{"id":2086,"date":"2024-07-03T20:40:26","date_gmt":"2024-07-03T20:40:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2086"},"modified":"2024-11-21T21:57:51","modified_gmt":"2024-11-21T21:57:51","slug":"exponential-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponential-functions-learn-it-1\/","title":{"raw":"Exponential Functions: Learn It 1","rendered":"Exponential Functions: Learn It 1"},"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\nWhat exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.\r\n<ul>\r\n \t<li><strong>Percent change<\/strong> refers to a change based on a percent of the original amount.<\/li>\r\n \t<li><strong>Exponential growth<\/strong> refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.<\/li>\r\n \t<li><strong>Exponential decay<\/strong> refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.<\/li>\r\n<\/ul>\r\nFor us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth.\r\n<table style=\"border-collapse: collapse; width: 78.5374%; height: 132px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^x[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^0 = 1[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(0) = 0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^1 = 2[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(1) = 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^2 = 4[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(2) = 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^3 = 8[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(3) = 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 18.5692%; height: 22px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^4 = 16[\/latex]<\/td>\r\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(4) = 8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can infer that for these two functions, exponential growth dwarfs linear growth.\r\n<ul>\r\n \t<li><strong>Exponential growth<\/strong> refers to the original value from the range increases by the same percentage over equal increments found in the domain.<\/li>\r\n \t<li><strong>Linear growth<\/strong> refers to the original value from the range increases by the same amount over equal increments found in the domain.<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24699[\/ohm2_question]<\/section>Apparently, the difference between \u201cthe same percentage\u201d and \u201cthe same amount\u201d is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding [latex]2[\/latex] to the output whenever the input was increased by one.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>exponential function<\/h3>\r\nThe general form of the <strong>exponential formula<\/strong>\u00a0is\r\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\r\nwhere [latex]a[\/latex] is any nonzero number and [latex]b[\/latex] is a positive real number not equal to [latex]1[\/latex].\r\n<ul>\r\n \t<li>if [latex]b&gt;1[\/latex], the function grows at a rate proportional to its size.<\/li>\r\n \t<li>if [latex]0 \\lt b \\lt 1[\/latex], the function decays at a rate proportional to its size.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Why do we limit the base [latex]b[\/latex]\u00a0to positive values?<\/strong>\r\n\r\n<hr \/>\r\n\r\nThis is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:\r\n<ul>\r\n \t<li>Consider a base of \u20139 and exponent of [latex]\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\r\n<\/ul>\r\n<strong>Why do we limit the base to positive values other than 1?\r\n<\/strong>\r\n\r\n<hr \/>\r\n\r\nThis is because a base of 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:\r\n<ul>\r\n \t<li>Consider a base of 1.\u00a0Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\r\n<\/ul>\r\n<\/section>To evaluate an exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], we simply substitute [latex]x[\/latex] with the given value, and calculate the resulting power.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllll}f\\left(x\\right)\\hfill &amp; ={2}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; ={2}^{3}\\text{}\\hfill &amp; \\text{Substitute }x=3. \\hfill \\\\ \\hfill &amp; =8\\text{}\\hfill &amp; \\text{Evaluate the power}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/section>When evaluating an exponential function, it is important to follow the order of operations.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; =30{\\left(2\\right)}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; =30{\\left(2\\right)}^{3}\\hfill &amp; \\text{Substitute }x=3.\\hfill \\\\ \\hfill &amp; =30\\left(8\\right)\\text{ }\\hfill &amp; \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill &amp; =240\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\nNote that if the order of operations were not followed, the result would be incorrect:\r\n<p style=\"text-align: center;\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.[reveal-answer q=\"454575\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"454575\"]Follow the order of operations. Be sure to pay attention to the parentheses.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; =5{\\left(3\\right)}^{x+1}\\hfill &amp; \\hfill \\\\ f\\left(2\\right)\\hfill &amp; =5{\\left(3\\right)}^{2+1}\\hfill &amp; \\text{Substitute }x=2.\\hfill \\\\ \\hfill &amp; =5{\\left(3\\right)}^{3}\\hfill &amp; \\text{Add the exponents}.\\hfill \\\\ \\hfill &amp; =5\\left(27\\right)\\hfill &amp; \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill &amp; =135\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]73212[\/ohm_question]<\/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<p>What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.<\/p>\n<ul>\n<li><strong>Percent change<\/strong> refers to a change based on a percent of the original amount.<\/li>\n<li><strong>Exponential growth<\/strong> refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.<\/li>\n<li><strong>Exponential decay<\/strong> refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.<\/li>\n<\/ul>\n<p>For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth.<\/p>\n<table style=\"border-collapse: collapse; width: 78.5374%; height: 132px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^x[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^0 = 1[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(0) = 0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^1 = 2[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(1) = 2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^2 = 4[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(2) = 4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^3 = 8[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(3) = 6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 18.5692%; height: 22px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 29.6804%; height: 22px;\">[latex]y = 2^4 = 16[\/latex]<\/td>\n<td style=\"width: 30.7433%; height: 22px;\">[latex]y = 2(4) = 8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can infer that for these two functions, exponential growth dwarfs linear growth.<\/p>\n<ul>\n<li><strong>Exponential growth<\/strong> refers to the original value from the range increases by the same percentage over equal increments found in the domain.<\/li>\n<li><strong>Linear growth<\/strong> refers to the original value from the range increases by the same amount over equal increments found in the domain.<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24699\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24699&theme=lumen&iframe_resize_id=ohm24699&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Apparently, the difference between \u201cthe same percentage\u201d and \u201cthe same amount\u201d is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding [latex]2[\/latex] to the output whenever the input was increased by one.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>exponential function<\/h3>\n<p>The general form of the <strong>exponential formula<\/strong>\u00a0is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\n<p>where [latex]a[\/latex] is any nonzero number and [latex]b[\/latex] is a positive real number not equal to [latex]1[\/latex].<\/p>\n<ul>\n<li>if [latex]b>1[\/latex], the function grows at a rate proportional to its size.<\/li>\n<li>if [latex]0 \\lt b \\lt 1[\/latex], the function decays at a rate proportional to its size.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>Why do we limit the base [latex]b[\/latex]\u00a0to positive values?<\/strong><\/p>\n<hr \/>\n<p>This is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\n<ul>\n<li>Consider a base of \u20139 and exponent of [latex]\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\n<\/ul>\n<p><strong>Why do we limit the base to positive values other than 1?<br \/>\n<\/strong><\/p>\n<hr \/>\n<p>This is because a base of 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:<\/p>\n<ul>\n<li>Consider a base of 1.\u00a0Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\n<\/ul>\n<\/section>\n<p>To evaluate an exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], we simply substitute [latex]x[\/latex] with the given value, and calculate the resulting power.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllll}f\\left(x\\right)\\hfill & ={2}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & ={2}^{3}\\text{}\\hfill & \\text{Substitute }x=3. \\hfill \\\\ \\hfill & =8\\text{}\\hfill & \\text{Evaluate the power}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<p>When evaluating an exponential function, it is important to follow the order of operations.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & =30{\\left(2\\right)}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & =30{\\left(2\\right)}^{3}\\hfill & \\text{Substitute }x=3.\\hfill \\\\ \\hfill & =30\\left(8\\right)\\text{ }\\hfill & \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill & =240\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<p>Note that if the order of operations were not followed, the result would be incorrect:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Let [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q454575\">Show Solution<\/button><\/p>\n<div id=\"q454575\" class=\"hidden-answer\" style=\"display: none\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & =5{\\left(3\\right)}^{x+1}\\hfill & \\hfill \\\\ f\\left(2\\right)\\hfill & =5{\\left(3\\right)}^{2+1}\\hfill & \\text{Substitute }x=2.\\hfill \\\\ \\hfill & =5{\\left(3\\right)}^{3}\\hfill & \\text{Add the exponents}.\\hfill \\\\ \\hfill & =5\\left(27\\right)\\hfill & \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill & =135\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm73212\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=73212&theme=lumen&iframe_resize_id=ohm73212&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":255,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2086"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":19,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2086\/revisions"}],"predecessor-version":[{"id":4731,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2086\/revisions\/4731"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/255"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2086\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2086"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2086"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2086"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2086"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}