{"id":2059,"date":"2016-11-02T23:47:58","date_gmt":"2016-11-02T23:47:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2059"},"modified":"2017-07-07T17:14:41","modified_gmt":"2017-07-07T17:14:41","slug":"transformations-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/transformations-of-logarithmic-functions\/","title":{"raw":"Horizontal and Vertical Shifts of Logarithmic Functions","rendered":"Horizontal and Vertical Shifts of Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Graph horizontal and vertical shifts of logarithmic functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the <strong>parent function<\/strong> [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.\r\n<h2>Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\r\nWhen a constant <em>c<\/em>\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)[\/latex], the result is a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and for <em>c\u00a0<\/em>&gt; 0 alongside the shift left, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nIn the graphs below, you can use the slider for the variable <em>c<\/em> to investigate horizontal shifts that are produced by either adding or subtracting a constant from the input\u00a0of a logarithmic function. The function [latex]f(x) = \\log_{b}{x+c}[\/latex] represents adding c units to the input of the function, and the function [latex]g(x) = \\log_{b}{x-c}[\/latex] represents subtracting c units from the input of the function.\r\n\r\nInvestigate the following questions:\r\n<ul>\r\n \t<li>Adjust the slider for c to 4.<\/li>\r\n \t<li>Which direction does the graph of [latex]f(x)[\/latex]shift? What is the vertical asymptote, x-intercept, and equation for this new function? How do the domain and range change?<\/li>\r\n \t<li>Which direction does the graph of [latex]g(x)[\/latex] shift? What is the vertical asymptote, x-intercept, and equation for this new function? How do the domain and range change?<\/li>\r\n<\/ul>\r\nhttps:\/\/www.desmos.com\/calculator\/6drjq3bh0m\r\n\r\n<\/div>\r\nThe still graphs below summarize the changes in the x-intercepts, vertical asymptotes, and equations\u00a0of a logarithmic function that has been shifted either right or left.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233827\/CNX_Precalc_Figure_04_04_007n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"900\" height=\"526\" data-media-type=\"image\/jpg\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal Shifts of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex]\r\n<ul>\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\r\n \t<li>has the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\r\n \t<li>has domain [latex]\\left(-c,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the translation.<\/h3>\r\n<ol>\r\n \t<li>Identify the horizontal shift:\r\n<ol>\r\n \t<li>If <em>c<\/em> &gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units.<\/li>\r\n \t<li>If <em>c\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>\u00a0from the\u00a0<em>x<\/em>\u00a0coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The Domain is [latex]\\left(-c,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u2013c.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Graphing a Horizontal Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nSketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"368750\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"368750\"]\r\n\r\nSince the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2[\/latex].\r\n\r\nThus <em>c\u00a0<\/em>= \u20132, so <em>c\u00a0<\/em>&lt; 0. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right 2 units.\r\n\r\nThe vertical asymptote is [latex]x=-\\left(-2\\right)[\/latex] or <em>x\u00a0<\/em>= 2.\r\n\r\nConsider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].\r\n\r\nThe new coordinates are found by adding 2 to the <em>x<\/em>\u00a0coordinates.\r\n\r\nLabel the points [latex]\\left(\\frac{7}{3},-1\\right)[\/latex], [latex]\\left(3,0\\right)[\/latex], and [latex]\\left(5,1\\right)[\/latex].\r\n\r\nThe domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233829\/CNX_Precalc_Figure_04_04_0082.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).\" width=\"487\" height=\"363\" data-media-type=\"image\/jpg\" \/>\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\nSketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"779370\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"779370\"]\r\n\r\nThe domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x\u00a0<\/em>= \u20134.<span id=\"fs-id1165135209395\" data-type=\"media\" data-alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\">\r\n<\/span>\r\n\r\n<img class=\"aligncenter size-full wp-image-3106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16230941\/CNX_Precalc_Figure_04_04_0092.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\" width=\"487\" height=\"363\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom11\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74340&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"250\">\r\n<\/iframe>\r\n\r\n<\/div>\r\n<h2>Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h2>\r\nWhen a constant <em>d<\/em>\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], the result is a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02233831\/CNX_Precalc_Figure_04_04_010F2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"900\" height=\"684\" data-media-type=\"image\/jpg\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Vertical Shifts of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex]\r\n<ul>\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\r\n \t<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the translation.<\/h3>\r\n<ol>\r\n \t<li>Identify the vertical shift:\r\n<ol>\r\n \t<li>If <em>d\u00a0<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units.<\/li>\r\n \t<li>If <em>d\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d\u00a0<\/em>units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>\u00a0to the <em>y\u00a0<\/em>coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Vertical Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nSketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"43912\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"43912\"]\r\n\r\nSince the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex], we will notice <em>d\u00a0<\/em>= \u20132. Thus <em>d\u00a0<\/em>&lt; 0.\r\n\r\nThis means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.\r\n\r\nThe vertical asymptote is <em>x\u00a0<\/em>= 0.\r\n\r\nConsider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].\r\n\r\nThe new coordinates are found by subtracting 2 from the <em data-effect=\"italics\">y <\/em>coordinates.\r\n\r\nLabel the points [latex]\\left(\\frac{1}{3},-3\\right)[\/latex], [latex]\\left(1,-2\\right)[\/latex], and [latex]\\left(3,-1\\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\/02233834\/CNX_Precalc_Figure_04_04_0112.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).\" width=\"487\" height=\"516\" data-media-type=\"image\/jpg\" \/> The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x = 0.[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n[reveal-answer q=\"338440\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"338440\"]The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165137874471\" data-type=\"media\" data-alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\">\r\n<\/span>\r\n\r\n<img class=\"aligncenter size-full wp-image-3109\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16231838\/CNX_Precalc_Figure_04_04_0122.jpg\" alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\" width=\"487\" height=\"474\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom12\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74341&amp;theme=oea&amp;iframe_resize_id=mom12\" width=\"100%\" height=\"300\">\r\n<\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Graph horizontal and vertical shifts of logarithmic functions<\/li>\n<\/ul>\n<\/div>\n<p>As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the <strong>parent function<\/strong> [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.<\/p>\n<h2>Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When a constant <em>c<\/em>\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)[\/latex], the result is a <strong>horizontal shift<\/strong> <em>c<\/em>\u00a0units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and for <em>c\u00a0<\/em>&gt; 0 alongside the shift left, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>In the graphs below, you can use the slider for the variable <em>c<\/em> to investigate horizontal shifts that are produced by either adding or subtracting a constant from the input\u00a0of a logarithmic function. The function [latex]f(x) = \\log_{b}{x+c}[\/latex] represents adding c units to the input of the function, and the function [latex]g(x) = \\log_{b}{x-c}[\/latex] represents subtracting c units from the input of the function.<\/p>\n<p>Investigate the following questions:<\/p>\n<ul>\n<li>Adjust the slider for c to 4.<\/li>\n<li>Which direction does the graph of [latex]f(x)[\/latex]shift? What is the vertical asymptote, x-intercept, and equation for this new function? How do the domain and range change?<\/li>\n<li>Which direction does the graph of [latex]g(x)[\/latex] shift? What is the vertical asymptote, x-intercept, and equation for this new function? How do the domain and range change?<\/li>\n<\/ul>\n<p>https:\/\/www.desmos.com\/calculator\/6drjq3bh0m<\/p>\n<\/div>\n<p>The still graphs below summarize the changes in the x-intercepts, vertical asymptotes, and equations\u00a0of a logarithmic function that has been shifted either right or left.<\/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\/02233827\/CNX_Precalc_Figure_04_04_007n2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.\" width=\"900\" height=\"526\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Horizontal Shifts of the Parent Function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>For any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex]<\/p>\n<ul>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n<li>has domain [latex]\\left(-c,\\infty \\right)[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the translation.<\/h3>\n<ol>\n<li>Identify the horizontal shift:\n<ol>\n<li>If <em>c<\/em> &gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>\u00a0units.<\/li>\n<li>If <em>c\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>\u00a0units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= \u2013<em>c<\/em>.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>\u00a0from the\u00a0<em>x<\/em>\u00a0coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The Domain is [latex]\\left(-c,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= \u2013c.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Graphing a Horizontal Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>Sketch the horizontal shift [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. 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=\"q368750\">Solution<\/span><\/p>\n<div id=\"q368750\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2[\/latex].<\/p>\n<p>Thus <em>c\u00a0<\/em>= \u20132, so <em>c\u00a0<\/em>&lt; 0. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right 2 units.<\/p>\n<p>The vertical asymptote is [latex]x=-\\left(-2\\right)[\/latex] or <em>x\u00a0<\/em>= 2.<\/p>\n<p>Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/p>\n<p>The new coordinates are found by adding 2 to the <em>x<\/em>\u00a0coordinates.<\/p>\n<p>Label the points [latex]\\left(\\frac{7}{3},-1\\right)[\/latex], [latex]\\left(3,0\\right)[\/latex], and [latex]\\left(5,1\\right)[\/latex].<\/p>\n<p>The domain is [latex]\\left(2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 2.<\/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\/02233829\/CNX_Precalc_Figure_04_04_0082.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).\" width=\"487\" height=\"363\" data-media-type=\"image\/jpg\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. 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=\"q779370\">Solution<\/span><\/p>\n<div id=\"q779370\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x\u00a0<\/em>= \u20134.<span id=\"fs-id1165135209395\" data-type=\"media\" data-alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\"><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16230941\/CNX_Precalc_Figure_04_04_0092.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).\" width=\"487\" height=\"363\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom11\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74340&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<h2>Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When a constant <em>d<\/em>\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], the result is a <strong>vertical shift<\/strong> <em>d<\/em>\u00a0units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d[\/latex].<\/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\/02233831\/CNX_Precalc_Figure_04_04_010F2.jpg\" alt=\"Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.\" width=\"900\" height=\"684\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Shifts of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>For any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex]<\/p>\n<ul>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n<li>has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a logarithmic function with the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the translation.<\/h3>\n<ol>\n<li>Identify the vertical shift:\n<ol>\n<li>If <em>d\u00a0<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>\u00a0units.<\/li>\n<li>If <em>d\u00a0<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d\u00a0<\/em>units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote <em>x\u00a0<\/em>= 0.<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>\u00a0to the <em>y\u00a0<\/em>coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Vertical Shift of the Parent Function\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q43912\">Solution<\/span><\/p>\n<div id=\"q43912\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex], we will notice <em>d\u00a0<\/em>= \u20132. Thus <em>d\u00a0<\/em>&lt; 0.<\/p>\n<p>This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.<\/p>\n<p>The vertical asymptote is <em>x\u00a0<\/em>= 0.<\/p>\n<p>Consider the three key points from the parent function, [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/p>\n<p>The new coordinates are found by subtracting 2 from the <em data-effect=\"italics\">y <\/em>coordinates.<\/p>\n<p>Label the points [latex]\\left(\\frac{1}{3},-3\\right)[\/latex], [latex]\\left(1,-2\\right)[\/latex], and [latex]\\left(3,-1\\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\/02233834\/CNX_Precalc_Figure_04_04_0112.jpg\" alt=\"Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1\/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).\" width=\"487\" height=\"516\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\">The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x = 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)+2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. 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=\"q338440\">Solution<\/span><\/p>\n<div id=\"q338440\" class=\"hidden-answer\" style=\"display: none\">The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is <em>x\u00a0<\/em>= 0.<span id=\"fs-id1165137874471\" data-type=\"media\" data-alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\"><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3109\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16231838\/CNX_Precalc_Figure_04_04_0122.jpg\" alt=\"Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).\" width=\"487\" height=\"474\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom12\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=74341&amp;theme=oea&amp;iframe_resize_id=mom12\" width=\"100%\" height=\"300\"><br \/>\n<\/iframe><\/p>\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-2059\">\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 Shifts of Logarithmic Functions Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/6drjq3bh0m\">https:\/\/www.desmos.com\/calculator\/6drjq3bh0m<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 74340, 74341. <strong>Authored by<\/strong>: Nearing,Daniel, mb Meacham,William, mb Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY +GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 74340, 74341\",\"author\":\"Nearing,Daniel, mb Meacham,William, mb 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