{"id":928,"date":"2025-07-16T18:20:57","date_gmt":"2025-07-16T18:20:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=928"},"modified":"2026-01-13T16:18:59","modified_gmt":"2026-01-13T16:18:59","slug":"graphs-of-polynomial-functions-learn-it-5-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions-learn-it-5-2\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 6","rendered":"Graphs of Polynomial Functions: Learn It 6"},"content":{"raw":"<h2>Writing Formulas for Polynomial Functions<\/h2>\r\n<div class=\"page\" title=\"Page 532\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nNow that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an [latex]x[\/latex]-intercept where each factor is equal to zero, we can form a function that will pass through a set of [latex]x[\/latex]-intercepts by introducing a corresponding set of factors.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>factored form of polynomials<\/h3>\r\nIf a polynomial of lowest degree [latex]p[\/latex]\u00a0has zeros at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form:\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nwhere\r\n<ul>\r\n \t<li>the powers [latex]{p}_{i}[\/latex] on each factor can be determined by the behavior of the graph at the corresponding intercept<\/li>\r\n \t<li>the stretch factor [latex]a[\/latex] can be determined given a value of the function other than the [latex]x[\/latex]-intercept.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a graph of a polynomial function, write a formula for the function<\/strong>\r\n<ol>\r\n \t<li>Identify the [latex]x[\/latex]-intercepts of the graph to find the factors of the polynomial.<\/li>\r\n \t<li>Examine the behavior of the graph at the [latex]x[\/latex]-intercepts to determine the multiplicity of each factor.<\/li>\r\n \t<li>Find the polynomial of least degree containing all of the factors found in the previous step.<\/li>\r\n \t<li>Use any other point on the graph (the [latex]y[\/latex]-intercept may be easiest) to determine the stretch factor.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Write a formula for the polynomial function.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201627\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/>[reveal-answer q=\"338564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338564\"]This graph has three [latex]x[\/latex]-intercepts: [latex]x= \u20133, 2,[\/latex] and [latex]5[\/latex].\r\n<ul>\r\n \t<li>At [latex]x= \u20133[\/latex] and [latex]x= 5[\/latex], the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear with a degree of [latex]1[\/latex].<\/li>\r\n \t<li>At [latex]x= 2[\/latex], the graph bounces off the [latex]x[\/latex]-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree (quadratic).<\/li>\r\n<\/ul>\r\nTogether, this gives us\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\r\nTo determine the stretch factor, we utilize another point on the graph. We will use the [latex]y[\/latex]-intercept [latex](0, \u20132)[\/latex], to solve for [latex]a[\/latex].\r\n\r\n[latex]\\begin{array}{l}f\\left(0\\right)=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=-60a\\hfill \\\\ \\text{ }a=\\frac{1}{30}\\hfill \\end{array}[\/latex]\r\n\r\nThe graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318832[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318833[\/ohm_question]<\/section><section>\r\n<h2>Local and Global Extrema<\/h2>\r\nWith quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.\r\n\r\nEach turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a <strong>global maximum <\/strong>or a <strong>global minimum<\/strong>. These are also referred to as the absolute maximum and absolute minimum values of the function.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>local and global extrema<\/h3>\r\nA <strong>local maximum<\/strong> or <strong>local minimum<\/strong> at [latex]x\u00a0= a[\/latex]\u00a0(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around [latex]x\u00a0= a[\/latex].\r\n<ul>\r\n \t<li>If a function has a local maximum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex]\u00a0in an open interval around [latex]x =\u00a0a[\/latex].<\/li>\r\n \t<li>If a function has a local minimum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex]\u00a0in an open interval around [latex]x\u00a0= a[\/latex].<\/li>\r\n<\/ul>\r\nA <strong>global maximum<\/strong> or <strong>global minimum<\/strong> is the output at the highest or lowest point of the function.\r\n<ul>\r\n \t<li>If a function has a global maximum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex].<\/li>\r\n \t<li>If a function has a global minimum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex].<\/li>\r\n<\/ul>\r\n<img class=\"wp-image-4963 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_-291x300.png\" alt=\"Graph of an even-degree polynomial that denotes the local maximum and minimum and the global maximum.\" width=\"363\" height=\"375\" \/>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Do all polynomial functions have a global minimum or maximum?<\/strong>\r\n\r\n<hr \/>\r\n\r\nNo. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\\left(x\\right)=x[\/latex] has neither a global maximum nor a global minimum.\r\n\r\n<\/section><section class=\"textbox example\">State the local and global extrema of the graph below.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201629\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/>[reveal-answer q=\"776863\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"776863\"]<strong>Local Extrema<\/strong>\r\n<ul>\r\n \t<li>Local maximum value is [latex]0[\/latex] at [latex]x=-1[\/latex].<\/li>\r\n \t<li>Local minimum value is [latex]-4[\/latex] at [latex]x = 0[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Global Extrema<\/strong>\r\n<ul>\r\n \t<li>Global maximum value is at approximately [latex]4.2[\/latex] at [latex]x \\approx 3.5[\/latex].<\/li>\r\n \t<li>There is no global minimum.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox interact\">Use an online graphing tool (such as <a href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>) to find the maximum and minimum values on the interval [latex]\\left[-2,7\\right][\/latex] of the function [latex]f\\left(x\\right)=0.1{\\left(x - \\frac{5}{3}\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 7\\right)[\/latex].Now change the value of the leading coefficient ([latex]a[\/latex]) to see how it affects the end behavior and y-intercept of the graph.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x - \\frac{5}{3}\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 7\\right)[\/latex].<\/p>\r\nMake sure to observe both positive and negative [latex]a[\/latex]-values, and large and small [latex]a[\/latex]-values.\r\n\r\n[reveal-answer q=\"43463\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"43463\"]\r\n\r\nThe minimum occurs at approximately the point [latex]\\left(5.98,-398.8\\right)[\/latex], and the maximum occurs at approximately the point [latex]\\left(0.02,3.24\\right)[\/latex].[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318834[\/ohm_question]<\/section><\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"<h2>Writing Formulas for Polynomial Functions<\/h2>\n<div class=\"page\" title=\"Page 532\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an [latex]x[\/latex]-intercept where each factor is equal to zero, we can form a function that will pass through a set of [latex]x[\/latex]-intercepts by introducing a corresponding set of factors.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>factored form of polynomials<\/h3>\n<p>If a polynomial of lowest degree [latex]p[\/latex]\u00a0has zeros at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>where<\/p>\n<ul>\n<li>the powers [latex]{p}_{i}[\/latex] on each factor can be determined by the behavior of the graph at the corresponding intercept<\/li>\n<li>the stretch factor [latex]a[\/latex] can be determined given a value of the function other than the [latex]x[\/latex]-intercept.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a graph of a polynomial function, write a formula for the function<\/strong><\/p>\n<ol>\n<li>Identify the [latex]x[\/latex]-intercepts of the graph to find the factors of the polynomial.<\/li>\n<li>Examine the behavior of the graph at the [latex]x[\/latex]-intercepts to determine the multiplicity of each factor.<\/li>\n<li>Find the polynomial of least degree containing all of the factors found in the previous step.<\/li>\n<li>Use any other point on the graph (the [latex]y[\/latex]-intercept may be easiest) to determine the stretch factor.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Write a formula for the polynomial function.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201627\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q338564\">Show Solution<\/button><\/p>\n<div id=\"q338564\" class=\"hidden-answer\" style=\"display: none\">This graph has three [latex]x[\/latex]-intercepts: [latex]x= \u20133, 2,[\/latex] and [latex]5[\/latex].<\/p>\n<ul>\n<li>At [latex]x= \u20133[\/latex] and [latex]x= 5[\/latex], the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear with a degree of [latex]1[\/latex].<\/li>\n<li>At [latex]x= 2[\/latex], the graph bounces off the [latex]x[\/latex]-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree (quadratic).<\/li>\n<\/ul>\n<p>Together, this gives us<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\n<p>To determine the stretch factor, we utilize another point on the graph. We will use the [latex]y[\/latex]-intercept [latex](0, \u20132)[\/latex], to solve for [latex]a[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}f\\left(0\\right)=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=-60a\\hfill \\\\ \\text{ }a=\\frac{1}{30}\\hfill \\end{array}[\/latex]<\/p>\n<p>The graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318832\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318832&theme=lumen&iframe_resize_id=ohm318832&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318833\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318833&theme=lumen&iframe_resize_id=ohm318833&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Local and Global Extrema<\/h2>\n<p>With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.<\/p>\n<p>Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a <strong>global maximum <\/strong>or a <strong>global minimum<\/strong>. These are also referred to as the absolute maximum and absolute minimum values of the function.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>local and global extrema<\/h3>\n<p>A <strong>local maximum<\/strong> or <strong>local minimum<\/strong> at [latex]x\u00a0= a[\/latex]\u00a0(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around [latex]x\u00a0= a[\/latex].<\/p>\n<ul>\n<li>If a function has a local maximum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex]\u00a0in an open interval around [latex]x =\u00a0a[\/latex].<\/li>\n<li>If a function has a local minimum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex]\u00a0in an open interval around [latex]x\u00a0= a[\/latex].<\/li>\n<\/ul>\n<p>A <strong>global maximum<\/strong> or <strong>global minimum<\/strong> is the output at the highest or lowest point of the function.<\/p>\n<ul>\n<li>If a function has a global maximum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex].<\/li>\n<li>If a function has a global minimum at [latex]a[\/latex], then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex].<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4963 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_-291x300.png\" alt=\"Graph of an even-degree polynomial that denotes the local maximum and minimum and the global maximum.\" width=\"363\" height=\"375\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_-291x300.png 291w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_-65x67.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_-225x232.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_-350x361.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/02222848\/4.4.L.6.Graph_.png 451w\" sizes=\"(max-width: 363px) 100vw, 363px\" \/><\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Do all polynomial functions have a global minimum or maximum?<\/strong><\/p>\n<hr \/>\n<p>No. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\\left(x\\right)=x[\/latex] has neither a global maximum nor a global minimum.<\/p>\n<\/section>\n<section class=\"textbox example\">State the local and global extrema of the graph below.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201629\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q776863\">Show Answer<\/button><\/p>\n<div id=\"q776863\" class=\"hidden-answer\" style=\"display: none\"><strong>Local Extrema<\/strong><\/p>\n<ul>\n<li>Local maximum value is [latex]0[\/latex] at [latex]x=-1[\/latex].<\/li>\n<li>Local minimum value is [latex]-4[\/latex] at [latex]x = 0[\/latex].<\/li>\n<\/ul>\n<p><strong>Global Extrema<\/strong><\/p>\n<ul>\n<li>Global maximum value is at approximately [latex]4.2[\/latex] at [latex]x \\approx 3.5[\/latex].<\/li>\n<li>There is no global minimum.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox interact\">Use an online graphing tool (such as <a href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>) to find the maximum and minimum values on the interval [latex]\\left[-2,7\\right][\/latex] of the function [latex]f\\left(x\\right)=0.1{\\left(x - \\frac{5}{3}\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 7\\right)[\/latex].Now change the value of the leading coefficient ([latex]a[\/latex]) to see how it affects the end behavior and y-intercept of the graph.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x - \\frac{5}{3}\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 7\\right)[\/latex].<\/p>\n<p>Make sure to observe both positive and negative [latex]a[\/latex]-values, and large and small [latex]a[\/latex]-values.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q43463\">Show Solution<\/button><\/p>\n<div id=\"q43463\" class=\"hidden-answer\" style=\"display: none\">\n<p>The minimum occurs at approximately the point [latex]\\left(5.98,-398.8\\right)[\/latex], and the maximum occurs at approximately the point [latex]\\left(0.02,3.24\\right)[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318834\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318834&theme=lumen&iframe_resize_id=ohm318834&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":13,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/928"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/928\/revisions"}],"predecessor-version":[{"id":5316,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/928\/revisions\/5316"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/928\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=928"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=928"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=928"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=928"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}