{"id":216,"date":"2023-11-08T16:10:28","date_gmt":"2023-11-08T16:10:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/introduction-exponential-functions\/"},"modified":"2024-08-01T00:58:05","modified_gmt":"2024-08-01T00:58:05","slug":"7-3-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/7-3-exponential-functions\/","title":{"raw":"7.3 Exponential Functions","rendered":"7.3 Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate exponential functions.<\/li>\r\n \t<li>Graph exponential functions by creating a table of values.<\/li>\r\n \t<li>Find the equation of the asymptote of an exponential function.<\/li>\r\n \t<li>Find the domain and the range of an exponential function using its graph and write them in interval notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\nLinear functions have a\u00a0constant rate of change \u2013 a constant number that the output increases for each increase in input. For example, in the equation [latex]f(x)=3x+4[\/latex]\u00a0, the slope tells us the output increases by three each time the input increases by one. Sometimes, on the other hand, quantities grow by a percent rate of change rather than by a fixed amount. In this lesson, we will define a function whose rate of change increases by a percent of the current value rather than a fixed quantity.\r\n\r\nTo illustrate\u00a0this difference consider two companies whose business is expanding: Company A has\u00a0[latex]100[\/latex] stores and expands by opening\u00a0[latex]50[\/latex] new stores a year, while Company B has\u00a0[latex]100[\/latex] stores and expands by increasing the number of stores by\u00a0[latex]50\\%[\/latex] of their total each year.\r\n\r\nThe table below compares\u00a0the growth of each company where company A increases the number of stores linearly, and company B increases the number of stores by a rate of\u00a0[latex]50\\%[\/latex] each year.\r\n<table style=\"width: 60%;\">\r\n<thead>\r\n<tr>\r\n<td style=\"width: 15.7534%;\">Year<\/td>\r\n<td style=\"width: 26.8836%;\">Stores, Company A<\/td>\r\n<td style=\"width: 27.0548%;\">\u00a0Description of Growth<\/td>\r\n<td style=\"width: 30.3082%;\">Stores, Company B<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 15.7534%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 26.8836%;\">[latex]100[\/latex]<\/td>\r\n<td style=\"width: 27.0548%;\">Starting with [latex]100[\/latex] each<\/td>\r\n<td style=\"width: 30.3082%;\">[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.7534%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 26.8836%;\">[latex]100+50=150[\/latex]<\/td>\r\n<td style=\"width: 27.0548%;\">They both grow by\u00a0[latex]50[\/latex] stores in the first year.<\/td>\r\n<td style=\"width: 30.3082%;\">[latex]100[\/latex][latex]+50\\%[\/latex] of [latex]100[\/latex] [latex]100 + 0.50(100) = 150[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.7534%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 26.8836%;\">[latex]150+50=200[\/latex]<\/td>\r\n<td style=\"width: 27.0548%;\">Store A grows by\u00a0[latex]50[\/latex], Store B grows by\u00a0[latex]75[\/latex]<\/td>\r\n<td style=\"width: 30.3082%;\">[latex]150[\/latex][latex]+ 50\\%[\/latex] of\u00a0[latex]150[\/latex] [latex]150 + 0.50(150) = 225[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.7534%;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 26.8836%;\">[latex]200+50=250[\/latex]<\/td>\r\n<td style=\"width: 27.0548%;\">Store A grows by\u00a0[latex]50[\/latex], Store B grows by\u00a0[latex]112.5[\/latex]<\/td>\r\n<td style=\"width: 30.3082%;\">[latex]225 + 50\\%[\/latex] of\u00a0[latex]225[\/latex] [latex]225 + 0.50(225) = 337.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCompany A has\u00a0[latex]100[\/latex] stores and expands by opening\u00a0[latex]50[\/latex] new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex]. Company B has\u00a0[latex]100[\/latex] stores and expands by increasing the number of stores by\u00a0[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].\r\n\r\nThe graphs comparing the number of stores for each company over a five-year period are shown below<strong>.<\/strong> 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=\"\" align=\"aligncenter\" width=\"338\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051913\/CNX_Precalc_Figure_04_01_0012.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"338\" height=\"586\" \/> The graph shows the number of stores Companies A and B opened over a five-year period.[\/caption]\r\n<p id=\"fs-id1165135209682\">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\u00a0[latex]1[\/latex], Company B always has more stores than Company A.<\/p>\r\n<p id=\"fs-id1165137836429\">Consider\u00a0the function representing the number of stores for Company B:<\/p>\r\n<p style=\"text-align: center;\">[latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex]<\/p>\r\nIn this exponential function,\u00a0[latex]100[\/latex] represents the initial number of stores,\u00a0[latex]0.50[\/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 [latex]B\\left(x\\right)=100{\\left(1.5\\right)}^{x}[\/latex], where\u00a0[latex]100[\/latex] is the initial value,\u00a0[latex]1.5[\/latex] is called the <em>base<\/em>, and <em>x<\/em>\u00a0is called the <em>exponent<\/em>. This is an exponential function.\r\n<div id=\"fs-id1165137564690\" class=\"note textbox\">\r\n<h3 class=\"title\">Exponential Growth<\/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<p id=\"fs-id1165137644244\">To evaluate an exponential function of the form [latex]f\\left(x\\right)={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\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\r\n\r\n<div id=\"eip-id1165137643186\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; ={2}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; ={2}^{3}\\text{ }\\hfill &amp; \\text{Substitute }x=3.\\hfill \\\\ \\hfill &amp; =8\\text{ }\\hfill &amp; \\text{Evaluate the power}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137849020\">To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:<\/p>\r\n<p id=\"fs-id1165137849024\">Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\r\n\r\n<div id=\"eip-id1165134086025\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; =30{\\left(2\\right)}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; =30{\\left(2\\right)}^{3}\\hfill &amp; \\text{Substitute }x=3.\\hfill \\\\ \\hfill &amp; =30\\left(8\\right)\\text{ }\\hfill &amp; \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill &amp; =240\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137841073\">Note that if the order of operations were not followed, the result would be incorrect:<\/p>\r\n\r\n<div id=\"eip-id1165135320147\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">In our first example, we will evaluate an exponential function without the aid of a calculator.<\/div>\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\r\nIn the next example, we will revisit the population of India.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAt the beginning of this section, we learned that the population of India was about\u00a0[latex]1.25[\/latex] billion in the year\u00a0[latex]2013[\/latex], with an annual growth rate of about\u00a0[latex]1.2\\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex], where <em>t<\/em>\u00a0is the number of years since\u00a0[latex]2013[\/latex]. To the nearest thousandth, what will the population of India be in\u00a0[latex]2031[\/latex]?\r\n[reveal-answer q=\"385742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385742\"]\r\n<p id=\"fs-id1165137786635\">To estimate the population in\u00a0[latex]2031[\/latex], we evaluate the model for [latex]t=18[\/latex], because\u00a0[latex]2031[\/latex] is\u00a0[latex]18[\/latex] years after\u00a0[latex]2013[\/latex]. Rounding to the nearest thousandth,<\/p>\r\n\r\n<div id=\"eip-id1165135657117\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/div>\r\n<p id=\"fs-id1165135394343\">There will be about\u00a0[latex]1.549[\/latex] billion people in India in the year\u00a0[latex]2031[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of using an exponential function to predict the population of a small town.\r\n\r\nhttps:\/\/youtu.be\/SbIydBmJePE\r\n<h2>Summary<\/h2>\r\nExponential growth grows by a rate proportional to the current amount.\u00a0For 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\u00a0[latex]f\\left(x\\right)=a{b}^{x}[\/latex]. \u00a0Evaluating exponential functions requires careful attention to the order of operations. Compound interest is an example of exponential growth.\r\n<p id=\"fs-id1165137592823\">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. Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex] whose base is greater than one. We will use the function [latex]f\\left(x\\right)={2}^{x}[\/latex]. Observe how the output values in the table below\u00a0change as the input increases by\u00a0[latex]1[\/latex].<\/p>\r\n\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>[latex]\u20133[\/latex]<\/td>\r\n<td>[latex]\u20132[\/latex]<\/td>\r\n<td>[latex]\u20131[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137432031\">Each output value is the product of the previous output and the base,\u00a0[latex]2[\/latex]. We call the base\u00a0[latex]2[\/latex] the <em>constant ratio<\/em>. In fact, for any exponential function of the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex], <em>b<\/em>\u00a0is the constant ratio of the function. This means that as the input increases by\u00a0[latex]1[\/latex], the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.<\/p>\r\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\r\n\r\n<ul id=\"fs-id1165137658509\">\r\n \t<li>the output values are positive for all values of <em>x<\/em>;<\/li>\r\n \t<li>as <em>x<\/em>\u00a0increases, the output values increase without bound; and<\/li>\r\n \t<li>as <em>x<\/em>\u00a0decreases, the output values grow smaller, approaching zero.<\/li>\r\n<\/ul>\r\nThe graph below shows the exponential growth function [latex]f\\left(x\\right)={2}^{x}[\/latex].\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\/121\/2016\/08\/05051915\/CNX_Precalc_Figure_04_02_0012.jpg\" alt=\"Graph of the exponential function, 2^(x), with labeled points at (-3, 1\/8), (-2, \u00bc), (-1, \u00bd), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\" \/> In the graph, notice that the graph gets close to the x-axis, but never touches it.[\/caption]\r\n<p id=\"fs-id1165137459614\">The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165137838249\">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]f\\left(x\\right)={b}^{x}[\/latex] whose base is between zero and one. We will use the function [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. Observe how the output values in the table below\u00a0change as the input increases by\u00a0[latex]1[\/latex].<\/p>\r\n\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>-[latex]3[\/latex]<\/td>\r\n<td>-[latex]2[\/latex]<\/td>\r\n<td>-[latex]1[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135347846\">Again, because the input is increasing by\u00a0[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>\r\n<p id=\"fs-id1165137452063\">Notice from the table that:<\/p>\r\n\r\n<ul id=\"fs-id1165135499992\">\r\n \t<li>the output values are positive for all values of <em>x<\/em>;<\/li>\r\n \t<li>as <em>x<\/em>\u00a0increases, the output values grow smaller, approaching zero; and<\/li>\r\n \t<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137405421\">The graph shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex].<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051917\/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=\"487\" height=\"520\" \/>\r\n<p id=\"fs-id1165137723586\"><strong>\u00a0<\/strong>The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165135571835\" class=\"note textbox\">\r\n<h3 class=\"title\">Characteristics of the Graph of\u00a0 [latex]f(x) = b^{x}[\/latex]<\/h3>\r\n<p id=\"fs-id1165137848929\">An exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], [latex]b&gt;0[\/latex], [latex]b\\ne 1[\/latex], has these characteristics:<\/p>\r\n\r\n<ul id=\"fs-id1165135186684\">\r\n \t<li><strong>one-to-one<\/strong> function<\/li>\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><em>x-<\/em>intercept: none<\/li>\r\n \t<li><em>y-<\/em>intercept: [latex]\\left(0,1\\right)[\/latex]<\/li>\r\n \t<li>increasing if [latex]b&gt;1[\/latex]<\/li>\r\n \t<li>decreasing if [latex]b&lt;1[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137471878\">Compare the graphs of <strong>exponential growth<\/strong> and decay functions below.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051919\/CNX_Precalc_Figure_04_02_003new2.jpg\" alt=\"Graph of two functions where the first graph is of a function of f(x) = b^x when b&gt;1 and the second graph is of the same function when b is 0&lt;b&lt;1. Both graphs have the points (0, 1) and (1, b) labeled.\" width=\"731\" height=\"407\" \/>\r\n\r\n<\/div>\r\nIn our first example, we will plot an exponential decay function where the base is between 0 and 1.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSketch a graph of [latex]f\\left(x\\right)={0.25}^{x}[\/latex]. State the domain, range.\r\n[reveal-answer q=\"203605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"203605\"]\r\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\r\n\r\n<ul>\r\n \t<li>Since <em>b\u00a0<\/em>=[latex]0.25[\/latex] is between zero and one, we know the function is decreasing, and we can verify this by creating a table of values. The left tail of the graph will increase without bound and the right tail will get really close to the x-axis.<\/li>\r\n \t<li>Create a table of points.\r\n<table id=\"Table_04_02_03\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>[latex]\u20133[\/latex]<\/td>\r\n<td>[latex]\u20132[\/latex]<\/td>\r\n<td>[latex]\u20131[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]64[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]0.25[\/latex]<\/td>\r\n<td>[latex]0.0625[\/latex]<\/td>\r\n<td>[latex]0.015625[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Plot the <em>y<\/em>-intercept, [latex]\\left(0,1\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(1,0.25\\right)[\/latex].<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points.<span id=\"fs-id1165137940681\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051920\/CNX_Precalc_Figure_04_02_0042.jpg\" alt=\"Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137548870\"><strong>\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of graphing an exponential function. The base of the exponential term is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex], so this graph will represent decay.\r\n\r\nhttps:\/\/youtu.be\/FMzZB9Ve-1U\r\n<div id=\"fs-id1165134195243\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135194093\">How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function<\/h3>\r\n<ol id=\"fs-id1165137435782\">\r\n \t<li>Create a table of points.<\/li>\r\n \t<li>Plot at least\u00a0[latex]3[\/latex]\u00a0point from the table, including the <em>y<\/em>-intercept [latex]\\left(0,1\\right)[\/latex].<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the range, [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn our next example, we will plot an exponential growth function where the base is greater than\u00a0[latex]1[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSketch a graph of [latex]f(x)={\\sqrt{2}(\\sqrt{2})}^{x}[\/latex].\u00a0State the domain and range.\r\n[reveal-answer q=\"334418\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"334418\"]\r\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\r\n\r\n<ul>\r\n \t<li>Since <em>b\u00a0<\/em>= [latex]\\sqrt{2}[\/latex], which is greater than\u00a0one, we know the function is increasing, and we can verify this by creating a table of values. The left tail of the graph will\u00a0get really close to the x-axis and the right tail will increase without bound.<\/li>\r\n \t<li>Create a table of points.\r\n<table id=\"Table_04_02_03\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>[latex]\u20133[\/latex]<\/td>\r\n<td>[latex]\u20132[\/latex]<\/td>\r\n<td>[latex]\u20131[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)=\\sqrt{2}{(\\sqrt{2})}^{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]0.5[\/latex]<\/td>\r\n<td>[latex]0.71[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1.41[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]2.83[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Plot the <em>y<\/em>-intercept, [latex]\\left(0,1.41\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points.<\/p>\r\n<img class=\" wp-image-3623 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05185425\/Screen-Shot-2016-08-05-at-11.53.45-AM.png\" alt=\"Screen Shot 2016-08-05 at 11.53.45 AM\" width=\"326\" height=\"231\" \/>\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nOur next video example includes graphing an exponential growth function and defining the domain and range of the function.\r\n\r\nhttps:\/\/youtu.be\/M6bpp0BRIf0\r\n<h2>Summary<\/h2>\r\nGraphs of exponential growth functions will have a right tail that increases without bound and a left tail that gets really close to the x-axis. On the other hand, graphs of exponential decay functions will have a left tail that increases without bound and a right tail that gets really close to the x-axis. Points can be generated with a table of values which can then be used to graph the function.\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate exponential functions.<\/li>\n<li>Graph exponential functions by creating a table of values.<\/li>\n<li>Find the equation of the asymptote of an exponential function.<\/li>\n<li>Find the domain and the range of an exponential function using its graph and write them in interval notation.<\/li>\n<\/ul>\n<\/div>\n<p>Linear functions have a\u00a0constant rate of change \u2013 a constant number that the output increases for each increase in input. For example, in the equation [latex]f(x)=3x+4[\/latex]\u00a0, the slope tells us the output increases by three each time the input increases by one. Sometimes, on the other hand, quantities grow by a percent rate of change rather than by a fixed amount. In this lesson, we will define a function whose rate of change increases by a percent of the current value rather than a fixed quantity.<\/p>\n<p>To illustrate\u00a0this difference consider two companies whose business is expanding: Company A has\u00a0[latex]100[\/latex] stores and expands by opening\u00a0[latex]50[\/latex] new stores a year, while Company B has\u00a0[latex]100[\/latex] stores and expands by increasing the number of stores by\u00a0[latex]50\\%[\/latex] of their total each year.<\/p>\n<p>The table below compares\u00a0the growth of each company where company A increases the number of stores linearly, and company B increases the number of stores by a rate of\u00a0[latex]50\\%[\/latex] each year.<\/p>\n<table style=\"width: 60%;\">\n<thead>\n<tr>\n<td style=\"width: 15.7534%;\">Year<\/td>\n<td style=\"width: 26.8836%;\">Stores, Company A<\/td>\n<td style=\"width: 27.0548%;\">\u00a0Description of Growth<\/td>\n<td style=\"width: 30.3082%;\">Stores, Company B<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 15.7534%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 26.8836%;\">[latex]100[\/latex]<\/td>\n<td style=\"width: 27.0548%;\">Starting with [latex]100[\/latex] each<\/td>\n<td style=\"width: 30.3082%;\">[latex]100[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.7534%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 26.8836%;\">[latex]100+50=150[\/latex]<\/td>\n<td style=\"width: 27.0548%;\">They both grow by\u00a0[latex]50[\/latex] stores in the first year.<\/td>\n<td style=\"width: 30.3082%;\">[latex]100[\/latex][latex]+50\\%[\/latex] of [latex]100[\/latex] [latex]100 + 0.50(100) = 150[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.7534%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 26.8836%;\">[latex]150+50=200[\/latex]<\/td>\n<td style=\"width: 27.0548%;\">Store A grows by\u00a0[latex]50[\/latex], Store B grows by\u00a0[latex]75[\/latex]<\/td>\n<td style=\"width: 30.3082%;\">[latex]150[\/latex][latex]+ 50\\%[\/latex] of\u00a0[latex]150[\/latex] [latex]150 + 0.50(150) = 225[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.7534%;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 26.8836%;\">[latex]200+50=250[\/latex]<\/td>\n<td style=\"width: 27.0548%;\">Store A grows by\u00a0[latex]50[\/latex], Store B grows by\u00a0[latex]112.5[\/latex]<\/td>\n<td style=\"width: 30.3082%;\">[latex]225 + 50\\%[\/latex] of\u00a0[latex]225[\/latex] [latex]225 + 0.50(225) = 337.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Company A has\u00a0[latex]100[\/latex] stores and expands by opening\u00a0[latex]50[\/latex] new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex]. Company B has\u00a0[latex]100[\/latex] stores and expands by increasing the number of stores by\u00a0[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].<\/p>\n<p>The graphs comparing the number of stores for each company over a five-year period are shown below<strong>.<\/strong> We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.<\/p>\n<div style=\"width: 348px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051913\/CNX_Precalc_Figure_04_01_0012.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"338\" height=\"586\" \/><\/p>\n<p class=\"wp-caption-text\">The graph shows the number of stores Companies A and B opened over a five-year period.<\/p>\n<\/div>\n<p id=\"fs-id1165135209682\">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\u00a0[latex]1[\/latex], Company B always has more stores than Company A.<\/p>\n<p id=\"fs-id1165137836429\">Consider\u00a0the function representing the number of stores for Company B:<\/p>\n<p style=\"text-align: center;\">[latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex]<\/p>\n<p>In this exponential function,\u00a0[latex]100[\/latex] represents the initial number of stores,\u00a0[latex]0.50[\/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 [latex]B\\left(x\\right)=100{\\left(1.5\\right)}^{x}[\/latex], where\u00a0[latex]100[\/latex] is the initial value,\u00a0[latex]1.5[\/latex] is called the <em>base<\/em>, and <em>x<\/em>\u00a0is called the <em>exponent<\/em>. This is an exponential function.<\/p>\n<div id=\"fs-id1165137564690\" class=\"note textbox\">\n<h3 class=\"title\">Exponential Growth<\/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<p id=\"fs-id1165137644244\">To evaluate an exponential function of the form [latex]f\\left(x\\right)={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\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<div id=\"eip-id1165137643186\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & ={2}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & ={2}^{3}\\text{ }\\hfill & \\text{Substitute }x=3.\\hfill \\\\ \\hfill & =8\\text{ }\\hfill & \\text{Evaluate the power}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137849020\">To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:<\/p>\n<p id=\"fs-id1165137849024\">Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<div id=\"eip-id1165134086025\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & =30{\\left(2\\right)}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & =30{\\left(2\\right)}^{3}\\hfill & \\text{Substitute }x=3.\\hfill \\\\ \\hfill & =30\\left(8\\right)\\text{ }\\hfill & \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill & =240\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137841073\">Note that if the order of operations were not followed, the result would be incorrect:<\/p>\n<div id=\"eip-id1165135320147\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">In our first example, we will evaluate an exponential function without the aid of a calculator.<\/div>\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<p>In the next example, we will revisit the population of India.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>At the beginning of this section, we learned that the population of India was about\u00a0[latex]1.25[\/latex] billion in the year\u00a0[latex]2013[\/latex], with an annual growth rate of about\u00a0[latex]1.2\\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.25{\\left(1.012\\right)}^{t}[\/latex], where <em>t<\/em>\u00a0is the number of years since\u00a0[latex]2013[\/latex]. To the nearest thousandth, what will the population of India be in\u00a0[latex]2031[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385742\">Show Solution<\/span><\/p>\n<div id=\"q385742\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137786635\">To estimate the population in\u00a0[latex]2031[\/latex], we evaluate the model for [latex]t=18[\/latex], because\u00a0[latex]2031[\/latex] is\u00a0[latex]18[\/latex] years after\u00a0[latex]2013[\/latex]. Rounding to the nearest thousandth,<\/p>\n<div id=\"eip-id1165135657117\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P\\left(18\\right)=1.25{\\left(1.012\\right)}^{18}\\approx 1.549[\/latex]<\/div>\n<p id=\"fs-id1165135394343\">There will be about\u00a0[latex]1.549[\/latex] billion people in India in the year\u00a0[latex]2031[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of using an exponential function to predict the population of a small town.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Evaluate a Given Exponential Function to Predict a Future Population\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SbIydBmJePE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Exponential growth grows by a rate proportional to the current amount.\u00a0For 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\u00a0[latex]f\\left(x\\right)=a{b}^{x}[\/latex]. \u00a0Evaluating exponential functions requires careful attention to the order of operations. Compound interest is an example of exponential growth.<\/p>\n<p id=\"fs-id1165137592823\">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. Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex] whose base is greater than one. We will use the function [latex]f\\left(x\\right)={2}^{x}[\/latex]. Observe how the output values in the table below\u00a0change as the input increases by\u00a0[latex]1[\/latex].<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>[latex]\u20133[\/latex]<\/td>\n<td>[latex]\u20132[\/latex]<\/td>\n<td>[latex]\u20131[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137432031\">Each output value is the product of the previous output and the base,\u00a0[latex]2[\/latex]. We call the base\u00a0[latex]2[\/latex] the <em>constant ratio<\/em>. In fact, for any exponential function of the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex], <em>b<\/em>\u00a0is the constant ratio of the function. This means that as the input increases by\u00a0[latex]1[\/latex], the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.<\/p>\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\n<ul id=\"fs-id1165137658509\">\n<li>the output values are positive for all values of <em>x<\/em>;<\/li>\n<li>as <em>x<\/em>\u00a0increases, the output values increase without bound; and<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow smaller, approaching zero.<\/li>\n<\/ul>\n<p>The graph below shows the exponential growth function [latex]f\\left(x\\right)={2}^{x}[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051915\/CNX_Precalc_Figure_04_02_0012.jpg\" alt=\"Graph of the exponential function, 2^(x), with labeled points at (-3, 1\/8), (-2, \u00bc), (-1, \u00bd), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\" \/><\/p>\n<p class=\"wp-caption-text\">In the graph, notice that the graph gets close to the x-axis, but never touches it.<\/p>\n<\/div>\n<p id=\"fs-id1165137459614\">The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\n<p id=\"fs-id1165137838249\">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]f\\left(x\\right)={b}^{x}[\/latex] whose base is between zero and one. We will use the function [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. Observe how the output values in the table below\u00a0change as the input increases by\u00a0[latex]1[\/latex].<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>&#8211;[latex]3[\/latex]<\/td>\n<td>&#8211;[latex]2[\/latex]<\/td>\n<td>&#8211;[latex]1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135347846\">Again, because the input is increasing by\u00a0[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 id=\"fs-id1165137452063\">Notice from the table that:<\/p>\n<ul id=\"fs-id1165135499992\">\n<li>the output values are positive for all values of <em>x<\/em>;<\/li>\n<li>as <em>x<\/em>\u00a0increases, the output values grow smaller, approaching zero; and<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p id=\"fs-id1165137405421\">The graph shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051917\/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=\"487\" height=\"520\" \/><\/p>\n<p id=\"fs-id1165137723586\"><strong>\u00a0<\/strong>The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\n<div id=\"fs-id1165135571835\" class=\"note textbox\">\n<h3 class=\"title\">Characteristics of the Graph of\u00a0 [latex]f(x) = b^{x}[\/latex]<\/h3>\n<p id=\"fs-id1165137848929\">An exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], [latex]b>0[\/latex], [latex]b\\ne 1[\/latex], has these characteristics:<\/p>\n<ul id=\"fs-id1165135186684\">\n<li><strong>one-to-one<\/strong> function<\/li>\n<li>domain: [latex]\\left(-\\infty , \\infty \\right)[\/latex]<\/li>\n<li>range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n<li><em>x-<\/em>intercept: none<\/li>\n<li><em>y-<\/em>intercept: [latex]\\left(0,1\\right)[\/latex]<\/li>\n<li>increasing if [latex]b>1[\/latex]<\/li>\n<li>decreasing if [latex]b<1[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137471878\">Compare the graphs of <strong>exponential growth<\/strong> and decay functions below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051919\/CNX_Precalc_Figure_04_02_003new2.jpg\" alt=\"Graph of two functions where the first graph is of a function of f(x) = b^x when b&gt;1 and the second graph is of the same function when b is 0&lt;b&lt;1. Both graphs have the points (0, 1) and (1, b) labeled.\" width=\"731\" height=\"407\" \/><\/p>\n<\/div>\n<p>In our first example, we will plot an exponential decay function where the base is between 0 and 1.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)={0.25}^{x}[\/latex]. State the domain, range.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q203605\">Show Solution<\/span><\/p>\n<div id=\"q203605\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\n<ul>\n<li>Since <em>b\u00a0<\/em>=[latex]0.25[\/latex] is between zero and one, we know the function is decreasing, and we can verify this by creating a table of values. The left tail of the graph will increase without bound and the right tail will get really close to the x-axis.<\/li>\n<li>Create a table of points.<br \/>\n<table id=\"Table_04_02_03\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>[latex]\u20133[\/latex]<\/td>\n<td>[latex]\u20132[\/latex]<\/td>\n<td>[latex]\u20131[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]0.0625[\/latex]<\/td>\n<td>[latex]0.015625[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y<\/em>-intercept, [latex]\\left(0,1\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(1,0.25\\right)[\/latex].<\/li>\n<\/ul>\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points.<span id=\"fs-id1165137940681\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051920\/CNX_Precalc_Figure_04_02_0042.jpg\" alt=\"Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).\" \/><\/span><\/p>\n<p id=\"fs-id1165137548870\"><strong>\u00a0<\/strong>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of graphing an exponential function. The base of the exponential term is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex], so this graph will represent decay.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Graph a Basic Exponential Function Using a Table of Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/FMzZB9Ve-1U?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"fs-id1165134195243\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135194093\">How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function<\/h3>\n<ol id=\"fs-id1165137435782\">\n<li>Create a table of points.<\/li>\n<li>Plot at least\u00a0[latex]3[\/latex]\u00a0point from the table, including the <em>y<\/em>-intercept [latex]\\left(0,1\\right)[\/latex].<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the range, [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<p>In our next example, we will plot an exponential growth function where the base is greater than\u00a0[latex]1[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Sketch a graph of [latex]f(x)={\\sqrt{2}(\\sqrt{2})}^{x}[\/latex].\u00a0State the domain and range.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q334418\">Show Solution<\/span><\/p>\n<div id=\"q334418\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\n<ul>\n<li>Since <em>b\u00a0<\/em>= [latex]\\sqrt{2}[\/latex], which is greater than\u00a0one, we know the function is increasing, and we can verify this by creating a table of values. The left tail of the graph will\u00a0get really close to the x-axis and the right tail will increase without bound.<\/li>\n<li>Create a table of points.<br \/>\n<table id=\"Table_04_02_03\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>[latex]\u20133[\/latex]<\/td>\n<td>[latex]\u20132[\/latex]<\/td>\n<td>[latex]\u20131[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)=\\sqrt{2}{(\\sqrt{2})}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]0.5[\/latex]<\/td>\n<td>[latex]0.71[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1.41[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2.83[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y<\/em>-intercept, [latex]\\left(0,1.41\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\n<\/ul>\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3623 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05185425\/Screen-Shot-2016-08-05-at-11.53.45-AM.png\" alt=\"Screen Shot 2016-08-05 at 11.53.45 AM\" width=\"326\" height=\"231\" \/><\/p>\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Our next video example includes graphing an exponential growth function and defining the domain and range of the function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Graph an Exponential Function Using a Table of Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/M6bpp0BRIf0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Graphs of exponential growth functions will have a right tail that increases without bound and a left tail that gets really close to the x-axis. On the other hand, graphs of exponential decay functions will have a left tail that increases without bound and a right tail that gets really close to the x-axis. Points can be generated with a table of values which can then be used to graph the function.<\/p>\n<\/div>\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-216\">\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>Evaluate a Given Exponential Function to Predict a Future Population. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/SbIydBmJePE\">https:\/\/youtu.be\/SbIydBmJePE<\/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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Compounded Interest Formula - Quarterly. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3az4AKvUmmI\">https:\/\/youtu.be\/3az4AKvUmmI<\/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>Ex: Compounded Interest Formula - Determine Deposit Needed (Present Value). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/saq9dF7a4r8\">https:\/\/youtu.be\/saq9dF7a4r8<\/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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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>: Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Determine Exponential Function Values and Graph the Function\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QFFAoX0We34\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Evaluate a Given Exponential Function to Predict a Future Population\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/SbIydBmJePE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Compounded Interest Formula - 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