To find the average rate of change of the function $g(x) = -x^2 + 8x + 24$ over the interval $[2, 10]$, we'll use the formula:

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

where $a = 2$ and $b = 10$.

Step 1: Evaluate the function at the endpoints of the interval.

1. Find $g(2)$:

$g(2) = -(2)^2 + 8(2) + 24$ $g(2) = -4 + 16 + 24 = 36$

2. Find $g(10)$:

$g(10) = -(10)^2 + 8(10) + 24$ $g(10) = -100 + 80 + 24 = 4$

Step 2: Calculate the average rate of change.

$\text{Average Rate of Change} = \frac{g(10) - g(2)}{10 - 2}$ $\text{Average Rate of Change} = \frac{4 - 36}{10 - 2}$ $\text{Average Rate of Change} = \frac{-32}{8} = -4$

So, the average rate of change of the function $g(x)$ over the interval $[2, 10]$ is $-4$.

Would you like any further explanation on these steps? Here are some related questions:

1. How does the sign of the average rate of change affect the interpretation of the function's behavior?
2. What is the significance of the average rate of change in a real-world context?
3. How can the vertex of the parabola described by $g(x)$ affect the average rate of change?
4. What would the average rate of change be if the interval were different, say from $4$ to $8$?
5. How can you use the derivative to find the instantaneous rate of change at any point within the interval?

Tip: When dealing with quadratic functions, the average rate of change can help you understand the overall behavior of the function over a specified interval, including whether it is generally increasing or decreasing.