{"id":42,"date":"2025-02-13T22:43:05","date_gmt":"2025-02-13T22:43:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rates-of-change-and-behavior-of-graphs\/"},"modified":"2026-04-01T08:59:07","modified_gmt":"2026-04-01T08:59:07","slug":"rates-of-change-and-behavior-of-graphs","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rates-of-change-and-behavior-of-graphs\/","title":{"raw":"Rates of Change and Behavior of Graphs: Learn It 1","rendered":"Rates of Change and Behavior of Graphs: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the average rate of change<\/li>\r\n \t<li>Use a graph to see where a function is going up, going down, or staying flat<\/li>\r\n \t<li>Use a graph to identify the highest and lowest points<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Rates of Change<\/h2>\r\nGasoline costs have experienced some wild fluctuations over the last several decades. The table below[footnote]\"Petroleum & Other Liquids,\" U.S. Energy Information Administration, March 31, 2026, https:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=pet&amp;s=emm_epmr_pte_nus_dpg&amp;f=a[\/footnote] lists the average cost, in dollars, of a gallon of gasoline for the years 2014\u20132023. The cost of gasoline can be considered as a function of year.\r\n<table style=\"width: 31.3474%;\" summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 25.4648%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td style=\"width: 5.1442%;\">2014<\/td>\r\n<td style=\"width: 7.69231%;\">2015<\/td>\r\n<td style=\"width: 7.21154%;\">2016<\/td>\r\n<td style=\"width: 7.21154%;\">2017<\/td>\r\n<td style=\"width: 7.05128%;\">2018<\/td>\r\n<td style=\"width: 7.69231%;\">2019<\/td>\r\n<td style=\"width: 7.21154%;\">2020<\/td>\r\n<td style=\"width: 7.85256%;\">2021<\/td>\r\n<td style=\"width: 7.37179%;\">2022<\/td>\r\n<td style=\"width: 7.37179%;\">2023<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25.4648%;\"><strong>[latex]C(y)[\/latex]<\/strong><\/td>\r\n<td style=\"width: 5.1442%;\">[latex]3.358[\/latex]<\/td>\r\n<td style=\"width: 7.69231%;\">[latex]2.429[\/latex]<\/td>\r\n<td style=\"width: 7.21154%;\">[latex]2.143[\/latex]<\/td>\r\n<td style=\"width: 7.21154%;\">[latex]2.415[\/latex]<\/td>\r\n<td style=\"width: 7.05128%;\">[latex]2.719[\/latex]<\/td>\r\n<td style=\"width: 7.69231%;\">[latex]2.604[\/latex]<\/td>\r\n<td style=\"width: 7.21154%;\">[latex]2.168[\/latex]<\/td>\r\n<td style=\"width: 7.85256%;\">[latex]3.008[\/latex]<\/td>\r\n<td style=\"width: 7.37179%;\">[latex]3.951[\/latex]<\/td>\r\n<td style=\"width: 7.37179%;\">[latex]3.519[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf we were interested only in how the gasoline prices changed between 2014 and 2023, we could compute that the cost per gallon had increased from [latex]$3.358[\/latex] to [latex]$3.519[\/latex], an increase of [latex]$0.161[\/latex]. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.\r\n\r\nThe price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change} &amp;=\\frac{\\text{Change in output}}{\\text{Change in input}}\\\\[2mm] &amp;=\\frac{\\Delta y}{\\Delta x}\\\\[2mm] &amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\\[2mm] &amp;=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/p>\r\nThe Greek letter [latex]\\Delta [\/latex] (delta) signifies the change in a quantity; we read the ratio as \"delta-[latex]y[\/latex] over delta-[latex]x[\/latex]\" or \"the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].\" Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value.\r\n<p class=\"whitespace-pre-wrap break-words\">In our example, the gasoline price increased by [latex]$0.161[\/latex] from 2014 to 2023. Over [latex]9[\/latex] years, the average rate of change was:<\/p>\r\n\r\n<center>[latex] \\frac{\\Delta y}{\\Delta x} = \\frac{$0.161}{9 \\text{ years}} \\approx 0.0179 \\text{ dollars per year} [\/latex]<\/center>\r\n<p class=\"whitespace-pre-wrap break-words\">On average, the price of gas increased by about [latex]1.79[\/latex] cents each year.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Other examples of rates of change include:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A population of rats increasing by [latex]40[\/latex] rats per week<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A car traveling [latex]68[\/latex] miles per hour (distance traveled changes by [latex]68[\/latex] miles each hour as time passes)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A car driving [latex]27[\/latex] miles per gallon (distance traveled changes by [latex]27[\/latex] miles for each gallon)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The current through an electrical circuit increasing by [latex]0.125[\/latex] amperes for every volt of increased voltage<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The amount of money in a college account decreasing by [latex]$4,000[\/latex] per quarter<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>Rate of Change<\/h3>\r\nA rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are \"output units per input units.\"\r\n[latex]\\\\[\/latex]\r\n\r\nThe average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/strong>\r\n<ol>\r\n \t<li>Calculate the difference [latex]{y}_{2}-{y}_{1}=\\Delta y[\/latex].<\/li>\r\n \t<li>Calculate the difference [latex]{x}_{2}-{x}_{1}=\\Delta x[\/latex].<\/li>\r\n \t<li>Find the ratio [latex]\\dfrac{\\Delta y}{\\Delta x}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Using the data in the table below, find the average rate of change of the price of gasoline between 2020 and 2022.\r\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2014<\/td>\r\n<td>2015<\/td>\r\n<td>2016<\/td>\r\n<td>2017<\/td>\r\n<td>2018<\/td>\r\n<td>2019<\/td>\r\n<td>2020<\/td>\r\n<td>2021<\/td>\r\n<td>2022<\/td>\r\n<td>2023<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>[latex]3.358[\/latex]<\/td>\r\n<td>[latex]2.429[\/latex]<\/td>\r\n<td>[latex]2.143[\/latex]<\/td>\r\n<td>[latex]2.415[\/latex]<\/td>\r\n<td>[latex]2.719[\/latex]<\/td>\r\n<td>[latex]2.604[\/latex]<\/td>\r\n<td>[latex]2.168[\/latex]<\/td>\r\n<td>[latex]3.008[\/latex]<\/td>\r\n<td>[latex]3.951[\/latex]<\/td>\r\n<td>[latex]3.519[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"576472\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"576472\"]\r\n<p class=\"whitespace-pre-wrap break-words\">In 2020, the price of gasoline was [latex]$2.168[\/latex]. In 2022, the cost was [latex]$3.951[\/latex]. The average rate of change is:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{align*} \\frac{\\Delta y}{\\Delta x} &amp;= \\frac{f(x_2) - f(x_1)}{x_2 - x_1} \\\\ &amp;= \\frac{$3.951 - $2.168}{2022 - 2020} \\\\ &amp;= \\frac{$1.783}{2 \\text{ years}} \\\\ &amp;= $0.8915 \\text{ per year} \\end{align*} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Analysis of the Solution<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Note that an increase is expressed by a positive change or \"positive increase.\" A rate of change is positive when the output increases as the input increases. In this case, we see a significant positive rate of change, indicating a sharp increase in gasoline prices between 2020 and 2022.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317880[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">After picking up a friend who lives [latex]10[\/latex] miles away, Anna records her distance from home over time. The values are shown in the table below.\u00a0Find her average speed over the first [latex]6[\/latex] hours.\r\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong><em>t<\/em> (hours)<\/strong><\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]55[\/latex]<\/td>\r\n<td>[latex]90[\/latex]<\/td>\r\n<td>[latex]153[\/latex]<\/td>\r\n<td>[latex]214[\/latex]<\/td>\r\n<td>[latex]240[\/latex]<\/td>\r\n<td>[latex]292[\/latex]<\/td>\r\n<td>[latex]300[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"462426\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"462426\"]\r\n\r\nHere, the average speed is the average rate of change. She traveled [latex]282 [\/latex] miles in [latex]6[\/latex] hours, for an average speed of\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{292 - 10}{6 - 0}&amp;=\\dfrac{282}{6}\\\\[2mm]&amp;=47 \\end{align}[\/latex]<\/p>\r\nThe average speed is [latex]47[\/latex] miles per hour.\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nBecause the speed is not constant, the average speed depends on the interval chosen. For the interval [latex][2,3][\/latex], the average speed is [latex]63[\/latex] miles per hour.\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]317881[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]g\\left(t\\right)[\/latex], find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194746\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/>\r\n[reveal-answer q=\"429524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"429524\"]\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194748\/CNX_Precalc_Figure_01_03_0022.jpg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/>At [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of\r\n<p style=\"text-align: center;\">[latex]\\dfrac{1 - 4}{2-\\left(-1\\right)}=\\dfrac{-3}{3}=-1[\/latex]<\/p>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nNote that the order we choose is very important. If, for example, we use [latex]\\dfrac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[\/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex].\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]317882[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{4].}[\/latex][reveal-answer q=\"263304\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"263304\"]We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f(2)&amp;=2^{2}-\\frac{1}{2} &amp;\\,\\,f(4)&amp;=4^{2}-\\frac{1}{4}\\\\[2mm]&amp;=4-\\frac{1}{2}&amp;\\,\\,&amp;=16-\\frac{1}{4}\\\\[2mm]&amp;=\\frac{7}{2}&amp;\\,\\,&amp;=\\frac{63}{4}\\end{align}[\/latex]<\/p>\r\nNow we compute the average rate of change.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\dfrac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[2mm]&amp;=\\dfrac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[2mm]&amp;=\\dfrac{\\frac{49}{4}}{2}\\\\[2mm]&amp;=\\dfrac{49}{8}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317883[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317884[\/ohm_question]<\/section><section class=\"textbox example\" aria-label=\"Example\">The <strong>electrostatic force<\/strong> [latex]F[\/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[\/latex], in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex]. Find the average rate of change of force if the distance between the particles is increased from [latex]2[\/latex] cm to [latex]6[\/latex] cm.[reveal-answer q=\"159564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"159564\"]We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right][\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\dfrac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2}\\\\[2mm]&amp;=\\dfrac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2}&amp;\\text{Simplify} \\\\[2mm]&amp;=\\dfrac{\\frac{2}{36}-\\frac{2}{4}}{4}\\\\[2mm]&amp;=\\dfrac{-\\frac{16}{36}}{4}&amp;\\text{Combine numerator terms}\\\\[2mm]&amp;=-\\dfrac{1}{9}&amp;\\text{Simplify}&amp;\\end{align}[\/latex]<\/p>\r\nThe average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].[reveal-answer q=\"427790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"427790\"]We use the average rate of change formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\dfrac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&amp;\\text{Evaluate}\\\\[2mm]&amp;=\\dfrac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&amp;\\text{Simplify}\\\\[2mm]&amp;=\\dfrac{{a}^{2}+3a+1 - 1}{a}&amp;\\text{Simplify and factor}\\\\[2mm]&amp;=\\dfrac{a\\left(a+3\\right)}{a}&amp;\\text{Divide by the common factor }a\\\\[2mm]&amp;=a+3\\end{align}[\/latex]<\/p>\r\nThis result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a[\/latex]. For example, on the interval [latex]\\left[0,5\\right][\/latex], the average rate of change would be [latex]5+3=8[\/latex].\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]317885[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<h3>Calculus Notation<\/h3>\r\n<p style=\"text-align: left;\">In Calculus, average rate of change over the interval [latex][a,b][\/latex] may also be presented as the formula:<\/p>\r\n<p style=\"text-align: center;\">[latex]f_{\\text{avg}} = \\dfrac{f\\left(b\\right) - f\\left(a\\right)}{b - a}[\/latex]<\/p>\r\n\r\n<\/section><\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the average rate of change<\/li>\n<li>Use a graph to see where a function is going up, going down, or staying flat<\/li>\n<li>Use a graph to identify the highest and lowest points<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Rates of Change<\/h2>\n<p>Gasoline costs have experienced some wild fluctuations over the last several decades. The table below<a class=\"footnote\" title=\"&quot;Petroleum &amp; Other Liquids,&quot; U.S. Energy Information Administration, March 31, 2026, https:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=pet&amp;s=emm_epmr_pte_nus_dpg&amp;f=a\" id=\"return-footnote-42-1\" href=\"#footnote-42-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> lists the average cost, in dollars, of a gallon of gasoline for the years 2014\u20132023. The cost of gasoline can be considered as a function of year.<\/p>\n<table style=\"width: 31.3474%;\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td style=\"width: 25.4648%;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<td style=\"width: 5.1442%;\">2014<\/td>\n<td style=\"width: 7.69231%;\">2015<\/td>\n<td style=\"width: 7.21154%;\">2016<\/td>\n<td style=\"width: 7.21154%;\">2017<\/td>\n<td style=\"width: 7.05128%;\">2018<\/td>\n<td style=\"width: 7.69231%;\">2019<\/td>\n<td style=\"width: 7.21154%;\">2020<\/td>\n<td style=\"width: 7.85256%;\">2021<\/td>\n<td style=\"width: 7.37179%;\">2022<\/td>\n<td style=\"width: 7.37179%;\">2023<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25.4648%;\"><strong>[latex]C(y)[\/latex]<\/strong><\/td>\n<td style=\"width: 5.1442%;\">[latex]3.358[\/latex]<\/td>\n<td style=\"width: 7.69231%;\">[latex]2.429[\/latex]<\/td>\n<td style=\"width: 7.21154%;\">[latex]2.143[\/latex]<\/td>\n<td style=\"width: 7.21154%;\">[latex]2.415[\/latex]<\/td>\n<td style=\"width: 7.05128%;\">[latex]2.719[\/latex]<\/td>\n<td style=\"width: 7.69231%;\">[latex]2.604[\/latex]<\/td>\n<td style=\"width: 7.21154%;\">[latex]2.168[\/latex]<\/td>\n<td style=\"width: 7.85256%;\">[latex]3.008[\/latex]<\/td>\n<td style=\"width: 7.37179%;\">[latex]3.951[\/latex]<\/td>\n<td style=\"width: 7.37179%;\">[latex]3.519[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If we were interested only in how the gasoline prices changed between 2014 and 2023, we could compute that the cost per gallon had increased from [latex]$3.358[\/latex] to [latex]$3.519[\/latex], an increase of [latex]$0.161[\/latex]. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.<\/p>\n<p>The price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change} &=\\frac{\\text{Change in output}}{\\text{Change in input}}\\\\[2mm] &=\\frac{\\Delta y}{\\Delta x}\\\\[2mm] &=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\\[2mm] &=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/p>\n<p>The Greek letter [latex]\\Delta[\/latex] (delta) signifies the change in a quantity; we read the ratio as &#8220;delta-[latex]y[\/latex] over delta-[latex]x[\/latex]&#8221; or &#8220;the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].&#8221; Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">In our example, the gasoline price increased by [latex]$0.161[\/latex] from 2014 to 2023. Over [latex]9[\/latex] years, the average rate of change was:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x} = \\frac{$0.161}{9 \\text{ years}} \\approx 0.0179 \\text{ dollars per year}[\/latex]<\/div>\n<p class=\"whitespace-pre-wrap break-words\">On average, the price of gas increased by about [latex]1.79[\/latex] cents each year.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Other examples of rates of change include:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A population of rats increasing by [latex]40[\/latex] rats per week<\/li>\n<li class=\"whitespace-normal break-words\">A car traveling [latex]68[\/latex] miles per hour (distance traveled changes by [latex]68[\/latex] miles each hour as time passes)<\/li>\n<li class=\"whitespace-normal break-words\">A car driving [latex]27[\/latex] miles per gallon (distance traveled changes by [latex]27[\/latex] miles for each gallon)<\/li>\n<li class=\"whitespace-normal break-words\">The current through an electrical circuit increasing by [latex]0.125[\/latex] amperes for every volt of increased voltage<\/li>\n<li class=\"whitespace-normal break-words\">The amount of money in a college account decreasing by [latex]$4,000[\/latex] per quarter<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>Rate of Change<\/h3>\n<p>A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are &#8220;output units per input units.&#8221;<br \/>\n[latex]\\\\[\/latex]<\/p>\n<p>The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/strong><\/p>\n<ol>\n<li>Calculate the difference [latex]{y}_{2}-{y}_{1}=\\Delta y[\/latex].<\/li>\n<li>Calculate the difference [latex]{x}_{2}-{x}_{1}=\\Delta x[\/latex].<\/li>\n<li>Find the ratio [latex]\\dfrac{\\Delta y}{\\Delta x}[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Using the data in the table below, find the average rate of change of the price of gasoline between 2020 and 2022.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2014<\/td>\n<td>2015<\/td>\n<td>2016<\/td>\n<td>2017<\/td>\n<td>2018<\/td>\n<td>2019<\/td>\n<td>2020<\/td>\n<td>2021<\/td>\n<td>2022<\/td>\n<td>2023<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>[latex]3.358[\/latex]<\/td>\n<td>[latex]2.429[\/latex]<\/td>\n<td>[latex]2.143[\/latex]<\/td>\n<td>[latex]2.415[\/latex]<\/td>\n<td>[latex]2.719[\/latex]<\/td>\n<td>[latex]2.604[\/latex]<\/td>\n<td>[latex]2.168[\/latex]<\/td>\n<td>[latex]3.008[\/latex]<\/td>\n<td>[latex]3.951[\/latex]<\/td>\n<td>[latex]3.519[\/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=\"q576472\">Show Answer<\/button><\/p>\n<div id=\"q576472\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">In 2020, the price of gasoline was [latex]$2.168[\/latex]. In 2022, the cost was [latex]$3.951[\/latex]. The average rate of change is:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{align*} \\frac{\\Delta y}{\\Delta x} &= \\frac{f(x_2) - f(x_1)}{x_2 - x_1} \\\\ &= \\frac{$3.951 - $2.168}{2022 - 2020} \\\\ &= \\frac{$1.783}{2 \\text{ years}} \\\\ &= $0.8915 \\text{ per year} \\end{align*}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><strong>Analysis of the Solution<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">Note that an increase is expressed by a positive change or &#8220;positive increase.&#8221; A rate of change is positive when the output increases as the input increases. In this case, we see a significant positive rate of change, indicating a sharp increase in gasoline prices between 2020 and 2022.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317880\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317880&theme=lumen&iframe_resize_id=ohm317880&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">After picking up a friend who lives [latex]10[\/latex] miles away, Anna records her distance from home over time. The values are shown in the table below.\u00a0Find her average speed over the first [latex]6[\/latex] hours.<\/p>\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\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><em>t<\/em> (hours)<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]55[\/latex]<\/td>\n<td>[latex]90[\/latex]<\/td>\n<td>[latex]153[\/latex]<\/td>\n<td>[latex]214[\/latex]<\/td>\n<td>[latex]240[\/latex]<\/td>\n<td>[latex]292[\/latex]<\/td>\n<td>[latex]300[\/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=\"q462426\">Show Solution<\/button><\/p>\n<div id=\"q462426\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here, the average speed is the average rate of change. She traveled [latex]282[\/latex] miles in [latex]6[\/latex] hours, for an average speed of<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{292 - 10}{6 - 0}&=\\dfrac{282}{6}\\\\[2mm]&=47 \\end{align}[\/latex]<\/p>\n<p>The average speed is [latex]47[\/latex] miles per hour.<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Because the speed is not constant, the average speed depends on the interval chosen. For the interval [latex][2,3][\/latex], the average speed is [latex]63[\/latex] miles per hour.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317881\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317881&theme=lumen&iframe_resize_id=ohm317881&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]g\\left(t\\right)[\/latex], find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194746\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q429524\">Show Solution<\/button><\/p>\n<div id=\"q429524\" class=\"hidden-answer\" style=\"display: none\">\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194748\/CNX_Precalc_Figure_01_03_0022.jpg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/>At [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{1 - 4}{2-\\left(-1\\right)}=\\dfrac{-3}{3}=-1[\/latex]<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Note that the order we choose is very important. If, for example, we use [latex]\\dfrac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[\/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317882\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317882&theme=lumen&iframe_resize_id=ohm317882&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{4].}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q263304\">Show Solution<\/button><\/p>\n<div id=\"q263304\" class=\"hidden-answer\" style=\"display: none\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f(2)&=2^{2}-\\frac{1}{2} &\\,\\,f(4)&=4^{2}-\\frac{1}{4}\\\\[2mm]&=4-\\frac{1}{2}&\\,\\,&=16-\\frac{1}{4}\\\\[2mm]&=\\frac{7}{2}&\\,\\,&=\\frac{63}{4}\\end{align}[\/latex]<\/p>\n<p>Now we compute the average rate of change.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\dfrac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[2mm]&=\\dfrac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[2mm]&=\\dfrac{\\frac{49}{4}}{2}\\\\[2mm]&=\\dfrac{49}{8}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317883\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317883&theme=lumen&iframe_resize_id=ohm317883&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317884\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317884&theme=lumen&iframe_resize_id=ohm317884&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The <strong>electrostatic force<\/strong> [latex]F[\/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[\/latex], in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex]. Find the average rate of change of force if the distance between the particles is increased from [latex]2[\/latex] cm to [latex]6[\/latex] cm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q159564\">Show Solution<\/button><\/p>\n<div id=\"q159564\" class=\"hidden-answer\" style=\"display: none\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right][\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\dfrac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2}\\\\[2mm]&=\\dfrac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2}&\\text{Simplify} \\\\[2mm]&=\\dfrac{\\frac{2}{36}-\\frac{2}{4}}{4}\\\\[2mm]&=\\dfrac{-\\frac{16}{36}}{4}&\\text{Combine numerator terms}\\\\[2mm]&=-\\dfrac{1}{9}&\\text{Simplify}&\\end{align}[\/latex]<\/p>\n<p>The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q427790\">Show Solution<\/button><\/p>\n<div id=\"q427790\" class=\"hidden-answer\" style=\"display: none\">We use the average rate of change formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\dfrac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&\\text{Evaluate}\\\\[2mm]&=\\dfrac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&\\text{Simplify}\\\\[2mm]&=\\dfrac{{a}^{2}+3a+1 - 1}{a}&\\text{Simplify and factor}\\\\[2mm]&=\\dfrac{a\\left(a+3\\right)}{a}&\\text{Divide by the common factor }a\\\\[2mm]&=a+3\\end{align}[\/latex]<\/p>\n<p>This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a[\/latex]. For example, on the interval [latex]\\left[0,5\\right][\/latex], the average rate of change would be [latex]5+3=8[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317885\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317885&theme=lumen&iframe_resize_id=ohm317885&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<h3>Calculus Notation<\/h3>\n<p style=\"text-align: left;\">In Calculus, average rate of change over the interval [latex][a,b][\/latex] may also be presented as the formula:<\/p>\n<p style=\"text-align: center;\">[latex]f_{\\text{avg}} = \\dfrac{f\\left(b\\right) - f\\left(a\\right)}{b - a}[\/latex]<\/p>\n<\/section>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-42-1\">\"Petroleum &amp; Other Liquids,\" U.S. Energy Information Administration, March 31, 2026, https:\/\/www.eia.gov\/dnav\/pet\/hist\/LeafHandler.ashx?n=pet&amp;s=emm_epmr_pte_nus_dpg&amp;f=a <a href=\"#return-footnote-42-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":6,"menu_order":21,"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":36,"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\/42"}],"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":16,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions"}],"predecessor-version":[{"id":6100,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/42\/revisions\/6100"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/42\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=42"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=42"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=42"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}