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Given the function h(x)=-x^(2)-9x+23, determine the average rate of change of the function over the interval -7 <= x <= 1. Answer:

Given the function h(x)=−x2−9x+23 h(x)=-x^{2}-9 x+23 , determine the average rate of change of the function over the interval −7≤x≤1 -7 \leq x \leq 1 .\newlineAnswer:

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Q. Given the function h(x)=−x2−9x+23 h(x)=-x^{2}-9 x+23 , determine the average rate of change of the function over the interval −7≤x≤1 -7 \leq x \leq 1 .\newlineAnswer:
  1. Given Function: We have the function h(x)=−x2−9x+23h(x) = -x^2 - 9x + 23. To find the average rate of change over the interval [−7,1][-7, 1], we will use the formula for the average rate of change, which is (h(b)−h(a))/(b−a)(h(b) - h(a)) / (b - a), where aa and bb are the endpoints of the interval.
  2. Calculate h(−7)h(-7): First, we need to find the value of h(−7)h(-7). We substitute x=−7x = -7 into the function h(x)h(x).
    h(−7)=−(−7)2−9(−7)+23h(-7) = -(-7)^2 - 9(-7) + 23
    h(−7)=−(49)+63+23h(-7) = -(49) + 63 + 23
    h(−7)=−49+63+23h(-7) = -49 + 63 + 23
    h(−7)=14+23h(-7) = 14 + 23
    h(−7)=37h(-7) = 37
  3. Calculate h(1)h(1): Next, we need to find the value of h(1)h(1). We substitute x=1x = 1 into the function h(x)h(x).
    h(1)=−(1)2−9(1)+23h(1) = -(1)^2 - 9(1) + 23
    h(1)=−1−9+23h(1) = -1 - 9 + 23
    h(1)=−10+23h(1) = -10 + 23
    h(1)=13h(1) = 13
  4. Average Rate of Change: Now we have h(−7)=37h(-7) = 37 and h(1)=13h(1) = 13. We can use these values to calculate the average rate of change over the interval [−7,1][-7, 1].\newlineAverage rate of change = (h(1)−h(−7))/(1−(−7))(h(1) - h(-7)) / (1 - (-7))\newlineAverage rate of change = (13−37)/(1+7)(13 - 37) / (1 + 7)\newlineAverage rate of change = (−24)/8(-24) / 8\newlineAverage rate of change = −3-3

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