To determine the average rate of change of the function $h(x) = -x^2 - 5x + 9$ over the interval $[-8, 2]$, we use the formula for the average rate of change:

$\text{Average rate of change} = \frac{h(b) - h(a)}{b - a}$

where $a = -8$ and $b = 2$.

Step 1: Calculate $h(a)$ and $h(b)$

1. Calculate $h(-8)$:

   $h(-8) = -(-8)^2 - 5(-8) + 9$ $= -64 + 40 + 9$ $= -64 + 49$ $= -15$

2. Calculate $h(2)$:

   $h(2) = -(2)^2 - 5(2) + 9$ $= -4 - 10 + 9$ $= -14 + 9$ $= -5$

Step 2: Apply the Average Rate of Change Formula

$\text{Average rate of change} = \frac{h(2) - h(-8)}{2 - (-8)}$ $= \frac{-5 - (-15)}{2 + 8}$ $= \frac{-5 + 15}{10}$ $= \frac{10}{10}$ $= 1$

The average rate of change of the function $h(x) = -x^2 - 5x + 9$ over the interval $[-8, 2]$ is $\boxed{1}$.

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. What is the significance of the average rate of change for a quadratic function over a specific interval?
2. How can you determine if a function is increasing or decreasing over a given interval using the first derivative?
3. What are other methods to find the average rate of change for non-linear functions?
4. How does the average rate of change relate to the slope of a secant line on a graph?
5. How would the average rate of change differ if the interval was $[2, 8]$ instead of $[-8, 2]$?

Tip: The average rate of change is essentially the slope of the line that connects two points on a curve, providing a simple way to understand how a function behaves over a specific interval.