{"id":1026,"date":"2024-04-10T15:32:11","date_gmt":"2024-04-10T15:32:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1026"},"modified":"2024-08-05T12:24:48","modified_gmt":"2024-08-05T12:24:48","slug":"more-basic-functions-and-graphs-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/more-basic-functions-and-graphs-background-youll-need-3\/","title":{"raw":"More Basic Functions and Graphs: Background You\u2019ll Need 3","rendered":"More Basic Functions and Graphs: Background You\u2019ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Explain the difference between exponential growth and decay<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identify Exponential Growth and Decay<\/h2>\r\n<p>In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth (or decay), we may choose to model the given scenario using the following function:<\/p>\r\n<p style=\"text-align: center;\">[latex]y={A}_{0}{b}^{x}[\/latex]<\/p>\r\n<p>where [latex]{A}_{0}[\/latex] is equal to the value at [latex]x=0[\/latex],\u00a0[latex]b[\/latex] is the base, and [latex]x[\/latex]\u00a0is the exponent. Note that the variable is in the exponent which makes the function exponential.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>exponential function<\/h3>\r\n<p>For any real number [latex]x[\/latex], an exponential function is a function with the form<\/p>\r\n<center>[latex]y={A}_{0}{b}^{x}[\/latex]<\/center>\r\n<p>where<\/p>\r\n<ul>\r\n\t<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\r\n\t<li>[latex]b[\/latex] is any positive real number such that [latex]b\u22601[\/latex].<\/li>\r\n\t<li>The domain is [latex]\\left(-\\infty , \\infty \\right)[\/latex], or all real numbers<\/li>\r\n\t<li>The range is all positive real numbers if [latex]a &gt; 0[\/latex]<\/li>\r\n\t<li>The range is all negative real numbers if [latex]a &lt; 0[\/latex]<\/li>\r\n\t<li>The y-intercept is\u00a0 [latex]\\left(0,{A}_{0}\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex]<\/li>\r\n<\/ul>\r\n<p>An exponential function models <strong>exponential growth<\/strong> when [latex]b &gt; 1[\/latex] and <strong>exponential decay<\/strong> when [latex]b &lt; 1[\/latex].<\/p>\r\n<\/section>\r\n<p>When [latex]b&gt;1[\/latex], the exponential function represents <strong>exponential growt<\/strong><strong>h<\/strong>. Common applications of exponential growth include\u00a0doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.<\/p>\r\n<p>When [latex]b&lt;1[\/latex], the exponential function represents <strong>exponential decay<\/strong>. One common application of exponential decay includes\u00a0calculating half-life,\u00a0or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.<\/p>\r\n<p>Exponential growth and decay graphs have a distinctive shape, as we can see in the graphs below. It is important to remember that, although parts of each of the two graphs seem to lie on the [latex]x[\/latex]-axis, they are really a tiny distance above the [latex]x[\/latex]-axis.<\/p>\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\/03181337\/CNX_Precalc_Figure_04_07_0022.jpg\" alt=\"Graph of y=2e^(3x) with the labeled points (-1\/3, 2\/e), (0, 2), and (1\/3, 2e) and with the asymptote at y=0.\" width=\"487\" height=\"326\" \/> A graph showing exponential growth. The equation is [latex]y=2{e}^{3x}[\/latex].[\/caption] [caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03181339\/CNX_Precalc_Figure_04_07_0032.jpg\" alt=\"Graph of y=3e^(-2x) with the labeled points (-1\/2, 3e), (0, 3), and (1\/2, 3\/e) and with the asymptote at y=0.\" width=\"487\" height=\"438\" \/> A graph showing exponential decay. The equation is [latex]y=3{e}^{-2x}[\/latex].[\/caption]\r\n\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]218951[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]218952[\/ohm_question]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Explain the difference between exponential growth and decay<\/li>\n<\/ul>\n<\/section>\n<h2>Identify Exponential Growth and Decay<\/h2>\n<p>In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth (or decay), we may choose to model the given scenario using the following function:<\/p>\n<p style=\"text-align: center;\">[latex]y={A}_{0}{b}^{x}[\/latex]<\/p>\n<p>where [latex]{A}_{0}[\/latex] is equal to the value at [latex]x=0[\/latex],\u00a0[latex]b[\/latex] is the base, and [latex]x[\/latex]\u00a0is the exponent. Note that the variable is in the exponent which makes the function exponential.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>exponential function<\/h3>\n<p>For any real number [latex]x[\/latex], an exponential function is a function with the form<\/p>\n<div style=\"text-align: center;\">[latex]y={A}_{0}{b}^{x}[\/latex]<\/div>\n<p>where<\/p>\n<ul>\n<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\n<li>[latex]b[\/latex] is any positive real number such that [latex]b\u22601[\/latex].<\/li>\n<li>The domain is [latex]\\left(-\\infty , \\infty \\right)[\/latex], or all real numbers<\/li>\n<li>The range is all positive real numbers if [latex]a > 0[\/latex]<\/li>\n<li>The range is all negative real numbers if [latex]a < 0[\/latex]<\/li>\n<li>The y-intercept is\u00a0 [latex]\\left(0,{A}_{0}\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex]<\/li>\n<\/ul>\n<p>An exponential function models <strong>exponential growth<\/strong> when [latex]b > 1[\/latex] and <strong>exponential decay<\/strong> when [latex]b < 1[\/latex].<\/p>\n<\/section>\n<p>When [latex]b>1[\/latex], the exponential function represents <strong>exponential growt<\/strong><strong>h<\/strong>. Common applications of exponential growth include\u00a0doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.<\/p>\n<p>When [latex]b<1[\/latex], the exponential function represents <strong>exponential decay<\/strong>. One common application of exponential decay includes\u00a0calculating half-life,\u00a0or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.<\/p>\n<p>Exponential growth and decay graphs have a distinctive shape, as we can see in the graphs below. It is important to remember that, although parts of each of the two graphs seem to lie on the [latex]x[\/latex]-axis, they are really a tiny distance above the [latex]x[\/latex]-axis.<\/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\/03181337\/CNX_Precalc_Figure_04_07_0022.jpg\" alt=\"Graph of y=2e^(3x) with the labeled points (-1\/3, 2\/e), (0, 2), and (1\/3, 2e) and with the asymptote at y=0.\" width=\"487\" height=\"326\" \/><figcaption class=\"wp-caption-text\">A graph showing exponential growth. The equation is [latex]y=2{e}^{3x}[\/latex].<\/figcaption><\/figure>\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\/03181339\/CNX_Precalc_Figure_04_07_0032.jpg\" alt=\"Graph of y=3e^(-2x) with the labeled points (-1\/2, 3e), (0, 3), and (1\/2, 3\/e) and with the asymptote at y=0.\" width=\"487\" height=\"438\" \/><figcaption class=\"wp-caption-text\">A graph showing exponential decay. The equation is [latex]y=3{e}^{-2x}[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm218951\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218951&theme=lumen&iframe_resize_id=ohm218951&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm218952\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218952&theme=lumen&iframe_resize_id=ohm218952&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"background_you_need","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1026"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1026\/revisions"}],"predecessor-version":[{"id":4468,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1026\/revisions\/4468"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1026\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1026"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1026"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1026"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}