{"id":262,"date":"2023-06-21T13:22:49","date_gmt":"2023-06-21T13:22:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/characteristics-of-graphs-of-exponential-functions\/"},"modified":"2023-10-19T22:44:32","modified_gmt":"2023-10-19T22:44:32","slug":"characteristics-of-graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/characteristics-of-graphs-of-exponential-functions\/","title":{"raw":"\u25aa   Characteristics of Graphs of Exponential Functions","rendered":"\u25aa   Characteristics of Graphs of Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether an exponential function and its associated graph represents growth or decay.<\/li>\r\n \t<li>Sketch a graph of an exponential function.<\/li>\r\n<\/ul>\r\n<\/div>\r\nBefore 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\u2019ll 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 1.\r\n<table id=\"Table_04_02_01\" summary=\"Two rows and eight columns. The first row is labeled,\"><colgroup> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/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>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEach output value is the product of the previous output and the base, 2. We call the base 2 the <em>constant ratio<\/em>. In fact, for any exponential function with 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 1, the output value will be the product of the base and the previous output, regardless of the value of <em>a<\/em>.\r\n\r\nNotice from the table that:\r\n<ul>\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<\/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\u00a0shows 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\/896\/2016\/11\/02231129\/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\" \/> Notice that the graph gets close to the x-axis but never touches it.[\/caption]\r\n\r\nThe domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].\r\n\r\nTo get a sense of the behavior of <strong>exponential decay<\/strong>, we can create a table of values for a function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex] whose base is between zero and one. We\u2019ll 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 1.\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>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/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>8<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/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\nAgain, because the input is increasing by 1, each output value is the product of the previous output and the base or constant ratio [latex]\\frac{1}{2}[\/latex].\r\n\r\nNotice from the table that:\r\n<ul>\r\n \t<li>the output values are positive for all values of <em>x<\/em><\/li>\r\n \t<li>as <em>x<\/em>\u00a0increases, the output values grow smaller, approaching zero<\/li>\r\n \t<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound<\/li>\r\n<\/ul>\r\nThe graph below shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{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\/896\/2016\/11\/02231133\/CNX_Precalc_Figure_04_02_0022.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\" \/> 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], and the horizontal asymptote is [latex]y=0[\/latex].[\/caption]\r\n<div class=\"textbox\">\r\n<h3>A General Note: Characteristics of the Graph of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nAn exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], [latex]b&gt;0[\/latex], [latex]b\\ne 1[\/latex], has these characteristics:\r\n<ul>\r\n \t<li><strong>one-to-one<\/strong> function<\/li>\r\n \t<li>horizontal asymptote: [latex]y=0[\/latex]<\/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\nUse an online graphing tool to graph\u00a0[latex]f\\left(x\\right)={b}^{x}[\/latex].\r\n\r\nAdjust the [latex]b[\/latex] values to see various graphs of\u00a0<strong>exponential growth<\/strong> and <strong>decay<\/strong> functions.\r\n\r\nTry decimal values between [latex]0[\/latex] and [latex]1[\/latex], and also values greater than [latex]1[\/latex]. Which one is growth and which one is decay?\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function<\/h3>\r\n<ol>\r\n \t<li>Create a table of points.<\/li>\r\n \t<li>Plot at least 3\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], the range, [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote, [latex]y=0[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nWhen sketching the graph of an exponential function by plotting points, include a few input values left and right of zero as well as zero itself.\r\n\r\nWith few exceptions, such as functions that would be undefined at zero or negative input like the radical or (as you'll see soon) the logarithmic function, it is good practice to let the input equal [latex]-3, -2, -1, 0, 1, 2, \\text{ and } 3[\/latex] to get the idea of the shape of the graph.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Sketching the Graph of an Exponential Function of the Form [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nSketch a graph of [latex]f\\left(x\\right)={0.25}^{x}[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"410947\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"410947\"]\r\n\r\nBefore graphing, identify the behavior and create a table of points for the graph.\r\n<ul>\r\n \t<li>Since <em>b\u00a0<\/em>= 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote <em>y\u00a0<\/em>= 0.<\/li>\r\n \t<li>Create a table of points.\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>\u20133<\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/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>64<\/td>\r\n<td>16<\/td>\r\n<td>4<\/td>\r\n<td>1<\/td>\r\n<td>0.25<\/td>\r\n<td>0.0625<\/td>\r\n<td>0.015625<\/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\nDraw a smooth curve connecting the points.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231140\/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).\" width=\"487\" height=\"332\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/div>\r\nWatch the following video for 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\r\nThe next example shows how to 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=\" 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\nThe 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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSketch the graph of [latex]f\\left(x\\right)={4}^{x}[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"192861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"192861\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\"><img class=\"aligncenter wp-image-3353 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\" width=\"487\" height=\"332\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]3607[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine whether an exponential function and its associated graph represents growth or decay.<\/li>\n<li>Sketch a graph of an exponential function.<\/li>\n<\/ul>\n<\/div>\n<p>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\u2019ll 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 1.<\/p>\n<table id=\"Table_04_02_01\" summary=\"Two rows and eight columns. The first row is labeled,\">\n<colgroup> <\/colgroup>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/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>1<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Each output value is the product of the previous output and the base, 2. We call the base 2 the <em>constant ratio<\/em>. In fact, for any exponential function with 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 1, 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>Notice from the table that:<\/p>\n<ul>\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<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow smaller, approaching zero<\/li>\n<\/ul>\n<p>The graph below\u00a0shows 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\/896\/2016\/11\/02231129\/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\">Notice that the graph gets close to the x-axis but never touches it.<\/p>\n<\/div>\n<p>The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<p>To get a sense of the behavior of <strong>exponential decay<\/strong>, we can create a table of values for a function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex] whose base is between zero and one. We\u2019ll 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 1.<\/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>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td>8<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/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>Again, because the input is increasing by 1, each output value is the product of the previous output and the base or constant ratio [latex]\\frac{1}{2}[\/latex].<\/p>\n<p>Notice from the table that:<\/p>\n<ul>\n<li>the output values are positive for all values of <em>x<\/em><\/li>\n<li>as <em>x<\/em>\u00a0increases, the output values grow smaller, approaching zero<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound<\/li>\n<\/ul>\n<p>The graph below shows the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{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\/896\/2016\/11\/02231133\/CNX_Precalc_Figure_04_02_0022.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\" \/><\/p>\n<p class=\"wp-caption-text\">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], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Characteristics of the Graph of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>An exponential function with 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>\n<li><strong>one-to-one<\/strong> function<\/li>\n<li>horizontal asymptote: [latex]y=0[\/latex]<\/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>Use an online graphing tool to graph\u00a0[latex]f\\left(x\\right)={b}^{x}[\/latex].<\/p>\n<p>Adjust the [latex]b[\/latex] values to see various graphs of\u00a0<strong>exponential growth<\/strong> and <strong>decay<\/strong> functions.<\/p>\n<p>Try decimal values between [latex]0[\/latex] and [latex]1[\/latex], and also values greater than [latex]1[\/latex]. Which one is growth and which one is decay?<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function<\/h3>\n<ol>\n<li>Create a table of points.<\/li>\n<li>Plot at least 3\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], the range, [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote, [latex]y=0[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>When sketching the graph of an exponential function by plotting points, include a few input values left and right of zero as well as zero itself.<\/p>\n<p>With few exceptions, such as functions that would be undefined at zero or negative input like the radical or (as you&#8217;ll see soon) the logarithmic function, it is good practice to let the input equal [latex]-3, -2, -1, 0, 1, 2, \\text{ and } 3[\/latex] to get the idea of the shape of the graph.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Sketching the Graph of an Exponential Function of the Form [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)={0.25}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q410947\">Show Solution<\/span><\/p>\n<div id=\"q410947\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before graphing, identify the behavior and create a table of points for the graph.<\/p>\n<ul>\n<li>Since <em>b\u00a0<\/em>= 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote <em>y\u00a0<\/em>= 0.<\/li>\n<li>Create a table of points.<br \/>\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>\u20133<\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td>64<\/td>\n<td>16<\/td>\n<td>4<\/td>\n<td>1<\/td>\n<td>0.25<\/td>\n<td>0.0625<\/td>\n<td>0.015625<\/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>Draw a smooth curve connecting the points.<\/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\/896\/2016\/11\/02231140\/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).\" width=\"487\" height=\"332\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for 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<p>The next example shows how to 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=\"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>The 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Sketch the graph of [latex]f\\left(x\\right)={4}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q192861\">Show Solution<\/span><\/p>\n<div id=\"q192861\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3353 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\" width=\"487\" height=\"332\" alt=\"image\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3607\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3607&theme=oea&iframe_resize_id=ohm3607&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/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-262\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Characteristics of Graphs of Exponential Functions Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/3pqyjsunkg\">https:\/\/www.desmos.com\/calculator\/3pqyjsunkg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 3607. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: Reidel,Jessica. <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>: IMathAS Community License CC-BY + GPL<\/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@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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@5.2<\/li><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 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