{"id":263,"date":"2023-06-21T13:22:49","date_gmt":"2023-06-21T13:22:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/horizontal-and-vertical-translations-of-exponential-functions\/"},"modified":"2023-10-20T01:06:42","modified_gmt":"2023-10-20T01:06:42","slug":"horizontal-and-vertical-translations-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/horizontal-and-vertical-translations-of-exponential-functions\/","title":{"raw":"\u25aa   Horizontal and Vertical Shift of Exponential Functions","rendered":"\u25aa   Horizontal and Vertical Shift of Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/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<div class=\"textbox examples\">\r\n<h3>Tip for success<\/h3>\r\nTranslating exponential functions follows the same ideas you've used to translate other functions. Add or subtract a value inside the function argument (in the exponent) to shift horizontally, and add or subtract a value outside the function argument to shift vertically.\r\n\r\n<\/div>\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\" \/>\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<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n<ol>\r\n \t<li>Use an online graphing calculator to plot\u00a0[latex]f(x) = 2^x+a[\/latex]<\/li>\r\n \t<li>Adjust the value of [latex]a[\/latex] until the graph has been shifted 4 units up.<\/li>\r\n \t<li>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 its 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 its domain and range?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"619964\"]Show Solution[\/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 \t<li>Verify above answers using this Desmos activity:\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\/tbtmwfoflm\" target=\"_blank\" rel=\"noopener\">Vertical Shift of Exponential Functions<\/a><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\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 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 key-takeaways\">\r\n<h3>try it<\/h3>\r\n<ol>\r\n \t<li>Using an online graphing calculator, plot\u00a0[latex]f(x) = 2^{(x+a)}[\/latex]<\/li>\r\n \t<li>Adjust the value of [latex]a[\/latex] until the graph is shifted 4 units to the right. What is the equation for this function? What is the new y-intercept? What are its domain and range?<\/li>\r\n \t<li>Now adjust the value of [latex]a[\/latex] until the graph has been shifted 3 units to the left. What is the equation for this function? What is the new y-intercept? What are its domain and range?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"94739\"]Show Solution[\/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}{16}), [\/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 \t<li>Verify above answers using this Desmos activity:\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\/rlvklokhbk\" target=\"_blank\" rel=\"noopener\">Horizontal Shift of Exponential Functions<\/a><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse an online graphing calculator to plot the function\u00a0[latex]f\\left(x\\right)={2}^{x-1}+3[\/latex]. State domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"699634\"]Show 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], and 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\nWatch the following video for 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<h2>Using a Graph to Approximate a Solution to an Exponential Equation<\/h2>\r\nAs we discussed in the previous section, 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]. However, [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], which is [latex](2, 4)[\/latex]. You can\u00a0see they cross at [latex]y=4[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-1451\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation.png\" alt=\"Graphs of y=2^x and y=4\" width=\"337\" height=\"302\" \/>\r\n\r\nIn the following example you can try this yourself.\r\n<div class=\"textbox exercises\">\r\n<h3>Example : Approximating the Solution of an Exponential Equation<\/h3>\r\nUse an online graphing calculator to solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically.\r\n\r\n[reveal-answer q=\"89148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"89148\"]\r\n\r\n[Case 1] Using online graphing calculators\r\n\r\nFirst, graph the function [latex]f(x)=1.2{\\left(5\\right)}^{x}+2.8[\/latex] and graph [latex]f(x) = 42[\/latex].\r\n\r\nOnline graphing calculators automatically calculate points of interest including intersections. Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the x and y values for the point.\r\n\r\n[Case 2] Using TI-84 CE\r\n\r\nFirst, graph the function\u00a0[latex]f(x)=1.2{\\left(5\\right)}^{x}+2.8[\/latex] and graph [latex]f(x) = 42[\/latex] by typing in [latex]Y1=1.2{\\left(5\\right)}^{x}+2.8[\/latex] and [latex]Y2= 42[\/latex], respectively.\r\n\r\nSecond, click \"calc (2nd+trace)\" and then \"5: intersect.\" Now TI-84 will ask you the \"first curve\" and \"second curve.\" Since TI-84 chooses the first and second curve automatically, just click \"enter.\"\r\n\r\nFinally, we can find the solution [latex]x\\approx. 2.166[\/latex], which is the x value of the intersecting point.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 258px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 129px;\">\r\n<td style=\"width: 170.969px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1452\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-1-Y-.png\" alt=\"TI-84 Y=\" width=\"260\" height=\"192\" \/><\/td>\r\n<td style=\"width: 170.969px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1453\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-2-Graph.png\" alt=\"Graphs of y=1.2(5)^x+2.8 and y=42\" width=\"261\" height=\"197\" \/><\/td>\r\n<td style=\"width: 170.969px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1454\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-3-intersect.png\" alt=\"5:intersect under CALCULATE\" width=\"260\" height=\"197\" \/><\/td>\r\n<td style=\"width: 171px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1455\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-4-First-Curve.png\" alt=\"First Curve\" width=\"262\" height=\"194\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 129px;\">\r\n<td style=\"width: 170.969px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1456\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-5-Second-Curve.png\" alt=\"Second Curve\" width=\"259\" height=\"195\" \/><\/td>\r\n<td style=\"width: 170.969px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1457\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-6-Guess.png\" alt=\"Guess\" width=\"261\" height=\"197\" \/><\/td>\r\n<td style=\"width: 170.969px; height: 129px;\"><img class=\"aligncenter size-full wp-image-1458\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-7-Intersection.png\" alt=\"Intersection\" width=\"260\" height=\"196\" \/><\/td>\r\n<td style=\"width: 171px; height: 129px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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\"]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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/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<div class=\"textbox examples\">\n<h3>Tip for success<\/h3>\n<p>Translating exponential functions follows the same ideas you&#8217;ve used to translate other functions. Add or subtract a value inside the function argument (in the exponent) to shift horizontally, and add or subtract a value outside the function argument to shift vertically.<\/p>\n<\/div>\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\" \/><\/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<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<ol>\n<li>Use an online graphing calculator to plot\u00a0[latex]f(x) = 2^x+a[\/latex]<\/li>\n<li>Adjust the value of [latex]a[\/latex] until the graph has been shifted 4 units up.<\/li>\n<li>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 its 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 its domain and range?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q619964\">Show Solution<\/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<li>Verify above answers using this Desmos activity:\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\/tbtmwfoflm\" target=\"_blank\" rel=\"noopener\">Vertical Shift of Exponential Functions<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\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 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 key-takeaways\">\n<h3>try it<\/h3>\n<ol>\n<li>Using an online graphing calculator, plot\u00a0[latex]f(x) = 2^{(x+a)}[\/latex]<\/li>\n<li>Adjust the value of [latex]a[\/latex] until the graph is shifted 4 units to the right. What is the equation for this function? What is the new y-intercept? What are its domain and range?<\/li>\n<li>Now adjust the value of [latex]a[\/latex] until the graph has been shifted 3 units to the left. What is the equation for this function? What is the new y-intercept? What are its domain and range?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q94739\">Show Solution<\/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}{16}),[\/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<li>Verify above answers using this Desmos activity:\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\/rlvklokhbk\" target=\"_blank\" rel=\"noopener\">Horizontal Shift of Exponential Functions<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use an online graphing calculator to plot the function\u00a0[latex]f\\left(x\\right)={2}^{x-1}+3[\/latex]. State domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q699634\">Show 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], and 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>Watch the following video for 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<h2>Using a Graph to Approximate a Solution to an Exponential Equation<\/h2>\n<p>As we discussed in the previous section, 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]. However, [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], which is [latex](2, 4)[\/latex]. You can\u00a0see they cross at [latex]y=4[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1451\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation.png\" alt=\"Graphs of y=2^x and y=4\" width=\"337\" height=\"302\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation.png 405w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation-300x269.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation-65x58.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation-225x202.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Using-Graph-to-Approximate-a-Solution-to-an-Exponential-Equation-350x314.png 350w\" sizes=\"auto, (max-width: 337px) 100vw, 337px\" \/><\/p>\n<p>In the following example you can try this yourself.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example : Approximating the Solution of an Exponential Equation<\/h3>\n<p>Use an online graphing calculator to solve [latex]42=1.2{\\left(5\\right)}^{x}+2.8[\/latex] graphically.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q89148\">Show Solution<\/span><\/p>\n<div id=\"q89148\" class=\"hidden-answer\" style=\"display: none\">\n<p>[Case 1] Using online graphing calculators<\/p>\n<p>First, graph the function [latex]f(x)=1.2{\\left(5\\right)}^{x}+2.8[\/latex] and graph [latex]f(x) = 42[\/latex].<\/p>\n<p>Online graphing calculators automatically calculate points of interest including intersections. Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the x and y values for the point.<\/p>\n<p>[Case 2] Using TI-84 CE<\/p>\n<p>First, graph the function\u00a0[latex]f(x)=1.2{\\left(5\\right)}^{x}+2.8[\/latex] and graph [latex]f(x) = 42[\/latex] by typing in [latex]Y1=1.2{\\left(5\\right)}^{x}+2.8[\/latex] and [latex]Y2= 42[\/latex], respectively.<\/p>\n<p>Second, click &#8220;calc (2nd+trace)&#8221; and then &#8220;5: intersect.&#8221; Now TI-84 will ask you the &#8220;first curve&#8221; and &#8220;second curve.&#8221; Since TI-84 chooses the first and second curve automatically, just click &#8220;enter.&#8221;<\/p>\n<p>Finally, we can find the solution [latex]x\\approx. 2.166[\/latex], which is the x value of the intersecting point.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 258px;\">\n<tbody>\n<tr style=\"height: 129px;\">\n<td style=\"width: 170.969px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1452\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-1-Y-.png\" alt=\"TI-84 Y=\" width=\"260\" height=\"192\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-1-Y-.png 260w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-1-Y--65x48.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-1-Y--225x166.png 225w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/td>\n<td style=\"width: 170.969px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1453\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-2-Graph.png\" alt=\"Graphs of y=1.2(5)^x+2.8 and y=42\" width=\"261\" height=\"197\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-2-Graph.png 261w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-2-Graph-65x49.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-2-Graph-225x170.png 225w\" sizes=\"auto, (max-width: 261px) 100vw, 261px\" \/><\/td>\n<td style=\"width: 170.969px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1454\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-3-intersect.png\" alt=\"5:intersect under CALCULATE\" width=\"260\" height=\"197\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-3-intersect.png 260w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-3-intersect-65x49.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-3-intersect-225x170.png 225w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/td>\n<td style=\"width: 171px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1455\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-4-First-Curve.png\" alt=\"First Curve\" width=\"262\" height=\"194\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-4-First-Curve.png 262w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-4-First-Curve-65x48.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-4-First-Curve-225x167.png 225w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/td>\n<\/tr>\n<tr style=\"height: 129px;\">\n<td style=\"width: 170.969px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1456\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-5-Second-Curve.png\" alt=\"Second Curve\" width=\"259\" height=\"195\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-5-Second-Curve.png 259w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-5-Second-Curve-65x49.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-5-Second-Curve-225x169.png 225w\" sizes=\"auto, (max-width: 259px) 100vw, 259px\" \/><\/td>\n<td style=\"width: 170.969px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1457\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-6-Guess.png\" alt=\"Guess\" width=\"261\" height=\"197\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-6-Guess.png 261w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-6-Guess-65x49.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-6-Guess-225x170.png 225w\" sizes=\"auto, (max-width: 261px) 100vw, 261px\" \/><\/td>\n<td style=\"width: 170.969px; height: 129px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1458\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-7-Intersection.png\" alt=\"Intersection\" width=\"260\" height=\"196\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-7-Intersection.png 260w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-7-Intersection-65x49.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/06\/7.4-Example_Approximating-the-Solution-of-an-Exponential-Equation-7-Intersection-225x170.png 225w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/td>\n<td style=\"width: 171px; height: 129px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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\">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\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-263\">\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>Authored by<\/strong>: Michelle Chung. <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":395986,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Michelle Chung\",\"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 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