{"id":15987,"date":"2019-09-26T16:05:37","date_gmt":"2019-09-26T16:05:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/graph-exponential-functions\/"},"modified":"2024-05-02T16:00:57","modified_gmt":"2024-05-02T16:00:57","slug":"graph-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/graph-exponential-functions\/","title":{"raw":"Graphing Exponential Functions","rendered":"Graphing Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Graph exponential functions<\/li>\r\n<\/ul>\r\n<\/div>\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 style=\"width: 60%;\" 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]\u00a0<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\nAs x decreases, the output values grow smaller and smaller, getting closer and closer to zero but never actually reaching zero or crossing the x-axis.\u00a0 This is a special property of this type of graph.\u00a0 We say that the x-axis is an <em>asymptote<\/em> of an exponential growth function.\r\n\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 style=\"width: 62.2509%; height: 78px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 26.259%;\"><em><strong>x<\/strong><\/em><\/td>\r\n<td style=\"width: 10.6715%;\">-[latex]3[\/latex]<\/td>\r\n<td style=\"width: 10.086%;\">-[latex]2[\/latex]<\/td>\r\n<td style=\"width: 9.49966%;\">-[latex]1[\/latex]<\/td>\r\n<td style=\"width: 6.72131%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 10.1918%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 10.1918%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.6508%;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.259%;\"><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 10.6715%;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 10.086%;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 9.49966%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 6.72131%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 10.1918%;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 10.1918%;\">[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.6508%;\">[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\nSimilar to the exponential growth functions, a special property of exponential decay functions is that as the input values get larger and larger, the output values get closer and closer to zero without every actually touching or crossing the x-axis.\u00a0 As a result, we say that the x-axis is an asymptote of an exponential decay function.\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\" style=\"text-align: center;\"><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\" style=\"width: 928.672px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 135px;\"><em><strong>x<\/strong><\/em><\/td>\r\n<td style=\"width: 55px;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 53px;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 47px;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 47px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 67px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 83px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 99.6719px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 135px;\"><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 55px;\">[latex]64[\/latex]<\/td>\r\n<td style=\"width: 53px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 47px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 47px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 67px;\">[latex]0.25[\/latex]<\/td>\r\n<td style=\"width: 83px;\">[latex]0.0625[\/latex]<\/td>\r\n<td style=\"width: 99.6719px;\">[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\" style=\"text-align: center;\"><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.\u00a0 Make sure that, in the direction the function is decreasing, the function will approach the x-axis but never actually cross is.<\/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\" style=\"width: 778px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 148.542px;\"><em><strong>x<\/strong><\/em><\/td>\r\n<td style=\"width: 80.7639px;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 87.4306px;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 71.875px;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 87.4306px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 71.875px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 87.4306px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 71.875px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 148.542px;\"><strong>[latex]f\\left(x\\right)=\\sqrt{2}{(\\sqrt{2})}^{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 80.7639px;\">[latex]0.5[\/latex]<\/td>\r\n<td style=\"width: 87.4306px;\">[latex]0.71[\/latex]<\/td>\r\n<td style=\"width: 71.875px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 87.4306px;\">[latex]1.41[\/latex]<\/td>\r\n<td style=\"width: 71.875px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 87.4306px;\">[latex]2.83[\/latex]<\/td>\r\n<td style=\"width: 71.875px;\">[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=\"aligncenter wp-image-3623\" 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=\"Graph of the function f of x equals the squareroot of 2 times the squareroot of 2 to the x power with points at (-3, 0.5), (-1, 1), (1, 2), and (3, 4).\" 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","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Graph exponential functions<\/li>\n<\/ul>\n<\/div>\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 style=\"width: 60%;\" 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]\u00a0<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>As x decreases, the output values grow smaller and smaller, getting closer and closer to zero but never actually reaching zero or crossing the x-axis.\u00a0 This is a special property of this type of graph.\u00a0 We say that the x-axis is an <em>asymptote<\/em> of an exponential growth function.<\/p>\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 style=\"width: 62.2509%; height: 78px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td style=\"width: 26.259%;\"><em><strong>x<\/strong><\/em><\/td>\n<td style=\"width: 10.6715%;\">&#8211;[latex]3[\/latex]<\/td>\n<td style=\"width: 10.086%;\">&#8211;[latex]2[\/latex]<\/td>\n<td style=\"width: 9.49966%;\">&#8211;[latex]1[\/latex]<\/td>\n<td style=\"width: 6.72131%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 10.1918%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 10.1918%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.6508%;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.259%;\"><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 10.6715%;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 10.086%;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 9.49966%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 6.72131%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 10.1918%;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 10.1918%;\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.6508%;\">[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>Similar to the exponential growth functions, a special property of exponential decay functions is that as the input values get larger and larger, the output values get closer and closer to zero without every actually touching or crossing the x-axis.\u00a0 As a result, we say that the x-axis is an asymptote of an exponential decay function.<\/p>\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\" style=\"text-align: center;\"><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\" style=\"width: 928.672px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td style=\"width: 135px;\"><em><strong>x<\/strong><\/em><\/td>\n<td style=\"width: 55px;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 53px;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 47px;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 47px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 67px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 83px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 99.6719px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 135px;\"><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 55px;\">[latex]64[\/latex]<\/td>\n<td style=\"width: 53px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 47px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 47px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 67px;\">[latex]0.25[\/latex]<\/td>\n<td style=\"width: 83px;\">[latex]0.0625[\/latex]<\/td>\n<td style=\"width: 99.6719px;\">[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\" style=\"text-align: center;\"><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-1\" 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.\u00a0 Make sure that, in the direction the function is decreasing, the function will approach the x-axis but never actually cross is.<\/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\" style=\"width: 778px;\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td style=\"width: 148.542px;\"><em><strong>x<\/strong><\/em><\/td>\n<td style=\"width: 80.7639px;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 87.4306px;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 71.875px;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 87.4306px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 71.875px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 87.4306px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 71.875px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 148.542px;\"><strong>[latex]f\\left(x\\right)=\\sqrt{2}{(\\sqrt{2})}^{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 80.7639px;\">[latex]0.5[\/latex]<\/td>\n<td style=\"width: 87.4306px;\">[latex]0.71[\/latex]<\/td>\n<td style=\"width: 71.875px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 87.4306px;\">[latex]1.41[\/latex]<\/td>\n<td style=\"width: 71.875px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 87.4306px;\">[latex]2.83[\/latex]<\/td>\n<td style=\"width: 71.875px;\">[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=\"aligncenter wp-image-3623\" 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=\"Graph of the function f of x equals the squareroot of 2 times the squareroot of 2 to the x power with points at (-3, 0.5), (-1, 1), (1, 2), and (3, 4).\" 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-2\" 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\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-15987\">\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>Graph a Basic Exponential Function Using a Table of Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/FMzZB9Ve-1U\">https:\/\/youtu.be\/FMzZB9Ve-1U<\/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>Graph an Exponential Function Using a Table of Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/M6bpp0BRIf0\">https:\/\/youtu.be\/M6bpp0BRIf0<\/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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":169554,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Graph a Basic Exponential Function Using a Table of Values\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/FMzZB9Ve-1U\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Graph an Exponential Function Using a Table of Values\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/M6bpp0BRIf0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"f2a2ebc8661f4c09a4dd15b875789e9d, e540ebce15744efdb270c8cda10bab83 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