{"id":2314,"date":"2025-08-12T21:40:51","date_gmt":"2025-08-12T21:40:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2314"},"modified":"2025-10-21T18:15:38","modified_gmt":"2025-10-21T18:15:38","slug":"graphing-parametric-equations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphing-parametric-equations-learn-it-2\/","title":{"raw":"Graphing Parametric Equations: Learn It 2","rendered":"Graphing Parametric Equations: Learn It 2"},"content":{"raw":"<h2>Rectangular vs. Parametric Graphs<\/h2>\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph the parametric equations [latex]x=5\\cos t[\/latex] and [latex]y=2\\sin t[\/latex]. First, construct the graph using data points generated from the parametric form. Then graph the rectangular form of the equation. Compare the two graphs.[reveal-answer q=\"651424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"651424\"]\r\n\r\nConstruct a table of values like the table below.\r\n<table id=\"Table_08_07_03\" summary=\"Twelve rows and three columns. First column is labeled t, second column is labeled x(t)=5cos(t), third column is labeled y(t) = 2sin(t). The table has ordered triples of each of these row values: (0, x=5cos(0)=5, y=2sin(0)=0), (1, x=5cos(1) =approx 2.7, y=2sin(1) =approx 1.7), (2, x=5cos(2) =approx -2.1, y=2sin(2) =approx 1.8), (3, x=5cos(3) =approx -4.95, y=2sin(3) =approx 0.28), (4, x=5cos(4) =approx -3.3, y=2sin(4) =approx -1.5), (5, x=5cos(5) =approx 1.4, y=2sin(5) =approx -1.9), (-1, x=5cos(-1) =approx 2.7, y=2sin(-1) =approx -1.7), (-2, x=5cos(-2) =approx -2.1, y=2sin(-2) =approx -1.8), (-3, x=5cos(-3) =approx -4.95, y=2sin(-3) =approx -0.28), (-4, x=5cos(-4) =approx -3.3, y=2sin(-4) =approx 1.5), (-5, x=5cos(-5) =approx 1.4, y=2sin(-5) =approx 1.9).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x=5\\cos t[\/latex]<\/th>\r\n<th>[latex]y=2\\sin t[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\text{0}[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(0\\right)=5[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(0\\right)=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{1}[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(1\\right)\\approx 2.7[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(1\\right)\\approx 1.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{2}[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(2\\right)\\approx -2.1[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(2\\right)\\approx 1.8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{3}[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(3\\right)\\approx -4.95[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(3\\right)\\approx 0.28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{4}[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(4\\right)\\approx -3.3[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(4\\right)\\approx -1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\text{5}[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(5\\right)\\approx 1.4[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(5\\right)\\approx -1.9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(-1\\right)\\approx 2.7[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(-1\\right)\\approx -1.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(-2\\right)\\approx -2.1[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(-2\\right)\\approx -1.8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(-3\\right)\\approx -4.95[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(-3\\right)\\approx -0.28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(-4\\right)\\approx -3.3[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(-4\\right)\\approx 1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]x=5\\cos \\left(-5\\right)\\approx 1.4[\/latex]<\/td>\r\n<td>[latex]y=2\\sin \\left(-5\\right)\\approx 1.9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the [latex]\\left(x,y\\right)[\/latex] values from the table. See Figure 4.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180951\/CNX_Precalc_Figure_08_07_0062.jpg\" alt=\"Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.\" width=\"975\" height=\"290\" \/>\r\n\r\nNext, translate the parametric equations to rectangular form. To do this, we solve for [latex]t[\/latex] in either [latex]x\\left(t\\right)[\/latex] or [latex]y\\left(t\\right)[\/latex], and then substitute the expression for [latex]t[\/latex] in the other equation. The result will be a function [latex]y\\left(x\\right)[\/latex] if solving for [latex]t[\/latex] as a function of [latex]x[\/latex], or [latex]x\\left(y\\right)[\/latex] if solving for [latex]t[\/latex] as a function of [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;x=5\\cos t \\\\ &amp;\\frac{x}{5}=\\cos t&amp;&amp; \\text{Solve for }\\cos t. \\\\ &amp;y=2\\sin t&amp;&amp; \\text{Solve for }\\sin t. \\\\ &amp;\\frac{y}{2}=\\sin t \\end{align}[\/latex]<\/p>\r\nThen, use the Pythagorean Theorem.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {\\cos }^{2}t+{\\sin }^{2}t=1\\\\ {\\left(\\frac{x}{5}\\right)}^{2}+{\\left(\\frac{y}{2}\\right)}^{2}=1\\\\ \\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\end{gathered}[\/latex]<\/p>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nIn Figure 5, the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180953\/CNX_Precalc_Figure_08_07_0072.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"487\" height=\"290\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph the parametric equations [latex]x=t+1[\/latex] and [latex]y=\\sqrt{t},t\\ge 0[\/latex], and the rectangular equivalent [latex]y=\\sqrt{x - 1}[\/latex] on the same coordinate system.\r\n\r\n[reveal-answer q=\"771276\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"771276\"]\r\n\r\nConstruct a table of values for the parametric equations, as we did in the previous example, and graph [latex]y=\\sqrt{t},t\\ge 0[\/latex] on the same grid.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180957\/CNX_Precalc_Figure_08_07_0082.jpg\" alt=\"Overlayed graph of the two versions of the given function, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"488\" height=\"291\" \/>\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nWith the domain on [latex]t[\/latex] restricted, we only plot positive values of [latex]t[\/latex]. The parametric data is graphed in blue and the graph of the rectangular equation is dashed in red. Once again, we see that the two forms overlap.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nSketch the graph of the parametric equations [latex]x=2\\cos \\theta \\text{ and }y=4\\sin \\theta [\/latex], along with the rectangular equation on the same grid.\r\n\r\n[reveal-answer q=\"857454\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"857454\"]\r\n\r\nThe graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181016\/CNX_Precalc_Figure_08_07_0092.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]173887[\/ohm_question]<\/section><\/div>","rendered":"<h2>Rectangular vs. Parametric Graphs<\/h2>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the parametric equations [latex]x=5\\cos t[\/latex] and [latex]y=2\\sin t[\/latex]. First, construct the graph using data points generated from the parametric form. Then graph the rectangular form of the equation. Compare the two graphs.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q651424\">Show Solution<\/button><\/p>\n<div id=\"q651424\" class=\"hidden-answer\" style=\"display: none\">\n<p>Construct a table of values like the table below.<\/p>\n<table id=\"Table_08_07_03\" summary=\"Twelve rows and three columns. First column is labeled t, second column is labeled x(t)=5cos(t), third column is labeled y(t) = 2sin(t). The table has ordered triples of each of these row values: (0, x=5cos(0)=5, y=2sin(0)=0), (1, x=5cos(1) =approx 2.7, y=2sin(1) =approx 1.7), (2, x=5cos(2) =approx -2.1, y=2sin(2) =approx 1.8), (3, x=5cos(3) =approx -4.95, y=2sin(3) =approx 0.28), (4, x=5cos(4) =approx -3.3, y=2sin(4) =approx -1.5), (5, x=5cos(5) =approx 1.4, y=2sin(5) =approx -1.9), (-1, x=5cos(-1) =approx 2.7, y=2sin(-1) =approx -1.7), (-2, x=5cos(-2) =approx -2.1, y=2sin(-2) =approx -1.8), (-3, x=5cos(-3) =approx -4.95, y=2sin(-3) =approx -0.28), (-4, x=5cos(-4) =approx -3.3, y=2sin(-4) =approx 1.5), (-5, x=5cos(-5) =approx 1.4, y=2sin(-5) =approx 1.9).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x=5\\cos t[\/latex]<\/th>\n<th>[latex]y=2\\sin t[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\text{0}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(0\\right)=5[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(0\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{1}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(1\\right)\\approx 2.7[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(1\\right)\\approx 1.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{2}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(2\\right)\\approx -2.1[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(2\\right)\\approx 1.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{3}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(3\\right)\\approx -4.95[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(3\\right)\\approx 0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{4}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(4\\right)\\approx -3.3[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(4\\right)\\approx -1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{5}[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(5\\right)\\approx 1.4[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(5\\right)\\approx -1.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-1\\right)\\approx 2.7[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-1\\right)\\approx -1.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-2\\right)\\approx -2.1[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-2\\right)\\approx -1.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-3\\right)\\approx -4.95[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-3\\right)\\approx -0.28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-4\\right)\\approx -3.3[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-4\\right)\\approx 1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]x=5\\cos \\left(-5\\right)\\approx 1.4[\/latex]<\/td>\n<td>[latex]y=2\\sin \\left(-5\\right)\\approx 1.9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plot the [latex]\\left(x,y\\right)[\/latex] values from the table. See Figure 4.<\/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\/27180951\/CNX_Precalc_Figure_08_07_0062.jpg\" alt=\"Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.\" width=\"975\" height=\"290\" \/><\/p>\n<p>Next, translate the parametric equations to rectangular form. To do this, we solve for [latex]t[\/latex] in either [latex]x\\left(t\\right)[\/latex] or [latex]y\\left(t\\right)[\/latex], and then substitute the expression for [latex]t[\/latex] in the other equation. The result will be a function [latex]y\\left(x\\right)[\/latex] if solving for [latex]t[\/latex] as a function of [latex]x[\/latex], or [latex]x\\left(y\\right)[\/latex] if solving for [latex]t[\/latex] as a function of [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&x=5\\cos t \\\\ &\\frac{x}{5}=\\cos t&& \\text{Solve for }\\cos t. \\\\ &y=2\\sin t&& \\text{Solve for }\\sin t. \\\\ &\\frac{y}{2}=\\sin t \\end{align}[\/latex]<\/p>\n<p>Then, use the Pythagorean Theorem.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {\\cos }^{2}t+{\\sin }^{2}t=1\\\\ {\\left(\\frac{x}{5}\\right)}^{2}+{\\left(\\frac{y}{2}\\right)}^{2}=1\\\\ \\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{4}=1\\end{gathered}[\/latex]<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>In Figure 5, the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.<\/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\/27180953\/CNX_Precalc_Figure_08_07_0072.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"487\" height=\"290\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the parametric equations [latex]x=t+1[\/latex] and [latex]y=\\sqrt{t},t\\ge 0[\/latex], and the rectangular equivalent [latex]y=\\sqrt{x - 1}[\/latex] on the same coordinate system.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q771276\">Show Solution<\/button><\/p>\n<div id=\"q771276\" class=\"hidden-answer\" style=\"display: none\">\n<p>Construct a table of values for the parametric equations, as we did in the previous example, and graph [latex]y=\\sqrt{t},t\\ge 0[\/latex] on the same grid.<\/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\/27180957\/CNX_Precalc_Figure_08_07_0082.jpg\" alt=\"Overlayed graph of the two versions of the given function, showing that they are the same whether they are given in parametric or rectangular coordinates.\" width=\"488\" height=\"291\" \/><\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>With the domain on [latex]t[\/latex] restricted, we only plot positive values of [latex]t[\/latex]. The parametric data is graphed in blue and the graph of the rectangular equation is dashed in red. Once again, we see that the two forms overlap.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>Sketch the graph of the parametric equations [latex]x=2\\cos \\theta \\text{ and }y=4\\sin \\theta[\/latex], along with the rectangular equation on the same grid.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q857454\">Show Solution<\/button><\/p>\n<div id=\"q857454\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181016\/CNX_Precalc_Figure_08_07_0092.jpg\" alt=\"Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm173887\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173887&theme=lumen&iframe_resize_id=ohm173887&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":13,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"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\/2314"}],"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":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2314\/revisions"}],"predecessor-version":[{"id":4799,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2314\/revisions\/4799"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2314\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2314"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2314"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2314"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}