To determine the average rate of change of the function $h(x) = -x^2 - 9x + 26$ over the interval $[-9, 2]$, we can use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a}$

Here, $a = -9$ and $b = 2$.

1. Calculate $h(-9)$:

$h(-9) = -(-9)^2 - 9(-9) + 26$

$h(-9) = -81 + 81 + 26 = 26$

2. Calculate $h(2)$:

$h(2) = -(2)^2 - 9(2) + 26$

$h(2) = -4 - 18 + 26 = 4$

3. Find the average rate of change:

$\text{Average Rate of Change} = \frac{h(2) - h(-9)}{2 - (-9)} = \frac{4 - 26}{2 + 9} = \frac{-22}{11} = -2$

Thus, the average rate of change of the function $h(x)$ over the interval $[-9, 2]$ is $-2$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the significance of the average rate of change in the context of a function?
2. How does the average rate of change relate to the slope of the secant line?
3. Can the average rate of change be positive, and what would that indicate?
4. How would the average rate of change differ if the interval was $[-9, 9]$?
5. How can we interpret the results geometrically on the graph of the function?

Tip: The average rate of change can give you an idea of the general behavior of a function over a specific interval, similar to the concept of a slope in linear functions.