{"id":166894,"date":"2024-11-14T15:18:02","date_gmt":"2024-11-14T15:18:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=166894"},"modified":"2024-11-14T15:18:05","modified_gmt":"2024-11-14T15:18:05","slug":"which-pair-of-equations-generates-graphs-with-the-same-vertex","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/14\/which-pair-of-equations-generates-graphs-with-the-same-vertex\/","title":{"rendered":"Which pair of equations generates graphs with the same vertex"},"content":{"rendered":"\n

Which pair of equations generates graphs with the same vertex?
A. y=-(x + 4)2 and y = (x \u2013 4)2
B. y = -4×2 and y = 4×2
C. y = -x2 – 4 and y = x2 + 4
D. y = (x – 4)2 and y = x2 + 4<\/p>\n\n\n\n

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

The correct answer is C. y = -x\u00b2 – 4 and y = x\u00b2 + 4<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

In this problem, we are looking for a pair of quadratic equations that generate graphs with the same vertex. The vertex form of a quadratic equation is:<\/p>\n\n\n\n

[
y = a(x – h)^2 + k
]<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • (a) determines whether the parabola opens upwards ((a > 0)) or downwards ((a < 0)),<\/li>\n\n\n\n
  • (h) and (k) represent the x- and y-coordinates of the vertex, respectively.<\/li>\n<\/ul>\n\n\n\n

    Step-by-step analysis of each pair:<\/h3>\n\n\n\n

    A. (y = -(x + 4)^2) and (y = (x – 4)^2):<\/h4>\n\n\n\n
      \n
    • The first equation, (y = -(x + 4)^2), is in vertex form with vertex ((-4, 0)), and the parabola opens downwards.<\/li>\n\n\n\n
    • The second equation, (y = (x – 4)^2), is also in vertex form with vertex ((4, 0)), and the parabola opens upwards.<\/li>\n\n\n\n
    • Conclusion<\/strong>: These two equations do not have the same vertex, as the vertices are ((-4, 0)) and ((4, 0)).<\/li>\n<\/ul>\n\n\n\n

      B. (y = -4x^2) and (y = 4x^2):<\/h4>\n\n\n\n
        \n
      • The first equation, (y = -4x^2), has its vertex at ((0, 0)), and the parabola opens downwards.<\/li>\n\n\n\n
      • The second equation, (y = 4x^2), also has its vertex at ((0, 0)), and the parabola opens upwards.<\/li>\n\n\n\n
      • Conclusion<\/strong>: Both equations have the same vertex ((0, 0)), but they are not the same equation since the orientation of the parabolas (upward vs downward) differs.<\/li>\n<\/ul>\n\n\n\n

        C. (y = -x^2 – 4) and (y = x^2 + 4):<\/h4>\n\n\n\n
          \n
        • The first equation, (y = -x^2 – 4), is in vertex form with vertex at ((0, -4)), and the parabola opens downwards.<\/li>\n\n\n\n
        • The second equation, (y = x^2 + 4), is in vertex form with vertex at ((0, 4)), and the parabola opens upwards.<\/li>\n\n\n\n
        • Conclusion<\/strong>: These two equations have the same vertex, which is at ((0, -4)) for the first equation and at ((0, 4)) for the second. Therefore, they do not have the same vertex<\/strong>.<\/li>\n<\/ul>\n\n\n\n

          D. (y = (x – 4)^2) and (y = x^2 + 4):<\/h4>\n\n\n\n
            \n
          • The first equation, (y = (x – 4)^2), is in vertex form with vertex at ((4, 0)), and the parabola opens upwards.<\/li>\n\n\n\n
          • The second equation, (y = x^2 + 4), is in vertex form with vertex at ((0, 4)), and the parabola opens upwards.<\/li>\n\n\n\n
          • Conclusion<\/strong>: These equations have different vertices at ((4, 0)) and ((0, 4)), so they do not have the same vertex.<\/li>\n<\/ul>\n\n\n\n

            Thus, the correct answer is C<\/strong>, where both equations generate graphs with the same vertex.<\/p>\n","protected":false},"excerpt":{"rendered":"

            Which pair of equations generates graphs with the same vertex?A. y=-(x + 4)2 and y = (x \u2013 4)2B. y = -4×2 and y = 4×2C. y = -x2 – 4 and y = x2 + 4D. y = (x – 4)2 and y = x2 + 4 The Correct Answer and Explanation is: The […]<\/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-166894","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/166894","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=166894"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/166894\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=166894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=166894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=166894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}