{"id":251,"date":"2025-02-13T22:45:35","date_gmt":"2025-02-13T22:45:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rotation-of-axes\/"},"modified":"2025-10-22T23:16:53","modified_gmt":"2025-10-22T23:16:53","slug":"rotation-of-axes","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rotation-of-axes\/","title":{"raw":"Rotation of Axes: Learn It 1","rendered":"Rotation of Axes: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify nondegenerate conic sections given their general form equations.<\/li>\r\n \t<li>Write equations of rotated conics in standard form.<\/li>\r\n \t<li>Identify conics without rotating axes.<\/li>\r\n<\/ul>\r\n<\/section>As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a <em>cone<\/em>. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183021\/CNX_Precalc_Figure_10_04_0012.jpg\" alt=\"\" width=\"975\" height=\"650\" \/>\r\n\r\nEllipses, circles, hyperbolas, and parabolas are sometimes called the <strong>nondegenerate conic sections<\/strong>, in contrast to the <strong>degenerate conic sections<\/strong>. A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183025\/CNX_Precalc_Figure_10_04_002n2.jpg\" alt=\"\" width=\"975\" height=\"719\" \/>\r\n<h2>Identifying Nondegenerate Conics in General Form<\/h2>\r\nIn previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.\r\n<div style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\r\nwhere [latex]A,B[\/latex], and [latex]C[\/latex] are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.\r\n\r\nYou may notice that the general form equation has an [latex]xy[\/latex] term that we have not seen in any of the standard form equations. As we will discuss later, the [latex]xy[\/latex] term rotates the conic whenever [latex]\\text{ }B\\text{ }[\/latex] is not equal to zero.\r\n<table id=\"Table_10_04_01\" summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th><strong>Conic Sections<\/strong><\/th>\r\n<th><strong>Example<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>ellipse<\/td>\r\n<td>[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>circle<\/td>\r\n<td>[latex]4{x}^{2}+4{y}^{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>hyperbola<\/td>\r\n<td>[latex]4{x}^{2}-9{y}^{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>parabola<\/td>\r\n<td>[latex]4{x}^{2}=9y\\text{ or }4{y}^{2}=9x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>one line<\/td>\r\n<td>[latex]4x+9y=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>intersecting lines<\/td>\r\n<td>[latex]\\left(x - 4\\right)\\left(y+4\\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>parallel lines<\/td>\r\n<td>[latex]\\left(x - 4\\right)\\left(x - 9\\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>a point<\/td>\r\n<td>[latex]4{x}^{2}+4{y}^{2}=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>no graph<\/td>\r\n<td>[latex]4{x}^{2}+4{y}^{2}=-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>general form of conic sections<\/h3>\r\nA <strong>nondegenerate conic section<\/strong> has the general form\r\n<p style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/p>\r\nwhere [latex]A,B[\/latex], and [latex]C[\/latex] are not all zero.\r\n\r\nThe table below summarizes the different conic sections where [latex]B=0[\/latex], and [latex]A[\/latex] and [latex]C[\/latex] are nonzero real numbers. This indicates that the conic has not been rotated.\r\n<table id=\"Table_10_04_02\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>ellipse<\/strong><\/td>\r\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A\\ne C\\text{ and }AC&gt;0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>circle<\/strong><\/td>\r\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A=C[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>hyperbola<\/strong><\/td>\r\n<td>[latex]A{x}^{2}-C{y}^{2}+Dx+Ey+F=0\\text{ or }-A{x}^{2}+C{y}^{2}+Dx+Ey+F=0[\/latex], where [latex]A[\/latex] and [latex]C[\/latex] are positive<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>parabola<\/strong><\/td>\r\n<td>[latex]A{x}^{2}+Dx+Ey+F=0\\text{ or }C{y}^{2}+Dx+Ey+F=0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the equation of a conic, identify the type of conic.\r\n<\/strong>\r\n<ol>\r\n \t<li>Rewrite the equation in the general form, [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex].<\/li>\r\n \t<li>Identify the values of [latex]A[\/latex] and [latex]C[\/latex] from the general form.\r\n<ol>\r\n \t<li>If [latex]A[\/latex] and [latex]C[\/latex] are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse.<\/li>\r\n \t<li>If [latex]A[\/latex] and [latex]C[\/latex] are equal and nonzero and have the same sign, then the graph is a circle.<\/li>\r\n \t<li>If [latex]A[\/latex] and [latex]C[\/latex] are nonzero and have opposite signs, then the graph is a hyperbola.<\/li>\r\n \t<li>If either [latex]A[\/latex] or [latex]C[\/latex] is zero, then the graph is a parabola.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Identify the graph of each of the following nondegenerate conic sections.\r\n<ol>\r\n \t<li>[latex]4{x}^{2}-9{y}^{2}+36x+36y - 125=0[\/latex]<\/li>\r\n \t<li>[latex]9{y}^{2}+16x+36y - 10=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}+3{y}^{2}-2x - 6y - 4=0[\/latex]<\/li>\r\n \t<li>[latex]-25{x}^{2}-4{y}^{2}+100x+16y+20=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"722128\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"722128\"]\r\n<ol>\r\n \t<li>Rewriting the general form, we have\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183027\/eq1_n2.jpg\" alt=\"\" \/>\r\n[latex]A=4[\/latex] and [latex]C=-9[\/latex], so we observe that [latex]A[\/latex] and [latex]C[\/latex] have opposite signs. The graph of this equation is a hyperbola.<\/li>\r\n \t<li>Rewriting the general form, we have\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183029\/eq2_n2.jpg\" alt=\"\" \/>[latex]A=0[\/latex] and [latex]C=9[\/latex]. We can determine that the equation is a parabola, since [latex]A[\/latex] is zero.<\/li>\r\n \t<li>Rewriting the general form, we have\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183031\/eq3_n2.jpg\" alt=\"\" \/>[latex]A=3[\/latex] and [latex]C=3[\/latex]. Because [latex]A=C[\/latex], the graph of this equation is a circle.<\/li>\r\n \t<li>Rewriting the general form, we have <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183033\/eq42.jpg\" alt=\"\" \/>[latex]A=-25[\/latex] and [latex]C=-4[\/latex]. Because [latex]AC&gt;0[\/latex] and [latex]A\\ne C[\/latex], the graph of this equation is an ellipse.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Identify the graph of each of the following nondegenerate conic sections.\r\n<ol>\r\n \t<li>[latex]16{y}^{2}-{x}^{2}+x - 4y - 9=0[\/latex]<\/li>\r\n \t<li>[latex]16{x}^{2}+4{y}^{2}+16x+49y - 81=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"185596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"185596\"]\r\n<ol>\r\n \t<li>hyperbola<\/li>\r\n \t<li>ellipse<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<dl id=\"fs-id1840460\" class=\"definition\">\r\n \t<dd id=\"fs-id1840465\"><\/dd>\r\n<\/dl>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify nondegenerate conic sections given their general form equations.<\/li>\n<li>Write equations of rotated conics in standard form.<\/li>\n<li>Identify conics without rotating axes.<\/li>\n<\/ul>\n<\/section>\n<p>As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a <em>cone<\/em>. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183021\/CNX_Precalc_Figure_10_04_0012.jpg\" alt=\"\" width=\"975\" height=\"650\" \/><\/p>\n<p>Ellipses, circles, hyperbolas, and parabolas are sometimes called the <strong>nondegenerate conic sections<\/strong>, in contrast to the <strong>degenerate conic sections<\/strong>. A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183025\/CNX_Precalc_Figure_10_04_002n2.jpg\" alt=\"\" width=\"975\" height=\"719\" \/><\/p>\n<h2>Identifying Nondegenerate Conics in General Form<\/h2>\n<p>In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.<\/p>\n<div style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p>where [latex]A,B[\/latex], and [latex]C[\/latex] are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.<\/p>\n<p>You may notice that the general form equation has an [latex]xy[\/latex] term that we have not seen in any of the standard form equations. As we will discuss later, the [latex]xy[\/latex] term rotates the conic whenever [latex]\\text{ }B\\text{ }[\/latex] is not equal to zero.<\/p>\n<table id=\"Table_10_04_01\" summary=\"..\">\n<thead>\n<tr>\n<th><strong>Conic Sections<\/strong><\/th>\n<th><strong>Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>ellipse<\/td>\n<td>[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>circle<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>hyperbola<\/td>\n<td>[latex]4{x}^{2}-9{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>parabola<\/td>\n<td>[latex]4{x}^{2}=9y\\text{ or }4{y}^{2}=9x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>one line<\/td>\n<td>[latex]4x+9y=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>intersecting lines<\/td>\n<td>[latex]\\left(x - 4\\right)\\left(y+4\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>parallel lines<\/td>\n<td>[latex]\\left(x - 4\\right)\\left(x - 9\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>a point<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>no graph<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>general form of conic sections<\/h3>\n<p>A <strong>nondegenerate conic section<\/strong> has the general form<\/p>\n<p style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/p>\n<p>where [latex]A,B[\/latex], and [latex]C[\/latex] are not all zero.<\/p>\n<p>The table below summarizes the different conic sections where [latex]B=0[\/latex], and [latex]A[\/latex] and [latex]C[\/latex] are nonzero real numbers. This indicates that the conic has not been rotated.<\/p>\n<table id=\"Table_10_04_02\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>ellipse<\/strong><\/td>\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A\\ne C\\text{ and }AC>0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>circle<\/strong><\/td>\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A=C[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>hyperbola<\/strong><\/td>\n<td>[latex]A{x}^{2}-C{y}^{2}+Dx+Ey+F=0\\text{ or }-A{x}^{2}+C{y}^{2}+Dx+Ey+F=0[\/latex], where [latex]A[\/latex] and [latex]C[\/latex] are positive<\/td>\n<\/tr>\n<tr>\n<td><strong>parabola<\/strong><\/td>\n<td>[latex]A{x}^{2}+Dx+Ey+F=0\\text{ or }C{y}^{2}+Dx+Ey+F=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the equation of a conic, identify the type of conic.<br \/>\n<\/strong><\/p>\n<ol>\n<li>Rewrite the equation in the general form, [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex].<\/li>\n<li>Identify the values of [latex]A[\/latex] and [latex]C[\/latex] from the general form.\n<ol>\n<li>If [latex]A[\/latex] and [latex]C[\/latex] are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse.<\/li>\n<li>If [latex]A[\/latex] and [latex]C[\/latex] are equal and nonzero and have the same sign, then the graph is a circle.<\/li>\n<li>If [latex]A[\/latex] and [latex]C[\/latex] are nonzero and have opposite signs, then the graph is a hyperbola.<\/li>\n<li>If either [latex]A[\/latex] or [latex]C[\/latex] is zero, then the graph is a parabola.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Identify the graph of each of the following nondegenerate conic sections.<\/p>\n<ol>\n<li>[latex]4{x}^{2}-9{y}^{2}+36x+36y - 125=0[\/latex]<\/li>\n<li>[latex]9{y}^{2}+16x+36y - 10=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}+3{y}^{2}-2x - 6y - 4=0[\/latex]<\/li>\n<li>[latex]-25{x}^{2}-4{y}^{2}+100x+16y+20=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q722128\">Show Solution<\/button><\/p>\n<div id=\"q722128\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Rewriting the general form, we have<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183027\/eq1_n2.jpg\" alt=\"\" \/><br \/>\n[latex]A=4[\/latex] and [latex]C=-9[\/latex], so we observe that [latex]A[\/latex] and [latex]C[\/latex] have opposite signs. The graph of this equation is a hyperbola.<\/li>\n<li>Rewriting the general form, we have<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183029\/eq2_n2.jpg\" alt=\"\" \/>[latex]A=0[\/latex] and [latex]C=9[\/latex]. We can determine that the equation is a parabola, since [latex]A[\/latex] is zero.<\/li>\n<li>Rewriting the general form, we have<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183031\/eq3_n2.jpg\" alt=\"\" \/>[latex]A=3[\/latex] and [latex]C=3[\/latex]. Because [latex]A=C[\/latex], the graph of this equation is a circle.<\/li>\n<li>Rewriting the general form, we have <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183033\/eq42.jpg\" alt=\"\" \/>[latex]A=-25[\/latex] and [latex]C=-4[\/latex]. Because [latex]AC>0[\/latex] and [latex]A\\ne C[\/latex], the graph of this equation is an ellipse.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Identify the graph of each of the following nondegenerate conic sections.<\/p>\n<ol>\n<li>[latex]16{y}^{2}-{x}^{2}+x - 4y - 9=0[\/latex]<\/li>\n<li>[latex]16{x}^{2}+4{y}^{2}+16x+49y - 81=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q185596\">Show Solution<\/button><\/p>\n<div id=\"q185596\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>hyperbola<\/li>\n<li>ellipse<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<dl id=\"fs-id1840460\" class=\"definition\">\n<dd id=\"fs-id1840465\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"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\/251"}],"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\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/251\/revisions"}],"predecessor-version":[{"id":4843,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/251\/revisions\/4843"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/251\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=251"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=251"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=251"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}