To determine the average rate of change of the function $f(x) = x^2 - x - 8$ over the interval $[-1, 5]$, we can use the formula for the average rate of change of a function over an interval $[a, b]$:

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

Here, $a = -1$ and $b = 5$.

Step 1: Find $f(-1)$ and $f(5)$

1. Calculate $f(-1)$: $f(-1) = (-1)^2 - (-1) - 8 = 1 + 1 - 8 = -6$

2. Calculate $f(5)$: $f(5) = 5^2 - 5 - 8 = 25 - 5 - 8 = 12$

Step 2: Apply the formula

Substitute $f(-1)$, $f(5)$, $a = -1$, and $b = 5$ into the formula: $\text{Average Rate of Change} = \frac{f(5) - f(-1)}{5 - (-1)} = \frac{12 - (-6)}{5 - (-1)} = \frac{12 + 6}{6} = \frac{18}{6} = 3$

Conclusion:

The average rate of change of the function $f(x) = x^2 - x - 8$ over the interval $[-1, 5]$ is $\boxed{3}$.

Would you like any more details, or do you have any questions?

Here are some related questions you might find interesting:

1. How would the average rate of change differ if the interval was $[0, 4]$?
2. What is the significance of the average rate of change in terms of the function's graph?
3. Can you find the instantaneous rate of change of $f(x)$ at $x = 2$?
4. How does the average rate of change compare to the slope of the secant line between $(-1, f(-1))$ and $(5, f(5))$?
5. How would you determine if the function $f(x)$ is increasing or decreasing over a specific interval?

Tip: The average rate of change of a function over an interval gives the slope of the secant line connecting the endpoints of the interval on the function's graph.