To determine the average rate of change of the function $f(x)$ on the interval $-8 \leq x \leq -6$, we apply the formula for the average rate of change:

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

Here:

• $a = -8$
• $b = -6$

From the graph:

• $f(-8) = 0$ (the $y$-coordinate of the point at $x = -8$),
• $f(-6) = 16$ (the $y$-coordinate of the point at $x = -6$).

Substitute these values into the formula:

$\text{Average Rate of Change} = \frac{f(-6) - f(-8)}{-6 - (-8)} = \frac{16 - 0}{-6 + 8} = \frac{16}{2} = 8$

Final Answer:

The average rate of change of $f(x)$ on the interval $-8 \leq x \leq -6$ is 8.

Would you like a deeper explanation, or do you have additional questions?

Follow-Up Questions:

1. What does the average rate of change represent graphically?
2. How would the calculation change for the interval $-6 \leq x \leq -4$?
3. Can the average rate of change be negative? If so, what does that indicate?
4. How does the average rate of change compare to the instantaneous rate of change?
5. Could we estimate the rate of change over a different interval from the same graph?

Tip:

The average rate of change on an interval represents the slope of the secant line connecting the two endpoints of the interval on the graph.