{"id":183442,"date":"2025-01-16T13:02:34","date_gmt":"2025-01-16T13:02:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183442"},"modified":"2025-01-16T13:02:35","modified_gmt":"2025-01-16T13:02:35","slug":"use-the-given-transformation-to-graph-the-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/16\/use-the-given-transformation-to-graph-the-function\/","title":{"rendered":"Use the given transformation to graph the function"},"content":{"rendered":"\n<p>Use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal squared function f (x) = 2 shifted to the right 2 units. Hint: The graph of the reciprocal square function is shown in Figure 1 in the Reading and Participation Activities section on Rational Functions. Select the correct graph of the function.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-267-1024x783.png\" alt=\"\" class=\"wp-image-183443\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To graph the given function, we start with the reciprocal squared function ( f(x) = \\frac{1}{x^2} ), and apply the specified transformations:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Start with the reciprocal squared function<\/strong>: The base function is ( f(x) = \\frac{1}{x^2} ). It has:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A vertical asymptote at ( x = 0 ), since ( \\frac{1}{x^2} ) becomes undefined at ( x = 0 ).<\/li>\n\n\n\n<li>A horizontal asymptote at ( y = 0 ), since as ( |x| \\to \\infty ), ( \\frac{1}{x^2} \\to 0 ).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Shift the function to the right by 2 units<\/strong>: To shift the function ( f(x) = \\frac{1}{x^2} ) to the right by 2 units, replace ( x ) with ( x &#8211; 2 ). This gives the transformed function:<br>[<br>f(x) = \\frac{1}{(x &#8211; 2)^2}.<br>]<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The vertical asymptote moves from ( x = 0 ) to ( x = 2 ).<\/li>\n\n\n\n<li>The horizontal asymptote remains at ( y = 0 ), as the shift does not affect the long-term behavior of the function.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Multiply by 2<\/strong>: Finally, multiply the function by 2:<br>[<br>f(x) = \\frac{2}{(x &#8211; 2)^2}.<br>]<br>This stretches the graph vertically by a factor of 2, making it rise faster as ( x ) approaches the vertical asymptote from either side.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Key Features of the Final Graph:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vertical Asymptote<\/strong>: ( x = 2 ).<\/li>\n\n\n\n<li><strong>Horizontal Asymptote<\/strong>: ( y = 0 ).<\/li>\n\n\n\n<li><strong>Behavior<\/strong>:<\/li>\n\n\n\n<li>As ( x \\to 2^+ ) or ( x \\to 2^- ), ( f(x) \\to \\infty ).<\/li>\n\n\n\n<li>As ( x \\to \\pm \\infty ), ( f(x) \\to 0^+ ).<\/li>\n<\/ul>\n\n\n\n<p>The correct graph will show a hyperbolic curve with the specified asymptotes and a steeper rise due to the vertical stretch. Based on the provided figure, match these characteristics to the correct graph.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal squared function f (x) = 2 shifted to the right 2 units. Hint: The graph of the reciprocal square function is shown in Figure 1 in the Reading and Participation Activities section on Rational Functions. Select the correct graph [&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 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-183442","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183442","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=183442"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183442\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183442"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183442"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}