{"id":1430,"date":"2025-07-25T01:17:24","date_gmt":"2025-07-25T01:17:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1430"},"modified":"2026-03-24T07:44:10","modified_gmt":"2026-03-24T07:44:10","slug":"exponential-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions-fresh-take\/","title":{"raw":"Exponential Functions: Fresh Take","rendered":"Exponential Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate exponential functions.<\/li>\r\n \t<li>Find the equation of an exponential function.<\/li>\r\n \t<li>Use compound interest formulas.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining Exponential Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Exponential Growth and Decay:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Exponential growth: Increase based on a constant multiplicative rate over equal time increments<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exponential decay: Decrease based on a constant multiplicative rate over equal time increments<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Contrast with Linear Growth:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Exponential: Changes by the same percentage over equal increments<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Linear: Changes by the same amount over equal increments<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form of Exponential Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = ab^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is any nonzero number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex] is a positive real number, [latex]b \\neq 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]b &gt; 1[\/latex]: function grows<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]0 &lt; b &lt; 1[\/latex]: function decays<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluating Exponential Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute the given value for [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Follow the order of operations carefully<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Which of the following equations represent exponential functions?<\/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\">[latex]f(x) = 2x^2 - 3x + 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]h(x) = 1.75x + 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"536253\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"536253\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.[reveal-answer q=\"860098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"860098\"][latex]5.5556[\/latex][\/hidden-answer]<\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggdaggeg-nqpn0SQB5ds\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/nqpn0SQB5ds?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ggdaggeg-nqpn0SQB5ds\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780749&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ggdaggeg-nqpn0SQB5ds&amp;vembed=0&amp;video_id=nqpn0SQB5ds&amp;video_target=tpm-plugin-ggdaggeg-nqpn0SQB5ds\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Exponential+Functions+-+Nerdstudy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Exponential Functions - Nerdstudy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<\/section>\r\n<h2 aria-label=\"Example\">Exponential Growth and Decay<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Exponential Growth:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Output increases by a constant factor over equal intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]b &gt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = 2^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exponential Decay:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Output decreases by a constant factor over equal intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]0 &lt; b &lt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]g(x) = (\\frac{1}{2})^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Characteristics of Exponential Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: [latex](0, 1)[\/latex] for [latex]f(x) = b^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No [latex]x[\/latex]-intercept<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">The population of China was about [latex]1.39[\/latex] billion in the year 2013 with an annual growth rate of about [latex]0.6 \\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.39{\\left(1.006\\right)}^{t}[\/latex] where [latex]t[\/latex]\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of China be in the year 2031? How does this compare to the population prediction we made for India in the previous example?[reveal-answer q=\"891037\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"891037\"]About [latex]1.548[\/latex] billion people; by the year 2031, India\u2019s population will exceed China\u2019s by about [latex]0.001[\/latex] billion, or [latex]1[\/latex] million people.[\/hidden-answer]<\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/m5Tf6vgoJtQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850239&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ&amp;vembed=0&amp;video_id=m5Tf6vgoJtQ&amp;video_target=tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Exponential+growth+and+decay+word+problems+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential growth and decay word problems | Algebra II | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Finding Equations of Exponential Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">General Form of Exponential Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = ab^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: initial value<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: growth factor (if [latex]b &gt; 1[\/latex]) or decay factor (if [latex]0 &lt; b &lt; 1[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Methods for Finding Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using two points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using a graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Principle:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Every point on the graph satisfies the equation of the function<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategies<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">When Given Two Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If one point is [latex](0, a)[\/latex], use it as the initial value<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If no [latex](0, a)[\/latex] point, set up a system of equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When Given a Graph:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose the [latex]y[\/latex]-intercept as one point if possible<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Select a second point with integer coordinates<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use points far apart to minimize rounding errors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Steps:\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify or calculate the initial value [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the second point to solve for [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the equation in the form [latex]f(x) = ab^x[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population <em>N<\/em>\u00a0of wolves over time <em>t<\/em>.[reveal-answer q=\"222558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222558\"][latex]\\left(0,129\\right)[\/latex] and [latex]\\left(2,236\\right);N\\left(t\\right)=129{\\left(\\text{1}\\text{.3526}\\right)}^{t}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the two points [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(2,4.5\\right)[\/latex], find the equation of the exponential function that passes through these two points.[reveal-answer q=\"40110\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"40110\"][latex]f\\left(x\\right)=2{\\left(1.5\\right)}^{x}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find an equation for the exponential function graphed below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225457\/CNX_Precalc_Figure_04_01_0052.jpg\" alt=\"Graph of an increasing function with a labeled point at (0, sqrt(2)).\" width=\"487\" height=\"294\" \/> Graph of f(x) with y-intercept labeled[\/caption]\r\n\r\n[reveal-answer q=\"564720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"564720\"][latex]f\\left(x\\right)=\\sqrt{2}{\\left(\\sqrt{2}\\right)}^{x}[\/latex]. Answers may vary due to round-off error. The answer should be very close to [latex]1.4142{\\left(1.4142\\right)}^{x}[\/latex].[\/hidden-answer]\r\n\r\n<\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adhcdgab-1IE5jNudELQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1IE5jNudELQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-adhcdgab-1IE5jNudELQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850238&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-adhcdgab-1IE5jNudELQ&amp;vembed=0&amp;video_id=1IE5jNudELQ&amp;video_target=tpm-plugin-adhcdgab-1IE5jNudELQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+an+Exponential+Function+Given+Two+Points+-+Initial+Value+Not+Given_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find an Exponential Function Given Two Points - Initial Value Not Given\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbefchae-ueCcMc1FUsw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ueCcMc1FUsw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbefchae-ueCcMc1FUsw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780750&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbefchae-ueCcMc1FUsw&amp;vembed=0&amp;video_id=ueCcMc1FUsw&amp;video_target=tpm-plugin-bbefchae-ueCcMc1FUsw\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Writing+Exponential+Functions+from+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Exponential Functions from a Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate exponential functions.<\/li>\n<li>Find the equation of an exponential function.<\/li>\n<li>Use compound interest formulas.<\/li>\n<\/ul>\n<\/section>\n<h2>Defining Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Exponential Growth and Decay:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential growth: Increase based on a constant multiplicative rate over equal time increments<\/li>\n<li class=\"whitespace-normal break-words\">Exponential decay: Decrease based on a constant multiplicative rate over equal time increments<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Contrast with Linear Growth:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential: Changes by the same percentage over equal increments<\/li>\n<li class=\"whitespace-normal break-words\">Linear: Changes by the same amount over equal increments<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Form of Exponential Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = ab^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is any nonzero number<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex] is a positive real number, [latex]b \\neq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]b > 1[\/latex]: function grows<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]0 < b < 1[\/latex]: function decays<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute the given value for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Follow the order of operations carefully<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Which of the following equations represent exponential functions?<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = 2x^2 - 3x + 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]h(x) = 1.75x + 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q536253\">Show Answer<\/button><\/p>\n<div id=\"q536253\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q860098\">Show Solution<\/button><\/p>\n<div id=\"q860098\" class=\"hidden-answer\" style=\"display: none\">[latex]5.5556[\/latex]<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggdaggeg-nqpn0SQB5ds\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/nqpn0SQB5ds?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ggdaggeg-nqpn0SQB5ds\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780749&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ggdaggeg-nqpn0SQB5ds&amp;vembed=0&amp;video_id=nqpn0SQB5ds&amp;video_target=tpm-plugin-ggdaggeg-nqpn0SQB5ds\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Exponential+Functions+-+Nerdstudy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Exponential Functions &#8211; Nerdstudy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<\/section>\n<h2 aria-label=\"Example\">Exponential Growth and Decay<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Exponential Growth:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Output increases by a constant factor over equal intervals<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]b > 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = 2^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Exponential Decay:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Output decreases by a constant factor over equal intervals<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]0 < b < 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]g(x) = (\\frac{1}{2})^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Characteristics of Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: [latex](0, 1)[\/latex] for [latex]f(x) = b^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">No [latex]x[\/latex]-intercept<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">The population of China was about [latex]1.39[\/latex] billion in the year 2013 with an annual growth rate of about [latex]0.6 \\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.39{\\left(1.006\\right)}^{t}[\/latex] where [latex]t[\/latex]\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of China be in the year 2031? How does this compare to the population prediction we made for India in the previous example?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q891037\">Show Solution<\/button><\/p>\n<div id=\"q891037\" class=\"hidden-answer\" style=\"display: none\">About [latex]1.548[\/latex] billion people; by the year 2031, India\u2019s population will exceed China\u2019s by about [latex]0.001[\/latex] billion, or [latex]1[\/latex] million people.<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/m5Tf6vgoJtQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850239&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ&amp;vembed=0&amp;video_id=m5Tf6vgoJtQ&amp;video_target=tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Exponential+growth+and+decay+word+problems+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential growth and decay word problems | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<\/section>\n<h2 data-type=\"title\">Finding Equations of Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">General Form of Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = ab^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: initial value<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: growth factor (if [latex]b > 1[\/latex]) or decay factor (if [latex]0 < b < 1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Methods for Finding Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using two points<\/li>\n<li class=\"whitespace-normal break-words\">Using a graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Every point on the graph satisfies the equation of the function<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategies<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">When Given Two Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If one point is [latex](0, a)[\/latex], use it as the initial value<\/li>\n<li class=\"whitespace-normal break-words\">If no [latex](0, a)[\/latex] point, set up a system of equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">When Given a Graph:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose the [latex]y[\/latex]-intercept as one point if possible<\/li>\n<li class=\"whitespace-normal break-words\">Select a second point with integer coordinates<\/li>\n<li class=\"whitespace-normal break-words\">Use points far apart to minimize rounding errors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Steps:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify or calculate the initial value [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use the second point to solve for [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write the equation in the form [latex]f(x) = ab^x[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population <em>N<\/em>\u00a0of wolves over time <em>t<\/em>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q222558\">Show Solution<\/button><\/p>\n<div id=\"q222558\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(0,129\\right)[\/latex] and [latex]\\left(2,236\\right);N\\left(t\\right)=129{\\left(\\text{1}\\text{.3526}\\right)}^{t}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the two points [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(2,4.5\\right)[\/latex], find the equation of the exponential function that passes through these two points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q40110\">Show Solution<\/button><\/p>\n<div id=\"q40110\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=2{\\left(1.5\\right)}^{x}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find an equation for the exponential function graphed below.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225457\/CNX_Precalc_Figure_04_01_0052.jpg\" alt=\"Graph of an increasing function with a labeled point at (0, sqrt(2)).\" width=\"487\" height=\"294\" \/><figcaption class=\"wp-caption-text\">Graph of f(x) with y-intercept labeled<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q564720\">Show Solution<\/button><\/p>\n<div id=\"q564720\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=\\sqrt{2}{\\left(\\sqrt{2}\\right)}^{x}[\/latex]. Answers may vary due to round-off error. The answer should be very close to [latex]1.4142{\\left(1.4142\\right)}^{x}[\/latex].<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adhcdgab-1IE5jNudELQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1IE5jNudELQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-adhcdgab-1IE5jNudELQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850238&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-adhcdgab-1IE5jNudELQ&amp;vembed=0&amp;video_id=1IE5jNudELQ&amp;video_target=tpm-plugin-adhcdgab-1IE5jNudELQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+an+Exponential+Function+Given+Two+Points+-+Initial+Value+Not+Given_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find an Exponential Function Given Two Points &#8211; Initial Value Not Given\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbefchae-ueCcMc1FUsw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ueCcMc1FUsw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbefchae-ueCcMc1FUsw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780750&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbefchae-ueCcMc1FUsw&amp;vembed=0&amp;video_id=ueCcMc1FUsw&amp;video_target=tpm-plugin-bbefchae-ueCcMc1FUsw\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Writing+Exponential+Functions+from+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Exponential Functions from a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"author":67,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Exponential Functions - Nerdstudy\",\"author\":\"\",\"organization\":\"Nerdstudy\",\"url\":\"https:\/\/youtu.be\/nqpn0SQB5ds\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Exponential growth and decay word problems | Algebra II | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/m5Tf6vgoJtQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find an Exponential Function Given Two Points - Initial Value Not Given\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/1IE5jNudELQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Writing Exponential Functions from a Graph\",\"author\":\"\",\"organization\":\"Rice Math\",\"url\":\"https:\/\/youtu.be\/ueCcMc1FUsw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Introduction to Exponential Functions - Nerdstudy","author":"","organization":"Nerdstudy","url":"https:\/\/youtu.be\/nqpn0SQB5ds","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Exponential growth and decay word problems | Algebra II | Khan Academy","author":"","organization":"Khan Academy","url":"https:\/\/youtu.be\/m5Tf6vgoJtQ","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex: Find an Exponential Function Given Two Points - Initial Value Not Given","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/1IE5jNudELQ","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Writing Exponential Functions from a Graph","author":"","organization":"Rice Math","url":"https:\/\/youtu.be\/ueCcMc1FUsw","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12780749&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggdaggeg-nqpn0SQB5ds&vembed=0&video_id=nqpn0SQB5ds&video_target=tpm-plugin-ggdaggeg-nqpn0SQB5ds'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850239&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ&vembed=0&video_id=m5Tf6vgoJtQ&video_target=tpm-plugin-dgbbhehh-m5Tf6vgoJtQ'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850238&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-adhcdgab-1IE5jNudELQ&vembed=0&video_id=1IE5jNudELQ&video_target=tpm-plugin-adhcdgab-1IE5jNudELQ'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12780750&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbefchae-ueCcMc1FUsw&vembed=0&video_id=ueCcMc1FUsw&video_target=tpm-plugin-bbefchae-ueCcMc1FUsw'><\/script>\n","media_targets":["tpm-plugin-ggdaggeg-nqpn0SQB5ds","tpm-plugin-dgbbhehh-m5Tf6vgoJtQ","tpm-plugin-adhcdgab-1IE5jNudELQ","tpm-plugin-bbefchae-ueCcMc1FUsw"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1430"}],"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\/67"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1430\/revisions"}],"predecessor-version":[{"id":5524,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1430\/revisions\/5524"}],"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\/1430\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1430"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1430"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1430"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1430"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}