{"id":2008,"date":"2016-11-02T23:19:56","date_gmt":"2016-11-02T23:19:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2008"},"modified":"2017-07-07T17:09:51","modified_gmt":"2017-07-07T17:09:51","slug":"horizontal-and-vertical-translations-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/horizontal-and-vertical-translations-of-exponential-functions\/","title":{"raw":"Horizontal and Vertical Translations of Exponential Functions","rendered":"Horizontal and Vertical Translations of Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Graph exponential functions shifted horizontally or vertically and write the associated equation<\/li>\r\n<\/ul>\r\n<\/div>\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<h2>Graphing a Vertical Shift<\/h2>\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\" data-media-type=\"image\/jpg\" \/>\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 to [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 to [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<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n<ol>\r\n \t<li>Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[\/latex] that has been shifted 4 units up. Add a line that represents the horizontal asymptote for this function. What is the equation for this function? What is the new y-intercept? What is it's domain and range?<\/li>\r\n \t<li>Now create a graph of the function [latex]f(x) = 2^x[\/latex] that has been shifted down 2 units. Add a line that represents the horizontal asymptote.What is the equation for this function? What is the new y-intercept? What is it's domain and range?<\/li>\r\n<\/ol>\r\nhttps:\/\/www.desmos.com\/calculator\/5mrjqegkxk\r\n\r\n[reveal-answer q=\"619964\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"619964\"]\r\n<ol>\r\n \t<li>Equation: [latex]f(x) = 2^x+4[\/latex], Horizontal Asymptote: [latex]y = 4[\/latex], y-intercept: [latex](0,5)[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](4,\\infty)[\/latex]<\/li>\r\n \t<li>Equation: [latex]f(x) = 2^x-2[\/latex], Horizontal Asymptote: [latex]y = -2[\/latex], y-intercept: [latex](0,-1)[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](-2,\\infty)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Graphing a Horizontal Shift<\/h2>\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 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\/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\" data-media-type=\"image\/jpg\" \/>\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(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(\\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 key-takeaways\">\r\n<h3>try it<\/h3>\r\n<ol>\r\n \t<li>Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[\/latex] that has been shifted 4 units to the right. What is the equation for this function? What is the new y-intercept? What are it's domain and range?<\/li>\r\n \t<li>Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[\/latex] that has been shifted 3 units to the left. What is the equation for this function? What is the new y-intercept? What are it's domain and range?<\/li>\r\n<\/ol>\r\nhttps:\/\/www.desmos.com\/calculator\/rpv1kea0pz\r\n\r\n[reveal-answer q=\"94739\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"94739\"]\r\n<ol>\r\n \t<li>Equation: [latex]f(x) = 2^{x-4}[\/latex], y-intercept: [latex](0,\\frac{1}{32})[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](0,\\infty)[\/latex]<\/li>\r\n \t<li>Equation: [latex]f(x) = 2^{x+3}[\/latex], y-intercept: [latex](0,8)[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](0,\\infty)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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>vertically <em>d<\/em>\u00a0units, in the <em>same<\/em> direction of the sign of <em>d<\/em>.<\/li>\r\n \t<li>horizontally <em>c<\/em>\u00a0units, in the <em>opposite<\/em> direction of the sign of <em>c<\/em>.<\/li>\r\n \t<li>The <em>y<\/em>-intercept becomes [latex]\\left(0,{b}^{c}+d\\right)[\/latex].<\/li>\r\n \t<li>The horizontal asymptote becomes <em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\r\n \t<li>The range becomes [latex]\\left(d,\\infty \\right)[\/latex].<\/li>\r\n \t<li>The domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], 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>Identify the shift as [latex]\\left(-c,d\\right)[\/latex]. 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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"344344\"]\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 as [latex]\\left(-c,d\\right)[\/latex], so the shift 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\" data-media-type=\"image\/jpg\" \/> 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].[\/caption]Shift the graph of [latex]f\\left(x\\right)={b}^{x}[\/latex] left 1 units and down 3 units.<span id=\"fs-id1165137591826\" data-type=\"media\" data-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).\">\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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the sliders in the graph below to create a graph of the function [latex]f\\left(x\\right)={2}^{x - 1}+3[\/latex]. State domain, range, and asymptote.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/e5l4eca3ob\r\n\r\n[reveal-answer q=\"699634\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"699634\"]\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(3,\\infty \\right)[\/latex]; the horizontal asymptote is <em>y\u00a0<\/em>= 3.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/05004257\/CNX_Precalc_Figure_04_02_0092.jpg\"><img class=\"size-full wp-image-3016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/05004257\/CNX_Precalc_Figure_04_02_0092.jpg\" alt=\"\" width=\"487\" height=\"490\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of the difference between horizontal and vertical translations of exponential functions and the resultant graphs and equations.\r\n\r\nhttps:\/\/youtu.be\/phYxEeJ7ZW4\r\n<h2>Use 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. An exponential equation is different from a function because a function is a large collection of points made of inputs and corresponding outputs, whereas equations that you have seen typically have one, two, or no solutions. \u00a0For example, [latex]f(x)=2^{x}[\/latex] is a function and is comprised of many points [latex](x,f(x))[\/latex], and [latex]4=2^{x}[\/latex] can be solved to find the specific value for x that makes it a true statement. The\u00a0graph below shows the intersection of the line [latex]f(x)=4[\/latex], and [latex]f(x)=2^{x}[\/latex], you can\u00a0see they cross at [latex]y=4[\/latex].\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/lhmpdkbjt0\r\n\r\n&nbsp;\r\n\r\nIn the next example, you can try this for yourself.\r\n<div class=\"textbox exercises\">\r\n<h3>Example : Approximating the Solution of an Exponential Equation<\/h3>\r\nUse Desmos to solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically.\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\" rel=\"noopener\"><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\r\n[reveal-answer q=\"89148\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"89148\"]\r\n\r\nFirst, graph the function [latex]f(x)=1.2{\\left(5\\right)}^{x}+2.8[\/latex] , then add another function [latex]f(x) = 42[\/latex].\r\n\r\nDesmos automatically calculates points of interest \u00a0including intersections. \u00a0Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the the x and y values for the point.\r\n\r\nYour graph will look like this:\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/ozaejvejqn\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\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"407425\"][latex]x\\approx -1.608[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Graph exponential functions shifted horizontally or vertically and write the associated equation<\/li>\n<\/ul>\n<\/div>\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<h2>Graphing a Vertical Shift<\/h2>\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\" data-media-type=\"image\/jpg\" \/><\/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 to [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 to [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<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<ol>\n<li>Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[\/latex] that has been shifted 4 units up. Add a line that represents the horizontal asymptote for this function. What is the equation for this function? What is the new y-intercept? What is it&#8217;s domain and range?<\/li>\n<li>Now create a graph of the function [latex]f(x) = 2^x[\/latex] that has been shifted down 2 units. Add a line that represents the horizontal asymptote.What is the equation for this function? What is the new y-intercept? What is it&#8217;s domain and range?<\/li>\n<\/ol>\n<p>https:\/\/www.desmos.com\/calculator\/5mrjqegkxk<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q619964\">Show Answer<\/span><\/p>\n<div id=\"q619964\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Equation: [latex]f(x) = 2^x+4[\/latex], Horizontal Asymptote: [latex]y = 4[\/latex], y-intercept: [latex](0,5)[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](4,\\infty)[\/latex]<\/li>\n<li>Equation: [latex]f(x) = 2^x-2[\/latex], Horizontal Asymptote: [latex]y = -2[\/latex], y-intercept: [latex](0,-1)[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](-2,\\infty)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Graphing a Horizontal Shift<\/h2>\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 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\/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\" data-media-type=\"image\/jpg\" \/><\/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(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(\\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 key-takeaways\">\n<h3>try it<\/h3>\n<ol>\n<li>Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[\/latex] that has been shifted 4 units to the right. What is the equation for this function? What is the new y-intercept? What are it&#8217;s domain and range?<\/li>\n<li>Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[\/latex] that has been shifted 3 units to the left. What is the equation for this function? What is the new y-intercept? What are it&#8217;s domain and range?<\/li>\n<\/ol>\n<p>https:\/\/www.desmos.com\/calculator\/rpv1kea0pz<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q94739\">Show Answer<\/span><\/p>\n<div id=\"q94739\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Equation: [latex]f(x) = 2^{x-4}[\/latex], y-intercept: [latex](0,\\frac{1}{32})[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](0,\\infty)[\/latex]<\/li>\n<li>Equation: [latex]f(x) = 2^{x+3}[\/latex], y-intercept: [latex](0,8)[\/latex]Domain: [latex](-\\infty,\\infty)[\/latex], Range: [latex](0,\\infty)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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>vertically <em>d<\/em>\u00a0units, in the <em>same<\/em> direction of the sign of <em>d<\/em>.<\/li>\n<li>horizontally <em>c<\/em>\u00a0units, in the <em>opposite<\/em> direction of the sign of <em>c<\/em>.<\/li>\n<li>The <em>y<\/em>-intercept becomes [latex]\\left(0,{b}^{c}+d\\right)[\/latex].<\/li>\n<li>The horizontal asymptote becomes <em>y<\/em> =\u00a0<em>d<\/em>.<\/li>\n<li>The range becomes [latex]\\left(d,\\infty \\right)[\/latex].<\/li>\n<li>The domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], 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>Identify the shift as [latex]\\left(-c,d\\right)[\/latex]. 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\">Solution<\/span><\/p>\n<div id=\"q344344\" class=\"hidden-answer\" style=\"display: none\">\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].<\/p>\n<p>Draw the horizontal asymptote [latex]y=d[\/latex], so draw [latex]y=-3[\/latex].<\/p>\n<p>Identify the shift as [latex]\\left(-c,d\\right)[\/latex], so the shift 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\" 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(-3,\\infty \\right)[\/latex]; 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 units and down 3 units.<span id=\"fs-id1165137591826\" data-type=\"media\" data-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).\"><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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the sliders in the graph below to create a graph of the function [latex]f\\left(x\\right)={2}^{x - 1}+3[\/latex]. State domain, range, and asymptote.<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/e5l4eca3ob<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q699634\">Solution<\/span><\/p>\n<div id=\"q699634\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(3,\\infty \\right)[\/latex]; the horizontal asymptote is <em>y\u00a0<\/em>= 3.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/05004257\/CNX_Precalc_Figure_04_02_0092.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3016 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/05004257\/CNX_Precalc_Figure_04_02_0092.jpg\" alt=\"\" width=\"487\" height=\"490\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of the difference between horizontal and vertical translations of exponential functions and the resultant 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<h2>Use 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. An exponential equation is different from a function because a function is a large collection of points made of inputs and corresponding outputs, whereas equations that you have seen typically have one, two, or no solutions. \u00a0For example, [latex]f(x)=2^{x}[\/latex] is a function and is comprised of many points [latex](x,f(x))[\/latex], and [latex]4=2^{x}[\/latex] can be solved to find the specific value for x that makes it a true statement. The\u00a0graph below shows the intersection of the line [latex]f(x)=4[\/latex], and [latex]f(x)=2^{x}[\/latex], you can\u00a0see they cross at [latex]y=4[\/latex].<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/lhmpdkbjt0<\/p>\n<p>&nbsp;<\/p>\n<p>In the next example, you can try this for yourself.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example : Approximating the Solution of an Exponential Equation<\/h3>\n<p>Use Desmos to solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically.<br \/>\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\" rel=\"noopener\"><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=\"q89148\">Solution<\/span><\/p>\n<div id=\"q89148\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, graph the function [latex]f(x)=1.2{\\left(5\\right)}^{x}+2.8[\/latex] , then add another function [latex]f(x) = 42[\/latex].<\/p>\n<p>Desmos automatically calculates points of interest \u00a0including intersections. \u00a0Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the the x and y values for the point.<\/p>\n<p>Your graph will look like this:<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/ozaejvejqn<\/p>\n<\/div>\n<\/div>\n<\/div>\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\">Solution<\/span><\/p>\n<div id=\"q407425\" class=\"hidden-answer\" style=\"display: none\">[latex]x\\approx -1.608[\/latex]<\/div>\n<\/div>\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-2008\">\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>Horizontal and Vertical Translations of Exponential Functions Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/5mrjqegkxk\">https:\/\/www.desmos.com\/calculator\/5mrjqegkxk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Horizontal and Vertical Translations of Exponential Functions 2 Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/rpv1kea0pz\">https:\/\/www.desmos.com\/calculator\/rpv1kea0pz<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Horizontal and Vertical Translations of Exponential Functions 3 Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/e5l4eca3ob\">https:\/\/www.desmos.com\/calculator\/e5l4eca3ob<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Solve Exponential Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/lhmpdkbjt0\">https:\/\/www.desmos.com\/calculator\/lhmpdkbjt0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Horizontal and Vertical Translations of Exponential Functions 4 Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/ozaejvejqn\">https:\/\/www.desmos.com\/calculator\/ozaejvejqn<\/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>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>Ex: Match the Graphs of Translated Exponential Function to Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/phYxEeJ7ZW4\">https:\/\/youtu.be\/phYxEeJ7ZW4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\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":8,"template":"","meta":{"_candela_citation":"[{\"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\":\"Ex: Match the Graphs of Translated Exponential Function to Equations\",\"author\":\"James Sousa 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