The function is given to be:
[tex]f(x)=-0.5(x-8)^2-2[/tex]
To calculate the average rate of change between two points, we can use the formula:
[tex]\Rightarrow\frac{f(a)-f(b)}{a-b}[/tex]
where a and b are the points.
For the function provided, we are to find the rate of change between -4 and 8. Hence, our formula will be:
[tex]\frac{f(-4)-f(8)}{-4-8}=\frac{f(-4)-f(8)}{-12}[/tex]
We can evaluate f(-4) and f(8) to be:
[tex]\begin{gathered} f(-4)=-0.5(-4-8)^2-2=-0.5(-12)^2-2=-0.5(144)-2=-72-2=-74 \\ f(8)=-0.5(8-8)^2-2=-2 \end{gathered}[/tex]
Therefore, the average rate of change is calculated to be:
[tex]\Rightarrow\frac{-74-(-2)}{-12}=\frac{-74+2}{-12}=\frac{-72}{-12}=6[/tex]
The average rate of change is 6.
