To solve the problem, we will find the average rate of change of the function $f(x)$ from $x = -1$ to $x = 7$. The formula for the average rate of change is:

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

Here, $a = -1$ and $b = 7$.

From the graph:

• At $x = -1$, $f(-1) = -2$ (y-coordinate at $x = -1$).
• At $x = 7$, $f(7) = -2$ (y-coordinate at $x = 7$).

Substitute these values into the formula:

$\text{Average Rate of Change} = \frac{f(7) - f(-1)}{7 - (-1)} = \frac{-2 - (-2)}{7 + 1} = \frac{-2 + 2}{8} = \frac{0}{8} = 0$

Final Answer:

The average rate of change is 0.

Would you like a detailed explanation of why the result is zero? Let me know if you have any further questions!

Related Questions:

1. What does it mean when the average rate of change is zero?
2. How do you calculate average rate of change for nonlinear functions?
3. Can you explain why $f(7)$ and $f(-1)$ are the same in this graph?
4. How would this calculation change for a different interval?
5. Can the average rate of change be negative or positive, and what does that signify?

Tip:

The average rate of change corresponds to the slope of the secant line between two points on the graph. If the points have the same y-value, the slope will always be zero!