even functions are symmetric about the [latex]y\\text{-}[\/latex] axis, whereas odd functions are symmetric about the origin.<\/li>\n<li>Even functions satisfy the condition [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].<\/li>\n<li>Odd functions satisfy the condition [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex].<\/li>\n<li>A function can be odd, even, or neither.<\/li>\n<li>A function can be compressed or stretched vertically by multiplying the output by a constant.<\/li>\n<li>A function can be compressed or stretched horizontally by multiplying the input by a constant.<\/li>\n<li>The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137448239\" class=\"definition\">\n<dt>even function<\/dt>\n<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex], and is symmetric about the [latex]y\\text{-}[\/latex] axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133242964\" class=\"definition\">\n<dt>horizontal compression<\/dt>\n<dd id=\"fs-id1165137833874\">a transformation that compresses a function\u2019s graph horizontally, by multiplying the input by a constant [latex]b>1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135440170\" class=\"definition\">\n<dt>horizontal reflection<\/dt>\n<dd id=\"fs-id1165137602051\">a transformation that reflects a function\u2019s graph across the <em>y<\/em>-axis by multiplying the input by [latex]-1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137922367\" class=\"definition\">\n<dt>horizontal shift<\/dt>\n<dd id=\"fs-id1165137922373\">a transformation that shifts a function\u2019s graph left or right by adding a positive or negative constant to the input<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137922379\" class=\"definition\">\n<dt>horizontal stretch<\/dt>\n<dd id=\"fs-id1165135675238\">a transformation that stretches a function\u2019s graph horizontally by multiplying the input by a constant [latex]0<b<1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134259240\" class=\"definition\">\n<dt>odd function<\/dt>\n<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex], and is symmetric about the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137676545\" class=\"definition\">\n<dt>vertical compression<\/dt>\n<dd id=\"fs-id1165137676551\">a function transformation that compresses the function\u2019s graph vertically by multiplying the output by a constant [latex]0<a<1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137662611\" class=\"definition\">\n<dt>vertical reflection<\/dt>\n<dd id=\"fs-id1165137834403\">a transformation that reflects a function\u2019s graph across the <em>x<\/em>-axis by multiplying the output by [latex]-1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135580354\" class=\"definition\">\n<dt>vertical shift<\/dt>\n<dd id=\"fs-id1165137862443\">a transformation that shifts a function\u2019s graph up or down by adding a positive or negative constant to the output<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137862450\" class=\"definition\">\n<dt>vertical stretch<\/dt>\n<dd id=\"fs-id1165132971698\">a transformation that stretches a function\u2019s graph vertically by multiplying the output by a constant [latex]a>1[\/latex]<\/dd>\n<\/dl>\n<\/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-154\">\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><\/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><\/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":395986,"menu_order":31,"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\"}]","CANDELA_OUTCOMES_GUID":"24cd2249-f76f-43f0-bac2-0caf5f1602d5","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-154","chapter","type-chapter","status-publish","hentry"],"part":115,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/154","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/154\/revisions"}],"predecessor-version":[{"id":986,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/154\/revisions\/986"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/115"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/154\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=154"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=154"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=154"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}