{"id":42744,"date":"2025-06-29T09:59:15","date_gmt":"2025-06-29T09:59:15","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=42744"},"modified":"2025-06-29T09:59:16","modified_gmt":"2025-06-29T09:59:16","slug":"determine-the-amplitude-period-and-midline-for-the-graph-shown-in-the-figure-below","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/determine-the-amplitude-period-and-midline-for-the-graph-shown-in-the-figure-below\/","title":{"rendered":"Determine the amplitude, period, and midline for the graph shown in the figure below."},"content":{"rendered":"\n

Determine the amplitude, period, and midline for the graph shown in the figure below. f(x) 9 8 7 6 5 4 3 2 1 -10 -8 -6 -4 -2 2 4 6 8 10 -1 amplitude 2 period 7 midline y = 5 Determine an equation for the graph involving the cosine function. f(x) =<\/p>\n\n\n\n

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

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

Here are the correct values and the equation for the given graph, along with a step-by-step explanation.<\/p>\n\n\n\n

1. Midline<\/strong><\/h3>\n\n\n\n

The midline is the horizontal line that runs exactly halfway between the maximum and minimum values of the graph.<\/p>\n\n\n\n

    \n
  • From the graph, the maximum value (the highest point) is\u00a0y = 8<\/strong>.<\/li>\n\n\n\n
  • The minimum value (the lowest point) is\u00a0y = 2<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    To find the midline, we calculate the average of the maximum and minimum values:
    Midline = (Maximum Value + Minimum Value) \/ 2
    Midline = (8 + 2) \/ 2 = 10 \/ 2 = 5<\/p>\n\n\n\n

    So, the midline<\/strong> is the line y = 5<\/strong>.<\/p>\n\n\n\n

    2. Amplitude<\/strong><\/h3>\n\n\n\n

    The amplitude is the distance from the midline to either a maximum or minimum point. It is half of the total height of the wave.<\/p>\n\n\n\n

    Amplitude = (Maximum Value – Minimum Value) \/ 2
    Amplitude = (8 – 2) \/ 2 = 6 \/ 2 = 3<\/p>\n\n\n\n

    Thus, the amplitude<\/strong> is 3<\/strong>.
    (Note: The value of 2 entered in the image for the amplitude is incorrect).<\/p>\n\n\n\n

    3. Period<\/strong><\/h3>\n\n\n\n

    The period is the horizontal length of one complete cycle. We can find this by measuring the horizontal distance between two consecutive minimums or maximums.<\/p>\n\n\n\n

      \n
    • The graph has a minimum at x = 0.<\/li>\n\n\n\n
    • The next minimum occurs at x = 7.<\/li>\n<\/ul>\n\n\n\n

      Period = (x-coordinate of second minimum) – (x-coordinate of first minimum)
      Period = 7 – 0 = 7<\/p>\n\n\n\n

      Therefore, the period<\/strong> is 7<\/strong>.<\/p>\n\n\n\n

      4. Equation of the Function<\/strong><\/h3>\n\n\n\n

      We need to find an equation in the form f(x) = Acos(B(x – C)) + D.<\/p>\n\n\n\n

        \n
      • A is the amplitude.<\/strong>\u00a0We found the amplitude is 3. However, the graph has a minimum at x=0, while a standard cosine function starts at a maximum. This indicates a vertical reflection, so we use\u00a0A = -3.<\/li>\n\n\n\n
      • D is the midline.<\/strong>\u00a0We found the midline is\u00a0y = 5, so\u00a0D = 5.<\/li>\n\n\n\n
      • B relates to the period.<\/strong>\u00a0The formula is Period = 2\u03c0\/B. We know the period is 7, so\u00a07 = 2\u03c0\/B. Solving for B gives\u00a0B = 2\u03c0\/7.<\/li>\n\n\n\n
      • C is the phase shift.<\/strong>\u00a0Since we used a reflected cosine (A = -3) which starts at a minimum, and our graph also starts at a minimum at x=0, there is no horizontal shift. So,\u00a0C = 0.<\/li>\n<\/ul>\n\n\n\n

        Plugging these values into the general equation:
        f(x) = -3cos((2\u03c0\/7)(x – 0)) + 5<\/p>\n\n\n\n

        This simplifies to the final equation:
        f(x) = -3cos((2\u03c0\/7)x) + 5<\/strong><\/p>\n\n\n\n

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

        Determine the amplitude, period, and midline for the graph shown in the figure below. f(x) 9 8 7 6 5 4 3 2 1 -10 -8 -6 -4 -2 2 4 6 8 10 -1 amplitude 2 period 7 midline y = 5 Determine an equation for the graph involving the cosine function. f(x) = […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-42744","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42744","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=42744"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42744\/revisions"}],"predecessor-version":[{"id":42749,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42744\/revisions\/42749"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=42744"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=42744"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=42744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}