{"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":"2025-10-13T20:51:09","modified_gmt":"2025-10-13T20:51:09","slug":"horizontal-and-vertical-translations-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/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 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<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<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\">\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\nSketch a graph of the function [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\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<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>","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<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<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\">\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>Sketch a graph of the function [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>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<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\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><\/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 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