{"id":1795,"date":"2024-06-06T22:52:48","date_gmt":"2024-06-06T22:52:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1795"},"modified":"2025-01-22T20:55:26","modified_gmt":"2025-01-22T20:55:26","slug":"module-8-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-8-cheat-sheet\/","title":{"raw":"Quadratic Functions: Cheat Sheet","rendered":"Quadratic Functions: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+LOHM\/Cheat+Sheets\/College+Algebra+Cheat+Sheet+M8_+Quadratic+Functions.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n\r\n<h2>Essential Concepts<\/h2>\r\n<h3><span data-sheets-root=\"1\">Introduction to Quadratic Functions and Parabolas<\/span><\/h3>\r\n<ul>\r\n \t<li>A quadratic function is a second-degree polynomial function<\/li>\r\n \t<li>The direction of the parabola depends on the sign of\r\n<ul>\r\n \t<li>If [latex]a &gt; 0[\/latex], parabola opens upward<\/li>\r\n \t<li>If [latex]a &lt; 0[\/latex], parabola opens downward<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Every parabola has an axis of symmetry that passes through the vertex<\/li>\r\n \t<li>Domain of all quadratic functions is all real numbers<\/li>\r\n \t<li>Range depends on direction of opening and vertex location<\/li>\r\n \t<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.<\/li>\r\n \t<li>The axis of symmetry is the vertical line passing through the vertex.<\/li>\r\n \t<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\r\n \t<li>Quadratic functions of form [latex]f(x)=ax^2+bx+c[\/latex] may be graphed by evaluating the function at various values of the input variable [latex]x[\/latex] to find each coordinating output [latex]f(x)[\/latex]. Plot enough points to obtain the shape of the graph, then draw a smooth curve between them.<\/li>\r\n \t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\r\n \t<li>The vertex (the turning point) of the graph of a parabola may be obtained using the formula [latex]\\left( -\\dfrac{b}{2a}, f\\left(-\\dfrac{b}{2a}\\right)\\right)[\/latex]<\/li>\r\n \t<li>The graph of a quadratic function opens up if the leading coefficient [latex]a[\/latex] is positive, and opens down if [latex]a[\/latex] is negative.<\/li>\r\n \t<li>Standard or vertex form, [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], is useful to easily identify the vertex of a parabola, where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Complex Numbers and Operations<\/span><\/h3>\r\n<ul id=\"fs-id1165135261454\">\r\n \t<li>The square root of any negative number can be written as a multiple of [latex]i[\/latex].<\/li>\r\n \t<li>To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.<\/li>\r\n \t<li>Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.<\/li>\r\n \t<li>Complex numbers can be multiplied and divided.<\/li>\r\n \t<li>To multiply complex numbers, distribute just as with polynomials.<\/li>\r\n \t<li>To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.<\/li>\r\n \t<li>The powers of [latex]i[\/latex]\u00a0are cyclic, repeating every fourth one.<\/li>\r\n \t<li>A quadratic equation can have real or complex roots<\/li>\r\n \t<li>The quadratic formula provides all solutions to a quadratic equation<\/li>\r\n \t<li>The discriminant determines the nature and number of solutions<\/li>\r\n \t<li>Complex roots always come in conjugate pairs<\/li>\r\n \t<li>When a parabola doesn't cross the [latex]x[\/latex]-axis, it has complex roots<\/li>\r\n<\/ul>\r\n<h3><span data-sheets-root=\"1\">Application of Quadratic Functions<\/span><\/h3>\r\n<ul>\r\n \t<li>The zeros, or [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]<em>-<\/em>axis.<\/li>\r\n \t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\r\n \t<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\r\n \t<li>The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\r\n \t<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165134570662\"><\/ul>\r\n<h2>Key Equations<\/h2>\r\n<table style=\"width: 84.2358%;\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 35.8491%;\">Type<\/th>\r\n<th style=\"width: 175.786%;\">Equation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 35.8491%;\"><strong>difference of squares formula<\/strong><\/td>\r\n<td style=\"width: 175.786%;\">[latex]a^2 - b^2 = (a+b)(a-b)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>general form of a quadratic function<\/strong><\/td>\r\n<td>[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>standard form of a quadratic function<\/strong><\/td>\r\n<td>[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 35.8491%;\"><strong>factor by grouping<\/strong><\/td>\r\n<td style=\"width: 175.786%;\">[latex]ax^2 + bx + c = (px + q)(rx + s)[\/latex]\r\nwhere [latex]pr = a[\/latex] and [latex]qs = c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>discriminant<\/strong><\/td>\r\n<td>[latex]b^2 - 4ac[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>discriminant Cases<\/strong><\/td>\r\n<td>[latex]b^2 - 4ac &gt; 0[\/latex]: Two real solutions\r\n[latex]b^2 - 4ac = 0[\/latex]: One real solution\r\n[latex]b^2 - 4ac &lt; 0[\/latex]: Two complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1165135449657\" class=\"definition\">\r\n \t<dt><strong>axis of symmetry<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\\frac{b}{2a}[\/latex].<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1165135320095\" class=\"definition\">\r\n \t<dt><strong>complex conjugate<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135320101\">the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135320107\" class=\"definition\">\r\n \t<dt><strong>complex number<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135320112\">the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[\/latex], where [latex]a[\/latex] is the real part, and [latex]bi[\/latex] is the imaginary part<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133260439\" class=\"definition\">\r\n \t<dt><strong>complex plane<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133260444\">a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290002\" class=\"definition\">\r\n \t<dt><strong>complex roots<\/strong><\/dt>\r\n \t<dd>Solutions to a quadratic equation that contain imaginary numbers, occurring when the discriminant is negative<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290001\" class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1165135623614\" class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1165131290003\" class=\"definition\">\r\n \t<dt><strong>conjugate pair<\/strong><\/dt>\r\n \t<dd>Two complex numbers in the form [latex]a + bi[\/latex] and [latex]a - bi[\/latex] that occur as solutions to quadratic equations with complex roots<\/dd>\r\n<\/dl>\r\n<strong>discriminant<\/strong><\/dt>\r\n \t<dd>the value under the radical in the quadratic formula, [latex]b^2-4ac[\/latex], which tells whether the quadratic has real or complex roots<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1165135502777\" class=\"definition\">\r\n \t<dt><strong>general form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133260450\" class=\"definition\">\r\n \t<dt><strong>imaginary number<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133260456\">a number in the form [latex]bi[\/latex]\u00a0where [latex]i=\\sqrt{-1}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931314\" class=\"definition\">\r\n \t<dt><strong>standard form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290008\" class=\"definition\">\r\n \t<dt><strong>parabola<\/strong><\/dt>\r\n \t<dd>The U-shaped graph of a quadratic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290001\" class=\"definition\">\r\n \t<dt><strong>quadratic function<\/strong><\/dt>\r\n \t<dd>A polynomial function of degree 2<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131290004\" class=\"definition\">\r\n \t<dt><strong>vertex<\/strong><\/dt>\r\n \t<dd>The highest or lowest point of a parabola [latex](h,k)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+LOHM\/Cheat+Sheets\/College+Algebra+Cheat+Sheet+M8_+Quadratic+Functions.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<h3><span data-sheets-root=\"1\">Introduction to Quadratic Functions and Parabolas<\/span><\/h3>\n<ul>\n<li>A quadratic function is a second-degree polynomial function<\/li>\n<li>The direction of the parabola depends on the sign of\n<ul>\n<li>If [latex]a > 0[\/latex], parabola opens upward<\/li>\n<li>If [latex]a < 0[\/latex], parabola opens downward<\/li>\n<\/ul>\n<\/li>\n<li>Every parabola has an axis of symmetry that passes through the vertex<\/li>\n<li>Domain of all quadratic functions is all real numbers<\/li>\n<li>Range depends on direction of opening and vertex location<\/li>\n<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.<\/li>\n<li>The axis of symmetry is the vertical line passing through the vertex.<\/li>\n<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\n<li>Quadratic functions of form [latex]f(x)=ax^2+bx+c[\/latex] may be graphed by evaluating the function at various values of the input variable [latex]x[\/latex] to find each coordinating output [latex]f(x)[\/latex]. Plot enough points to obtain the shape of the graph, then draw a smooth curve between them.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>The vertex (the turning point) of the graph of a parabola may be obtained using the formula [latex]\\left( -\\dfrac{b}{2a}, f\\left(-\\dfrac{b}{2a}\\right)\\right)[\/latex]<\/li>\n<li>The graph of a quadratic function opens up if the leading coefficient [latex]a[\/latex] is positive, and opens down if [latex]a[\/latex] is negative.<\/li>\n<li>Standard or vertex form, [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], is useful to easily identify the vertex of a parabola, where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Complex Numbers and Operations<\/span><\/h3>\n<ul id=\"fs-id1165135261454\">\n<li>The square root of any negative number can be written as a multiple of [latex]i[\/latex].<\/li>\n<li>To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.<\/li>\n<li>Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.<\/li>\n<li>Complex numbers can be multiplied and divided.<\/li>\n<li>To multiply complex numbers, distribute just as with polynomials.<\/li>\n<li>To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.<\/li>\n<li>The powers of [latex]i[\/latex]\u00a0are cyclic, repeating every fourth one.<\/li>\n<li>A quadratic equation can have real or complex roots<\/li>\n<li>The quadratic formula provides all solutions to a quadratic equation<\/li>\n<li>The discriminant determines the nature and number of solutions<\/li>\n<li>Complex roots always come in conjugate pairs<\/li>\n<li>When a parabola doesn&#8217;t cross the [latex]x[\/latex]-axis, it has complex roots<\/li>\n<\/ul>\n<h3><span data-sheets-root=\"1\">Application of Quadratic Functions<\/span><\/h3>\n<ul>\n<li>The zeros, or [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]<em>&#8211;<\/em>axis.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\n<li>The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\n<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\n<\/ul>\n<ul id=\"fs-id1165134570662\"><\/ul>\n<h2>Key Equations<\/h2>\n<table style=\"width: 84.2358%;\">\n<thead>\n<tr>\n<th style=\"width: 35.8491%;\">Type<\/th>\n<th style=\"width: 175.786%;\">Equation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 35.8491%;\"><strong>difference of squares formula<\/strong><\/td>\n<td style=\"width: 175.786%;\">[latex]a^2 - b^2 = (a+b)(a-b)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>general form of a quadratic function<\/strong><\/td>\n<td>[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>standard form of a quadratic function<\/strong><\/td>\n<td>[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 35.8491%;\"><strong>factor by grouping<\/strong><\/td>\n<td style=\"width: 175.786%;\">[latex]ax^2 + bx + c = (px + q)(rx + s)[\/latex]<br \/>\nwhere [latex]pr = a[\/latex] and [latex]qs = c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>discriminant<\/strong><\/td>\n<td>[latex]b^2 - 4ac[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>discriminant Cases<\/strong><\/td>\n<td>[latex]b^2 - 4ac > 0[\/latex]: Two real solutions<br \/>\n[latex]b^2 - 4ac = 0[\/latex]: One real solution<br \/>\n[latex]b^2 - 4ac < 0[\/latex]: Two complex solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>axis of symmetry<\/strong><\/dt>\n<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\\frac{b}{2a}[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135320095\" class=\"definition\">\n<dt><strong>complex conjugate<\/strong><\/dt>\n<dd id=\"fs-id1165135320101\">the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135320107\" class=\"definition\">\n<dt><strong>complex number<\/strong><\/dt>\n<dd id=\"fs-id1165135320112\">the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[\/latex], where [latex]a[\/latex] is the real part, and [latex]bi[\/latex] is the imaginary part<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260439\" class=\"definition\">\n<dt><strong>complex plane<\/strong><\/dt>\n<dd id=\"fs-id1165133260444\">a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290002\" class=\"definition\">\n<dt><strong>complex roots<\/strong><\/dt>\n<dd>Solutions to a quadratic equation that contain imaginary numbers, occurring when the discriminant is negative<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290001\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt><strong>conjugate pair<\/strong><\/dt>\n<dd>Two complex numbers in the form [latex]a + bi[\/latex] and [latex]a - bi[\/latex] that occur as solutions to quadratic equations with complex roots<\/dd>\n<\/dl>\n<p><strong>discriminant<\/strong><br \/>\n \tthe value under the radical in the quadratic formula, [latex]b^2-4ac[\/latex], which tells whether the quadratic has real or complex roots<\/p>\n<dl id=\"fs-id1165135502777\" class=\"definition\">\n<dt><strong>general form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133260450\" class=\"definition\">\n<dt><strong>imaginary number<\/strong><\/dt>\n<dd id=\"fs-id1165133260456\">a number in the form [latex]bi[\/latex]\u00a0where [latex]i=\\sqrt{-1}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931314\" class=\"definition\">\n<dt><strong>standard form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290008\" class=\"definition\">\n<dt><strong>parabola<\/strong><\/dt>\n<dd>The U-shaped graph of a quadratic function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>quadratic function<\/strong><\/dt>\n<dd>A polynomial function of degree 2<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131290004\" class=\"definition\">\n<dt><strong>vertex<\/strong><\/dt>\n<dd>The highest or lowest point of a parabola [latex](h,k)[\/latex]<\/dd>\n<\/dl>\n","protected":false},"author":12,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"cheat_sheet","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\/1795"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1795\/revisions"}],"predecessor-version":[{"id":7344,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1795\/revisions\/7344"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1795\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1795"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1795"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1795"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}