{"id":1996,"date":"2016-11-02T23:16:43","date_gmt":"2016-11-02T23:16:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1996"},"modified":"2017-07-07T17:00:21","modified_gmt":"2017-07-07T17:00:21","slug":"characteristics-of-graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/characteristics-of-graphs-of-exponential-functions\/","title":{"raw":"Characteristics of Graphs of Exponential Functions","rendered":"Characteristics of Graphs of Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Determine whether an exponential function and it's 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; 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\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\" data-media-type=\"image\/jpg\" \/> 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; and<\/li>\r\n \t<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound.<\/li>\r\n<\/ul>\r\nThe graph 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\" data-media-type=\"image\/jpg\" \/> 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 <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/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 the sliders in the graph below to compare the graphs of <strong>exponential growth<\/strong> and decay functions. Which one is growth and which one is decay?\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/3pqyjsunkg\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 by hand.<\/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 exercises\">\r\n<h3>Example: Sketching the Graph of an Exponential Function of the Form <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/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\"]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\" data-media-type=\"image\/jpg\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/div>\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<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\n[reveal-answer q=\"192861\"]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]; the horizontal asymptote is [latex]y=0[\/latex].<span id=\"fs-id1165137437648\" data-type=\"media\" data-alt=\"Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).\">\r\n<\/span>\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<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3607&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\">\r\n<\/iframe>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Determine whether an exponential function and it&#8217;s 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; and<\/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\" data-media-type=\"image\/jpg\" \/><\/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; and<\/li>\n<li>as <em>x<\/em>\u00a0decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p>The graph 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\" data-media-type=\"image\/jpg\" \/><\/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 <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/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 the sliders in the graph below to compare the graphs of <strong>exponential growth<\/strong> and decay functions. Which one is growth and which one is decay?<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/3pqyjsunkg<\/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 by hand.<\/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 exercises\">\n<h3>Example: Sketching the Graph of an Exponential Function of the Form <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/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\">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\" data-media-type=\"image\/jpg\" \/><\/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]; the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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.<br \/>\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q192861\">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]; the horizontal asymptote is [latex]y=0[\/latex].<span id=\"fs-id1165137437648\" data-type=\"media\" data-alt=\"Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).\"><br \/>\n<\/span><\/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=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3607&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/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-1996\">\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><\/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><\/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":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 3607\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"Reidel,Jessica\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"original\",\"description\":\"Characteristics of Graphs of Exponential Functions Interactive\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/www.desmos.com\/calculator\/3pqyjsunkg\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"54b66693-25a3-4a4e-9f20-bff31656af29","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1996","chapter","type-chapter","status-publish","hentry"],"part":1964,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1996","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1996\/revisions"}],"predecessor-version":[{"id":4260,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1996\/revisions\/4260"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1964"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1996\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=1996"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1996"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1996"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=1996"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}