{"id":2125,"date":"2024-07-09T18:40:47","date_gmt":"2024-07-09T18:40:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2125"},"modified":"2025-08-15T14:20:07","modified_gmt":"2025-08-15T14:20:07","slug":"exponential-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponential-functions-fresh-take\/","title":{"raw":"Exponential Functions: Fresh Take","rendered":"Exponential Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Understand what exponential functions are and learn their main features<\/li>\r\n \t<li>Write the equation for an exponential function<\/li>\r\n \t<li>Draw graphs of exponential functions<\/li>\r\n \t<li>Modify graphs of exponential functions using shifts, stretches, and reflections<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining 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\">Exponential Growth and Decay:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Exponential growth: Increase based on a constant multiplicative rate over equal time increments<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exponential decay: Decrease based on a constant multiplicative rate over equal time increments<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Contrast with Linear Growth:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Exponential: Changes by the same percentage over equal increments<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Linear: Changes by the same amount over equal increments<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form of Exponential Function:\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]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is any nonzero number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex] is a positive real number, [latex]b \\neq 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]b &gt; 1[\/latex]: function grows<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]0 &lt; b &lt; 1[\/latex]: function decays<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluating Exponential Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute the given value for [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Follow the order of operations carefully<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Which of the following equations represent exponential functions?<\/p>\r\n\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) = 2x^2 - 3x + 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]h(x) = 1.75x + 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"536253\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"536253\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.[reveal-answer q=\"860098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"860098\"][latex]5.5556[\/latex][\/hidden-answer]<\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggdaggeg-nqpn0SQB5ds\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/nqpn0SQB5ds?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ggdaggeg-nqpn0SQB5ds\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12780749&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggdaggeg-nqpn0SQB5ds&vembed=0&video_id=nqpn0SQB5ds&video_target=tpm-plugin-ggdaggeg-nqpn0SQB5ds'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Exponential+Functions+-+Nerdstudy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Exponential Functions - Nerdstudy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<\/section>\r\n<h2 aria-label=\"Example\">Exponential Growth and Decay<\/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\">Exponential Growth:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Output increases by a constant factor over equal intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]b &gt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = 2^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exponential Decay:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Output decreases by a constant factor over equal intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]0 &lt; b &lt; 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]g(x) = (\\frac{1}{2})^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Characteristics of Exponential Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Range: [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\">[latex]y[\/latex]-intercept: [latex](0, 1)[\/latex] for [latex]f(x) = b^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No [latex]x[\/latex]-intercept<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">The population of China was about [latex]1.39[\/latex] billion in the year 2013 with an annual growth rate of about [latex]0.6 \\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.39{\\left(1.006\\right)}^{t}[\/latex] where [latex]t[\/latex]\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of China be in the year 2031? How does this compare to the population prediction we made for India in the previous example?[reveal-answer q=\"891037\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"891037\"]About [latex]1.548[\/latex] billion people; by the year 2031, India\u2019s population will exceed China\u2019s by about [latex]0.001[\/latex] billion, or [latex]1[\/latex] million people.[\/hidden-answer]<\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/m5Tf6vgoJtQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850239&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ&vembed=0&video_id=m5Tf6vgoJtQ&video_target=tpm-plugin-dgbbhehh-m5Tf6vgoJtQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Exponential+growth+and+decay+word+problems+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential growth and decay word problems | Algebra II | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Finding Equations 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\">General Form of Exponential Functions:\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]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: initial value<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: growth factor (if [latex]b &gt; 1[\/latex]) or decay factor (if [latex]0 &lt; b &lt; 1[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Methods for Finding Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using two points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using a graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Principle:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Every point on the graph satisfies the equation of the function<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategies<\/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\">When Given Two Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If one point is [latex](0, a)[\/latex], use it as the initial value<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If no [latex](0, a)[\/latex] point, set up a system of equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When Given a Graph:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Choose the [latex]y[\/latex]-intercept as one point if possible<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Select a second point with integer coordinates<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use points far apart to minimize rounding errors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Steps:\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify or calculate the initial value [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the second point to solve for [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the equation in the form [latex]f(x) = ab^x[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population <em>N<\/em>\u00a0of wolves over time <em>t<\/em>.[reveal-answer q=\"222558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222558\"][latex]\\left(0,129\\right)[\/latex] and [latex]\\left(2,236\\right);N\\left(t\\right)=129{\\left(\\text{1}\\text{.3526}\\right)}^{t}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the two points [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(2,4.5\\right)[\/latex], find the equation of the exponential function that passes through these two points.[reveal-answer q=\"40110\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"40110\"][latex]f\\left(x\\right)=2{\\left(1.5\\right)}^{x}[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find an equation for the exponential function graphed below.\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\/02225457\/CNX_Precalc_Figure_04_01_0052.jpg\" alt=\"Graph of an increasing function with a labeled point at (0, sqrt(2)).\" width=\"487\" height=\"294\" \/> Graph of f(x) with y-intercept labeled[\/caption]\r\n\r\n[reveal-answer q=\"564720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"564720\"][latex]f\\left(x\\right)=\\sqrt{2}{\\left(\\sqrt{2}\\right)}^{x}[\/latex]. Answers may vary due to round-off error. The answer should be very close to [latex]1.4142{\\left(1.4142\\right)}^{x}[\/latex].[\/hidden-answer]<\/section><section aria-label=\"Example\"><\/section><section aria-label=\"Example\"><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adhcdgab-1IE5jNudELQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1IE5jNudELQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-adhcdgab-1IE5jNudELQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850238&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-adhcdgab-1IE5jNudELQ&vembed=0&video_id=1IE5jNudELQ&video_target=tpm-plugin-adhcdgab-1IE5jNudELQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+an+Exponential+Function+Given+Two+Points+-+Initial+Value+Not+Given_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find an Exponential Function Given Two Points - Initial Value Not Given\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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbefchae-ueCcMc1FUsw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ueCcMc1FUsw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbefchae-ueCcMc1FUsw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12780750&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbefchae-ueCcMc1FUsw&vembed=0&video_id=ueCcMc1FUsw&video_target=tpm-plugin-bbefchae-ueCcMc1FUsw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Writing+Exponential+Functions+from+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Exponential Functions from a Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\"><\/section><\/section><\/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]<\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/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 type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850237&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hcfdheee-FMzZB9Ve-1U&vembed=0&video_id=FMzZB9Ve-1U&video_target=tpm-plugin-hcfdheee-FMzZB9Ve-1U'><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/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 type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850236&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gdecfcac-M6bpp0BRIf0&vembed=0&video_id=M6bpp0BRIf0&video_target=tpm-plugin-gdecfcac-M6bpp0BRIf0'><\/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]<\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/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 type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850235&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cedaggbg-phYxEeJ7ZW4&vembed=0&video_id=phYxEeJ7ZW4&video_target=tpm-plugin-cedaggbg-phYxEeJ7ZW4'><\/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_transcript.txt\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]<\/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]<\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/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 type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12780751&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-abcgfaeb-yFejFtUtQAA&vembed=0&video_id=yFejFtUtQAA&video_target=tpm-plugin-abcgfaeb-yFejFtUtQAA'><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/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 type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12780752&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fbaegcdb-gVkONK_G8bg&vembed=0&video_id=gVkONK_G8bg&video_target=tpm-plugin-fbaegcdb-gVkONK_G8bg'><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/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 type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12850234&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bfgafhbe-bhGXeSiwnf4&vembed=0&video_id=bhGXeSiwnf4&video_target=tpm-plugin-bfgafhbe-bhGXeSiwnf4'><\/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>Understand what exponential functions are and learn their main features<\/li>\n<li>Write the equation for an exponential function<\/li>\n<li>Draw graphs of exponential functions<\/li>\n<li>Modify graphs of exponential functions using shifts, stretches, and reflections<\/li>\n<\/ul>\n<\/section>\n<h2>Defining 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\">Exponential Growth and Decay:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential growth: Increase based on a constant multiplicative rate over equal time increments<\/li>\n<li class=\"whitespace-normal break-words\">Exponential decay: Decrease based on a constant multiplicative rate over equal time increments<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Contrast with Linear Growth:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential: Changes by the same percentage over equal increments<\/li>\n<li class=\"whitespace-normal break-words\">Linear: Changes by the same amount over equal increments<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Form of Exponential Function:\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]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is any nonzero number<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex] is a positive real number, [latex]b \\neq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]b > 1[\/latex]: function grows<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]0 < b < 1[\/latex]: function decays<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute the given value for [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Follow the order of operations carefully<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Which of the following equations represent exponential functions?<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = 2x^2 - 3x + 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]h(x) = 1.75x + 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q536253\">Show Answer<\/button><\/p>\n<div id=\"q536253\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]g(x) = 0.875^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]j(x) = 1095.6^{2x}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q860098\">Show Solution<\/button><\/p>\n<div id=\"q860098\" class=\"hidden-answer\" style=\"display: none\">[latex]5.5556[\/latex]<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggdaggeg-nqpn0SQB5ds\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/nqpn0SQB5ds?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ggdaggeg-nqpn0SQB5ds\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12780749&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ggdaggeg-nqpn0SQB5ds&#38;vembed=0&#38;video_id=nqpn0SQB5ds&#38;video_target=tpm-plugin-ggdaggeg-nqpn0SQB5ds\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Exponential+Functions+-+Nerdstudy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Exponential Functions &#8211; Nerdstudy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<\/section>\n<h2 aria-label=\"Example\">Exponential Growth and Decay<\/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\">Exponential Growth:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Output increases by a constant factor over equal intervals<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]b > 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = 2^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Exponential Decay:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Output decreases by a constant factor over equal intervals<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]f(x) = ab^x[\/latex], where [latex]0 < b < 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]g(x) = (\\frac{1}{2})^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Characteristics of Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Range: [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\">[latex]y[\/latex]-intercept: [latex](0, 1)[\/latex] for [latex]f(x) = b^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">No [latex]x[\/latex]-intercept<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">The population of China was about [latex]1.39[\/latex] billion in the year 2013 with an annual growth rate of about [latex]0.6 \\%[\/latex]. This situation is represented by the growth function [latex]P\\left(t\\right)=1.39{\\left(1.006\\right)}^{t}[\/latex] where [latex]t[\/latex]\u00a0is the number of years since 2013. To the nearest thousandth, what will the population of China be in the year 2031? How does this compare to the population prediction we made for India in the previous example?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q891037\">Show Solution<\/button><\/p>\n<div id=\"q891037\" class=\"hidden-answer\" style=\"display: none\">About [latex]1.548[\/latex] billion people; by the year 2031, India\u2019s population will exceed China\u2019s by about [latex]0.001[\/latex] billion, or [latex]1[\/latex] million people.<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/m5Tf6vgoJtQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12850239&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dgbbhehh-m5Tf6vgoJtQ&#38;vembed=0&#38;video_id=m5Tf6vgoJtQ&#38;video_target=tpm-plugin-dgbbhehh-m5Tf6vgoJtQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Exponential+growth+and+decay+word+problems+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExponential growth and decay word problems | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<\/section>\n<h2 data-type=\"title\">Finding Equations 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\">General Form of Exponential Functions:\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]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex]: initial value<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b[\/latex]: growth factor (if [latex]b > 1[\/latex]) or decay factor (if [latex]0 < b < 1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Methods for Finding Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using two points<\/li>\n<li class=\"whitespace-normal break-words\">Using a graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Every point on the graph satisfies the equation of the function<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Problem-Solving Strategies<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">When Given Two Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If one point is [latex](0, a)[\/latex], use it as the initial value<\/li>\n<li class=\"whitespace-normal break-words\">If no [latex](0, a)[\/latex] point, set up a system of equations<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">When Given a Graph:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose the [latex]y[\/latex]-intercept as one point if possible<\/li>\n<li class=\"whitespace-normal break-words\">Select a second point with integer coordinates<\/li>\n<li class=\"whitespace-normal break-words\">Use points far apart to minimize rounding errors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Steps:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify or calculate the initial value [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Use the second point to solve for [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write the equation in the form [latex]f(x) = ab^x[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population <em>N<\/em>\u00a0of wolves over time <em>t<\/em>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q222558\">Show Solution<\/button><\/p>\n<div id=\"q222558\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(0,129\\right)[\/latex] and [latex]\\left(2,236\\right);N\\left(t\\right)=129{\\left(\\text{1}\\text{.3526}\\right)}^{t}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the two points [latex]\\left(1,3\\right)[\/latex] and [latex]\\left(2,4.5\\right)[\/latex], find the equation of the exponential function that passes through these two points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q40110\">Show Solution<\/button><\/p>\n<div id=\"q40110\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=2{\\left(1.5\\right)}^{x}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find an equation for the exponential function graphed below.<\/p>\n<figure style=\"width: 487px\" 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\/02225457\/CNX_Precalc_Figure_04_01_0052.jpg\" alt=\"Graph of an increasing function with a labeled point at (0, sqrt(2)).\" width=\"487\" height=\"294\" \/><figcaption class=\"wp-caption-text\">Graph of f(x) with y-intercept labeled<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q564720\">Show Solution<\/button><\/p>\n<div id=\"q564720\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=\\sqrt{2}{\\left(\\sqrt{2}\\right)}^{x}[\/latex]. Answers may vary due to round-off error. The answer should be very close to [latex]1.4142{\\left(1.4142\\right)}^{x}[\/latex].<\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Example\"><\/section>\n<section aria-label=\"Example\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-adhcdgab-1IE5jNudELQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1IE5jNudELQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-adhcdgab-1IE5jNudELQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12850238&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-adhcdgab-1IE5jNudELQ&#38;vembed=0&#38;video_id=1IE5jNudELQ&#38;video_target=tpm-plugin-adhcdgab-1IE5jNudELQ\"><\/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+-+Find+an+Exponential+Function+Given+Two+Points+-+Initial+Value+Not+Given_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find an Exponential Function Given Two Points &#8211; Initial Value Not Given\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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbefchae-ueCcMc1FUsw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ueCcMc1FUsw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbefchae-ueCcMc1FUsw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12780750&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbefchae-ueCcMc1FUsw&#38;vembed=0&#38;video_id=ueCcMc1FUsw&#38;video_target=tpm-plugin-bbefchae-ueCcMc1FUsw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Writing+Exponential+Functions+from+a+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Exponential Functions from a Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\"><\/section>\n<\/section>\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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/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 type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12850237&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hcfdheee-FMzZB9Ve-1U&#38;vembed=0&#38;video_id=FMzZB9Ve-1U&#38;video_target=tpm-plugin-hcfdheee-FMzZB9Ve-1U\"><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/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 type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12850236&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gdecfcac-M6bpp0BRIf0&#38;vembed=0&#38;video_id=M6bpp0BRIf0&#38;video_target=tpm-plugin-gdecfcac-M6bpp0BRIf0\"><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/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 type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12850235&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cedaggbg-phYxEeJ7ZW4&#38;vembed=0&#38;video_id=phYxEeJ7ZW4&#38;video_target=tpm-plugin-cedaggbg-phYxEeJ7ZW4\"><\/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 &#8211; Match the Graphs of Translated Exponential Function to Equations_transcript.txt\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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/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 type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12780751&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-abcgfaeb-yFejFtUtQAA&#38;vembed=0&#38;video_id=yFejFtUtQAA&#38;video_target=tpm-plugin-abcgfaeb-yFejFtUtQAA\"><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/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 type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12780752&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fbaegcdb-gVkONK_G8bg&#38;vembed=0&#38;video_id=gVkONK_G8bg&#38;video_target=tpm-plugin-fbaegcdb-gVkONK_G8bg\"><\/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 type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/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 type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12850234&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bfgafhbe-bhGXeSiwnf4&#38;vembed=0&#38;video_id=bhGXeSiwnf4&#38;video_target=tpm-plugin-bfgafhbe-bhGXeSiwnf4\"><\/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":12,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Exponential Functions - Nerdstudy\",\"author\":\"\",\"organization\":\"Nerdstudy\",\"url\":\"https:\/\/youtu.be\/nqpn0SQB5ds\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Exponential growth and decay word problems | Algebra II | Khan Academy\",\"author\":\"\",\"organization\":\"Khan 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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\":\"\",\"organization\":\"Brian McLogan\",\"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":255,"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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