To find the range of the quadratic function f(x)=2×2+6x\u22128f(x) = 2x^2 + 6x – 8f(x)=2×2+6x\u22128, we can follow a few key steps:<\/p>\n\n\n\n
Step 1: Identify the shape of the graph<\/h3>\n\n\n\n
The function is a quadratic equation, and since the coefficient of x2x^2×2 (which is 2) is positive, the graph is a parabola that opens upward<\/strong>.<\/p>\n\n\n\n The range of a parabola depends on its vertex. The vertex of a parabola represented by f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c can be found using the formula for the x-coordinate of the vertex:xvertex=\u2212b2ax_{\\text{vertex}} = \\frac{-b}{2a}xvertex\u200b=2a\u2212b\u200b<\/p>\n\n\n\n For the function f(x)=2×2+6x\u22128f(x) = 2x^2 + 6x – 8f(x)=2×2+6x\u22128:<\/p>\n\n\n\n Thus, the x-coordinate of the vertex is:xvertex=\u221262(2)=\u221264=\u221232x_{\\text{vertex}} = \\frac{-6}{2(2)} = \\frac{-6}{4} = -\\frac{3}{2}xvertex\u200b=2(2)\u22126\u200b=4\u22126\u200b=\u221223\u200b<\/p>\n\n\n\n Substitute x=\u221232x = -\\frac{3}{2}x=\u221223\u200b back into the original function to find the y-coordinate of the vertex.f(\u221232)=2(\u221232)2+6(\u221232)\u22128f\\left(-\\frac{3}{2}\\right) = 2\\left(-\\frac{3}{2}\\right)^2 + 6\\left(-\\frac{3}{2}\\right) – 8f(\u221223\u200b)=2(\u221223\u200b)2+6(\u221223\u200b)\u22128<\/p>\n\n\n\n First, square \u221232-\\frac{3}{2}\u221223\u200b:(\u221232)2=94\\left(-\\frac{3}{2}\\right)^2 = \\frac{9}{4}(\u221223\u200b)2=49\u200b<\/p>\n\n\n\n Now substitute into the function:f(\u221232)=2\u00d794+6\u00d7(\u221232)\u22128f\\left(-\\frac{3}{2}\\right) = 2 \\times \\frac{9}{4} + 6 \\times \\left(-\\frac{3}{2}\\right) – 8f(\u221223\u200b)=2\u00d749\u200b+6\u00d7(\u221223\u200b)\u22128f(\u221232)=184\u22129\u22128f\\left(-\\frac{3}{2}\\right) = \\frac{18}{4} – 9 – 8f(\u221223\u200b)=418\u200b\u22129\u22128f(\u221232)=184\u221217=184\u2212684=\u2212504=\u221212.5f\\left(-\\frac{3}{2}\\right) = \\frac{18}{4} – 17 = \\frac{18}{4} – \\frac{68}{4} = \\frac{-50}{4} = -12.5f(\u221223\u200b)=418\u200b\u221217=418\u200b\u2212468\u200b=4\u221250\u200b=\u221212.5<\/p>\n\n\n\n So, the vertex is at (\u221232,\u221212.5)\\left(-\\frac{3}{2}, -12.5\\right)(\u221223\u200b,\u221212.5).<\/p>\n\n\n\n Since the parabola opens upward, the lowest value of the function is the y-coordinate of the vertex, \u221212.5-12.5\u221212.5, and the function increases without bound as xxx moves away from the vertex. Therefore, the range of the function is:Range=[\u221212.5,\u221e)\\text{Range} = [-12.5, \\infty)Range=[\u221212.5,\u221e)<\/p>\n\n\n\n The range of f(x)=2×2+6x\u22128f(x) = 2x^2 + 6x – 8f(x)=2×2+6x\u22128 is:[\u221212.5,\u221e)[-12.5, \\infty)[\u221212.5,\u221e)<\/p>\n\n\n\n Find the range of f(x) = 2x^2 + 6x – 8. (Enter your answer using interval notation The Correct Answer and Explanation is: To find the range of the quadratic function f(x)=2×2+6x\u22128f(x) = 2x^2 + 6x – 8f(x)=2×2+6x\u22128, we can follow a few key steps: Step 1: Identify the shape of the graph The function […]<\/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-244555","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244555","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=244555"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244555\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244555"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244555"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244555"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 2: Find the vertex<\/h3>\n\n\n\n
\n
Step 3: Find the y-coordinate of the vertex<\/h3>\n\n\n\n
Step 4: Determine the range<\/h3>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
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