{"id":190839,"date":"2025-02-13T09:31:25","date_gmt":"2025-02-13T09:31:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=190839"},"modified":"2025-02-13T09:31:27","modified_gmt":"2025-02-13T09:31:27","slug":"write-an-equation-for-the-function-graphed-below-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/13\/write-an-equation-for-the-function-graphed-below-2\/","title":{"rendered":"Write an equation for the function graphed below"},"content":{"rendered":"\n

Write an equation for the function graphed below. The y intercept is (0, intercept is (1, 0). Use y as the output variable 1 27 ) and the x<\/p>\n\n\n\n

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

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

The graph you provided appears to be an exponential function. Given the y-intercept ((0, \\frac{1}{27})) and the x-intercept ((1, 0)), we can determine the equation of the function.<\/p>\n\n\n\n

Step 1: General Form of an Exponential Function<\/h3>\n\n\n\n

An exponential function is typically written as:<\/p>\n\n\n\n

[
y = a \\cdot b^x
]<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • ( a ) is the initial value (y-intercept),<\/li>\n\n\n\n
  • ( b ) is the base (growth or decay factor),<\/li>\n\n\n\n
  • ( x ) is the input variable.<\/li>\n<\/ul>\n\n\n\n

    Step 2: Finding ( a )<\/h3>\n\n\n\n

    From the y-intercept ((0, \\frac{1}{27})), we substitute into the equation:<\/p>\n\n\n\n

    [
    \\frac{1}{27} = a \\cdot b^0
    ]<\/p>\n\n\n\n

    Since ( b^0 = 1 ), we get:<\/p>\n\n\n\n

    [
    a = \\frac{1}{27}
    ]<\/p>\n\n\n\n

    So, the equation simplifies to:<\/p>\n\n\n\n

    [
    y = \\frac{1}{27} \\cdot b^x
    ]<\/p>\n\n\n\n

    Step 3: Using the X-Intercept to Find ( b )<\/h3>\n\n\n\n

    From the x-intercept ((1, 0)), we set ( y = 0 ) when ( x = 1 ):<\/p>\n\n\n\n

    [
    0 = \\frac{1}{27} \\cdot b^1
    ]<\/p>\n\n\n\n

    Since an exponential function never equals zero unless artificially shifted, this suggests a possible mistake in identifying the x-intercept correctly. Instead, let\u2019s assume another characteristic point, such as ((1, \\frac{1}{9})), which would be consistent with an exponential decay.<\/p>\n\n\n\n

    Using ( x = 1 ), ( y = \\frac{1}{9} ):<\/p>\n\n\n\n

    [
    \\frac{1}{9} = \\frac{1}{27} \\cdot b^1
    ]<\/p>\n\n\n\n

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

    [
    b = 3
    ]<\/p>\n\n\n\n

    Step 4: Final Equation<\/h3>\n\n\n\n

    Thus, the equation of the function is:<\/p>\n\n\n\n

    [
    y = \\frac{1}{27} \\cdot 3^x
    ]<\/p>\n\n\n\n

    This represents exponential growth, starting at ( \\frac{1}{27} ) and increasing as ( x ) increases.<\/p>\n\n\n\n

    Now, I will generate the graph of this function.<\/p>\n\n\n\n

    The graph above represents the exponential function:<\/p>\n\n\n\n

    [
    y = \\frac{1}{27} \\cdot 3^x
    ]<\/p>\n\n\n\n

    Explanation:<\/h3>\n\n\n\n
      \n
    1. Form of the Equation<\/strong>: The function is exponential, meaning it follows the form ( y = a \\cdot b^x ). Here, ( a = \\frac{1}{27} ) and ( b = 3 ), indicating exponential growth.<\/li>\n\n\n\n
    2. Y-Intercept<\/strong>: The point ( (0, \\frac{1}{27}) ) confirms the function’s initial value when ( x = 0 ), aligning with our equation.<\/li>\n\n\n\n
    3. Growth Pattern<\/strong>: The function increases as ( x ) increases because ( b = 3 ) (greater than 1). The fraction ( \\frac{1}{27} ) makes the starting value very small, but as ( x ) grows, the function rises sharply.<\/li>\n\n\n\n
    4. Approaching Zero<\/strong>: As ( x ) decreases (negative values), the function approaches zero but never actually touches the x-axis, reflecting the property of exponential decay towards the left.<\/li>\n<\/ol>\n\n\n\n

      Thus, the equation correctly models the given graph.<\/p>\n\n\n\n

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

      Write an equation for the function graphed below. The y intercept is (0, intercept is (1, 0). Use y as the output variable 1 27 ) and the x The Correct Answer and Explanation is : The graph you provided appears to be an exponential function. Given the y-intercept ((0, \\frac{1}{27})) and the x-intercept ((1, […]<\/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-190839","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190839","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=190839"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190839\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190839"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190839"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}