{"id":1200,"date":"2025-07-23T20:15:36","date_gmt":"2025-07-23T20:15:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1200"},"modified":"2026-04-01T08:36:39","modified_gmt":"2026-04-01T08:36:39","slug":"fitting-exponential-models-to-data-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/fitting-exponential-models-to-data-learn-it-3\/","title":{"raw":"Fitting Exponential Models to Data: Learn It 3","rendered":"Fitting Exponential Models to Data: Learn It 3"},"content":{"raw":"<div id=\"Example_04_08_02\" class=\"example\">\r\n<h2 id=\"fs-id1616172\" class=\"exercise\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Build a logistic model from data<\/span><\/h2>\r\n<\/div>\r\n<p id=\"fs-id1316523\">Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or <em>limiting value<\/em>. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.<\/p>\r\n<p id=\"fs-id1677718\">It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic <strong>regression analysis<\/strong>, we use the form most commonly used on graphing utilities:<\/p>\r\n\r\n<div id=\"eip-154\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\r\n<p id=\"fs-id1310104\">Recall that:<\/p>\r\n\r\n<ul id=\"fs-id1294720\">\r\n \t<li>[latex]\\frac{c}{1+a}[\/latex] is the initial value of the model.<\/li>\r\n \t<li>when <em>b\u00a0<\/em>&gt; 0, the model increases rapidly at first until it reaches its point of maximum growth rate, [latex]\\left(\\frac{\\mathrm{ln}\\left(a\\right)}{b},\\frac{c}{2}\\right)[\/latex]. At that point, growth steadily slows and the function becomes asymptotic to the upper bound <em>y\u00a0<\/em>= <em>c<\/em>.<\/li>\r\n \t<li><em>c<\/em>\u00a0is the limiting value, sometimes called the <em>carrying capacity<\/em>, of the model.<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3 class=\"title\">logistic regression<\/h3>\r\n<p id=\"fs-id1583226\"><strong>Logistic regression <\/strong>is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command \"Logistic\" on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form<\/p>\r\n\r\n<div id=\"eip-491\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\r\n<p id=\"fs-id1701600\">Note that<\/p>\r\n\r\n<ul id=\"fs-id1358262\">\r\n \t<li>The initial value of the model is [latex]\\frac{c}{1+a}[\/latex].<\/li>\r\n \t<li>Output values for the model grow closer and closer to <em>y<\/em>\u00a0=\u00a0<em>c<\/em>\u00a0as time increases.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div><section class=\"textbox interact\" aria-label=\"Interact\">Given a set of data, perform logistic regression using a graphing utility.\r\n<ol id=\"fs-id1690756\">\r\n \t<li>Use the STAT then EDIT menu to enter given data.\r\n<ol id=\"fs-id1562359\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Clear any existing data from the lists.<\/li>\r\n \t<li>List the input values in the L1 column.<\/li>\r\n \t<li>List the output values in the L2 column.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\r\n<ol id=\"fs-id1431005\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\r\n \t<li>Verify the data follow a logistic pattern.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Find the equation that models the data.\r\n<ol id=\"fs-id1585412\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Select \"Logistic\" from the STAT then CALC menu.<\/li>\r\n \t<li>Use the values returned for <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>\u00a0to record the model, [latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n<div id=\"Example_04_08_03\" class=\"example\">\r\n<div id=\"fs-id1646727\" class=\"exercise\"><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1422087\">Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. The table below\u00a0shows the percentage of Americans with cellular service between the years 1995 and 2012.[footnote]Source: <em>The World Bank, 2013<\/em>.[\/footnote]<\/p>\r\n\r\n<table id=\"Table_04_08_05\" summary=\"Nineteen rows and two columns. The first column is labeled, \">\r\n<thead>\r\n<tr>\r\n<th>Year<\/th>\r\n<th>Americans with Cellular Service (%)<\/th>\r\n<th>Year<\/th>\r\n<th>Americans with Cellular Service (%)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1995<\/td>\r\n<td>12.69<\/td>\r\n<td>2004<\/td>\r\n<td>62.852<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1996<\/td>\r\n<td>16.35<\/td>\r\n<td>2005<\/td>\r\n<td>68.63<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1997<\/td>\r\n<td>20.29<\/td>\r\n<td>2006<\/td>\r\n<td>76.64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1998<\/td>\r\n<td>25.08<\/td>\r\n<td>2007<\/td>\r\n<td>82.47<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1999<\/td>\r\n<td>30.81<\/td>\r\n<td>2008<\/td>\r\n<td>85.68<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2000<\/td>\r\n<td>38.75<\/td>\r\n<td>2009<\/td>\r\n<td>89.14<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2001<\/td>\r\n<td>45.00<\/td>\r\n<td>2010<\/td>\r\n<td>91.86<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2002<\/td>\r\n<td>49.16<\/td>\r\n<td>2011<\/td>\r\n<td>95.28<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2003<\/td>\r\n<td>55.15<\/td>\r\n<td>2012<\/td>\r\n<td>98.17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol id=\"fs-id1624806\">\r\n \t<li>Let <em>x<\/em>\u00a0represent time in years starting with <em>x\u00a0<\/em>= 0 for the year 1995. Let <em>y<\/em>\u00a0represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.<\/li>\r\n \t<li>Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.<\/li>\r\n \t<li>Discuss the value returned for the upper limit, <em>c<\/em>. What does this tell you about the model? What would the limiting value be if the model were exact?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"581544\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"581544\"]\r\n<ol id=\"fs-id1455761\">\r\n \t<li>Using the STAT then EDIT menu on a graphing utility, list the years using values 0\u201315 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in Figure 5:\r\n<figure id=\"CNX_Precalc_Figure_04_08_005\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"805\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010837\/CNX_Precalc_Figure_04_08_0052.jpg\" alt=\"Graph of a scattered plot.\" width=\"805\" height=\"396\" \/> <b>Figure 5<\/b>[\/caption]<\/figure>\r\n<p id=\"fs-id1366003\">Use the \"Logistic\" command from the STAT then CALC menu to obtain the logistic model,<\/p>\r\n<p id=\"fs-id1366003\" style=\"text-align: center;\">[latex]y=\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"806\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010837\/CNX_Precalc_Figure_04_08_0062.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"806\" height=\"396\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1650001\">Next, graph the model in the same window as shown in Figure 6\u00a0to verify it is a good fit:<span id=\"fs-id1246113\">\r\n<\/span><\/p>\r\n<\/li>\r\n \t<li>To approximate the percentage of Americans with cellular service in the year 2013, substitute <em>x\u00a0<\/em>= 18 for the in the model and solve for\u00a0<em>y<\/em>:\r\n<p id=\"fs-id1410550\" style=\"text-align: center;\">[latex]\\begin{align}y&amp; =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}&amp;&amp; \\text{Use the regression model found in part (a)}.\\\\ &amp; =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013\\left(18\\right)}}&amp;&amp; \\text{Substitute 18 for }x. \\\\ &amp; \\approx \\text{99}\\text{.3 }&amp;&amp; \\text{Round to the nearest tenth} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1569617\">According to the model, about 98.8% of Americans had cellular service in 2013.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p id=\"fs-id1428422\">The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be <em>c\u00a0<\/em>= 100 and the model\u2019s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service!<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321575[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<div id=\"Example_04_08_02\" class=\"example\">\n<h2 id=\"fs-id1616172\" class=\"exercise\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Build a logistic model from data<\/span><\/h2>\n<\/div>\n<p id=\"fs-id1316523\">Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or <em>limiting value<\/em>. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.<\/p>\n<p id=\"fs-id1677718\">It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic <strong>regression analysis<\/strong>, we use the form most commonly used on graphing utilities:<\/p>\n<div id=\"eip-154\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\n<p id=\"fs-id1310104\">Recall that:<\/p>\n<ul id=\"fs-id1294720\">\n<li>[latex]\\frac{c}{1+a}[\/latex] is the initial value of the model.<\/li>\n<li>when <em>b\u00a0<\/em>&gt; 0, the model increases rapidly at first until it reaches its point of maximum growth rate, [latex]\\left(\\frac{\\mathrm{ln}\\left(a\\right)}{b},\\frac{c}{2}\\right)[\/latex]. At that point, growth steadily slows and the function becomes asymptotic to the upper bound <em>y\u00a0<\/em>= <em>c<\/em>.<\/li>\n<li><em>c<\/em>\u00a0is the limiting value, sometimes called the <em>carrying capacity<\/em>, of the model.<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3 class=\"title\">logistic regression<\/h3>\n<p id=\"fs-id1583226\"><strong>Logistic regression <\/strong>is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command &#8220;Logistic&#8221; on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form<\/p>\n<div id=\"eip-491\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex]<\/div>\n<p id=\"fs-id1701600\">Note that<\/p>\n<ul id=\"fs-id1358262\">\n<li>The initial value of the model is [latex]\\frac{c}{1+a}[\/latex].<\/li>\n<li>Output values for the model grow closer and closer to <em>y<\/em>\u00a0=\u00a0<em>c<\/em>\u00a0as time increases.<\/li>\n<\/ul>\n<\/section>\n<div>\n<section class=\"textbox interact\" aria-label=\"Interact\">Given a set of data, perform logistic regression using a graphing utility.<\/p>\n<ol id=\"fs-id1690756\">\n<li>Use the STAT then EDIT menu to enter given data.\n<ol id=\"fs-id1562359\" style=\"list-style-type: lower-alpha;\">\n<li>Clear any existing data from the lists.<\/li>\n<li>List the input values in the L1 column.<\/li>\n<li>List the output values in the L2 column.<\/li>\n<\/ol>\n<\/li>\n<li>Graph and observe a scatter plot of the data using the STATPLOT feature.\n<ol id=\"fs-id1431005\" style=\"list-style-type: lower-alpha;\">\n<li>Use ZOOM [9] to adjust axes to fit the data.<\/li>\n<li>Verify the data follow a logistic pattern.<\/li>\n<\/ol>\n<\/li>\n<li>Find the equation that models the data.\n<ol id=\"fs-id1585412\" style=\"list-style-type: lower-alpha;\">\n<li>Select &#8220;Logistic&#8221; from the STAT then CALC menu.<\/li>\n<li>Use the values returned for <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>\u00a0to record the model, [latex]y=\\frac{c}{1+a{e}^{-bx}}[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Graph the model in the same window as the scatterplot to verify it is a good fit for the data.<\/li>\n<\/ol>\n<\/section>\n<\/div>\n<div id=\"Example_04_08_03\" class=\"example\">\n<div id=\"fs-id1646727\" class=\"exercise\">\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1422087\">Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. The table below\u00a0shows the percentage of Americans with cellular service between the years 1995 and 2012.<a class=\"footnote\" title=\"Source: The World Bank, 2013.\" id=\"return-footnote-1200-1\" href=\"#footnote-1200-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<table id=\"Table_04_08_05\" summary=\"Nineteen rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Americans with Cellular Service (%)<\/th>\n<th>Year<\/th>\n<th>Americans with Cellular Service (%)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1995<\/td>\n<td>12.69<\/td>\n<td>2004<\/td>\n<td>62.852<\/td>\n<\/tr>\n<tr>\n<td>1996<\/td>\n<td>16.35<\/td>\n<td>2005<\/td>\n<td>68.63<\/td>\n<\/tr>\n<tr>\n<td>1997<\/td>\n<td>20.29<\/td>\n<td>2006<\/td>\n<td>76.64<\/td>\n<\/tr>\n<tr>\n<td>1998<\/td>\n<td>25.08<\/td>\n<td>2007<\/td>\n<td>82.47<\/td>\n<\/tr>\n<tr>\n<td>1999<\/td>\n<td>30.81<\/td>\n<td>2008<\/td>\n<td>85.68<\/td>\n<\/tr>\n<tr>\n<td>2000<\/td>\n<td>38.75<\/td>\n<td>2009<\/td>\n<td>89.14<\/td>\n<\/tr>\n<tr>\n<td>2001<\/td>\n<td>45.00<\/td>\n<td>2010<\/td>\n<td>91.86<\/td>\n<\/tr>\n<tr>\n<td>2002<\/td>\n<td>49.16<\/td>\n<td>2011<\/td>\n<td>95.28<\/td>\n<\/tr>\n<tr>\n<td>2003<\/td>\n<td>55.15<\/td>\n<td>2012<\/td>\n<td>98.17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1624806\">\n<li>Let <em>x<\/em>\u00a0represent time in years starting with <em>x\u00a0<\/em>= 0 for the year 1995. Let <em>y<\/em>\u00a0represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.<\/li>\n<li>Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.<\/li>\n<li>Discuss the value returned for the upper limit, <em>c<\/em>. What does this tell you about the model? What would the limiting value be if the model were exact?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q581544\">Show Solution<\/button><\/p>\n<div id=\"q581544\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1455761\">\n<li>Using the STAT then EDIT menu on a graphing utility, list the years using values 0\u201315 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in Figure 5:<br \/>\n<figure id=\"CNX_Precalc_Figure_04_08_005\">\n<figure style=\"width: 805px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010837\/CNX_Precalc_Figure_04_08_0052.jpg\" alt=\"Graph of a scattered plot.\" width=\"805\" height=\"396\" \/><figcaption class=\"wp-caption-text\"><b>Figure 5<\/b><\/figcaption><\/figure>\n<\/figure>\n<p id=\"fs-id1366003\">Use the &#8220;Logistic&#8221; command from the STAT then CALC menu to obtain the logistic model,<\/p>\n<p style=\"text-align: center;\">[latex]y=\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}[\/latex]<\/p>\n<figure style=\"width: 806px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010837\/CNX_Precalc_Figure_04_08_0062.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"806\" height=\"396\" \/><figcaption class=\"wp-caption-text\"><b>Figure 6<\/b><\/figcaption><\/figure>\n<p id=\"fs-id1650001\">Next, graph the model in the same window as shown in Figure 6\u00a0to verify it is a good fit:<span id=\"fs-id1246113\"><br \/>\n<\/span><\/p>\n<\/li>\n<li>To approximate the percentage of Americans with cellular service in the year 2013, substitute <em>x\u00a0<\/em>= 18 for the in the model and solve for\u00a0<em>y<\/em>:\n<p id=\"fs-id1410550\" style=\"text-align: center;\">[latex]\\begin{align}y& =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}&& \\text{Use the regression model found in part (a)}.\\\\ & =\\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013\\left(18\\right)}}&& \\text{Substitute 18 for }x. \\\\ & \\approx \\text{99}\\text{.3 }&& \\text{Round to the nearest tenth} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1569617\">According to the model, about 98.8% of Americans had cellular service in 2013.<\/p>\n<\/li>\n<li>\n<p id=\"fs-id1428422\">The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be <em>c\u00a0<\/em>= 100 and the model\u2019s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service!<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321575\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321575&theme=lumen&iframe_resize_id=ohm321575&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1200-1\">Source: <em>The World Bank, 2013<\/em>. <a href=\"#return-footnote-1200-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":13,"menu_order":27,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":510,"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\/1200"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1200\/revisions"}],"predecessor-version":[{"id":6086,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1200\/revisions\/6086"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1200\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1200"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1200"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1200"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}