{"id":1991,"date":"2016-11-02T23:25:25","date_gmt":"2016-11-02T23:25:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1991"},"modified":"2020-03-12T17:26:24","modified_gmt":"2020-03-12T17:26:24","slug":"introduction-graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/chapter\/introduction-graphs-of-exponential-functions\/","title":{"raw":"Graphs of Exponential Functions","rendered":"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 \t<li>Graph exponential functions shifted horizontally or vertically and write the associated equation.<\/li>\r\n \t<li>Graph a stretched or compressed exponential function.<\/li>\r\n \t<li>Graph a reflected exponential function.<\/li>\r\n \t<li>Write the equation of an exponential function that has been transformed.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their visual representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.\r\n<h2>Characteristics of Graphs of Exponential Functions<\/h2>\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\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 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\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].<span id=\"fs-id1165137437648\">\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:\/\/ohm.lumenlearning.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<h2>Graphing Exponential Functions Using Transformations<\/h2>\r\nTransformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.\r\n<h3>Graphing a Vertical Shift<\/h3>\r\nThe first transformation occurs when we add a constant <em>d<\/em>\u00a0to the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] giving us a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two vertical shifts alongside it using [latex]d=3[\/latex]: the upward shift, [latex]g\\left(x\\right)={2}^{x}+3[\/latex] and the downward shift, [latex]h\\left(x\\right)={2}^{x}-3[\/latex]. Both vertical shifts are shown in the figure below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231142\/CNX_Precalc_Figure_04_02_0062.jpg\" alt=\"Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions\u2019 transformations are described in the text.\" width=\"487\" height=\"628\" \/>\r\n\r\nObserve the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically:\r\n<ul>\r\n \t<li>The domain [latex]\\left(-\\infty ,\\infty \\right)[\/latex] remains unchanged.<\/li>\r\n \t<li>When the function is shifted up 3\u00a0units giving [latex]g\\left(x\\right)={2}^{x}+3[\/latex]:\r\n<ul>\r\n \t<li>The <em>y-<\/em>intercept shifts up 3\u00a0units to [latex]\\left(0,4\\right)[\/latex].<\/li>\r\n \t<li>The asymptote shifts up 3\u00a0units to [latex]y=3[\/latex].<\/li>\r\n \t<li>The range becomes [latex]\\left(3,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>When the function is shifted down 3\u00a0units giving [latex]h\\left(x\\right)={2}^{x}-3[\/latex]:\r\n<ul>\r\n \t<li>The <em>y-<\/em>intercept shifts down 3\u00a0units to [latex]\\left(0,-2\\right)[\/latex].<\/li>\r\n \t<li>The asymptote also shifts down 3\u00a0units to [latex]y=-3[\/latex].<\/li>\r\n \t<li>The range becomes [latex]\\left(-3,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h3>Graphing a Horizontal Shift<\/h3>\r\nThe next transformation occurs when we add a constant <em>c<\/em>\u00a0to the input of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] giving us a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em>opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two horizontal shifts alongside it using [latex]c=3[\/latex]: the shift left, [latex]g\\left(x\\right)={2}^{x+3}[\/latex], and the shift right, [latex]h\\left(x\\right)={2}^{x - 3}[\/latex]. Both horizontal shifts are shown in the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231145\/CNX_Precalc_Figure_04_02_0072.jpg\" alt=\"Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions\u2019 asymptotes are at y=0Note that each functions\u2019 transformations are described in the text.\" width=\"731\" height=\"478\" \/>\r\n\r\nObserve the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] horizontally:\r\n<ul>\r\n \t<li>The domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], remains unchanged.<\/li>\r\n \t<li>The asymptote, [latex]y=0[\/latex], remains unchanged.<\/li>\r\n \t<li>The <em>y-<\/em>intercept shifts such that:\r\n<ul>\r\n \t<li>When the function is shifted left 3\u00a0units to [latex]g\\left(x\\right)={2}^{x+3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,8\\right)[\/latex]. This is because [latex]{2}^{x+3}=\\left({2}^{3}\\right){2}^{x}=\\left(8\\right){2}^{x}[\/latex], so the initial value of the function is 8.<\/li>\r\n \t<li>When the function is shifted right 3\u00a0units to [latex]h\\left(x\\right)={2}^{x - 3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,\\frac{1}{8}\\right)[\/latex]. Again, see that [latex]{2}^{x-3}=\\left({2}^{-3}\\right){2}^{x}=\\left(\\frac{1}{8}\\right){2}^{x}[\/latex], so the initial value of the function is [latex]\\frac{1}{8}[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Shifts of the Parent Function\u00a0[latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nFor any constants <em>c<\/em>\u00a0and <em>d<\/em>, the function [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]\r\n<ul>\r\n \t<li>shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] vertically <em>d<\/em>\u00a0units, in the <em>same<\/em> direction as the sign of <em>d<\/em>.<\/li>\r\n \t<li>shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] horizontally <em>c<\/em>\u00a0units, in the <em>opposite<\/em> direction as the sign of <em>c<\/em>.<\/li>\r\n \t<li>has a\u00a0<em>y<\/em>-intercept of [latex]\\left(0,{b}^{c}+d\\right)[\/latex].<\/li>\r\n \t<li>has a horizontal asymptote of\u00a0<em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\r\n \t<li>has a range of [latex]\\left(d,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which remains unchanged.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an exponential function with the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex], graph the translation<\/h3>\r\n<ol>\r\n \t<li>Draw the horizontal asymptote <em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\r\n \t<li>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left <em>c<\/em>\u00a0units if <em>c<\/em>\u00a0is positive and right [latex]c[\/latex] units if <em>c<\/em>\u00a0is negative.<\/li>\r\n \t<li>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] up <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is positive and down <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is negative.<\/li>\r\n \t<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range, [latex]\\left(d,\\infty \\right)[\/latex], and the horizontal asymptote [latex]y=d[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Shift of an Exponential Function<\/h3>\r\nGraph [latex]f\\left(x\\right)={2}^{x+1}-3[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"344344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"344344\"]\r\n\r\nWe have an exponential equation of the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex], with [latex]b=2[\/latex], [latex]c=1[\/latex], and [latex]d=-3[\/latex].\r\n\r\nDraw the horizontal asymptote [latex]y=d[\/latex], so draw [latex]y=-3[\/latex].\r\n\r\nIdentify the shift; it is [latex]\\left(-1,-3\\right)[\/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\/02231148\/CNX_Precalc_Figure_04_02_0082.jpg\" alt=\"Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).\" width=\"487\" height=\"519\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-3,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=-3[\/latex].[\/caption]Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left 1 unit and down 3 units.<span id=\"fs-id1165137591826\">\r\n<\/span>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-3,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=-3[\/latex].[\/hidden-answer]<\/div>\r\nIn the following video, we show more examples of the difference between horizontal and vertical shifts of exponential functions and the resulting graphs and equations.\r\n\r\nhttps:\/\/youtu.be\/phYxEeJ7ZW4\r\n<h3>Stretching, Compressing, or Reflecting an Exponential Function<\/h3>\r\nWhile horizontal and vertical shifts involve adding constants to the input or to the function itself, a <strong>stretch<\/strong> or <strong>compression<\/strong> occurs when we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|&gt;0[\/latex]. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the stretch, using [latex]a=3[\/latex], to get [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231151\/CNX_Precalc_Figure_04_02_0102.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" width=\"975\" height=\"445\" \/> (a) [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] stretches the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of 3. (b) [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] compresses the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]\\frac{1}{3}[\/latex].[\/caption]\r\n<div class=\"textbox\">\r\n<h3>A General Note: Stretches and Compressions of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)=a{b}^{x}[\/latex]\r\n<ul>\r\n \t<li>is stretched vertically by a factor of <em>a\u00a0<\/em>if [latex]|a|&gt;1[\/latex].<\/li>\r\n \t<li>is compressed vertically by a factor of <em>a<\/em>\u00a0if [latex]|a|&lt;1[\/latex].<\/li>\r\n \t<li>has a\u00a0<em>y<\/em>-intercept is [latex]\\left(0,a\\right)[\/latex].<\/li>\r\n \t<li>has a horizontal asymptote of [latex]y=0[\/latex], range of [latex]\\left(0,\\infty \\right)[\/latex], and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are all unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing the Stretch of an Exponential Function<\/h3>\r\nSketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"418729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418729\"]\r\n\r\nBefore graphing, identify the behavior and key points on the graph.\r\n<ul>\r\n \t<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em>x<\/em>-axis as <em>x<\/em>\u00a0increases.<\/li>\r\n \t<li>Since <em>a\u00a0<\/em>= 4, the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched vertically by a factor of 4.<\/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)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n<td>32<\/td>\r\n<td>16<\/td>\r\n<td>8<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>0.5<\/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(0,4\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\r\n \t<li>Draw a smooth curve connecting the points.<\/li>\r\n<\/ul>\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\/02231155\/CNX_Precalc_Figure_04_02_0112.jpg\" alt=\"Graph of the function, f(x) = 4(1\/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).\" width=\"487\" height=\"482\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], the horizontal asymptote is y\u00a0= 0.[\/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)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"796634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796634\"]\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].\u00a0<span id=\"fs-id1165135417835\">\r\n<\/span>\r\n\r\n<img class=\"aligncenter wp-image-3081 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16190801\/CNX_Precalc_Figure_04_02_0122.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"488\" height=\"294\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129498&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\">\r\n<\/iframe>\r\n\r\n<\/div>\r\n<h3>Graphing Reflections<\/h3>\r\nIn addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em>x<\/em>-axis. When we multiply the input by \u20131, we get a reflection about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={2}^{-x}[\/latex], are both shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02231158\/CNX_Precalc_Figure_04_02_0132.jpg\" alt=\"Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.\" width=\"975\" height=\"628\" \/> (a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]h\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.[\/caption]\r\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><\/figcaption><\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Reflecting the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>x<\/em>-axis.<\/li>\r\n \t<li>has a <em>y<\/em>-intercept of [latex]\\left(0,-1\\right)[\/latex].<\/li>\r\n \t<li>has a range of [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\r\n \t<li>has a horizontal asymptote of [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex]f\\left(x\\right)={b}^{-x}[\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>y<\/em>-axis.<\/li>\r\n \t<li>has a <em>y<\/em>-intercept of [latex]\\left(0,1\\right)[\/latex], a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing and Graphing the Reflection of an Exponential Function<\/h3>\r\nFind and graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x<\/em>-axis. State its domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"91748\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"91748\"]\r\n\r\nSince we want to reflect the parent function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by \u20131 to get [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.\r\n<table summary=\"Two rows and eight columns. The first row is labeled,\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\r\n<td>\u201364<\/td>\r\n<td>\u201316<\/td>\r\n<td>\u20134<\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20130.25<\/td>\r\n<td>\u20130.0625<\/td>\r\n<td>\u20130.0156<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot 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].\r\n\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\/02231202\/CNX_Precalc_Figure_04_02_0142.jpg\" alt=\"Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).\" width=\"487\" height=\"407\" \/> The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,0\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<h2>Summarizing Transformations of the Exponential Function<\/h2>\r\nNow that we have worked with each type of translation for the exponential function, we can summarize them\u00a0to arrive at the general equation for transforming exponential functions.\r\n<table summary=\"Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Transformations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center;\">Transformation<\/th>\r\n<th style=\"text-align: center;\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Shift\r\n<ul>\r\n \t<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\r\n \t<li>Vertically <em>d<\/em>\u00a0units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Stretch and Compression\r\n<ul>\r\n \t<li>Stretch if |<em>a<\/em>|&gt;1<\/li>\r\n \t<li>Compression if 0&lt;|<em>a<\/em>|&lt;1<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflection about the <em>x<\/em>-axis<\/td>\r\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reflection about the <em>y<\/em>-axis<\/td>\r\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>General equation for all transformations<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Transformations of Exponential Functions<\/h3>\r\nA transformation of an exponential function has the form\r\n\r\n[latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex], where the parent function, [latex]y={b}^{x}[\/latex], [latex]b&gt;1[\/latex], is\r\n<ul>\r\n \t<li>shifted horizontally <em>c<\/em>\u00a0units to the left.<\/li>\r\n \t<li>stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a| &gt; 1[\/latex].<\/li>\r\n \t<li>compressed vertically by a factor of [latex]|a|[\/latex]\u00a0if [latex]0 &lt; |a| &lt; 1[\/latex].<\/li>\r\n \t<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\nNote the order of the shifts, transformations, and reflections follow the order of operations.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Function from a Description<\/h3>\r\nWrite the equation for the function described below. Give the horizontal asymptote, domain, and range.\r\n<ul>\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <em>y<\/em>-axis, and then shifted up 4\u00a0units.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"290621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"290621\"]\r\n\r\nWe want to find an equation of the general form [latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex]. We use the description provided to find <em>a<\/em>, <em>b<\/em>, <em>c<\/em>, and <em>d<\/em>.\r\n<ul>\r\n \t<li>We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so <em>b\u00a0<\/em>= <em>e<\/em>.<\/li>\r\n \t<li>The function is stretched by a factor of 2, so <em>a\u00a0<\/em>= 2.<\/li>\r\n \t<li>The function is reflected about the <em>y<\/em>-axis. We replace <em>x<\/em>\u00a0with \u2013<em>x<\/em>\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\r\n \t<li>The graph is shifted vertically 4 units, so <em>d\u00a0<\/em>= 4.<\/li>\r\n<\/ul>\r\nSubstituting in the general form, we get:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}f\\left(x\\right)\\hfill &amp; =a{b}^{x+c}+d\\hfill \\\\ \\hfill &amp; =2{e}^{-x+0}+4\\hfill \\\\ \\hfill &amp; =2{e}^{-x}+4\\hfill \\end{array}[\/latex]<\/p>\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(4,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite the equation for the function described below. Give the horizontal asymptote, the domain, and the range.\r\n<ul>\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <em>x<\/em>-axis, and then shifted down 2\u00a0units.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"525289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525289\"]\r\n\r\n[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2[\/latex]; the domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,2\\right)[\/latex]; the horizontal asymptote is [latex]y=2[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Using a Graph to Approximate a Solution to an Exponential Equation<\/h2>\r\nGraphing can help you confirm or find the solution to an exponential equation.\u00a0For example,[latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] can be solved to find the specific value for x that makes it a true statement. Graphing [latex]y=4[\/latex] along with [latex]y=2^{x}[\/latex] in the same window, the point(s) of intersection if any represent the solutions of the equation.\r\n\r\nTo use a calculator to solve this, press <strong>[Y=]<\/strong> and enter [latex]1.2(5)x+2.8 [\/latex] next to <strong>Y1=<\/strong>. Then enter 42 next to <strong>Y2=<\/strong>. For a window, use the values \u20133 to 3 for[latex] x[\/latex] and \u20135 to 55 for[latex]y[\/latex].Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near[latex]x=2[\/latex].\r\n\r\nFor a better approximation, press <strong>[2ND]<\/strong> then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The x-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth,x\u22482.166.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]4=7.85{\\left(1.15\\right)}^{x}-2.27[\/latex] graphically. Round to the nearest thousandth.\r\n\r\n[reveal-answer q=\"407425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"407425\"]\r\n\r\n[latex]x\\approx -1.608[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<table summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td>General Form for the Transformation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\r\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a <em>y-<\/em>intercept at [latex]\\left(0, 1\\right)[\/latex], domain of [latex]\\left(-\\infty , \\infty \\right)[\/latex], range of [latex]\\left(0, \\infty \\right)[\/latex], and horizontal asymptote of [latex]y=0[\/latex].<\/li>\r\n \t<li>If [latex]b&gt;1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\r\n \t<li>If 0 &lt;\u00a0<em>b<\/em> &lt; 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n \t<li>The equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], where [latex]a&gt;0[\/latex], represents a vertical stretch if [latex]|a|&gt;1[\/latex] or compression if [latex]0&lt;|a|&lt;1[\/latex] of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\r\n \t<li>When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the <em>x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\left(x\\right)={b}^{-x}[\/latex], is a reflection about the <em>y<\/em>-axis.<\/li>\r\n \t<li>All transformations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex].<\/li>\r\n \t<li>Using the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], we can write the equation of a function given its description.<\/li>\r\n \t<li>Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/li>\r\n<\/ul>","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<li>Graph exponential functions shifted horizontally or vertically and write the associated equation.<\/li>\n<li>Graph a stretched or compressed exponential function.<\/li>\n<li>Graph a reflected exponential function.<\/li>\n<li>Write the equation of an exponential function that has been transformed.<\/li>\n<\/ul>\n<\/div>\n<p>As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their visual representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\n<h2>Characteristics of Graphs of Exponential Functions<\/h2>\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<\/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 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<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].<span id=\"fs-id1165137437648\"><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:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3607&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<h2>Graphing Exponential Functions Using Transformations<\/h2>\n<p>Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.<\/p>\n<h3>Graphing a Vertical Shift<\/h3>\n<p>The first transformation occurs when we add a constant <em>d<\/em>\u00a0to the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] giving us a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two vertical shifts alongside it using [latex]d=3[\/latex]: the upward shift, [latex]g\\left(x\\right)={2}^{x}+3[\/latex] and the downward shift, [latex]h\\left(x\\right)={2}^{x}-3[\/latex]. Both vertical shifts are shown in the figure 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\/896\/2016\/11\/02231142\/CNX_Precalc_Figure_04_02_0062.jpg\" alt=\"Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions\u2019 transformations are described in the text.\" width=\"487\" height=\"628\" \/><\/p>\n<p>Observe the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically:<\/p>\n<ul>\n<li>The domain [latex]\\left(-\\infty ,\\infty \\right)[\/latex] remains unchanged.<\/li>\n<li>When the function is shifted up 3\u00a0units giving [latex]g\\left(x\\right)={2}^{x}+3[\/latex]:\n<ul>\n<li>The <em>y-<\/em>intercept shifts up 3\u00a0units to [latex]\\left(0,4\\right)[\/latex].<\/li>\n<li>The asymptote shifts up 3\u00a0units to [latex]y=3[\/latex].<\/li>\n<li>The range becomes [latex]\\left(3,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>When the function is shifted down 3\u00a0units giving [latex]h\\left(x\\right)={2}^{x}-3[\/latex]:\n<ul>\n<li>The <em>y-<\/em>intercept shifts down 3\u00a0units to [latex]\\left(0,-2\\right)[\/latex].<\/li>\n<li>The asymptote also shifts down 3\u00a0units to [latex]y=-3[\/latex].<\/li>\n<li>The range becomes [latex]\\left(-3,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3>Graphing a Horizontal Shift<\/h3>\n<p>The next transformation occurs when we add a constant <em>c<\/em>\u00a0to the input of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] giving us a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em>opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph two horizontal shifts alongside it using [latex]c=3[\/latex]: the shift left, [latex]g\\left(x\\right)={2}^{x+3}[\/latex], and the shift right, [latex]h\\left(x\\right)={2}^{x - 3}[\/latex]. Both horizontal shifts are shown in the graph 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\/896\/2016\/11\/02231145\/CNX_Precalc_Figure_04_02_0072.jpg\" alt=\"Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions\u2019 asymptotes are at y=0Note that each functions\u2019 transformations are described in the text.\" width=\"731\" height=\"478\" \/><\/p>\n<p>Observe the results of shifting [latex]f\\left(x\\right)={2}^{x}[\/latex] horizontally:<\/p>\n<ul>\n<li>The domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], remains unchanged.<\/li>\n<li>The asymptote, [latex]y=0[\/latex], remains unchanged.<\/li>\n<li>The <em>y-<\/em>intercept shifts such that:\n<ul>\n<li>When the function is shifted left 3\u00a0units to [latex]g\\left(x\\right)={2}^{x+3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,8\\right)[\/latex]. This is because [latex]{2}^{x+3}=\\left({2}^{3}\\right){2}^{x}=\\left(8\\right){2}^{x}[\/latex], so the initial value of the function is 8.<\/li>\n<li>When the function is shifted right 3\u00a0units to [latex]h\\left(x\\right)={2}^{x - 3}[\/latex], the <em>y<\/em>-intercept becomes [latex]\\left(0,\\frac{1}{8}\\right)[\/latex]. Again, see that [latex]{2}^{x-3}=\\left({2}^{-3}\\right){2}^{x}=\\left(\\frac{1}{8}\\right){2}^{x}[\/latex], so the initial value of the function is [latex]\\frac{1}{8}[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>A General Note: Shifts of the Parent Function\u00a0[latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>For any constants <em>c<\/em>\u00a0and <em>d<\/em>, the function [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/p>\n<ul>\n<li>shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] vertically <em>d<\/em>\u00a0units, in the <em>same<\/em> direction as the sign of <em>d<\/em>.<\/li>\n<li>shifts the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] horizontally <em>c<\/em>\u00a0units, in the <em>opposite<\/em> direction as the sign of <em>c<\/em>.<\/li>\n<li>has a\u00a0<em>y<\/em>-intercept of [latex]\\left(0,{b}^{c}+d\\right)[\/latex].<\/li>\n<li>has a horizontal asymptote of\u00a0<em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\n<li>has a range of [latex]\\left(d,\\infty \\right)[\/latex].<\/li>\n<li>has a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which remains unchanged.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an exponential function with the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex], graph the translation<\/h3>\n<ol>\n<li>Draw the horizontal asymptote <em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\n<li>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left <em>c<\/em>\u00a0units if <em>c<\/em>\u00a0is positive and right [latex]c[\/latex] units if <em>c<\/em>\u00a0is negative.<\/li>\n<li>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] up <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is positive and down <em>d<\/em>\u00a0units if <em>d<\/em>\u00a0is negative.<\/li>\n<li>State the domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range, [latex]\\left(d,\\infty \\right)[\/latex], and the horizontal asymptote [latex]y=d[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Shift of an Exponential Function<\/h3>\n<p>Graph [latex]f\\left(x\\right)={2}^{x+1}-3[\/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=\"q344344\">Show Solution<\/span><\/p>\n<div id=\"q344344\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have an exponential equation of the form [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex], with [latex]b=2[\/latex], [latex]c=1[\/latex], and [latex]d=-3[\/latex].<\/p>\n<p>Draw the horizontal asymptote [latex]y=d[\/latex], so draw [latex]y=-3[\/latex].<\/p>\n<p>Identify the shift; it is [latex]\\left(-1,-3\\right)[\/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\/02231148\/CNX_Precalc_Figure_04_02_0082.jpg\" alt=\"Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).\" width=\"487\" height=\"519\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-3,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=-3[\/latex].<\/p>\n<\/div>\n<p>Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left 1 unit and down 3 units.<span id=\"fs-id1165137591826\"><br \/>\n<\/span>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-3,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=-3[\/latex].<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of the difference between horizontal and vertical shifts of exponential functions and the resulting graphs and equations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Match the Graphs of Translated Exponential Function to Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/phYxEeJ7ZW4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Stretching, Compressing, or Reflecting an Exponential Function<\/h3>\n<p>While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <strong>stretch<\/strong> or <strong>compression<\/strong> occurs when we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|>0[\/latex]. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the stretch, using [latex]a=3[\/latex], to get [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex].<\/p>\n<div style=\"width: 985px\" 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\/02231151\/CNX_Precalc_Figure_04_02_0102.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" width=\"975\" height=\"445\" \/><\/p>\n<p class=\"wp-caption-text\">(a) [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] stretches the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of 3. (b) [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] compresses the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]\\frac{1}{3}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Stretches and Compressions of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>The function [latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\n<ul>\n<li>is stretched vertically by a factor of <em>a\u00a0<\/em>if [latex]|a|>1[\/latex].<\/li>\n<li>is compressed vertically by a factor of <em>a<\/em>\u00a0if [latex]|a|<1[\/latex].<\/li>\n<li>has a\u00a0<em>y<\/em>-intercept is [latex]\\left(0,a\\right)[\/latex].<\/li>\n<li>has a horizontal asymptote of [latex]y=0[\/latex], range of [latex]\\left(0,\\infty \\right)[\/latex], and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are all unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing the Stretch of an Exponential Function<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{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=\"q418729\">Show Solution<\/span><\/p>\n<div id=\"q418729\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before graphing, identify the behavior and key points on the graph.<\/p>\n<ul>\n<li>Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as <em>x<\/em>\u00a0decreases, and the right tail will approach the <em>x<\/em>-axis as <em>x<\/em>\u00a0increases.<\/li>\n<li>Since <em>a\u00a0<\/em>= 4, the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched vertically by a factor of 4.<\/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)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\n<td>32<\/td>\n<td>16<\/td>\n<td>8<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>0.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y-<\/em>intercept, [latex]\\left(0,4\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\n<li>Draw a smooth curve connecting the points.<\/li>\n<\/ul>\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\/02231155\/CNX_Precalc_Figure_04_02_0112.jpg\" alt=\"Graph of the function, f(x) = 4(1\/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).\" width=\"487\" height=\"482\" \/><\/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 y\u00a0= 0.<\/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)=\\frac{1}{2}{\\left(4\\right)}^{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=\"q796634\">Show Solution<\/span><\/p>\n<div id=\"q796634\" 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].\u00a0<span id=\"fs-id1165135417835\"><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3081 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16190801\/CNX_Precalc_Figure_04_02_0122.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"488\" height=\"294\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129498&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"550\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<h3>Graphing Reflections<\/h3>\n<p>In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by \u20131, we get a reflection about the <em>x<\/em>-axis. When we multiply the input by \u20131, we get a reflection about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], and the reflection about the <em>y<\/em>-axis, [latex]h\\left(x\\right)={2}^{-x}[\/latex], are both shown below.<\/p>\n<div style=\"width: 985px\" 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\/02231158\/CNX_Precalc_Figure_04_02_0132.jpg\" alt=\"Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.\" width=\"975\" height=\"628\" \/><\/p>\n<p class=\"wp-caption-text\">(a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]h\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.<\/p>\n<\/div>\n<figure id=\"CNX_Precalc_Figure_04_02_013\"><figcaption><\/figcaption><\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Reflecting the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n<p>The function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>x<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-intercept of [latex]\\left(0,-1\\right)[\/latex].<\/li>\n<li>has a range of [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\n<li>has a horizontal asymptote of [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>The function [latex]f\\left(x\\right)={b}^{-x}[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the <em>y<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-intercept of [latex]\\left(0,1\\right)[\/latex], a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing and Graphing the Reflection of an Exponential Function<\/h3>\n<p>Find and graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q91748\">Show Solution<\/span><\/p>\n<div id=\"q91748\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since we want to reflect the parent function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the <em>x-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by \u20131 to get [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\n<td>\u201364<\/td>\n<td>\u201316<\/td>\n<td>\u20134<\/td>\n<td>\u20131<\/td>\n<td>\u20130.25<\/td>\n<td>\u20130.0625<\/td>\n<td>\u20130.0156<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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].<\/p>\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\/02231202\/CNX_Precalc_Figure_04_02_0142.jpg\" alt=\"Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).\" width=\"487\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,0\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summarizing Transformations of the Exponential Function<\/h2>\n<p>Now that we have worked with each type of translation for the exponential function, we can summarize them\u00a0to arrive at the general equation for transforming exponential functions.<\/p>\n<table summary=\"Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Transformations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\">Transformation<\/th>\n<th style=\"text-align: center;\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul>\n<li>Horizontally <em>c<\/em>\u00a0units to the left<\/li>\n<li>Vertically <em>d<\/em>\u00a0units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compression<\/p>\n<ul>\n<li>Stretch if |<em>a<\/em>|&gt;1<\/li>\n<li>Compression if 0&lt;|<em>a<\/em>|&lt;1<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflection about the <em>x<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflection about the <em>y<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all transformations<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A General Note: Transformations of Exponential Functions<\/h3>\n<p>A transformation of an exponential function has the form<\/p>\n<p>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], where the parent function, [latex]y={b}^{x}[\/latex], [latex]b>1[\/latex], is<\/p>\n<ul>\n<li>shifted horizontally <em>c<\/em>\u00a0units to the left.<\/li>\n<li>stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a| > 1[\/latex].<\/li>\n<li>compressed vertically by a factor of [latex]|a|[\/latex]\u00a0if [latex]0 < |a| < 1[\/latex].<\/li>\n<li>shifted vertically <em>d<\/em>\u00a0units.<\/li>\n<li>reflected about the <em>x-<\/em>axis when <em>a\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<p>Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Function from a Description<\/h3>\n<p>Write the equation for the function described below. Give the horizontal asymptote, domain, and range.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of 2, reflected across the <em>y<\/em>-axis, and then shifted up 4\u00a0units.<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q290621\">Show Solution<\/span><\/p>\n<div id=\"q290621\" class=\"hidden-answer\" style=\"display: none\">\n<p>We want to find an equation of the general form [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]. We use the description provided to find <em>a<\/em>, <em>b<\/em>, <em>c<\/em>, and <em>d<\/em>.<\/p>\n<ul>\n<li>We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so <em>b\u00a0<\/em>= <em>e<\/em>.<\/li>\n<li>The function is stretched by a factor of 2, so <em>a\u00a0<\/em>= 2.<\/li>\n<li>The function is reflected about the <em>y<\/em>-axis. We replace <em>x<\/em>\u00a0with \u2013<em>x<\/em>\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\n<li>The graph is shifted vertically 4 units, so <em>d\u00a0<\/em>= 4.<\/li>\n<\/ul>\n<p>Substituting in the general form, we get:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llll}f\\left(x\\right)\\hfill & =a{b}^{x+c}+d\\hfill \\\\ \\hfill & =2{e}^{-x+0}+4\\hfill \\\\ \\hfill & =2{e}^{-x}+4\\hfill \\end{array}[\/latex]<\/p>\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(4,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the <em>x<\/em>-axis, and then shifted down 2\u00a0units.<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525289\">Show Solution<\/span><\/p>\n<div id=\"q525289\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2[\/latex]; the domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,2\\right)[\/latex]; the horizontal asymptote is [latex]y=2[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Using a Graph to Approximate a Solution to an Exponential Equation<\/h2>\n<p>Graphing can help you confirm or find the solution to an exponential equation.\u00a0For example,[latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] can be solved to find the specific value for x that makes it a true statement. Graphing [latex]y=4[\/latex] along with [latex]y=2^{x}[\/latex] in the same window, the point(s) of intersection if any represent the solutions of the equation.<\/p>\n<p>To use a calculator to solve this, press <strong>[Y=]<\/strong> and enter [latex]1.2(5)x+2.8[\/latex] next to <strong>Y1=<\/strong>. Then enter 42 next to <strong>Y2=<\/strong>. For a window, use the values \u20133 to 3 for[latex]x[\/latex] and \u20135 to 55 for[latex]y[\/latex].Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near[latex]x=2[\/latex].<\/p>\n<p>For a better approximation, press <strong>[2ND]<\/strong> then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The x-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth,x\u22482.166.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]4=7.85{\\left(1.15\\right)}^{x}-2.27[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q407425\">Show Solution<\/span><\/p>\n<div id=\"q407425\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x\\approx -1.608[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<table summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Transformation of the Parent Function [latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The graph of the function [latex]f\\left(x\\right)={b}^{x}[\/latex] has a <em>y-<\/em>intercept at [latex]\\left(0, 1\\right)[\/latex], domain of [latex]\\left(-\\infty , \\infty \\right)[\/latex], range of [latex]\\left(0, \\infty \\right)[\/latex], and horizontal asymptote of [latex]y=0[\/latex].<\/li>\n<li>If [latex]b>1[\/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[\/latex], and the right tail will increase without bound.<\/li>\n<li>If 0 &lt;\u00a0<em>b<\/em> &lt; 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y=0[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x}+d[\/latex] represents a vertical shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)={b}^{x+c}[\/latex] represents a horizontal shift of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>The equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], where [latex]a>0[\/latex], represents a vertical stretch if [latex]|a|>1[\/latex] or compression if [latex]0<|a|<1[\/latex] of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex].<\/li>\n<li>When the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] is multiplied by \u20131, the result, [latex]f\\left(x\\right)=-{b}^{x}[\/latex], is a reflection about the <em>x<\/em>-axis. When the input is multiplied by \u20131, the result, [latex]f\\left(x\\right)={b}^{-x}[\/latex], is a reflection about the <em>y<\/em>-axis.<\/li>\n<li>All transformations of the exponential function can be summarized by the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex].<\/li>\n<li>Using the general equation [latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex], we can write the equation of a function given its description.<\/li>\n<li>Approximate solutions of the equation [latex]f\\left(x\\right)={b}^{x+c}+d[\/latex] can be found using a graphing calculator.<\/li>\n<\/ul>\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-1991\">\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><\/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><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>Question ID 63064. <strong>Authored by<\/strong>: Brin, Leon. <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>Question ID 129498. <strong>Authored by<\/strong>: Day, Alyson. <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><\/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":4,"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\":\"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\":\"cc\",\"description\":\"Question ID 63064\",\"author\":\"Brin, Leon\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 129498\",\"author\":\"Day, Alyson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1991","chapter","type-chapter","status-publish","hentry"],"part":1964,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/1991","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":20,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/1991\/revisions"}],"predecessor-version":[{"id":4862,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/1991\/revisions\/4862"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/parts\/1964"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/1991\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/media?parent=1991"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapter-type?post=1991"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/contributor?post=1991"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/license?post=1991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}