{"id":1425,"date":"2025-07-25T01:16:00","date_gmt":"2025-07-25T01:16:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1425"},"modified":"2026-03-18T06:14:54","modified_gmt":"2026-03-18T06:14:54","slug":"graphs-of-exponential-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-exponential-functions-fresh-take\/","title":{"raw":"Graphs of Exponential Functions: Fresh Take","rendered":"Graphs of Exponential Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph exponential functions<\/li>\r\n \t<li>Graph exponential functions using transformations<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Graphing Exponential Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Parent Function: [latex]f(x) = b^x[\/latex], where [latex]b &gt; 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: All positive real numbers [latex](0, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">y-intercept: [latex](0, 1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No x-intercept<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Growth vs. Decay:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]b &gt; 1[\/latex]: Exponential growth (increasing function)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]0 &lt; b &lt; 1[\/latex]: Exponential decay (decreasing function)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Strategy<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Create a table of points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Include negative and positive [latex]x[\/latex]-values<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Always include the [latex]y[\/latex]-intercept [latex](0, 1)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot at least [latex]3[\/latex] points, including the y-intercept<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw a smooth curve through the points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Indicate the horizontal asymptote at [latex]y = 0[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the graph of [latex]f\\left(x\\right)={4}^{x}[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"192861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"192861\"]The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].\r\n\r\n[caption id=\"attachment_3353\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\"><img class=\"wp-image-3353 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\" alt=\"\" width=\"487\" height=\"332\" \/><\/a> Graph of f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video for another example of graphing an exponential function. The base of the exponential term is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex], so this graph will represent decay.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hcfdheee-FMzZB9Ve-1U\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FMzZB9Ve-1U?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hcfdheee-FMzZB9Ve-1U\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850237&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hcfdheee-FMzZB9Ve-1U&amp;vembed=0&amp;video_id=FMzZB9Ve-1U&amp;video_target=tpm-plugin-hcfdheee-FMzZB9Ve-1U\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Basic+Exponential+Function+Using+a+Table+of+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Basic Exponential Function Using a Table of Values\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>The next video example includes graphing an exponential growth function and defining the domain and range of the function.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gdecfcac-M6bpp0BRIf0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/M6bpp0BRIf0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gdecfcac-M6bpp0BRIf0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850236&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gdecfcac-M6bpp0BRIf0&amp;vembed=0&amp;video_id=M6bpp0BRIf0&amp;video_target=tpm-plugin-gdecfcac-M6bpp0BRIf0\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+an+Exponential+Function+Using+a+Table+of+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph an Exponential Function Using a Table of Values\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Horizontal and Vertical Translations of Exponential Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Parent Function: [latex]f(x) = b^x[\/latex], where [latex]b &gt; 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Transformed Function: [latex]f(x) = ab^{x-h} + k[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: Vertical stretch\/compression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]h[\/latex]: Horizontal shift<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]k[\/latex]: Vertical shift<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical Shift: [latex]f(x) = b^x + d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Shift: [latex]f(x) = b^{x-c}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Transformation Effects<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical Shift ([latex]+d[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Moves graph up [latex]d[\/latex] units if [latex]d &gt; 0[\/latex], down if [latex]d &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Changes [latex]y[\/latex]-intercept to [latex](0, 1+d)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shifts asymptote to [latex]y = d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New range: [latex](d, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Shift ([latex]-c[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Moves graph right [latex]c[\/latex] units if [latex]c &gt; 0[\/latex], left if [latex]c &lt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Changes [latex]y[\/latex]-intercept to [latex](0, b^c)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Asymptote remains at [latex]y = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain and range unchanged<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combined Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Apply horizontal shift first, then vertical shift<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New [latex]y[\/latex]-intercept: [latex](0, b^c + d)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New asymptote: [latex]y = d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New range: [latex](d, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Strategy<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]b[\/latex], [latex]c[\/latex], and [latex]d[\/latex] in [latex]f(x) = b^{x-c} + d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw the horizontal asymptote [latex]y = d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot the [latex]y[\/latex]-intercept [latex](0, b^c + d)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sketch the graph, shifting horizontally by [latex]c[\/latex] and vertically by [latex]d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">State the domain [latex](-\\infty, \\infty)[\/latex], range [latex](d, \\infty)[\/latex], and asymptote [latex]y = d[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to plot the function\u00a0[latex]f\\left(x\\right)={2}^{x-1}+3[\/latex]. State domain, range, and asymptote.[reveal-answer q=\"699634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"699634\"]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.\r\n\r\n[caption id=\"attachment_3016\" align=\"aligncenter\" width=\"487\"]<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=\"wp-image-3016 size-full\" 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> Graph of f(x) with the horizontal asymptote labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video for more examples of the difference between horizontal and vertical shifts of exponential functions and the resulting graphs and equations.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cedaggbg-phYxEeJ7ZW4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/phYxEeJ7ZW4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cedaggbg-phYxEeJ7ZW4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850235&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cedaggbg-phYxEeJ7ZW4&amp;vembed=0&amp;video_id=phYxEeJ7ZW4&amp;video_target=tpm-plugin-cedaggbg-phYxEeJ7ZW4\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Match+the+Graphs+of+Translated+Exponential+Function+to+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Match the Graphs of Translated Exponential Function to Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Stretching, Compressing, or Reflecting an Exponential Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">General Transformed Function: [latex]f(x) = ab^{x-h} + k[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: Vertical stretch\/compression and reflection<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: Base of exponential ([latex]b &gt; 0, b \\neq 1[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]h[\/latex]: Horizontal shift<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]k[\/latex]: Vertical shift<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Stretch\/Compression:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = ab^x[\/latex], where [latex]a \\neq 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stretch if [latex]|a| &gt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compress if [latex]0 &lt; |a| &lt; 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflections:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">About [latex]x[\/latex]-axis: [latex]f(x) = -b^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">About [latex]y[\/latex]-axis: [latex]f(x) = b^{-x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Transformation Effects<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical Stretch\/Compression ([latex]a[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiplies all [latex]y[\/latex]-values by [latex]|a|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New [latex]y[\/latex]-intercept: [latex](0, a)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain and horizontal asymptote unchanged<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflection about [latex]x[\/latex]-axis ([latex]-b^x[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Flips graph upside down<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New range: [latex](-\\infty, 0)[\/latex] if [latex]b &gt; 1[\/latex], [latex](0, -\\infty)[\/latex] if [latex]0 &lt; b &lt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New [latex]y[\/latex]-intercept: [latex](0, -1)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflection about [latex]y[\/latex]-axis ([latex]b^{-x}[\/latex]):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reverses left-right orientation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain, range, and [latex]y[\/latex]-intercept unchanged<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Growth becomes decay (and vice versa)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Strategy<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]a[\/latex], [latex]b[\/latex], [latex]h[\/latex], and [latex]k[\/latex] in [latex]f(x) = ab^{x-h} + k[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply transformations in this order:\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal shift<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflection about [latex]y[\/latex]-axis (if applicable)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical stretch\/compression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflection about [latex]x[\/latex]-axis (if applicable)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical shift<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot key points: [latex]y[\/latex]-intercept and a few others<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sketch the curve and asymptote<\/li>\r\n \t<li class=\"whitespace-normal break-words\">State domain, range, and asymptote<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing tool to sketch the graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"796634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796634\"]The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].\u00a0<span id=\"fs-id1165135417835\">\r\n<\/span>\r\n\r\n[caption id=\"attachment_3081\" align=\"aligncenter\" width=\"488\"]<img class=\"wp-image-3081 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16190801\/CNX_Precalc_Figure_04_02_0122.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"488\" height=\"294\" \/> Graph of f(x) with the horizontal asymptote labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to\u00a0graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={1.25}^{x}[\/latex] about the <em>y<\/em>-axis. State its domain, range, and asymptote.[reveal-answer q=\"845922\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"845922\"]The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].\r\n\r\n[caption id=\"attachment_3082\" align=\"aligncenter\" width=\"731\"]<img class=\"wp-image-3082 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16191705\/CNX_Precalc_Figure_04_02_0152.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" width=\"731\" height=\"482\" \/> Graph of g(x) with the horizontal asymptote labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.\r\n<ul>\r\n \t<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the [latex]x[\/latex]-axis, and then shifted down [latex]2[\/latex]\u00a0units.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"525289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525289\"]\r\n\r\n[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2[\/latex]; the domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,-2\\right)[\/latex]; the horizontal asymptote is [latex]y=-2[\/latex].[\/hidden-answer]\r\n\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abcgfaeb-yFejFtUtQAA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/yFejFtUtQAA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-abcgfaeb-yFejFtUtQAA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780751&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abcgfaeb-yFejFtUtQAA&amp;vembed=0&amp;video_id=yFejFtUtQAA&amp;video_target=tpm-plugin-abcgfaeb-yFejFtUtQAA\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graphing+exponential+functions+with+horizontal+and+vertical+transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing exponential functions with horizontal and vertical transformations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbaegcdb-gVkONK_G8bg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/gVkONK_G8bg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbaegcdb-gVkONK_G8bg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780752&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbaegcdb-gVkONK_G8bg&amp;vembed=0&amp;video_id=gVkONK_G8bg&amp;video_target=tpm-plugin-fbaegcdb-gVkONK_G8bg\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+Graph+Exponential+Functions+with+Transformations+(3+Examples)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Graph Exponential Functions with Transformations (3 Examples)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfgafhbe-bhGXeSiwnf4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/bhGXeSiwnf4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bfgafhbe-bhGXeSiwnf4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850234&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfgafhbe-bhGXeSiwnf4&amp;vembed=0&amp;video_id=bhGXeSiwnf4&amp;video_target=tpm-plugin-bfgafhbe-bhGXeSiwnf4\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Equations+of+a+Transformed+Exponential+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Equations of a Transformed Exponential Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph exponential functions<\/li>\n<li>Graph exponential functions using transformations<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Graphing Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Parent Function: [latex]f(x) = b^x[\/latex], where [latex]b > 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: All real numbers [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: All positive real numbers [latex](0, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal asymptote: [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">y-intercept: [latex](0, 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">No x-intercept<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Growth vs. Decay:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]b > 1[\/latex]: Exponential growth (increasing function)<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]0 < b < 1[\/latex]: Exponential decay (decreasing function)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Create a table of points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Include negative and positive [latex]x[\/latex]-values<\/li>\n<li class=\"whitespace-normal break-words\">Always include the [latex]y[\/latex]-intercept [latex](0, 1)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Plot at least [latex]3[\/latex] points, including the y-intercept<\/li>\n<li class=\"whitespace-normal break-words\">Draw a smooth curve through the points<\/li>\n<li class=\"whitespace-normal break-words\">Indicate the horizontal asymptote at [latex]y = 0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch the graph of [latex]f\\left(x\\right)={4}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q192861\">Show Solution<\/button><\/p>\n<div id=\"q192861\" class=\"hidden-answer\" style=\"display: none\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<figure id=\"attachment_3353\" aria-describedby=\"caption-attachment-3353\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3353 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/08000344\/CNX_Precalc_Figure_04_02_0052.jpg\" alt=\"\" width=\"487\" height=\"332\" \/><\/a><figcaption id=\"caption-attachment-3353\" class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for another example of graphing an exponential function. The base of the exponential term is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex], so this graph will represent decay.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hcfdheee-FMzZB9Ve-1U\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FMzZB9Ve-1U?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hcfdheee-FMzZB9Ve-1U\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850237&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hcfdheee-FMzZB9Ve-1U&amp;vembed=0&amp;video_id=FMzZB9Ve-1U&amp;video_target=tpm-plugin-hcfdheee-FMzZB9Ve-1U\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Basic+Exponential+Function+Using+a+Table+of+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Basic Exponential Function Using a Table of Values\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The next video example includes graphing an exponential growth function and defining the domain and range of the function.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gdecfcac-M6bpp0BRIf0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/M6bpp0BRIf0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gdecfcac-M6bpp0BRIf0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850236&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gdecfcac-M6bpp0BRIf0&amp;vembed=0&amp;video_id=M6bpp0BRIf0&amp;video_target=tpm-plugin-gdecfcac-M6bpp0BRIf0\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+an+Exponential+Function+Using+a+Table+of+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph an Exponential Function Using a Table of Values\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Horizontal and Vertical Translations of Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Parent Function: [latex]f(x) = b^x[\/latex], where [latex]b > 0[\/latex] and [latex]b \\neq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">General Transformed Function: [latex]f(x) = ab^{x-h} + k[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: Vertical stretch\/compression<\/li>\n<li class=\"whitespace-normal break-words\">[latex]h[\/latex]: Horizontal shift<\/li>\n<li class=\"whitespace-normal break-words\">[latex]k[\/latex]: Vertical shift<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical Shift: [latex]f(x) = b^x + d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Shift: [latex]f(x) = b^{x-c}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Transformation Effects<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical Shift ([latex]+d[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Moves graph up [latex]d[\/latex] units if [latex]d > 0[\/latex], down if [latex]d < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Changes [latex]y[\/latex]-intercept to [latex](0, 1+d)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shifts asymptote to [latex]y = d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">New range: [latex](d, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Shift ([latex]-c[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Moves graph right [latex]c[\/latex] units if [latex]c > 0[\/latex], left if [latex]c < 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Changes [latex]y[\/latex]-intercept to [latex](0, b^c)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Asymptote remains at [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain and range unchanged<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combined Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Apply horizontal shift first, then vertical shift<\/li>\n<li class=\"whitespace-normal break-words\">New [latex]y[\/latex]-intercept: [latex](0, b^c + d)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">New asymptote: [latex]y = d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">New range: [latex](d, \\infty)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify [latex]b[\/latex], [latex]c[\/latex], and [latex]d[\/latex] in [latex]f(x) = b^{x-c} + d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Draw the horizontal asymptote [latex]y = d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Plot the [latex]y[\/latex]-intercept [latex](0, b^c + d)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Sketch the graph, shifting horizontally by [latex]c[\/latex] and vertically by [latex]d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">State the domain [latex](-\\infty, \\infty)[\/latex], range [latex](d, \\infty)[\/latex], and asymptote [latex]y = d[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to plot the function\u00a0[latex]f\\left(x\\right)={2}^{x-1}+3[\/latex]. State domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q699634\">Show Solution<\/button><\/p>\n<div id=\"q699634\" class=\"hidden-answer\" style=\"display: none\">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<figure id=\"attachment_3016\" aria-describedby=\"caption-attachment-3016\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><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=\"wp-image-3016 size-full\" 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><figcaption id=\"caption-attachment-3016\" class=\"wp-caption-text\">Graph of f(x) with the horizontal asymptote labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more examples of the difference between horizontal and vertical shifts of exponential functions and the resulting graphs and equations.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cedaggbg-phYxEeJ7ZW4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/phYxEeJ7ZW4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cedaggbg-phYxEeJ7ZW4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850235&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cedaggbg-phYxEeJ7ZW4&amp;vembed=0&amp;video_id=phYxEeJ7ZW4&amp;video_target=tpm-plugin-cedaggbg-phYxEeJ7ZW4\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Match+the+Graphs+of+Translated+Exponential+Function+to+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Match the Graphs of Translated Exponential Function to Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Stretching, Compressing, or Reflecting an Exponential Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">General Transformed Function: [latex]f(x) = ab^{x-h} + k[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: Vertical stretch\/compression and reflection<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: Base of exponential ([latex]b > 0, b \\neq 1[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]h[\/latex]: Horizontal shift<\/li>\n<li class=\"whitespace-normal break-words\">[latex]k[\/latex]: Vertical shift<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Stretch\/Compression:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = ab^x[\/latex], where [latex]a \\neq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Stretch if [latex]|a| > 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Compress if [latex]0 < |a| < 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflections:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">About [latex]x[\/latex]-axis: [latex]f(x) = -b^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">About [latex]y[\/latex]-axis: [latex]f(x) = b^{-x}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Transformation Effects<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical Stretch\/Compression ([latex]a[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiplies all [latex]y[\/latex]-values by [latex]|a|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">New [latex]y[\/latex]-intercept: [latex](0, a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Domain and horizontal asymptote unchanged<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflection about [latex]x[\/latex]-axis ([latex]-b^x[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Flips graph upside down<\/li>\n<li class=\"whitespace-normal break-words\">New range: [latex](-\\infty, 0)[\/latex] if [latex]b > 1[\/latex], [latex](0, -\\infty)[\/latex] if [latex]0 < b < 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">New [latex]y[\/latex]-intercept: [latex](0, -1)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflection about [latex]y[\/latex]-axis ([latex]b^{-x}[\/latex]):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reverses left-right orientation<\/li>\n<li class=\"whitespace-normal break-words\">Domain, range, and [latex]y[\/latex]-intercept unchanged<\/li>\n<li class=\"whitespace-normal break-words\">Growth becomes decay (and vice versa)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Graphing Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify [latex]a[\/latex], [latex]b[\/latex], [latex]h[\/latex], and [latex]k[\/latex] in [latex]f(x) = ab^{x-h} + k[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply transformations in this order:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal shift<\/li>\n<li class=\"whitespace-normal break-words\">Reflection about [latex]y[\/latex]-axis (if applicable)<\/li>\n<li class=\"whitespace-normal break-words\">Vertical stretch\/compression<\/li>\n<li class=\"whitespace-normal break-words\">Reflection about [latex]x[\/latex]-axis (if applicable)<\/li>\n<li class=\"whitespace-normal break-words\">Vertical shift<\/li>\n<\/ol>\n<\/li>\n<li class=\"whitespace-normal break-words\">Plot key points: [latex]y[\/latex]-intercept and a few others<\/li>\n<li class=\"whitespace-normal break-words\">Sketch the curve and asymptote<\/li>\n<li class=\"whitespace-normal break-words\">State domain, range, and asymptote<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing tool to sketch the graph of [latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q796634\">Show Solution<\/button><\/p>\n<div id=\"q796634\" class=\"hidden-answer\" style=\"display: none\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].\u00a0<span id=\"fs-id1165135417835\"><br \/>\n<\/span><\/p>\n<figure id=\"attachment_3081\" aria-describedby=\"caption-attachment-3081\" style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3081 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16190801\/CNX_Precalc_Figure_04_02_0122.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" width=\"488\" height=\"294\" \/><figcaption id=\"caption-attachment-3081\" class=\"wp-caption-text\">Graph of f(x) with the horizontal asymptote labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to\u00a0graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={1.25}^{x}[\/latex] about the <em>y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q845922\">Show Solution<\/button><\/p>\n<div id=\"q845922\" class=\"hidden-answer\" style=\"display: none\">The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<figure id=\"attachment_3082\" aria-describedby=\"caption-attachment-3082\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3082 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16191705\/CNX_Precalc_Figure_04_02_0152.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" width=\"731\" height=\"482\" \/><figcaption id=\"caption-attachment-3082\" class=\"wp-caption-text\">Graph of g(x) with the horizontal asymptote labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul>\n<li>[latex]f\\left(x\\right)={e}^{x}[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{3}[\/latex], reflected across the [latex]x[\/latex]-axis, and then shifted down [latex]2[\/latex]\u00a0units.<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q525289\">Show Solution<\/button><\/p>\n<div id=\"q525289\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2[\/latex]; the domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(-\\infty ,-2\\right)[\/latex]; the horizontal asymptote is [latex]y=-2[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abcgfaeb-yFejFtUtQAA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/yFejFtUtQAA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-abcgfaeb-yFejFtUtQAA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780751&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-abcgfaeb-yFejFtUtQAA&amp;vembed=0&amp;video_id=yFejFtUtQAA&amp;video_target=tpm-plugin-abcgfaeb-yFejFtUtQAA\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graphing+exponential+functions+with+horizontal+and+vertical+transformations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing exponential functions with horizontal and vertical transformations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbaegcdb-gVkONK_G8bg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/gVkONK_G8bg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbaegcdb-gVkONK_G8bg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780752&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbaegcdb-gVkONK_G8bg&amp;vembed=0&amp;video_id=gVkONK_G8bg&amp;video_target=tpm-plugin-fbaegcdb-gVkONK_G8bg\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+Graph+Exponential+Functions+with+Transformations+(3+Examples)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Graph Exponential Functions with Transformations (3 Examples)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bfgafhbe-bhGXeSiwnf4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/bhGXeSiwnf4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bfgafhbe-bhGXeSiwnf4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850234&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bfgafhbe-bhGXeSiwnf4&amp;vembed=0&amp;video_id=bhGXeSiwnf4&amp;video_target=tpm-plugin-bfgafhbe-bhGXeSiwnf4\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Equations+of+a+Transformed+Exponential+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Equations of a Transformed Exponential Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"author":67,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Graph a Basic Exponential Function Using a Table of Values\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/FMzZB9Ve-1U\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graph an Exponential Function Using a Table of Values\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/M6bpp0BRIf0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Match the Graphs of Translated Exponential Function to Equations\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/phYxEeJ7ZW4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graphing exponential functions with horizontal and vertical transformations\",\"author\":\"Brian McLogan\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/yFejFtUtQAA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to Graph Exponential Functions with Transformations (3 Examples)\",\"author\":\"\",\"organization\":\"Mario\\'s Math Tutoring\",\"url\":\"https:\/\/youtu.be\/gVkONK_G8bg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Equations of a Transformed Exponential Function\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/bhGXeSiwnf4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"fresh_take","content_attributions":null,"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' 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