Question

Given the function f(x) = -x^2 - 9x + 25, determine the average rate of change of the function over the interval −6 ≤ x ≤ 2.

Answer 1

Step-by-step explanation

To find the average rate of change of the function, we have to apply the formula:

`(f(b)-f(a))/(b-a)`

for a ≤ x ≤ b

This implies that:

`\begin{gathered} a=-6 \\ b=2 \end{gathered}`

Therefore, the average rate of change of the function is:

`(f(2)-f(-6))/(2-(-6))`

To find f(2), we have to substitute 2 for x in the given function:

`\begin{gathered} f(2)=-(2)^2-9(2)+25 \\ f(2)=-4-18+25 \\ f(2)=3 \end{gathered}`

To find f(-6), substitute -6 for x in the given function:

`\begin{gathered} f(-6)=-(-6)^2-9(-6)+25 \\ f(-6)=-36+54+25 \\ f(-6)=43 \end{gathered}`

Therefore, the average rate of change of the function is:

`\begin{gathered} (3-43)/(2+6) \\ \Rightarrow(-40)/(8) \\ \Rightarrow-5 \end{gathered}`

That is the answer.