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

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

Where:

• $a = -1$
• $b = 5$

First, we calculate the values of $f(a)$ and $f(b)$.

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

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

Now, plug these values into the formula:

$\text{Average Rate of Change} = \frac{f(5) - f(-1)}{5 - (-1)} = \frac{12 - (-6)}{5 - (-1)} = \frac{12 + 6}{6} = \frac{18}{6} = 3$

So, the average rate of change of the function $f(x) = x^2 - x - 8$ over the interval $-1 \leq x \leq 5$ is $3$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What does the average rate of change represent in the context of a function?
2. How does the average rate of change differ from the instantaneous rate of change?
3. Can the average rate of change be negative? If so, what does it indicate?
4. How would you find the instantaneous rate of change of the function at $x = 3$?
5. What is the significance of the points where the average rate of change equals zero?

Tip: When solving problems involving functions, always check the interval carefully to ensure you are evaluating the function at the correct endpoints.