{"id":344,"date":"2019-07-15T22:44:20","date_gmt":"2019-07-15T22:44:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/transformations-of-logarithmic-functions\/"},"modified":"2019-07-15T22:44:20","modified_gmt":"2019-07-15T22:44:20","slug":"transformations-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/transformations-of-logarithmic-functions\/","title":{"raw":"Horizontal and Vertical Shifts of Logarithmic Functions","rendered":"Horizontal and Vertical Shifts of Logarithmic Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Graph horizontal and vertical shifts of logarithmic functions.<\/li>\n<\/ul>\n<\/div>\nAs we mentioned in the beginning of the section, transformations of logarithmic functions behave similar 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.\n<h2>Graphing a Horizontal Shift of&nbsp;[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\nWhen a constant <em>c<\/em>&nbsp;is 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>&nbsp;units in the <em>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] alongside the shift left,&nbsp;[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] where&nbsp;<em>c&nbsp;<\/em>&gt; 0.\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\nUsing an online graphing calculator, plot the functions&nbsp;[latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] and&nbsp;[latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex]\n\nInvestigate the following questions:\n<ul>\n \t<li>Adjust the [latex]c[\/latex] value to 4.<\/li>\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>\n \t<li>Which direction does the graph of [latex]h(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<\/div>\nThe graphs below summarize the changes in the x-intercepts, vertical asymptotes, and equations&nbsp;of a logarithmic function that has been shifted either right or left.\n\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\">\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>\nFor any constant <em>c<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex]\n<ul>\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left <em>c<\/em>&nbsp;units if <em>c&nbsp;<\/em>&gt; 0.<\/li>\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>&nbsp;units if <em>c&nbsp;<\/em>&lt; 0.<\/li>\n \t<li>has the vertical asymptote <em>x&nbsp;<\/em>= \u2013<em>c<\/em>.<\/li>\n \t<li>has domain [latex]\\left(-c,\\infty \\right)[\/latex].<\/li>\n \t<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 Of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the Horizontal Shift<\/h3>\n<ol>\n \t<li>Identify the horizontal shift:\n<ul>\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>&nbsp;units.<\/li>\n \t<li>If <em>c&nbsp;<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>&nbsp;units.<\/li>\n<\/ul>\n<\/li>\n \t<li>Draw the vertical asymptote <em>x&nbsp;<\/em>= \u2013<em>c<\/em>.<\/li>\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting <em>c<\/em>&nbsp;from the&nbsp;<em>x<\/em>&nbsp;coordinate in each point.<\/li>\n \t<li>Label the three points.<\/li>\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&nbsp;<\/em>= \u2013c.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\nLogarithm functions follow the same principles as the other toolkit functions with regard to translations.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:&nbsp;Graphing a Horizontal Shift of the Parent Function&nbsp;[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\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.\n\n[reveal-answer q=\"368750\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"368750\"]\n\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].\n\nThus <em>c&nbsp;<\/em>= \u20132, so <em>c&nbsp;<\/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.\n\nThe vertical asymptote is [latex]x=-\\left(-2\\right)[\/latex] or <em>x&nbsp;<\/em>= 2.\n\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].\n\nThe new coordinates are found by adding 2 to the <em>x<\/em>&nbsp;coordinates of each point.\n\nPlot and label the points [latex]\\left(\\frac{7}{3},-1\\right)[\/latex], [latex]\\left(3,0\\right)[\/latex], and [latex]\\left(5,1\\right)[\/latex].\n\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&nbsp;<\/em>= 2.\n\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\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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.\n\n[reveal-answer q=\"779370\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"779370\"]\n\nThe domain is [latex]\\left(-4,\\infty \\right)[\/latex], the range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the asymptote <em>x&nbsp;<\/em>= \u20134.\n\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\">\n\n[\/hidden-answer]\n\n[ohm_question]74340[\/ohm_question]\n\n<\/div>\n<h2>Graphing a Vertical Shift of&nbsp;[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h2>\nWhen a constant <em>d<\/em>&nbsp;is 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>&nbsp;units in the direction of the sign of&nbsp;<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].\n\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\">\n<div class=\"textbox\">\n<h3>A General Note: Vertical Shifts of the Parent Function&nbsp;[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\nFor any constant <em>d<\/em>, the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex]\n<ul>\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>&nbsp;units if <em>d&nbsp;<\/em>&gt; 0.<\/li>\n \t<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d<\/em>&nbsp;units if <em>d&nbsp;<\/em>&lt; 0.<\/li>\n \t<li>has the vertical asymptote <em>x&nbsp;<\/em>= 0.<\/li>\n \t<li>has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n \t<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 Of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the Vertical Shift<\/h3>\n<ol>\n \t<li>Identify the vertical shift:\n<ul>\n \t<li>If <em>d&nbsp;<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>&nbsp;units.<\/li>\n \t<li>If <em>d&nbsp;<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d&nbsp;<\/em>units.<\/li>\n<\/ul>\n<\/li>\n \t<li>Draw the vertical asymptote <em>x&nbsp;<\/em>= 0.<\/li>\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding <em>d<\/em>&nbsp;to the <em>y&nbsp;<\/em>coordinate of each point.<\/li>\n \t<li>Label the three points.<\/li>\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&nbsp;<\/em>= 0.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Vertical Shift of the Parent Function&nbsp;[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\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.\n\n[reveal-answer q=\"43912\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"43912\"]\n\nSince the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex], we notice <em>d&nbsp;<\/em>= \u20132. Thus <em>d&nbsp;<\/em>&lt; 0.\n\nThis means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down 2 units.\n\nThe vertical asymptote is <em>x&nbsp;<\/em>= 0.\n\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].\n\nThe new coordinates are found by subtracting 2 from the <em>y <\/em>coordinates of each point.\n\nLabel the points [latex]\\left(\\frac{1}{3},-3\\right)[\/latex], [latex]\\left(1,-2\\right)[\/latex], and [latex]\\left(3,-1\\right)[\/latex].\n\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\"> 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>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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.\n\n[reveal-answer q=\"338440\"]Show Solution[\/reveal-answer]\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&nbsp;<\/em>= 0.\n\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\">\n\n[\/hidden-answer]\n\n[ohm_question]74341[\/ohm_question]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/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 functions behave similar 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&nbsp;[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When a constant <em>c<\/em>&nbsp;is 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>&nbsp;units in the <em>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] alongside the shift left,&nbsp;[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] where&nbsp;<em>c&nbsp;<\/em>&gt; 0.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Using an online graphing calculator, plot the functions&nbsp;[latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] and&nbsp;[latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex]<\/p>\n<p>Investigate the following questions:<\/p>\n<ul>\n<li>Adjust the [latex]c[\/latex] value to 4.<\/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<li>Which direction does the graph of [latex]h(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<\/div>\n<p>The graphs below summarize the changes in the x-intercepts, vertical asymptotes, and equations&nbsp;of 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\" \/><\/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>&nbsp;units if <em>c&nbsp;<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>&nbsp;units if <em>c&nbsp;<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x&nbsp;<\/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 Of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the Horizontal Shift<\/h3>\n<ol>\n<li>Identify the horizontal shift:\n<ul>\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>&nbsp;units.<\/li>\n<li>If <em>c&nbsp;<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right <em>c<\/em>&nbsp;units.<\/li>\n<\/ul>\n<\/li>\n<li>Draw the vertical asymptote <em>x&nbsp;<\/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>&nbsp;from the&nbsp;<em>x<\/em>&nbsp;coordinate in each point.<\/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&nbsp;<\/em>= \u2013c.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Logarithm functions follow the same principles as the other toolkit functions with regard to translations.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:&nbsp;Graphing a Horizontal Shift of the Parent Function&nbsp;[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\">Show 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&nbsp;<\/em>= \u20132, so <em>c&nbsp;<\/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&nbsp;<\/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>&nbsp;coordinates of each point.<\/p>\n<p>Plot and 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&nbsp;<\/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\" \/><\/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\">Show 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&nbsp;<\/em>= \u20134.<\/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=\"ohm74340\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=74340&theme=oea&iframe_resize_id=ohm74340&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing a Vertical Shift of&nbsp;[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h2>\n<p>When a constant <em>d<\/em>&nbsp;is 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>&nbsp;units in the direction of the sign of&nbsp;<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\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Vertical Shifts of the Parent Function&nbsp;[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>&nbsp;units if <em>d&nbsp;<\/em>&gt; 0.<\/li>\n<li>shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d<\/em>&nbsp;units if <em>d&nbsp;<\/em>&lt; 0.<\/li>\n<li>has the vertical asymptote <em>x&nbsp;<\/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 Of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the Vertical Shift<\/h3>\n<ol>\n<li>Identify the vertical shift:\n<ul>\n<li>If <em>d&nbsp;<\/em>&gt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up <em>d<\/em>&nbsp;units.<\/li>\n<li>If <em>d&nbsp;<\/em>&lt; 0, shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down <em>d&nbsp;<\/em>units.<\/li>\n<\/ul>\n<\/li>\n<li>Draw the vertical asymptote <em>x&nbsp;<\/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>&nbsp;to the <em>y&nbsp;<\/em>coordinate of each point.<\/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&nbsp;<\/em>= 0.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Vertical Shift of the Parent Function&nbsp;[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\">Show 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 notice <em>d&nbsp;<\/em>= \u20132. Thus <em>d&nbsp;<\/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&nbsp;<\/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>y <\/em>coordinates of each point.<\/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\" \/><\/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\">Show 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&nbsp;<\/em>= 0.<\/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=\"ohm74341\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=74341&theme=oea&iframe_resize_id=ohm74341&show_question_numbers\" width=\"100%\" height=\"150\"><\/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-344\">\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":17533,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 74340, 74341\",\"author\":\"Nearing,Daniel, mb Meacham,William, mb 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