{"id":4054,"date":"2024-09-17T14:17:32","date_gmt":"2024-09-17T14:17:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=4054"},"modified":"2025-08-14T00:16:02","modified_gmt":"2025-08-14T00:16:02","slug":"fitting-linear-models-to-data-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/fitting-linear-models-to-data-learn-it-6\/","title":{"raw":"Fitting Linear Models to Data: Learn It 6","rendered":"Fitting Linear Models to Data: Learn It 6"},"content":{"raw":"<h2>Distinguishing Between Linear and Nonlinear Models<\/h2>\r\n<div class=\"page\" title=\"Page 449\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nAs we saw previously with the cricket-chirp model example, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <strong>correlation coefficient<\/strong>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a \"diagnostic on\" selection to find the correlation coefficient, which mathematicians label as [latex]r[\/latex]. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>correlation coefficient ([latex]r[\/latex])<\/h3>\r\nThe <strong>correlation coefficient<\/strong> is a value, [latex]r[\/latex], between [latex]\u20131[\/latex] and [latex]1[\/latex].\r\n<ul>\r\n \t<li>[latex]r &gt; 0[\/latex] suggests a positive (increasing) relationship<\/li>\r\n \t<li>[latex]r &lt; 0[\/latex] suggests a positive (increasing) relationship<\/li>\r\n \t<li>The closer the value is to [latex]0[\/latex], the more scattered the data.<\/li>\r\n \t<li>The closer the value is to [latex]1[\/latex] or [latex]\u20131[\/latex], the less scattered the data is:\r\n<ul>\r\n \t<li>[latex]|r| &lt; 0.3[\/latex] is weak<\/li>\r\n \t<li>[latex]0.3 \u2264 |r| &lt; 0.7[\/latex] is moderate<\/li>\r\n \t<li>[latex]|r| \u2265 0.7[\/latex] is strong<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nWe should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense of the relationship between the value of [latex]r[\/latex]\u00a0and the graph of the data, the image below\u00a0shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.\r\n\r\n<section class=\"textbox example\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"901\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014349\/CNX_Precalc_Figure_02_04_0072.jpg\" alt=\"A series of scatterplot graphs. Some are linear and some are not.\" width=\"901\" height=\"401\" \/> Plotted data and related correlation coefficients. (credit: \"DenisBoigelot,\" Wikimedia Commons)[\/caption]\r\n\r\n<\/section><section class=\"textbox interact\" aria-label=\"Interact\"><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" width=\"100%\" height=\"1100\" frameborder=\"no\" data-mce-fragment=\"1\"><\/iframe>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/section><section class=\"textbox example\">Calculate the correlation coefficient for cricket-chirp data using the statistical tool above. Interpret your correlation coefficient.\r\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps<\/strong><\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]20.4[\/latex]<\/td>\r\n<td>[latex]33[\/latex]<\/td>\r\n<td>[latex]31[\/latex]<\/td>\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]18.5[\/latex]<\/td>\r\n<td>[latex]37[\/latex]<\/td>\r\n<td>[latex]26[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Temperature<\/strong><\/td>\r\n<td>[latex]80.5[\/latex]<\/td>\r\n<td>[latex]70.5[\/latex]<\/td>\r\n<td>[latex]57[\/latex]<\/td>\r\n<td>[latex]66[\/latex]<\/td>\r\n<td>[latex]68[\/latex]<\/td>\r\n<td>[latex]72[\/latex]<\/td>\r\n<td>[latex]52[\/latex]<\/td>\r\n<td>[latex]73.5[\/latex]<\/td>\r\n<td>[latex]53[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"888312\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"888312\"]After entering your data set into the statistical tool, if you scroll down, under <strong>\"Model Summary\", <\/strong>you will find the value of \"Correlation Coefficient [latex]r[\/latex].\"\r\n\r\nYou should see that <strong>[latex]r = 0.951[\/latex]<\/strong>.\r\n[latex]\\\\[\/latex]\r\nThis value is very close to [latex]1[\/latex], which suggests a very strong increasing linear relationship.\r\n\r\n<strong>What this means:<\/strong>\r\n<ul>\r\n \t<li><strong>Strong Positive Linear Relationship<\/strong>: The high value of [latex]r[\/latex] (close to [latex]1[\/latex]) suggests that as the number of cricket chirps increases, the temperature also increases.<\/li>\r\n \t<li><strong>Consistency<\/strong>: The data points lie very close to the best-fit line, meaning the relationship between chirps and temperature is consistent and predictable.<\/li>\r\n<\/ul>\r\nIn summary, a correlation coefficient of [latex]r = 0.951[\/latex] means there is a strong, positive linear relationship between the number of cricket chirps and the temperature. This suggests that you can reliably predict the temperature based on the number of chirps, and vice versa.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"1139\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/17003803\/Picture17.png\" alt=\"From left to right, graphs showing perfect positive correlation, strong positive correlation, weak positive correlation, no correlation, weak negative correlation, strong negative correlation, and perfect negative correlation.\" width=\"1139\" height=\"235\" \/> Graphs with varying correlation[\/caption]\r\n\r\n[reveal-answer q=\"371551\"]General Interpretation[\/reveal-answer]\r\n[hidden-answer a=\"371551\"]\r\n<table style=\"width: 84.1432%;\">\r\n<tbody>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\"><strong>Correlation Coefficient, [latex]r[\/latex]<\/strong><\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\"><strong>General Interpretation<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-1[\/latex] to [latex]-0.7[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\">Strong negative linear relationship<\/td>\r\n<\/tr>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-0.7[\/latex] to [latex]-0.3[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\">Moderate negative linear relationship<\/td>\r\n<\/tr>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-0.3[\/latex] to [latex]-0.1[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\">Weak negative linear relationship<\/td>\r\n<\/tr>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-0.1[\/latex] to [latex]0.1[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\">Negligible or no linear relationship<\/td>\r\n<\/tr>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]0.1[\/latex] to [latex]0.3[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\">Weak positive linear relationship<\/td>\r\n<\/tr>\r\n<tr style=\"height: 55px;\">\r\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]0.3[\/latex] to [latex]0.7[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 55px;\">Moderate positive linear relationship<\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px;\">\r\n<td style=\"width: 43.071%; height: 36px; text-align: center;\">[latex]0.7[\/latex] to [latex]1[\/latex]<\/td>\r\n<td style=\"width: 78.7239%; height: 36px;\">Strong positive linear relationship<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]23241[\/ohm2_question]<\/section><section class=\"textbox proTip\"><strong>Association does not imply causation!!!<\/strong>\r\nDo not interpret a high correlation between the two variable in the data as a cause-and-effect relationship.<\/section><\/div>\r\n<\/div>\r\n<section><section class=\"textbox interact\" aria-label=\"Interact\"><section>Can you guess what the correlation coefficient for the scatterplot below?<\/section><section><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/guesscorr\/\" width=\"100%\" height=\"750\" data-mce-fragment=\"1\"><\/iframe>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/guesscorr\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">\u00a0<\/span><\/section><\/section><\/section>","rendered":"<h2>Distinguishing Between Linear and Nonlinear Models<\/h2>\n<div class=\"page\" title=\"Page 449\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>As we saw previously with the cricket-chirp model example, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the <strong>correlation coefficient<\/strong>, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a &#8220;diagnostic on&#8221; selection to find the correlation coefficient, which mathematicians label as [latex]r[\/latex]. The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>correlation coefficient ([latex]r[\/latex])<\/h3>\n<p>The <strong>correlation coefficient<\/strong> is a value, [latex]r[\/latex], between [latex]\u20131[\/latex] and [latex]1[\/latex].<\/p>\n<ul>\n<li>[latex]r > 0[\/latex] suggests a positive (increasing) relationship<\/li>\n<li>[latex]r < 0[\/latex] suggests a positive (increasing) relationship<\/li>\n<li>The closer the value is to [latex]0[\/latex], the more scattered the data.<\/li>\n<li>The closer the value is to [latex]1[\/latex] or [latex]\u20131[\/latex], the less scattered the data is:\n<ul>\n<li>[latex]|r| < 0.3[\/latex] is weak<\/li>\n<li>[latex]0.3 \u2264 |r| < 0.7[\/latex] is moderate<\/li>\n<li>[latex]|r| \u2265 0.7[\/latex] is strong<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p>We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense of the relationship between the value of [latex]r[\/latex]\u00a0and the graph of the data, the image below\u00a0shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.<\/p>\n<section class=\"textbox example\">\n<figure style=\"width: 901px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/19014349\/CNX_Precalc_Figure_02_04_0072.jpg\" alt=\"A series of scatterplot graphs. Some are linear and some are not.\" width=\"901\" height=\"401\" \/><figcaption class=\"wp-caption-text\">Plotted data and related correlation coefficients. (credit: &#8220;DenisBoigelot,&#8221; Wikimedia Commons)<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\"><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" width=\"100%\" height=\"1100\" frameborder=\"no\" data-mce-fragment=\"1\"><\/iframe>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/section>\n<section class=\"textbox example\">Calculate the correlation coefficient for cricket-chirp data using the statistical tool above. Interpret your correlation coefficient.<\/p>\n<table summary=\"Two rows and ten columns. The first row is labeled, 'chirps'. The second row is labeled is labeled, 'Temp'. Reading the remaining rows as ordered pairs (i.e., (chirps, Temp), we have the following values: (44, 80.5), (35, 70.5), (20.4, 57), (33, 66), (31, 68), (35, 72), (18.5, 52), (37, 73.5) and (26, 53).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>Chirps<\/strong><\/td>\n<td>[latex]44[\/latex]<\/td>\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]20.4[\/latex]<\/td>\n<td>[latex]33[\/latex]<\/td>\n<td>[latex]31[\/latex]<\/td>\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]18.5[\/latex]<\/td>\n<td>[latex]37[\/latex]<\/td>\n<td>[latex]26[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Temperature<\/strong><\/td>\n<td>[latex]80.5[\/latex]<\/td>\n<td>[latex]70.5[\/latex]<\/td>\n<td>[latex]57[\/latex]<\/td>\n<td>[latex]66[\/latex]<\/td>\n<td>[latex]68[\/latex]<\/td>\n<td>[latex]72[\/latex]<\/td>\n<td>[latex]52[\/latex]<\/td>\n<td>[latex]73.5[\/latex]<\/td>\n<td>[latex]53[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q888312\">Show Answer<\/button><\/p>\n<div id=\"q888312\" class=\"hidden-answer\" style=\"display: none\">After entering your data set into the statistical tool, if you scroll down, under <strong>&#8220;Model Summary&#8221;, <\/strong>you will find the value of &#8220;Correlation Coefficient [latex]r[\/latex].&#8221;<\/p>\n<p>You should see that <strong>[latex]r = 0.951[\/latex]<\/strong>.<br \/>\n[latex]\\\\[\/latex]<br \/>\nThis value is very close to [latex]1[\/latex], which suggests a very strong increasing linear relationship.<\/p>\n<p><strong>What this means:<\/strong><\/p>\n<ul>\n<li><strong>Strong Positive Linear Relationship<\/strong>: The high value of [latex]r[\/latex] (close to [latex]1[\/latex]) suggests that as the number of cricket chirps increases, the temperature also increases.<\/li>\n<li><strong>Consistency<\/strong>: The data points lie very close to the best-fit line, meaning the relationship between chirps and temperature is consistent and predictable.<\/li>\n<\/ul>\n<p>In summary, a correlation coefficient of [latex]r = 0.951[\/latex] means there is a strong, positive linear relationship between the number of cricket chirps and the temperature. This suggests that you can reliably predict the temperature based on the number of chirps, and vice versa.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<figure style=\"width: 1139px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/09\/17003803\/Picture17.png\" alt=\"From left to right, graphs showing perfect positive correlation, strong positive correlation, weak positive correlation, no correlation, weak negative correlation, strong negative correlation, and perfect negative correlation.\" width=\"1139\" height=\"235\" \/><figcaption class=\"wp-caption-text\">Graphs with varying correlation<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q371551\">General Interpretation<\/button><\/p>\n<div id=\"q371551\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 84.1432%;\">\n<tbody>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\"><strong>Correlation Coefficient, [latex]r[\/latex]<\/strong><\/td>\n<td style=\"width: 78.7239%; height: 55px;\"><strong>General Interpretation<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-1[\/latex] to [latex]-0.7[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 55px;\">Strong negative linear relationship<\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-0.7[\/latex] to [latex]-0.3[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 55px;\">Moderate negative linear relationship<\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-0.3[\/latex] to [latex]-0.1[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 55px;\">Weak negative linear relationship<\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]-0.1[\/latex] to [latex]0.1[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 55px;\">Negligible or no linear relationship<\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]0.1[\/latex] to [latex]0.3[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 55px;\">Weak positive linear relationship<\/td>\n<\/tr>\n<tr style=\"height: 55px;\">\n<td style=\"width: 43.071%; height: 55px; text-align: center;\">[latex]0.3[\/latex] to [latex]0.7[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 55px;\">Moderate positive linear relationship<\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"width: 43.071%; height: 36px; text-align: center;\">[latex]0.7[\/latex] to [latex]1[\/latex]<\/td>\n<td style=\"width: 78.7239%; height: 36px;\">Strong positive linear relationship<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm23241\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=23241&theme=lumen&iframe_resize_id=ohm23241&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\"><strong>Association does not imply causation!!!<\/strong><br \/>\nDo not interpret a high correlation between the two variable in the data as a cause-and-effect relationship.<\/section>\n<\/div>\n<\/div>\n<section>\n<section class=\"textbox interact\" aria-label=\"Interact\">\n<section>Can you guess what the correlation coefficient for the scatterplot below?<\/section>\n<section><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/guesscorr\/\" width=\"100%\" height=\"750\" data-mce-fragment=\"1\"><\/iframe>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/guesscorr\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">\u00a0<\/span><\/section>\n<\/section>\n<\/section>\n","protected":false},"author":15,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":164,"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4054"}],"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\/15"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4054\/revisions"}],"predecessor-version":[{"id":7713,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4054\/revisions\/7713"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/164"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4054\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=4054"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4054"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=4054"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=4054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}