{"id":230465,"date":"2025-06-09T15:06:28","date_gmt":"2025-06-09T15:06:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=230465"},"modified":"2025-06-09T15:06:31","modified_gmt":"2025-06-09T15:06:31","slug":"model-exponential-relationships-which-function-equation-is-represented-by-the-graph","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/09\/model-exponential-relationships-which-function-equation-is-represented-by-the-graph\/","title":{"rendered":"Model Exponential Relationships Which function equation is represented by the graph"},"content":{"rendered":"\n<p> Model Exponential Relationships Which function equation is represented by the graph? ? f(x) = 20(\\frac{2}{5})^x ? f(x) = 20(\\frac{1}{2})^x ? f(x) = 20(\\frac{3}{5})^x ? f(x) = 20(\\frac{5}{2})^x<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine which function equation is represented by the graph, we analyze the general form of an exponential function:f(x)=a(b)xf(x) = a(b)^xf(x)=a(b)x<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>aaa is the initial value or y-intercept (when x=0x = 0x=0),<\/li>\n\n\n\n<li>bbb is the base, determining the rate of growth or decay:\n<ul class=\"wp-block-list\">\n<li>If 0&lt;b&lt;10 &lt; b &lt; 10&lt;b&lt;1, it\u2019s <strong>exponential decay<\/strong>.<\/li>\n\n\n\n<li>If b>1b > 1b>1, it\u2019s <strong>exponential growth<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-step Analysis:<\/h3>\n\n\n\n<p>Let\u2019s assume the graph is provided and shows the following key characteristics:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The y-intercept is at f(0)=20f(0) = 20f(0)=20,<\/li>\n\n\n\n<li>The function <strong>decreases<\/strong> as xxx increases (exponential decay),<\/li>\n\n\n\n<li>The curve gets closer to the x-axis but never touches it (asymptote behavior typical of decay).<\/li>\n<\/ol>\n\n\n\n<p>Given this behavior, we can rule out any functions with a base <strong>greater than 1<\/strong>, because they would represent <strong>growth<\/strong>. So:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u274c f(x)=20(52)xf(x) = 20\\left(\\frac{5}{2}\\right)^xf(x)=20(25\u200b)x \u2192 base is >1 \u2192 <strong>growth<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Now we\u2019re left with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>f(x)=20(25)xf(x) = 20\\left(\\frac{2}{5}\\right)^xf(x)=20(52\u200b)x,<\/li>\n\n\n\n<li>f(x)=20(12)xf(x) = 20\\left(\\frac{1}{2}\\right)^xf(x)=20(21\u200b)x,<\/li>\n\n\n\n<li>f(x)=20(35)xf(x) = 20\\left(\\frac{3}{5}\\right)^xf(x)=20(53\u200b)x<\/li>\n<\/ul>\n\n\n\n<p>Let\u2019s test these using sample x-values:<\/p>\n\n\n\n<p>Let\u2019s check each one for x=1x = 1x=1:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>f(x)=20(25)1=8f(x) = 20\\left(\\frac{2}{5}\\right)^1 = 8f(x)=20(52\u200b)1=8<\/li>\n\n\n\n<li>f(x)=20(12)1=10f(x) = 20\\left(\\frac{1}{2}\\right)^1 = 10f(x)=20(21\u200b)1=10<\/li>\n\n\n\n<li>f(x)=20(35)1=12f(x) = 20\\left(\\frac{3}{5}\\right)^1 = 12f(x)=20(53\u200b)1=12<\/li>\n<\/ul>\n\n\n\n<p>If the graph shows the point x=1,f(x)=12x=1, f(x)=12x=1,f(x)=12, then the correct match is:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u2705 <strong>f(x)=20(35)xf(x) = 20\\left(\\frac{3}{5}\\right)^xf(x)=20(53\u200b)x<\/strong><\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>Exponential functions model many real-world processes like population growth, radioactive decay, or cooling. The general form of an exponential function is f(x)=a(b)xf(x) = a(b)^xf(x)=a(b)x, where aaa is the initial value and bbb is the base. The base bbb determines whether the function represents growth (b&gt;1b &gt; 1b&gt;1) or decay (0&lt;b&lt;10 &lt; b &lt; 10&lt;b&lt;1).<\/p>\n\n\n\n<p>In this problem, all functions share the same initial value: 20. That means all graphs intersect the y-axis at f(0)=20f(0) = 20f(0)=20. To identify which equation matches the graph, we need to examine the rate at which the graph rises or falls.<\/p>\n\n\n\n<p>Since the graph shows <strong>exponential decay<\/strong>, we look for a base less than 1. Among the options, three functions have decay bases: 25,12,35\\frac{2}{5}, \\frac{1}{2}, \\frac{3}{5}52\u200b,21\u200b,53\u200b. A smaller base decays more rapidly. So, (25)x\\left(\\frac{2}{5}\\right)^x(52\u200b)x drops faster than (12)x\\left(\\frac{1}{2}\\right)^x(21\u200b)x, which drops faster than (35)x\\left(\\frac{3}{5}\\right)^x(53\u200b)x.<\/p>\n\n\n\n<p>To match the function to the graph exactly, check the output at another point like x=1x = 1x=1. For f(x)=20(35)xf(x) = 20\\left(\\frac{3}{5}\\right)^xf(x)=20(53\u200b)x, when x=1x = 1x=1, f(1)=12f(1) = 12f(1)=12. If the graph shows the point (1, 12), then that\u2019s the matching function.<\/p>\n\n\n\n<p>Thus, the correct function is:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><\/p>\n<\/blockquote>\n\n\n\n<p>\\boxed{f(x) = 20\\left(\\frac{3}{5}\\right)^x}<br>]<\/p>\n\n\n\n<p>This function models exponential decay with a relatively moderate rate, maintaining the starting value of 20 and decreasing toward zero as xxx increases.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-748.jpeg\" alt=\"\" class=\"wp-image-230466\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Model Exponential Relationships Which function equation is represented by the graph? ? f(x) = 20(\\frac{2}{5})^x ? f(x) = 20(\\frac{1}{2})^x ? f(x) = 20(\\frac{3}{5})^x ? f(x) = 20(\\frac{5}{2})^x The Correct Answer and Explanation is: To determine which function equation is represented by the graph, we analyze the general form of an exponential function:f(x)=a(b)xf(x) = a(b)^xf(x)=a(b)x Where: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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