{"id":1194,"date":"2025-07-23T20:10:26","date_gmt":"2025-07-23T20:10:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1194"},"modified":"2026-04-01T08:37:06","modified_gmt":"2026-04-01T08:37:06","slug":"exponential-and-logarithmic-models-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-models-learn-it-5\/","title":{"raw":"Fitting Exponential Models to Data: Learn It 1","rendered":"Fitting Exponential Models to Data: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Build an exponential model from data.<\/li>\r\n \t<li>Build a logarithmic model from data.<\/li>\r\n \t<li>Build a logistic model from data.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Regression<\/h2>\r\nAs we have learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that\u2019s not the whole story. It\u2019s the <em>way<\/em> data increase or decrease that helps us determine whether it is best modeled by an exponential function. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let\u2019s review exponential growth and decay.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">Recall that exponential functions have the form [latex]y=a{b}^{x}[\/latex] or [latex]y={A}_{0}{e}^{kx}[\/latex]. When performing regression analysis, we use the form most commonly used on graphing utilities, [latex]y=a{b}^{x}[\/latex]. Take a moment to reflect on the characteristics we\u2019ve already learned about the exponential function [latex]y=a{b}^{x}[\/latex] (assume [latex]a &gt; 0[\/latex]):\r\n<ul>\r\n \t<li>[latex]b[\/latex]\u00a0must be greater than zero and not equal to one.<\/li>\r\n \t<li>The initial value of the model is\u00a0[latex]a[\/latex].\r\n<ul>\r\n \t<li>If [latex]b\u00a0&gt; 1[\/latex], the function models <strong>exponential growth<\/strong>. As [latex]x[\/latex]\u00a0increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.<\/li>\r\n \t<li>If [latex]0 &lt; b\u00a0&lt; 1[\/latex], the function models <strong>exponential decay<\/strong>. As [latex]x[\/latex]\u00a0increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the [latex]x[\/latex]-axis. In other words, the outputs never become equal to or less than zero.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>exponential regression<\/h3>\r\n<strong>Exponential regression<\/strong> is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.\u00a0 The exponential regression equation is of the form\r\n<p style=\"text-align: center;\">[latex]y=a{b}^{x}[\/latex]<\/p>\r\nNote that:\r\n<ul>\r\n \t<li>[latex]b[\/latex]\u00a0must be non-negative.<\/li>\r\n \t<li>When [latex]b \\gt 1[\/latex], we have an exponential growth model.<\/li>\r\n \t<li>When [latex]0 \\lt b \\lt 1[\/latex], we have an exponential decay model.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox interact\" aria-label=\"Interact\">Given a set of data, perform exponential regression using a graphing utility.\r\n<ol>\r\n \t<li>Use the STAT then EDIT menu to enter given data.\r\n<ol 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 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 an exponential pattern.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Find the equation that models the data.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Select \u201cExpReg\u201d from the STAT then CALC menu.<\/li>\r\n \t<li>Use the values returned for [latex]a[\/latex] and [latex]b[\/latex] to record the model.<\/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><section class=\"textbox example\" aria-label=\"Example\">In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from [latex]2,871[\/latex] crashes were used to measure the association of a person\u2019s blood alcohol level (BAC) with the risk of being in an accident.\r\n[latex]\\\\[\/latex]\r\nThe table below\u00a0shows results from the study.[footnote]Source: <em>Indiana University Center for Studies of Law in Action, 2007<\/em>.[\/footnote] The <em>relative risk<\/em> is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of [latex]0.09[\/latex] is [latex]3.54[\/latex] times as likely to crash as a person who has not been drinking alcohol.\r\n<table summary=\"Two rows and thirteen columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>BAC<\/strong><\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0.01[\/latex]<\/td>\r\n<td>[latex]0.03[\/latex]<\/td>\r\n<td>[latex]0.05[\/latex]<\/td>\r\n<td>[latex]0.07[\/latex]<\/td>\r\n<td>[latex]0.09[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1.03[\/latex]<\/td>\r\n<td>[latex]1.06[\/latex]<\/td>\r\n<td>[latex]1.38[\/latex]<\/td>\r\n<td>[latex]2.09[\/latex]<\/td>\r\n<td>[latex]3.54[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>BAC<\/strong><\/td>\r\n<td>[latex]0.11[\/latex]<\/td>\r\n<td>[latex]0.13[\/latex]<\/td>\r\n<td>[latex]0.15[\/latex]<\/td>\r\n<td>[latex]0.17[\/latex]<\/td>\r\n<td>[latex]0.19[\/latex]<\/td>\r\n<td>[latex]0.21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\r\n<td>[latex]6.41[\/latex]<\/td>\r\n<td>[latex]12.6[\/latex]<\/td>\r\n<td>[latex]22.1[\/latex]<\/td>\r\n<td>[latex]39.05[\/latex]<\/td>\r\n<td>[latex]65.32[\/latex]<\/td>\r\n<td>[latex]99.78[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>Let [latex]x[\/latex]\u00a0represent the BAC level and let [latex]y[\/latex]represent the corresponding relative risk. Use exponential regression to fit a model to these data.<\/li>\r\n \t<li>After [latex]6[\/latex] drinks, a person weighing [latex]160[\/latex] pounds will have a BAC of about [latex]0.16[\/latex]. How many times more likely is a person with this weight to crash if they drive after having a [latex]6[\/latex]-pack of beer? Round to the nearest hundredth.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"134040\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"134040\"]\r\n\r\n1. Using an online graphing tool, create a table by clicking on the + in the upper left and selecting the table icon. Enter the data.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03175712\/CNX_Precalc_Figure_04_08_0012.jpg\" alt=\"Graph of a scattered plot.\" width=\"487\" height=\"475\" \/>\r\n\r\nBelow your table enter\u00a0[latex]y_1[\/latex]~[latex]ab^{x_1}[\/latex].\r\n\r\nNotice that [latex]{r}^{2}[\/latex] is very close to 1 which indicates the model is a good fit to the data.\r\n\r\nIf using an online graphing tool, the model obtained is\u00a0[latex]y=0.55877{\\left(\\text{57,700,000,000}\\right)}^{x}[\/latex]\r\n\r\nIf using a graphing calculator, the model obtained is [latex]y=0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03175715\/CNX_Precalc_Figure_04_08_0022.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"487\" height=\"475\" \/>\r\n\r\n2. Use the model to estimate the risk associated with a BAC of [latex]0.16[\/latex]. Substitute [latex]0.16[\/latex] for [latex]x[\/latex]\u00a0in the model and solve for [latex]y[\/latex].\r\n\r\nUsing the online graphing tool model:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y\\hfill &amp; =0.55877{\\left(\\text{57,700,000,000}\\right)}^{x}\\hfill &amp; \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill &amp; =0.55877{\\left(\\text{57,700,000,000}\\right)}^{0.16}\\hfill &amp; \\text{Substitute 0}\\text{.16 for }x\\text{.}\\hfill \\\\ \\hfill &amp; \\approx \\text{29}\\text{.44}\\hfill &amp; \\text{Round to the nearest hundredth}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\nIf a [latex]160[\/latex]-pound person drives after having [latex]6[\/latex] drinks, he or she is about [latex]29.44[\/latex] times more likely to crash than if driving while sober.\r\n\r\nUsing the graphing calculator model:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y\\hfill &amp; =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}\\hfill &amp; \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill &amp; =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{0.16}\\hfill &amp; \\text{Substitute 0}\\text{.16 for }x\\text{.}\\hfill \\\\ \\hfill &amp; \\approx \\text{26}\\text{.35}\\hfill &amp; \\text{Round to the nearest hundredth}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\nIf a [latex]160[\/latex]-pound person drives after having [latex]6[\/latex] drinks, he or she is about [latex]26.35[\/latex] times more likely to crash than if driving while sober.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321570[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321571[\/ohm_question]\r\n\r\n<\/section><\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?\r\n<\/strong>\r\n\r\n<hr \/>\r\n\r\nNo. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation).\r\n\r\n<\/section>&nbsp;","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Build an exponential model from data.<\/li>\n<li>Build a logarithmic model from data.<\/li>\n<li>Build a logistic model from data.<\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Regression<\/h2>\n<p>As we have learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that\u2019s not the whole story. It\u2019s the <em>way<\/em> data increase or decrease that helps us determine whether it is best modeled by an exponential function. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let\u2019s review exponential growth and decay.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">Recall that exponential functions have the form [latex]y=a{b}^{x}[\/latex] or [latex]y={A}_{0}{e}^{kx}[\/latex]. When performing regression analysis, we use the form most commonly used on graphing utilities, [latex]y=a{b}^{x}[\/latex]. Take a moment to reflect on the characteristics we\u2019ve already learned about the exponential function [latex]y=a{b}^{x}[\/latex] (assume [latex]a > 0[\/latex]):<\/p>\n<ul>\n<li>[latex]b[\/latex]\u00a0must be greater than zero and not equal to one.<\/li>\n<li>The initial value of the model is\u00a0[latex]a[\/latex].\n<ul>\n<li>If [latex]b\u00a0> 1[\/latex], the function models <strong>exponential growth<\/strong>. As [latex]x[\/latex]\u00a0increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.<\/li>\n<li>If [latex]0 < b\u00a0< 1[\/latex], the function models <strong>exponential decay<\/strong>. As [latex]x[\/latex]\u00a0increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the [latex]x[\/latex]-axis. In other words, the outputs never become equal to or less than zero.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>exponential regression<\/h3>\n<p><strong>Exponential regression<\/strong> is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.\u00a0 The exponential regression equation is of the form<\/p>\n<p style=\"text-align: center;\">[latex]y=a{b}^{x}[\/latex]<\/p>\n<p>Note that:<\/p>\n<ul>\n<li>[latex]b[\/latex]\u00a0must be non-negative.<\/li>\n<li>When [latex]b \\gt 1[\/latex], we have an exponential growth model.<\/li>\n<li>When [latex]0 \\lt b \\lt 1[\/latex], we have an exponential decay model.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\">Given a set of data, perform exponential regression using a graphing utility.<\/p>\n<ol>\n<li>Use the STAT then EDIT menu to enter given data.\n<ol 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 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 an exponential pattern.<\/li>\n<\/ol>\n<\/li>\n<li>Find the equation that models the data.\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Select \u201cExpReg\u201d from the STAT then CALC menu.<\/li>\n<li>Use the values returned for [latex]a[\/latex] and [latex]b[\/latex] to record the model.<\/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<section class=\"textbox example\" aria-label=\"Example\">In 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from [latex]2,871[\/latex] crashes were used to measure the association of a person\u2019s blood alcohol level (BAC) with the risk of being in an accident.<br \/>\n[latex]\\\\[\/latex]<br \/>\nThe table below\u00a0shows results from the study.<a class=\"footnote\" title=\"Source: Indiana University Center for Studies of Law in Action, 2007.\" id=\"return-footnote-1194-1\" href=\"#footnote-1194-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> The <em>relative risk<\/em> is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of [latex]0.09[\/latex] is [latex]3.54[\/latex] times as likely to crash as a person who has not been drinking alcohol.<\/p>\n<table summary=\"Two rows and thirteen columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>BAC<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]0.03[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<td>[latex]0.07[\/latex]<\/td>\n<td>[latex]0.09[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1.03[\/latex]<\/td>\n<td>[latex]1.06[\/latex]<\/td>\n<td>[latex]1.38[\/latex]<\/td>\n<td>[latex]2.09[\/latex]<\/td>\n<td>[latex]3.54[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>BAC<\/strong><\/td>\n<td>[latex]0.11[\/latex]<\/td>\n<td>[latex]0.13[\/latex]<\/td>\n<td>[latex]0.15[\/latex]<\/td>\n<td>[latex]0.17[\/latex]<\/td>\n<td>[latex]0.19[\/latex]<\/td>\n<td>[latex]0.21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Relative Risk of Crashing<\/strong><\/td>\n<td>[latex]6.41[\/latex]<\/td>\n<td>[latex]12.6[\/latex]<\/td>\n<td>[latex]22.1[\/latex]<\/td>\n<td>[latex]39.05[\/latex]<\/td>\n<td>[latex]65.32[\/latex]<\/td>\n<td>[latex]99.78[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Let [latex]x[\/latex]\u00a0represent the BAC level and let [latex]y[\/latex]represent the corresponding relative risk. Use exponential regression to fit a model to these data.<\/li>\n<li>After [latex]6[\/latex] drinks, a person weighing [latex]160[\/latex] pounds will have a BAC of about [latex]0.16[\/latex]. How many times more likely is a person with this weight to crash if they drive after having a [latex]6[\/latex]-pack of beer? Round to the nearest hundredth.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q134040\">Show Solution<\/button><\/p>\n<div id=\"q134040\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. Using an online graphing tool, create a table by clicking on the + in the upper left and selecting the table icon. Enter the data.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03175712\/CNX_Precalc_Figure_04_08_0012.jpg\" alt=\"Graph of a scattered plot.\" width=\"487\" height=\"475\" \/><\/p>\n<p>Below your table enter\u00a0[latex]y_1[\/latex]~[latex]ab^{x_1}[\/latex].<\/p>\n<p>Notice that [latex]{r}^{2}[\/latex] is very close to 1 which indicates the model is a good fit to the data.<\/p>\n<p>If using an online graphing tool, the model obtained is\u00a0[latex]y=0.55877{\\left(\\text{57,700,000,000}\\right)}^{x}[\/latex]<\/p>\n<p>If using a graphing calculator, the model obtained is [latex]y=0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03175715\/CNX_Precalc_Figure_04_08_0022.jpg\" alt=\"Graph of a scattered plot with an estimation line.\" width=\"487\" height=\"475\" \/><\/p>\n<p>2. Use the model to estimate the risk associated with a BAC of [latex]0.16[\/latex]. Substitute [latex]0.16[\/latex] for [latex]x[\/latex]\u00a0in the model and solve for [latex]y[\/latex].<\/p>\n<p>Using the online graphing tool model:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y\\hfill & =0.55877{\\left(\\text{57,700,000,000}\\right)}^{x}\\hfill & \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill & =0.55877{\\left(\\text{57,700,000,000}\\right)}^{0.16}\\hfill & \\text{Substitute 0}\\text{.16 for }x\\text{.}\\hfill \\\\ \\hfill & \\approx \\text{29}\\text{.44}\\hfill & \\text{Round to the nearest hundredth}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<p>If a [latex]160[\/latex]-pound person drives after having [latex]6[\/latex] drinks, he or she is about [latex]29.44[\/latex] times more likely to crash than if driving while sober.<\/p>\n<p>Using the graphing calculator model:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}y\\hfill & =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{x}\\hfill & \\text{Use the regression model found in part (a)}\\text{.}\\hfill \\\\ \\hfill & =0.58304829{\\left(\\text{22,072,021,300}\\right)}^{0.16}\\hfill & \\text{Substitute 0}\\text{.16 for }x\\text{.}\\hfill \\\\ \\hfill & \\approx \\text{26}\\text{.35}\\hfill & \\text{Round to the nearest hundredth}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<p>If a [latex]160[\/latex]-pound person drives after having [latex]6[\/latex] drinks, he or she is about [latex]26.35[\/latex] times more likely to crash than if driving while sober.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321570\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321570&theme=lumen&iframe_resize_id=ohm321570&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321571\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321571&theme=lumen&iframe_resize_id=ohm321571&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Is it reasonable to assume that an exponential regression model will represent a situation indefinitely?<br \/>\n<\/strong><\/p>\n<hr \/>\n<p>No. Remember that models are formed by real-world data gathered for regression. It is usually reasonable to make estimates within the interval of original observation (interpolation). However, when a model is used to make predictions, it is important to use reasoning skills to determine whether the model makes sense for inputs far beyond the original observation interval (extrapolation).<\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1194-1\">Source: <em>Indiana University Center for Studies of Law in Action, 2007<\/em>. <a href=\"#return-footnote-1194-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":13,"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":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\/1194"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1194\/revisions"}],"predecessor-version":[{"id":6087,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1194\/revisions\/6087"}],"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\/1194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}