given the function ( h(x) = -x^{2}-5x + 9 ), determine the average rate of change of the function over the…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = h(x)) over the interval ([a,b]) is (\frac{h(b)-h(a)}{b - a}). Here (a=-8) and (b = 2).

Step2: Calculate (h(-8))

Substitute (x=-8) into (h(x)=-x^{2}-5x + 9). [ \begin{align*} h(-8)&=-(-8)^{2}-5\times(-8)+9\ &=-64 + 40+9\ &=-15 \end{align*} ]

Step3: Calculate (h(2))

Substitute (x = 2) into (h(x)=-x^{2}-5x + 9). [ \begin{align*} h(2)&=-(2)^{2}-5\times(2)+9\ &=-4-10 + 9\ &=-5 \end{align*} ]

Step4: Calculate the average rate of change

Substitute (h(-8)=-15), (h(2)=-5), (a=-8), and (b = 2) into the formula (\frac{h(b)-h(a)}{b - a}). [ \begin{align*} \frac{h(2)-h(-8)}{2-(-8)}&=\frac{-5-(-15)}{2 + 8}\ &=\frac{-5 + 15}{10}\ &=\frac{10}{10}\ &=1 \end{align*} ]

Answer:

(1)