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

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

where $a = -1$ and $b = 5$.

Step 1: Calculate $f(-1)$

$f(-1) = (-1)^2 - 2(-1) - 2 = 1 + 2 - 2 = 1$

Step 2: Calculate $f(5)$

$f(5) = (5)^2 - 2(5) - 2 = 25 - 10 - 2 = 13$

Step 3: Calculate the Average Rate of Change

$\text{Average Rate of Change} = \frac{f(5) - f(-1)}{5 - (-1)} = \frac{13 - 1}{5 + 1} = \frac{12}{6} = 2$

So, the average rate of change of the function over the interval $[-1, 5]$ is $2$.

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

Here are 5 related questions you might want to explore:

1. What is the significance of the average rate of change for a quadratic function?
2. How does the average rate of change relate to the derivative of the function?
3. Can you find the instantaneous rate of change at $x = 2$ for this function?
4. How would the average rate of change differ for different intervals, like $[0, 5]$?
5. How do the intercepts of the function affect its graph?

Tip: The average rate of change between two points gives the slope of the secant line between them, which can help in understanding the overall behavior of the function over that interval.