{"id":227440,"date":"2025-06-06T06:26:52","date_gmt":"2025-06-06T06:26:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=227440"},"modified":"2025-06-06T06:26:54","modified_gmt":"2025-06-06T06:26:54","slug":"understanding-dilations-of-ex-complete-the-table-to-investigate-dilations-of-exponential-functions","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/06\/understanding-dilations-of-ex-complete-the-table-to-investigate-dilations-of-exponential-functions\/","title":{"rendered":"Understanding Dilations of Ex Complete the table to investigate dilations of exponential functions"},"content":{"rendered":"\n

Understanding Dilations of Ex Complete the table to investigate dilations of exponential functions. x 2x 3\\cdot2x 23x -2 1\\ 4 3\\ 4 1\\ 64 -1 1\\ 2 3\\ 2 1\\ 8 0 a b c 1 d e f 2 4 12 64 a = b = c = d = e = f =<\/p>\n\n\n\n

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

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

Completed Table:<\/h3>\n\n\n\n

Let’s evaluate each function for the missing values.<\/p>\n\n\n\n

Given functions:<\/p>\n\n\n\n

    \n
  • 2×2^x2x<\/li>\n\n\n\n
  • 3\u22c52×3 \\cdot 2^x3\u22c52x<\/li>\n\n\n\n
  • 23×2^{3x}23x<\/li>\n<\/ul>\n\n\n\n

    When x=0x = 0x=0:<\/h4>\n\n\n\n
      \n
    • 20=1\u21d2a=12^0 = 1 \\Rightarrow a = 120=1\u21d2a=1<\/li>\n\n\n\n
    • 3\u22c520=3\u22c51=3\u21d2b=33 \\cdot 2^0 = 3 \\cdot 1 = 3 \\Rightarrow b = 33\u22c520=3\u22c51=3\u21d2b=3<\/li>\n\n\n\n
    • 23\u22c50=20=1\u21d2c=12^{3 \\cdot 0} = 2^0 = 1 \\Rightarrow c = 123\u22c50=20=1\u21d2c=1<\/li>\n<\/ul>\n\n\n\n

      When x=1x = 1x=1:<\/h4>\n\n\n\n
        \n
      • 21=2\u21d2d=22^1 = 2 \\Rightarrow d = 221=2\u21d2d=2<\/li>\n\n\n\n
      • 3\u22c521=3\u22c52=6\u21d2e=63 \\cdot 2^1 = 3 \\cdot 2 = 6 \\Rightarrow e = 63\u22c521=3\u22c52=6\u21d2e=6<\/li>\n\n\n\n
      • 23\u22c51=23=8\u21d2f=82^{3 \\cdot 1} = 2^3 = 8 \\Rightarrow f = 823\u22c51=23=8\u21d2f=8<\/li>\n<\/ul>\n\n\n\n
        \n\n\n\n

        Final Answers:<\/h3>\n\n\n\n
          \n
        • a=1a = 1a=1<\/li>\n\n\n\n
        • b=3b = 3b=3<\/li>\n\n\n\n
        • c=1c = 1c=1<\/li>\n\n\n\n
        • d=2d = 2d=2<\/li>\n\n\n\n
        • e=6e = 6e=6<\/li>\n\n\n\n
        • f=8f = 8f=8<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

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

          This table investigates how exponential functions change when dilated (stretched or compressed) in different ways. The three functions here\u20142×2^x2x, 3\u22c52×3 \\cdot 2^x3\u22c52x, and 23×2^{3x}23x\u2014each demonstrate a different kind of transformation of the base exponential function.<\/p>\n\n\n\n

            \n
          1. 2×2^x2x<\/strong> is the base exponential function. As xxx increases, the output doubles with each step. For example, 2\u22122=142^{-2} = \\frac{1}{4}2\u22122=41\u200b, 2\u22121=122^{-1} = \\frac{1}{2}2\u22121=21\u200b, 20=12^0 = 120=1, 21=22^1 = 221=2, and 22=42^2 = 422=4.<\/li>\n\n\n\n
          2. 3\u22c52×3 \\cdot 2^x3\u22c52x<\/strong> represents a vertical dilation<\/strong> by a factor of 3. This means every value of 2×2^x2x is multiplied by 3. So, the shape of the graph stays the same, but all y-values are scaled three times higher.<\/li>\n\n\n\n
          3. 23×2^{3x}23x<\/strong> shows a horizontal compression<\/strong>. The exponent 3x3x3x causes the function to grow much faster. For instance, at x=2x = 2x=2, instead of calculating 22=42^2 = 422=4, we compute 26=642^{6} = 6426=64. This means the graph of 23×2^{3x}23x increases more steeply, compressing it horizontally compared to 2×2^x2x.<\/li>\n<\/ol>\n\n\n\n

            In summary, these variations help students understand how exponential functions behave under dilations\u2014whether stretching or compressing vertically or horizontally\u2014shaping their graphs differently while preserving their general exponential nature.<\/p>\n\n\n\n

            \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

            Understanding Dilations of Ex Complete the table to investigate dilations of exponential functions. x 2x 3\\cdot2x 23x -2 1\\ 4 3\\ 4 1\\ 64 -1 1\\ 2 3\\ 2 1\\ 8 0 a b c 1 d e f 2 4 12 64 a = b = c = d = e = f = […]<\/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-227440","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227440","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=227440"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227440\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=227440"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=227440"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=227440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}