{"id":107,"date":"2025-02-13T22:43:51","date_gmt":"2025-02-13T22:43:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions\/"},"modified":"2026-03-16T17:39:36","modified_gmt":"2026-03-16T17:39:36","slug":"exponential-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions\/","title":{"raw":"Graphs of Exponential Functions: Learn It 1","rendered":"Graphs of Exponential Functions: 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>Graph exponential functions.<\/li>\r\n \t<li>Graph exponential functions using transformations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p id=\"fs-id1165137442020\">Exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\r\n\r\n<h2>Defining Exponential Functions<\/h2>\r\nWhat exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.\r\n<ul>\r\n \t<li><strong>Percent change<\/strong> refers to a change based on a percent of the original amount.<\/li>\r\n \t<li><strong>Exponential growth<\/strong> refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.<\/li>\r\n \t<li><strong>Exponential decay<\/strong> refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.<\/li>\r\n<\/ul>\r\nFor us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth.\r\n<table style=\"border-collapse: collapse; width: 78.5374%; height: 132px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^x[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^0 = 1[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(0) = 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^1 = 2[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(1) = 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^2 = 4[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(2) = 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^3 = 8[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(3) = 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^4 = 16[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(4) = 8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can infer that for these two functions, exponential growth dwarfs linear growth.\r\n<ul>\r\n \t<li><strong>Exponential growth<\/strong> refers to the original value from the range increases by the same percentage over equal increments found in the domain.<\/li>\r\n \t<li><strong>Linear growth<\/strong> refers to the original value from the range increases by the same amount over equal increments found in the domain.<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321379[\/ohm_question]<\/section>Apparently, the difference between \u201cthe same percentage\u201d and \u201cthe same amount\u201d is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding [latex]2[\/latex] to the output whenever the input was increased by one.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>exponential function<\/h3>\r\nThe general form of the <strong>exponential formula<\/strong>\u00a0is\r\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\r\nwhere [latex]a[\/latex] is any nonzero number and [latex]b[\/latex] is a positive real number not equal to [latex]1[\/latex].\r\n<ul>\r\n \t<li>if [latex]b&gt;1[\/latex], the function grows at a rate proportional to its size.<\/li>\r\n \t<li>if [latex]0 \\lt b \\lt 1[\/latex], the function decays at a rate proportional to its size.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Why do we limit the base [latex]b[\/latex]\u00a0to positive values?<\/strong>\r\n\r\n<hr \/>\r\n\r\nThis is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:\r\n<ul>\r\n \t<li>Consider a base of \u20139 and exponent of [latex]\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\r\n<\/ul>\r\n<strong>Why do we limit the base to positive values other than 1?\r\n<\/strong>\r\n\r\n<hr \/>\r\n\r\nThis is because a base of 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:\r\n<ul>\r\n \t<li>Consider a base of 1.\u00a0Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 aria-label=\"Example\">Exponential Growth<\/h2>\r\nBecause the output of exponential functions increases very rapidly, the term \"exponential growth\" is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.\r\n\r\nTo get a sense of the behavior of <strong>exponential growth<\/strong>, we can create a table of values for a function of the form [latex]f(x)={b}^{x}[\/latex], where [latex]f \\gt 1[\/latex].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Let's take a look at [latex]f(x)={2}^{x}[\/latex].\r\n<table id=\"Table_04_02_01\" style=\"width: 100%;\" summary=\"Two rows and eight columns. The first row is labeled,\"><colgroup> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 18.3633%;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 17.1657%;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 13.9721%;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 13.9055%;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 8.9155%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 8.9155%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 8.98204%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 8.98204%;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.3633%;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 17.1657%;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 13.9721%;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 13.9055%;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 8.9155%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 8.9155%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 8.98204%;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 8.98204%;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\" wp-image-2094 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/08181744\/Screenshot-2024-07-08-at-11.17.38%E2%80%AFAM.png\" alt=\"\" width=\"300\" height=\"296\" \/>We call the base [latex]2[\/latex] the <em>constant ratio<\/em>. This means that as the input increases by [latex]1[\/latex], the output value will be the product of the base and the previous output. Did you notice that the next number is [latex]2[\/latex] times the previous number?\r\n\r\nThis pattern shows <strong>exponential growth<\/strong> because the output value increases by a factor of [latex]2[\/latex] each time.\r\n\r\nCharacteristics:\r\n<ul>\r\n \t<li><strong>Domain: [latex](-\\infty,\\infty)[\/latex]<\/strong><\/li>\r\n \t<li><strong>Range: [latex](0,\\infty)[\/latex]<\/strong><\/li>\r\n \t<li>As [latex]x \\rightarrow \\infty, f(x) \\rightarrow \\infty[\/latex].<\/li>\r\n \t<li>As [latex]x \\rightarrow -\\infty, f(x) \\rightarrow 0[\/latex].<\/li>\r\n \t<li>the graph of [latex]f[\/latex] will never touch the [latex]x[\/latex]-axis because base two raised to any exponent never has the result of zero.<\/li>\r\n \t<li><strong>Horizontal Asymptote: [latex]y = 0[\/latex]<\/strong><\/li>\r\n \t<li>[latex]f(x)[\/latex] is always increasing.<\/li>\r\n \t<li>No [latex]x[\/latex]-intercept.<\/li>\r\n \t<li>[latex]y[\/latex]-intercept is [latex](0,1)[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>exponential growth<\/h3>\r\nA function that models <strong>exponential growth<\/strong> grows by a rate proportional to the amount present. For any real number [latex]x[\/latex]\u00a0and any positive real numbers <em>a\u00a0<\/em>and <em>b<\/em> such that [latex]b\\ne 1[\/latex], an exponential growth function has the form\r\n<p style=\"text-align: center;\">[latex]\\text{ }f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\r\nwhere\r\n<ul>\r\n \t<li>[latex]a[\/latex]\u00a0is the initial or starting value of the function.<\/li>\r\n \t<li>[latex]b[\/latex]\u00a0is the growth factor or growth multiplier per unit [latex]x[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>In more general terms, an <em>exponential function <\/em>consists of a\u00a0constant base raised to a variable exponent.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">To differentiate between linear and exponential functions, let\u2019s consider two companies, A and B.\r\n<ul>\r\n \t<li>Company A has [latex]100[\/latex] stores and expands by opening [latex]50[\/latex] new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex].<\/li>\r\n \t<li>Company B has [latex]100[\/latex] stores and expands by increasing the number of stores by [latex]50 \\%[\/latex] each year, so its growth can be represented by the function [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/li>\r\n<\/ul>\r\nA few years of growth for these companies are illustrated below.\r\n<table summary=\"Six rows and three columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th>Year,\u00a0[latex]x[\/latex]<\/th>\r\n<th>Stores, Company A<\/th>\r\n<th>Stores, Company B<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]100 + 50(0) = 100[\/latex]<\/td>\r\n<td>[latex]100(1 + 0.5)^0 = 100[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]100 + 50(1) = 150[\/latex]<\/td>\r\n<td>[latex]100(1 + 0.5)^1 = 150[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]100 + 50(2) = 200[\/latex]<\/td>\r\n<td>[latex]100(1 + 0.5)^2 = 225[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]100 + 50(3) = 250[\/latex]<\/td>\r\n<td>[latex]100(1 + 0.5)^3 =\u00a0337.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]A(x) = 100 + 50x[\/latex]<\/td>\r\n<td>[latex]B(x) = 100(1 + 0.5)^x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe graphs comparing the number of stores for each company over a five-year period are shown below. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.\r\n\r\n[caption id=\"attachment_4998\" align=\"aligncenter\" width=\"419\"]<img class=\"wp-image-4998 \" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_-173x300.png\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"419\" height=\"727\" \/> The graph shows the numbers of stores Companies A and B opened over a five-year period.[\/caption]\r\n\r\n<div class=\"mceTemp\"><\/div>\r\nNotice that the domain for both functions is [latex]\\left[0,\\infty \\right)[\/latex], and the range for both functions is [latex]\\left[100,\\infty \\right)[\/latex]. After year 1, Company B always has more stores than Company A.\r\n\r\nLet's more closely examine the function representing the number of stores for Company B,\r\n\r\n<center>[latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/center>In this exponential function, [latex]100[\/latex] represents the initial number of stores, [latex]0.5[\/latex] represents the growth rate, and [latex]1+0.5=1.5[\/latex] represents the growth factor. Generalizing further, we can write this function as\r\n\r\n<center>[latex]B\\left(x\\right)=100{\\left(1.5\\right)}^{x}[\/latex]<\/center>\r\nwhere [latex]100[\/latex] is the initial value, [latex]1.5[\/latex] is called the <em>base<\/em>, and [latex]x[\/latex]\u00a0is called the <em>exponent<\/em>.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The population of India was about [latex]1.25[\/latex] billion in the year 2013 with an annual growth rate of about [latex]1.2\\%[\/latex].\r\n[latex]\\\\[\/latex]\r\nThis situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex] where [latex]t[\/latex]\u00a0is the number of years since 2013.\r\n[latex]\\\\[\/latex]\r\nTo the nearest thousandth, what will the population of India be in 2031?[reveal-answer q=\"924755\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"924755\"]To estimate the population in 2031, we evaluate the models for [latex]t\u00a0= 18[\/latex], because 2031 is [latex]18[\/latex] years after 2013. Rounding to the nearest thousandth,\r\n<p style=\"text-align: center;\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/p>\r\nThere will be about [latex]1.549[\/latex] billion people in India in the year 2031.\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]321380[\/ohm_question]<\/section>\r\n<h2>Exponential Decay<\/h2>\r\nTo get a sense of the behavior of <strong>exponential decay<\/strong>, we can create a table of values for a function of the form [latex]g(x)={b}^{x}[\/latex], where [latex]0 \\lt b \\lt 1[\/latex].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Let's take a look at the function [latex]g(x)={(\\frac{1}{2})}^{x}[\/latex].Observe how the output values in the table below change as the input increases by [latex]1[\/latex].\r\n<table style=\"width: 100%;\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 37.6529%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 7.93341%;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 3.42556%;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 3.42556%;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 4.4796%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 16.996%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16.6008%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 7.50984%;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 37.6529%;\"><strong>[latex]g(x)=(\\frac{1}{2})^{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 7.93341%;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 3.42556%;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 3.42556%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 4.4796%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16.996%;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6008%;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 7.50984%;\">[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231133\/CNX_Precalc_Figure_04_02_0022.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"325\" height=\"347\" \/>When the input is increasing by [latex]1[\/latex], each output value is the product of the previous output and the base or constant ratio [latex]\\frac{1}{2}[\/latex].\r\n\r\nNotice from the table that:\r\n<ul>\r\n \t<li>the output values are positive for all values of [latex]x[\/latex].<\/li>\r\n \t<li>as [latex]x[\/latex] increases, the output values grow smaller, approaching zero.<\/li>\r\n \t<li>as [latex]x[\/latex] decreases, the output values grow without bound.<\/li>\r\n<\/ul>\r\nCharacteristics:\r\n<ul>\r\n \t<li>Domain: [latex]\\left(-\\infty , \\infty \\right)[\/latex]<\/li>\r\n \t<li>Range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\r\n \t<li>[latex]x[\/latex]<em>-<\/em>intercept: none<\/li>\r\n \t<li>[latex]y[\/latex]<em>-<\/em>intercept: [latex]\\left(0,1\\right)[\/latex]<\/li>\r\n \t<li>Horizontal asymptote: [latex]y=0[\/latex]<\/li>\r\n<\/ul>\r\nThis is an <strong>exponential decay<\/strong>.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321381[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321382[\/ohm_question]<\/section><section id=\"fs-id1165137447701\" class=\"key-concepts\">\u00a0<\/section><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph exponential functions.<\/li>\n<li>Graph exponential functions using transformations.<\/li>\n<\/ul>\n<\/section>\n<p id=\"fs-id1165137442020\">Exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\n<h2>Defining Exponential Functions<\/h2>\n<p>What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.<\/p>\n<ul>\n<li><strong>Percent change<\/strong> refers to a change based on a percent of the original amount.<\/li>\n<li><strong>Exponential growth<\/strong> refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.<\/li>\n<li><strong>Exponential decay<\/strong> refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.<\/li>\n<\/ul>\n<p>For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth.<\/p>\n<table style=\"border-collapse: collapse; width: 78.5374%; height: 132px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^x[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^0 = 1[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(0) = 0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^1 = 2[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(1) = 2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^2 = 4[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(2) = 4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^3 = 8[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(3) = 6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^4 = 16[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(4) = 8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can infer that for these two functions, exponential growth dwarfs linear growth.<\/p>\n<ul>\n<li><strong>Exponential growth<\/strong> refers to the original value from the range increases by the same percentage over equal increments found in the domain.<\/li>\n<li><strong>Linear growth<\/strong> refers to the original value from the range increases by the same amount over equal increments found in the domain.<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321379\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321379&theme=lumen&iframe_resize_id=ohm321379&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Apparently, the difference between \u201cthe same percentage\u201d and \u201cthe same amount\u201d is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding [latex]2[\/latex] to the output whenever the input was increased by one.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>exponential function<\/h3>\n<p>The general form of the <strong>exponential formula<\/strong>\u00a0is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\n<p>where [latex]a[\/latex] is any nonzero number and [latex]b[\/latex] is a positive real number not equal to [latex]1[\/latex].<\/p>\n<ul>\n<li>if [latex]b>1[\/latex], the function grows at a rate proportional to its size.<\/li>\n<li>if [latex]0 \\lt b \\lt 1[\/latex], the function decays at a rate proportional to its size.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Why do we limit the base [latex]b[\/latex]\u00a0to positive values?<\/strong><\/p>\n<hr \/>\n<p>This is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\n<ul>\n<li>Consider a base of \u20139 and exponent of [latex]\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\n<\/ul>\n<p><strong>Why do we limit the base to positive values other than 1?<br \/>\n<\/strong><\/p>\n<hr \/>\n<p>This is because a base of 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:<\/p>\n<ul>\n<li>Consider a base of 1.\u00a0Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\n<\/ul>\n<\/section>\n<h2 aria-label=\"Example\">Exponential Growth<\/h2>\n<p>Because the output of exponential functions increases very rapidly, the term &#8220;exponential growth&#8221; is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.<\/p>\n<p>To get a sense of the behavior of <strong>exponential growth<\/strong>, we can create a table of values for a function of the form [latex]f(x)={b}^{x}[\/latex], where [latex]f \\gt 1[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Let&#8217;s take a look at [latex]f(x)={2}^{x}[\/latex].<\/p>\n<table id=\"Table_04_02_01\" style=\"width: 100%;\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<colgroup> <\/colgroup>\n<tbody>\n<tr>\n<td style=\"width: 18.3633%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 17.1657%;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 13.9721%;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 13.9055%;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 8.9155%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 8.9155%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 8.98204%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 8.98204%;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.3633%;\"><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 17.1657%;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td style=\"width: 13.9721%;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 13.9055%;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 8.9155%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 8.9155%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 8.98204%;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 8.98204%;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2094 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/08181744\/Screenshot-2024-07-08-at-11.17.38%E2%80%AFAM.png\" alt=\"\" width=\"300\" height=\"296\" \/>We call the base [latex]2[\/latex] the <em>constant ratio<\/em>. This means that as the input increases by [latex]1[\/latex], the output value will be the product of the base and the previous output. Did you notice that the next number is [latex]2[\/latex] times the previous number?<\/p>\n<p>This pattern shows <strong>exponential growth<\/strong> because the output value increases by a factor of [latex]2[\/latex] each time.<\/p>\n<p>Characteristics:<\/p>\n<ul>\n<li><strong>Domain: [latex](-\\infty,\\infty)[\/latex]<\/strong><\/li>\n<li><strong>Range: [latex](0,\\infty)[\/latex]<\/strong><\/li>\n<li>As [latex]x \\rightarrow \\infty, f(x) \\rightarrow \\infty[\/latex].<\/li>\n<li>As [latex]x \\rightarrow -\\infty, f(x) \\rightarrow 0[\/latex].<\/li>\n<li>the graph of [latex]f[\/latex] will never touch the [latex]x[\/latex]-axis because base two raised to any exponent never has the result of zero.<\/li>\n<li><strong>Horizontal Asymptote: [latex]y = 0[\/latex]<\/strong><\/li>\n<li>[latex]f(x)[\/latex] is always increasing.<\/li>\n<li>No [latex]x[\/latex]-intercept.<\/li>\n<li>[latex]y[\/latex]-intercept is [latex](0,1)[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>exponential growth<\/h3>\n<p>A function that models <strong>exponential growth<\/strong> grows by a rate proportional to the amount present. For any real number [latex]x[\/latex]\u00a0and any positive real numbers <em>a\u00a0<\/em>and <em>b<\/em> such that [latex]b\\ne 1[\/latex], an exponential growth function has the form<\/p>\n<p style=\"text-align: center;\">[latex]\\text{ }f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\n<p>where<\/p>\n<ul>\n<li>[latex]a[\/latex]\u00a0is the initial or starting value of the function.<\/li>\n<li>[latex]b[\/latex]\u00a0is the growth factor or growth multiplier per unit [latex]x[\/latex].<\/li>\n<\/ul>\n<\/section>\n<p>In more general terms, an <em>exponential function <\/em>consists of a\u00a0constant base raised to a variable exponent.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">To differentiate between linear and exponential functions, let\u2019s consider two companies, A and B.<\/p>\n<ul>\n<li>Company A has [latex]100[\/latex] stores and expands by opening [latex]50[\/latex] new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex].<\/li>\n<li>Company B has [latex]100[\/latex] stores and expands by increasing the number of stores by [latex]50 \\%[\/latex] each year, so its growth can be represented by the function [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/li>\n<\/ul>\n<p>A few years of growth for these companies are illustrated below.<\/p>\n<table summary=\"Six rows and three columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Year,\u00a0[latex]x[\/latex]<\/th>\n<th>Stores, Company A<\/th>\n<th>Stores, Company B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]100 + 50(0) = 100[\/latex]<\/td>\n<td>[latex]100(1 + 0.5)^0 = 100[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]100 + 50(1) = 150[\/latex]<\/td>\n<td>[latex]100(1 + 0.5)^1 = 150[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]100 + 50(2) = 200[\/latex]<\/td>\n<td>[latex]100(1 + 0.5)^2 = 225[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]100 + 50(3) = 250[\/latex]<\/td>\n<td>[latex]100(1 + 0.5)^3 =\u00a0337.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]A(x) = 100 + 50x[\/latex]<\/td>\n<td>[latex]B(x) = 100(1 + 0.5)^x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The graphs comparing the number of stores for each company over a five-year period are shown below. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.<\/p>\n<figure id=\"attachment_4998\" aria-describedby=\"caption-attachment-4998\" style=\"width: 419px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4998\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_-173x300.png\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"419\" height=\"727\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_-173x300.png 173w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_-65x113.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_-225x390.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_-350x607.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/03175243\/7.1.L.1.Graph1_.png 487w\" sizes=\"(max-width: 419px) 100vw, 419px\" \/><figcaption id=\"caption-attachment-4998\" class=\"wp-caption-text\">The graph shows the numbers of stores Companies A and B opened over a five-year period.<\/figcaption><\/figure>\n<div class=\"mceTemp\"><\/div>\n<p>Notice that the domain for both functions is [latex]\\left[0,\\infty \\right)[\/latex], and the range for both functions is [latex]\\left[100,\\infty \\right)[\/latex]. After year 1, Company B always has more stores than Company A.<\/p>\n<p>Let&#8217;s more closely examine the function representing the number of stores for Company B,<\/p>\n<div style=\"text-align: center;\">[latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/div>\n<p>In this exponential function, [latex]100[\/latex] represents the initial number of stores, [latex]0.5[\/latex] represents the growth rate, and [latex]1+0.5=1.5[\/latex] represents the growth factor. Generalizing further, we can write this function as<\/p>\n<div style=\"text-align: center;\">[latex]B\\left(x\\right)=100{\\left(1.5\\right)}^{x}[\/latex]<\/div>\n<p>where [latex]100[\/latex] is the initial value, [latex]1.5[\/latex] is called the <em>base<\/em>, and [latex]x[\/latex]\u00a0is called the <em>exponent<\/em>.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The population of India was about [latex]1.25[\/latex] billion in the year 2013 with an annual growth rate of about [latex]1.2\\%[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nThis situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex] where [latex]t[\/latex]\u00a0is the number of years since 2013.<br \/>\n[latex]\\\\[\/latex]<br \/>\nTo the nearest thousandth, what will the population of India be in 2031?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q924755\">Show Solution<\/button><\/p>\n<div id=\"q924755\" class=\"hidden-answer\" style=\"display: none\">To estimate the population in 2031, we evaluate the models for [latex]t\u00a0= 18[\/latex], because 2031 is [latex]18[\/latex] years after 2013. Rounding to the nearest thousandth,<\/p>\n<p style=\"text-align: center;\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/p>\n<p>There will be about [latex]1.549[\/latex] billion people in India in the year 2031.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321380\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321380&theme=lumen&iframe_resize_id=ohm321380&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Exponential Decay<\/h2>\n<p>To get a sense of the behavior of <strong>exponential decay<\/strong>, we can create a table of values for a function of the form [latex]g(x)={b}^{x}[\/latex], where [latex]0 \\lt b \\lt 1[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Let&#8217;s take a look at the function [latex]g(x)={(\\frac{1}{2})}^{x}[\/latex].Observe how the output values in the table below change as the input increases by [latex]1[\/latex].<\/p>\n<table style=\"width: 100%;\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td style=\"width: 37.6529%;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 7.93341%;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 3.42556%;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 3.42556%;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 4.4796%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 16.996%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16.6008%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 7.50984%;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 37.6529%;\"><strong>[latex]g(x)=(\\frac{1}{2})^{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 7.93341%;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 3.42556%;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 3.42556%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 4.4796%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16.996%;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6008%;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 7.50984%;\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231133\/CNX_Precalc_Figure_04_02_0022.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"325\" height=\"347\" \/>When the input is increasing by [latex]1[\/latex], each output value is the product of the previous output and the base or constant ratio [latex]\\frac{1}{2}[\/latex].<\/p>\n<p>Notice from the table that:<\/p>\n<ul>\n<li>the output values are positive for all values of [latex]x[\/latex].<\/li>\n<li>as [latex]x[\/latex] increases, the output values grow smaller, approaching zero.<\/li>\n<li>as [latex]x[\/latex] decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p>Characteristics:<\/p>\n<ul>\n<li>Domain: [latex]\\left(-\\infty , \\infty \\right)[\/latex]<\/li>\n<li>Range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n<li>[latex]x[\/latex]<em>&#8211;<\/em>intercept: none<\/li>\n<li>[latex]y[\/latex]<em>&#8211;<\/em>intercept: [latex]\\left(0,1\\right)[\/latex]<\/li>\n<li>Horizontal asymptote: [latex]y=0[\/latex]<\/li>\n<\/ul>\n<p>This is an <strong>exponential decay<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321381\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321381&theme=lumen&iframe_resize_id=ohm321381&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321382\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321382&theme=lumen&iframe_resize_id=ohm321382&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section id=\"fs-id1165137447701\" class=\"key-concepts\">\u00a0<\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":5,"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":105,"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\/107"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/107\/revisions"}],"predecessor-version":[{"id":5871,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/107\/revisions\/5871"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/107\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=107"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=107"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=107"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}