{"id":186448,"date":"2025-01-25T07:51:45","date_gmt":"2025-01-25T07:51:45","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186448"},"modified":"2025-01-25T07:51:47","modified_gmt":"2025-01-25T07:51:47","slug":"department-of-mathematics","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/25\/department-of-mathematics\/","title":{"rendered":"Department of Mathematics"},"content":{"rendered":"\n

Department of Mathematics Course Help Hw19 Obj C4: Problem 5 Previous Problem List Next (1 point) Find the exponential function f(x)-a.b whose graph is shown below 13.6) Preview My Answers Submit Answers Page generated at 10\/17\/2018 at 01 58am EDT<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To determine the exponential function ( f(x) = a \\cdot b^x ) that matches the provided graph, we need to analyze the graph’s key features, such as the initial value and the growth rate. The parameter ( a ) represents the initial value of the function when ( x = 0 ), and ( b ) is the base that dictates the growth rate.<\/p>\n\n\n\n

Step 1: Identify the Initial Value<\/strong><\/p>\n\n\n\n

The initial value ( a ) is the value of ( f(x) ) when ( x = 0 ). From the graph, observe the point where the curve intersects the y-axis (i.e., when ( x = 0 )). This point provides the value of ( a ).<\/p>\n\n\n\n

Step 2: Determine the Growth Rate<\/strong><\/p>\n\n\n\n

The base ( b ) determines how the function grows as ( x ) increases. If the graph shows exponential growth, ( b ) will be greater than 1; if it shows exponential decay, ( b ) will be between 0 and 1. To find ( b ), select two points on the graph with known coordinates. Let these points be ( (x_1, y_1) ) and ( (x_2, y_2) ). Using the general form of the exponential function, we have:<\/p>\n\n\n\n

[ y_1 = a \\cdot b^{x_1} ]
[ y_2 = a \\cdot b^{x_2} ]<\/p>\n\n\n\n

Dividing the second equation by the first eliminates ( a ):<\/p>\n\n\n\n

[ \\frac{y_2}{y_1} = \\frac{a \\cdot b^{x_2}}{a \\cdot b^{x_1}} = b^{x_2 – x_1} ]<\/p>\n\n\n\n

Taking the natural logarithm of both sides:<\/p>\n\n\n\n

[ \\ln\\left( \\frac{y_2}{y_1} \\right) = (x_2 – x_1) \\cdot \\ln(b) ]<\/p>\n\n\n\n

Solving for ( \\ln(b) ):<\/p>\n\n\n\n

[ \\ln(b) = \\frac{\\ln\\left( \\frac{y_2}{y_1} \\right)}{x_2 – x_1} ]<\/p>\n\n\n\n

Finally, exponentiating both sides gives ( b ):<\/p>\n\n\n\n

[ b = \\exp\\left( \\frac{\\ln\\left( \\frac{y_2}{y_1} \\right)}{x_2 – x_1} \\right) ]<\/p>\n\n\n\n

Step 3: Construct the Exponential Function<\/strong><\/p>\n\n\n\n

With ( a ) and ( b ) determined, the exponential function is:\ue186<\/p>\n\n\n\n

[ f(x) = a \\cdot b^x ]<\/p>\n\n\n\n

Example:<\/strong><\/p>\n\n\n\n

Suppose the graph shows that when ( x = 0 ), ( f(x) = 3 ) (so ( a = 3 )), and when ( x = 2 ), ( f(x) = 12 ). Using the formula for ( b ):\ue186<\/p>\n\n\n\n

[ b = \\exp\\left( \\frac{\\ln\\left( \\frac{12}{3} \\right)}{2 – 0} \\right) = \\exp\\left( \\frac{\\ln(4)}{2} \\right) = \\exp\\left( \\frac{2 \\ln(2)}{2} \\right) = \\exp(\\ln(2)) = 2 ]<\/p>\n\n\n\n

Therefore, the exponential function is:\ue186<\/p>\n\n\n\n

[ f(x) = 3 \\cdot 2^x ]<\/p>\n\n\n\n

This function starts at 3 when ( x = 0 ) and doubles every time ( x ) increases by 1, matching the observed growth rate.<\/p>\n\n\n\n

For a visual explanation of graphing exponential functions, you might find the following video helpful:<\/p>\n\n\n\n

Graphing Exponential Functions With e, Transformations<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Department of Mathematics Course Help Hw19 Obj C4: Problem 5 Previous Problem List Next (1 point) Find the exponential function f(x)-a.b whose graph is shown below 13.6) Preview My Answers Submit Answers Page generated at 10\/17\/2018 at 01 58am EDT The Correct Answer and Explanation is : To determine the exponential function ( f(x) = […]<\/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 center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-186448","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186448","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=186448"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186448\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186448"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186448"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}