{"id":1418,"date":"2021-08-24T14:23:35","date_gmt":"2021-08-24T14:23:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1418"},"modified":"2022-01-31T21:38:26","modified_gmt":"2022-01-31T21:38:26","slug":"evaluating-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/evaluating-exponential-functions\/","title":{"raw":"Evaluating Exponential Functions","rendered":"Evaluating Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify and evaluate exponential functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nA function of the form [latex] f(x)=b^{x}, b&gt;0, b \\neq 1[\/latex] is called an exponential function.\u00a0In an exponential function the base is a positive constant other than [latex]1[\/latex], and the exponent is a variable expression. For example, if [latex]f(x)=3^{x}[\/latex], then [latex]f(2)=3^{2}=9, f(0)=3^{0}=1[\/latex] and [latex]f(-2)=3^{-2}=\\frac{1}{3^{2}}=\\frac{1}{9}[\/latex].\r\n<div id=\"fs-id1165137564690\" class=\"note textbox\">\r\n<h3 class=\"title\">Exponential Growth function<\/h3>\r\n<p id=\"fs-id1165137834019\">A function that models <strong>exponential growth<\/strong> grows by a rate proportional to the current amount. For any real number <em>x<\/em>\u00a0and any positive real numbers <em>a\u00a0<\/em>and <em>b<\/em>\u00a0such that [latex]b\\ne 1[\/latex], an exponential growth function has the form<\/p>\r\n\r\n<div id=\"fs-id1165137851784\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/div>\r\n<p id=\"eip-626\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137863819\">\r\n \t<li><em>a<\/em>\u00a0is the initial or starting value of the function.<\/li>\r\n \t<li><em>b<\/em>\u00a0is the growth factor or growth multiplier per unit <em>x<\/em><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Evaluating Exponential Functions<\/h2>\r\n<p id=\"fs-id1165137644244\">To evaluate an exponential function of the form [latex]f\\left(x\\right)=a \\cdot {b}^{x}[\/latex], we simply substitute <em>x<\/em>\u00a0with the given value, and calculate the resulting power. For example:<\/p>\r\n<p id=\"fs-id1165135403544\">Let [latex]f(x)=7 \\cdot 2^{x}[\/latex]<\/p>\r\nThen [latex]2[\/latex] is the base of the exponential function. Since [latex]f(x)=7 \\cdot 2^{0} = 7 \\cdot 1= 7[\/latex], the constant [latex]a=7[\/latex] in this example is the initial value of the function when the variable [latex]x[\/latex] is [latex]0[\/latex].\r\n<p style=\"text-align: center;\">If [latex]x[\/latex] is [latex]1[\/latex], then\r\n[latex]f(1)=7 \\cdot 2^{1}[\/latex]\r\n[latex]= 7 \\cdot 2[\/latex]\r\n[latex]= 14[\/latex].<\/p>\r\n<p style=\"text-align: center;\">If [latex]x[\/latex] is [latex]3[\/latex], then\r\n[latex]f(3)=7 \\cdot 2^{3}[\/latex]\r\n[latex]= 7 \\cdot 8[\/latex]\r\n[latex]= 56[\/latex].<\/p>\r\nTo evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. Consider the following example.\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nLet [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.\r\n[reveal-answer q=\"211228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211228\"]\r\n<p id=\"fs-id1165137598173\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\r\n\r\n<div id=\"eip-id1165135208555\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; =5{\\left(3\\right)}^{x+1}\\hfill &amp; \\hfill \\\\ f\\left(2\\right)\\hfill &amp; =5{\\left(3\\right)}^{2+1}\\hfill &amp; \\text{Substitute }x=2.\\hfill \\\\ \\hfill &amp; =5{\\left(3\\right)}^{3}\\hfill &amp; \\text{Add the exponents}.\\hfill \\\\ \\hfill &amp; =5\\left(27\\right)\\hfill &amp; \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill &amp; =135\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we present more examples of evaluating an exponential function at several different values.\r\n\r\nhttps:\/\/youtu.be\/QFFAoX0We34\r\n<div id=\"fs-id1165135511324\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Number [latex]e[\/latex]<\/h3>\r\n<p id=\"fs-id1165135511335\">The letter <em>e<\/em> represents the irrational number<\/p>\r\n\r\n<div id=\"eip-id1165135378658\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\left(1+\\frac{1}{n}\\right)}^{n}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">as n increases without bound<\/div>\r\n<p id=\"fs-id1165135369344\">The letter <em>e <\/em>is used as a base for many real-world exponential models. To work with base <em>e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137664673\" class=\"note textbox\">\r\n<h3 class=\"title\">The Continuous Growth\/Decay Formula<\/h3>\r\n<p id=\"fs-id1165135453868\">For all real numbers r,\u00a0<em>t<\/em>, and all positive numbers <em>a<\/em>, continuous growth or decay is represented by the formula<\/p>\r\n\r\n<div id=\"fs-id1165135536370\" class=\"equation\" style=\"text-align: center;\">[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/div>\r\n<p id=\"eip-101\">where<\/p>\r\n\r\n<ul id=\"fs-id1165135152052\">\r\n \t<li><em>a<\/em>\u00a0is the initial value,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the continuous growth or decay rate per unit time,<\/li>\r\n \t<li>and <em>t<\/em>\u00a0is the elapsed time.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135560686\">If <em>r\u00a0<\/em>&gt;[latex]0[\/latex], then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt;\u00a0[latex]0[\/latex], then the formula represents continuous decay.<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1165137812323\">For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\r\n\r\n<div id=\"eip-id1165134324899\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P{e}^{rt}[\/latex]<\/div>\r\n<p id=\"eip-962\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137827330\">\r\n \t<li><em>P<\/em>\u00a0is the principal or the initial amount invested,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the growth or interest rate per unit time,<\/li>\r\n \t<li>and <em>t<\/em>\u00a0is the period or term of the investment.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<span style=\"font-size: 1rem; text-align: initial; background-color: #e3eff6;\">The population of a town, in thousands, [latex]t[\/latex] years after [latex]2013[\/latex] is modeled by the function [latex]A(t)=13.2 \\cdot e^{0.07t}[\/latex]. Find the population of the town in the year...<\/span>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>2013<\/li>\r\n \t<li>2020<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"248731\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"248731\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li style=\"text-align: left;\">The year [latex]2013[\/latex] is the starting year. Since [latex]t[\/latex] represents the number of years after [latex]2013[\/latex], in this case [latex]t=2013-2013=0[\/latex]. The population in [latex]2013[\/latex] is found by evaluating [latex]A(0)[\/latex].\r\n<ul>\r\n \t<li style=\"text-align: left;\">[latex]A(0) = 13.2 \\cdot e^{0.07-0}[\/latex]\r\n[latex]= 13.2 \\cdot e^{0}[\/latex]\r\n[latex]= 13.2 \\cdot 1[\/latex]\r\n[latex]= 13.2[\/latex]<\/li>\r\n \t<li>Population is given in thousands, so the population in [latex]2013[\/latex] was [latex]13,200[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"text-align: left;\">The year [latex]2020[\/latex] corresponds to [latex]t=2020-2013=7[\/latex]. The population in [latex]2020[\/latex] is found by evaluating [latex]A(7)[\/latex].\r\n<ul>\r\n \t<li style=\"text-align: left;\">[latex]A(7)=13.2 \\cdot e^{0.07 \\cdot 7}[\/latex]\r\n[latex]\\approx 13.2 \\cdot 2.718282^{0.49}[\/latex]\r\n[latex]13.2 \\cdot 1.63231627[\/latex]\r\n[latex]=21.54657477[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">Since the population is given in thousands, the population in [latex]2020[\/latex] was approximately [latex]21,547[\/latex]. Since the population is a whole number, we round the solution to the nearest unit.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\nNote that constants [latex]\\pi[\/latex] and [latex]e[\/latex] are\u00a0built-in keys on many calculators. They are carried to greater accuracy, and so your answer using the [latex]e[\/latex]\u00a0button on a calculator will be slightly different. In this example, such a calculator will produce the result [latex]A(7)=21.5465741[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show an example of calculating the remaining amount of a radioactive substance after it decays for a length of time.\r\n\r\nhttps:\/\/youtu.be\/Vyl3NcTGRAo","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify and evaluate exponential functions<\/li>\n<\/ul>\n<\/div>\n<p>A function of the form [latex]f(x)=b^{x}, b>0, b \\neq 1[\/latex] is called an exponential function.\u00a0In an exponential function the base is a positive constant other than [latex]1[\/latex], and the exponent is a variable expression. For example, if [latex]f(x)=3^{x}[\/latex], then [latex]f(2)=3^{2}=9, f(0)=3^{0}=1[\/latex] and [latex]f(-2)=3^{-2}=\\frac{1}{3^{2}}=\\frac{1}{9}[\/latex].<\/p>\n<div id=\"fs-id1165137564690\" class=\"note textbox\">\n<h3 class=\"title\">Exponential Growth function<\/h3>\n<p id=\"fs-id1165137834019\">A function that models <strong>exponential growth<\/strong> grows by a rate proportional to the current amount. For any real number <em>x<\/em>\u00a0and any positive real numbers <em>a\u00a0<\/em>and <em>b<\/em>\u00a0such that [latex]b\\ne 1[\/latex], an exponential growth function has the form<\/p>\n<div id=\"fs-id1165137851784\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/div>\n<p id=\"eip-626\">where<\/p>\n<ul id=\"fs-id1165137863819\">\n<li><em>a<\/em>\u00a0is the initial or starting value of the function.<\/li>\n<li><em>b<\/em>\u00a0is the growth factor or growth multiplier per unit <em>x<\/em><\/li>\n<\/ul>\n<\/div>\n<h2>Evaluating Exponential Functions<\/h2>\n<p id=\"fs-id1165137644244\">To evaluate an exponential function of the form [latex]f\\left(x\\right)=a \\cdot {b}^{x}[\/latex], we simply substitute <em>x<\/em>\u00a0with the given value, and calculate the resulting power. For example:<\/p>\n<p id=\"fs-id1165135403544\">Let [latex]f(x)=7 \\cdot 2^{x}[\/latex]<\/p>\n<p>Then [latex]2[\/latex] is the base of the exponential function. Since [latex]f(x)=7 \\cdot 2^{0} = 7 \\cdot 1= 7[\/latex], the constant [latex]a=7[\/latex] in this example is the initial value of the function when the variable [latex]x[\/latex] is [latex]0[\/latex].<\/p>\n<p style=\"text-align: center;\">If [latex]x[\/latex] is [latex]1[\/latex], then<br \/>\n[latex]f(1)=7 \\cdot 2^{1}[\/latex]<br \/>\n[latex]= 7 \\cdot 2[\/latex]<br \/>\n[latex]= 14[\/latex].<\/p>\n<p style=\"text-align: center;\">If [latex]x[\/latex] is [latex]3[\/latex], then<br \/>\n[latex]f(3)=7 \\cdot 2^{3}[\/latex]<br \/>\n[latex]= 7 \\cdot 8[\/latex]<br \/>\n[latex]= 56[\/latex].<\/p>\n<p>To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. Consider the following example.<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Let [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211228\">Show Solution<\/span><\/p>\n<div id=\"q211228\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137598173\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\n<div id=\"eip-id1165135208555\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & =5{\\left(3\\right)}^{x+1}\\hfill & \\hfill \\\\ f\\left(2\\right)\\hfill & =5{\\left(3\\right)}^{2+1}\\hfill & \\text{Substitute }x=2.\\hfill \\\\ \\hfill & =5{\\left(3\\right)}^{3}\\hfill & \\text{Add the exponents}.\\hfill \\\\ \\hfill & =5\\left(27\\right)\\hfill & \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill & =135\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we present more examples of evaluating an exponential function at several different values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine Exponential Function Values and Graph the Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QFFAoX0We34?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"fs-id1165135511324\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Number [latex]e[\/latex]<\/h3>\n<p id=\"fs-id1165135511335\">The letter <em>e<\/em> represents the irrational number<\/p>\n<div id=\"eip-id1165135378658\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\left(1+\\frac{1}{n}\\right)}^{n}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">as n increases without bound<\/div>\n<p id=\"fs-id1165135369344\">The letter <em>e <\/em>is used as a base for many real-world exponential models. To work with base <em>e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.<\/p>\n<\/div>\n<div id=\"fs-id1165137664673\" class=\"note textbox\">\n<h3 class=\"title\">The Continuous Growth\/Decay Formula<\/h3>\n<p id=\"fs-id1165135453868\">For all real numbers r,\u00a0<em>t<\/em>, and all positive numbers <em>a<\/em>, continuous growth or decay is represented by the formula<\/p>\n<div id=\"fs-id1165135536370\" class=\"equation\" style=\"text-align: center;\">[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/div>\n<p id=\"eip-101\">where<\/p>\n<ul id=\"fs-id1165135152052\">\n<li><em>a<\/em>\u00a0is the initial value,<\/li>\n<li><em>r<\/em>\u00a0is the continuous growth or decay rate per unit time,<\/li>\n<li>and <em>t<\/em>\u00a0is the elapsed time.<\/li>\n<\/ul>\n<p id=\"fs-id1165135560686\">If <em>r\u00a0<\/em>&gt;[latex]0[\/latex], then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt;\u00a0[latex]0[\/latex], then the formula represents continuous decay.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137812323\">For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\n<div id=\"eip-id1165134324899\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P{e}^{rt}[\/latex]<\/div>\n<p id=\"eip-962\">where<\/p>\n<ul id=\"fs-id1165137827330\">\n<li><em>P<\/em>\u00a0is the principal or the initial amount invested,<\/li>\n<li><em>r<\/em>\u00a0is the growth or interest rate per unit time,<\/li>\n<li>and <em>t<\/em>\u00a0is the period or term of the investment.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p><span style=\"font-size: 1rem; text-align: initial; background-color: #e3eff6;\">The population of a town, in thousands, [latex]t[\/latex] years after [latex]2013[\/latex] is modeled by the function [latex]A(t)=13.2 \\cdot e^{0.07t}[\/latex]. Find the population of the town in the year&#8230;<\/span><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>2013<\/li>\n<li>2020<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q248731\">Show Answer<\/span><\/p>\n<div id=\"q248731\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li style=\"text-align: left;\">The year [latex]2013[\/latex] is the starting year. Since [latex]t[\/latex] represents the number of years after [latex]2013[\/latex], in this case [latex]t=2013-2013=0[\/latex]. The population in [latex]2013[\/latex] is found by evaluating [latex]A(0)[\/latex].\n<ul>\n<li style=\"text-align: left;\">[latex]A(0) = 13.2 \\cdot e^{0.07-0}[\/latex]<br \/>\n[latex]= 13.2 \\cdot e^{0}[\/latex]<br \/>\n[latex]= 13.2 \\cdot 1[\/latex]<br \/>\n[latex]= 13.2[\/latex]<\/li>\n<li>Population is given in thousands, so the population in [latex]2013[\/latex] was [latex]13,200[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li style=\"text-align: left;\">The year [latex]2020[\/latex] corresponds to [latex]t=2020-2013=7[\/latex]. The population in [latex]2020[\/latex] is found by evaluating [latex]A(7)[\/latex].\n<ul>\n<li style=\"text-align: left;\">[latex]A(7)=13.2 \\cdot e^{0.07 \\cdot 7}[\/latex]<br \/>\n[latex]\\approx 13.2 \\cdot 2.718282^{0.49}[\/latex]<br \/>\n[latex]13.2 \\cdot 1.63231627[\/latex]<br \/>\n[latex]=21.54657477[\/latex]<\/li>\n<li style=\"text-align: left;\">Since the population is given in thousands, the population in [latex]2020[\/latex] was approximately [latex]21,547[\/latex]. Since the population is a whole number, we round the solution to the nearest unit.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>Note that constants [latex]\\pi[\/latex] and [latex]e[\/latex] are\u00a0built-in keys on many calculators. They are carried to greater accuracy, and so your answer using the [latex]e[\/latex]\u00a0button on a calculator will be slightly different. In this example, such a calculator will produce the result [latex]A(7)=21.5465741[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show an example of calculating the remaining amount of a radioactive substance after it decays for a length of time.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Determine a Continuous Exponential Decay Function and Make a Prediction\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Vyl3NcTGRAo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1418\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Determine Exponential Function Values and Graph the Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QFFAoX0We34\">https:\/\/youtu.be\/QFFAoX0We34<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determine a Continuous Exponential Decay Function and Make a Prediction. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Vyl3NcTGRAo\">https:\/\/youtu.be\/Vyl3NcTGRAo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\">https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/li><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. 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