{"id":1928,"date":"2024-06-24T22:45:15","date_gmt":"2024-06-24T22:45:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1928"},"modified":"2025-08-15T02:40:44","modified_gmt":"2025-08-15T02:40:44","slug":"graphs-of-polynomial-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-of-polynomial-functions-learn-it-4\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 4","rendered":"Graphs of Polynomial Functions: Learn It 4"},"content":{"raw":"<h2>Graphing Polynomial Functions<\/h2>\r\nNow that we've mastered identifying zeros and understanding their multiplicities and behavior, it's time to bring everything together and graph polynomial functions. Graphing these functions allows us to visualize their behavior and understand the relationships between their algebraic expressions and their graphical representations.\r\n\r\nTo accurately graph a polynomial function, we\u2019ll consider several key elements:\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li><strong>Zeros and Their Behavior:<\/strong> Where the graph crosses or touches the [latex]x[\/latex]-axis.\r\n<ul>\r\n \t<li>For zeros with even multiplicities, the graphs touch or are tangent to the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values.<\/li>\r\n \t<li>For zeros with odd multiplicities, the graphs cross or intersect the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>[latex]y[\/latex] - intercepts:<\/strong> Points where the graph intersects the [latex]y[\/latex]-axis.<\/li>\r\n \t<li><strong>End Behavior:<\/strong> How the graph behaves as [latex]x[\/latex]\u00a0approaches positive or negative infinity.\r\n[reveal-answer q=\"558286\"]Determining End Behavior[\/reveal-answer]\r\n[hidden-answer a=\"558286\"]\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">Even Degree<\/th>\r\n<th style=\"text-align: center;\">Odd Degree<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\r\n\r\n[caption id=\"attachment_12504\" align=\"alignnone\" width=\"423\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/11.png\"><img class=\"wp-image-12504 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201602\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a> End behavior for positive leading coefficient[\/caption]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12505\" align=\"alignnone\" width=\"397\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/12.png\"><img class=\"wp-image-12505 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201605\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a> End behavior for positive leading coefficient[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n\r\n[caption id=\"attachment_12506\" align=\"alignnone\" width=\"387\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/13.png\"><img class=\"wp-image-12506 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201607\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a> End behavior for negative leading coefficient[\/caption]<\/td>\r\n<td>\r\n\r\n[caption id=\"attachment_12507\" align=\"alignnone\" width=\"404\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/14.png\"><img class=\"wp-image-12507 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201609\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a> End behavior for negative leading coefficient[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]<\/li>\r\n \t<li><strong>Local Behavior:\u00a0 <\/strong>How the graph behaves near critical points, including [pb_glossary id=\"4311\"]turning points[\/pb_glossary] and changes in concavity<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial function, sketch the graph<\/strong>\r\n<ol>\r\n \t<li>Find the intercepts.<\/li>\r\n \t<li>Check for symmetry. If the function is an even function, its graph is symmetric with respect to the\u00a0[latex]y[\/latex]-axis, that is,\u00a0[latex]f(\u2013x) = f(x)[\/latex].\r\nIf a function is an odd function, its graph is symmetric with respect to the origin, that is,\u00a0[latex]f(\u2013x) = \u2013f(x)[\/latex].<\/li>\r\n \t<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the [latex]x[\/latex]-intercepts.<\/li>\r\n \t<li>Determine the end behavior by examining the leading term.<\/li>\r\n \t<li>Use the end behavior and the behavior at the intercepts to sketch the graph.<\/li>\r\n \t<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\r\n \t<li>Optionally, use technology to check the graph.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Graph [latex]f(x)=-2(x+3)^{2}(x - 5)[\/latex].[reveal-answer q=\"707233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"707233\"]\r\n<ul>\r\n \t<li><strong>Zeros and Their Behavior<\/strong><\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 40px;\">This graph has two [latex]x[\/latex]<em>-<\/em>intercepts.<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>At [latex]x= \u20133[\/latex], the factor is squared, indicating a multiplicity of [latex]2[\/latex]. The graph will touch the [latex]x[\/latex]-intercept at this value.<\/li>\r\n \t<li>At [latex]x\u00a0= 5[\/latex], the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>[latex]y[\/latex] - intercepts: <\/strong>The [latex]y[\/latex]-intercept is found by evaluating [latex]f(0)[\/latex].<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ f\\left(0\\right)=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=-2\\cdot 9\\cdot \\left(-5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=90\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"padding-left: 40px;\">The <em>y<\/em>-intercept is [latex](0, 90)[\/latex].<\/p>\r\n\r\n<ul>\r\n \t<li>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"199\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201614\/CNX_Precalc_Figure_03_04_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"199\" height=\"148\" \/> Graph looking at end behavior[\/caption]\r\n\r\n<strong>End Behavior: <\/strong>We can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex], a negative odd degree polynomial. So the end behavior, as seen in the graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.<\/li>\r\n<\/ul>\r\nTo sketch the graph, we consider the following:\r\n<ul id=\"fs-id1165134374741\">\r\n \t<li>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"345\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201618\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"345\" height=\"256\" \/> Graph including end behavior and labeled points[\/caption]\r\n\r\nAs [latex]x\\to -\\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\infty [\/latex], so we know the graph starts in the second quadrant and is decreasing toward the [latex]x[\/latex]-axis.<\/li>\r\n \t<li>At [latex]\\left(-3,0\\right)[\/latex] the graph touch of the <em>x<\/em>-axis, so the function must start increasing after this point.<\/li>\r\n \t<li>At [latex](0, 90)[\/latex], the graph crosses the <em>y<\/em>-axis.<\/li>\r\n \t<li>At [latex]\\left(5,0\\right)[\/latex] the graph cross of the <em>x<\/em>-axis.<\/li>\r\n<\/ul>\r\nThe complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex] is as follows:\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\/896\/2016\/11\/02201620\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/> Graph of a polynomial[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]24612[\/ohm2_question]<\/section>","rendered":"<h2>Graphing Polynomial Functions<\/h2>\n<p>Now that we&#8217;ve mastered identifying zeros and understanding their multiplicities and behavior, it&#8217;s time to bring everything together and graph polynomial functions. Graphing these functions allows us to visualize their behavior and understand the relationships between their algebraic expressions and their graphical representations.<\/p>\n<p>To accurately graph a polynomial function, we\u2019ll consider several key elements:<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><strong>Zeros and Their Behavior:<\/strong> Where the graph crosses or touches the [latex]x[\/latex]-axis.\n<ul>\n<li>For zeros with even multiplicities, the graphs touch or are tangent to the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values.<\/li>\n<li>For zeros with odd multiplicities, the graphs cross or intersect the [latex]x[\/latex]-axis at these [latex]x[\/latex]-values.<\/li>\n<\/ul>\n<\/li>\n<li><strong>[latex]y[\/latex] &#8211; intercepts:<\/strong> Points where the graph intersects the [latex]y[\/latex]-axis.<\/li>\n<li><strong>End Behavior:<\/strong> How the graph behaves as [latex]x[\/latex]\u00a0approaches positive or negative infinity.\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q558286\">Determining End Behavior<\/button><\/p>\n<div id=\"q558286\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Even Degree<\/th>\n<th style=\"text-align: center;\">Odd Degree<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\n<figure id=\"attachment_12504\" aria-describedby=\"caption-attachment-12504\" style=\"width: 423px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12504 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201602\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><figcaption id=\"caption-attachment-12504\" class=\"wp-caption-text\">End behavior for positive leading coefficient<\/figcaption><\/figure>\n<\/td>\n<td>\n<figure id=\"attachment_12505\" aria-describedby=\"caption-attachment-12505\" style=\"width: 397px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/12.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12505 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201605\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><figcaption id=\"caption-attachment-12505\" class=\"wp-caption-text\">End behavior for positive leading coefficient<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<figure id=\"attachment_12506\" aria-describedby=\"caption-attachment-12506\" style=\"width: 387px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/13.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12506 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201607\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><figcaption id=\"caption-attachment-12506\" class=\"wp-caption-text\">End behavior for negative leading coefficient<\/figcaption><\/figure>\n<\/td>\n<td>\n<figure id=\"attachment_12507\" aria-describedby=\"caption-attachment-12507\" style=\"width: 404px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/14.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12507 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201609\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><figcaption id=\"caption-attachment-12507\" class=\"wp-caption-text\">End behavior for negative leading coefficient<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/li>\n<li><strong>Local Behavior:\u00a0 <\/strong>How the graph behaves near critical points, including <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1928_4311\">turning points<\/a> and changes in concavity<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a polynomial function, sketch the graph<\/strong><\/p>\n<ol>\n<li>Find the intercepts.<\/li>\n<li>Check for symmetry. If the function is an even function, its graph is symmetric with respect to the\u00a0[latex]y[\/latex]-axis, that is,\u00a0[latex]f(\u2013x) = f(x)[\/latex].<br \/>\nIf a function is an odd function, its graph is symmetric with respect to the origin, that is,\u00a0[latex]f(\u2013x) = \u2013f(x)[\/latex].<\/li>\n<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the [latex]x[\/latex]-intercepts.<\/li>\n<li>Determine the end behavior by examining the leading term.<\/li>\n<li>Use the end behavior and the behavior at the intercepts to sketch the graph.<\/li>\n<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\n<li>Optionally, use technology to check the graph.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Graph [latex]f(x)=-2(x+3)^{2}(x - 5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q707233\">Show Solution<\/button><\/p>\n<div id=\"q707233\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li><strong>Zeros and Their Behavior<\/strong><\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">This graph has two [latex]x[\/latex]<em>&#8211;<\/em>intercepts.<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>At [latex]x= \u20133[\/latex], the factor is squared, indicating a multiplicity of [latex]2[\/latex]. The graph will touch the [latex]x[\/latex]-intercept at this value.<\/li>\n<li>At [latex]x\u00a0= 5[\/latex], the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/li>\n<\/ul>\n<\/li>\n<li><strong>[latex]y[\/latex] &#8211; intercepts: <\/strong>The [latex]y[\/latex]-intercept is found by evaluating [latex]f(0)[\/latex].<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ f\\left(0\\right)=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=-2\\cdot 9\\cdot \\left(-5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=90\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"padding-left: 40px;\">The <em>y<\/em>-intercept is [latex](0, 90)[\/latex].<\/p>\n<ul>\n<li>\n<figure style=\"width: 199px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201614\/CNX_Precalc_Figure_03_04_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"199\" height=\"148\" \/><figcaption class=\"wp-caption-text\">Graph looking at end behavior<\/figcaption><\/figure>\n<p><strong>End Behavior: <\/strong>We can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex], a negative odd degree polynomial. So the end behavior, as seen in the graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.<\/li>\n<\/ul>\n<p>To sketch the graph, we consider the following:<\/p>\n<ul id=\"fs-id1165134374741\">\n<li>\n<figure style=\"width: 345px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201618\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"345\" height=\"256\" \/><figcaption class=\"wp-caption-text\">Graph including end behavior and labeled points<\/figcaption><\/figure>\n<p>As [latex]x\\to -\\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\infty[\/latex], so we know the graph starts in the second quadrant and is decreasing toward the [latex]x[\/latex]-axis.<\/li>\n<li>At [latex]\\left(-3,0\\right)[\/latex] the graph touch of the <em>x<\/em>-axis, so the function must start increasing after this point.<\/li>\n<li>At [latex](0, 90)[\/latex], the graph crosses the <em>y<\/em>-axis.<\/li>\n<li>At [latex]\\left(5,0\\right)[\/latex] the graph cross of the <em>x<\/em>-axis.<\/li>\n<\/ul>\n<p>The complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex] is as follows:<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201620\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/><figcaption class=\"wp-caption-text\">Graph of a polynomial<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm24612\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24612&theme=lumen&iframe_resize_id=ohm24612&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_1928_4311\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1928_4311\"><div tabindex=\"-1\"><p>A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).A polynomial of degree n will have at most n\u22121 turning points.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":206,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1928"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1928\/revisions"}],"predecessor-version":[{"id":7758,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1928\/revisions\/7758"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/206"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1928\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1928"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1928"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1928"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1928"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}