To solve the problem, we need to compute the average rate of change of $f(x)$ over the given intervals. The formula for the average rate of change is:

$\text{Average Rate of Change on } [a, b] = \frac{f(b) - f(a)}{b - a}.$

Let’s calculate the average rate of change for each interval:

A. $[0, 4]$

Using the graph:

• $f(0) = 9$ (from the y-coordinate of the graph at $x = 0$).
• $f(4) = 0$ (from the y-coordinate at $x = 4$).

$\text{Average Rate of Change} = \frac{f(4) - f(0)}{4 - 0} = \frac{0 - 9}{4} = -\frac{9}{4}.$

B. $[0, 2]$

Using the graph:

• $f(0) = 9$.
• $f(2) = 1$.

$\text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0} = \frac{1 - 9}{2} = -4.$

C. $[2, 4]$

Using the graph:

• $f(2) = 1$.
• $f(4) = 0$.

$\text{Average Rate of Change} = \frac{f(4) - f(2)}{4 - 2} = \frac{0 - 1}{2} = -\frac{1}{2}.$

D. $[2, 5]$

Using the graph:

• $f(2) = 1$.
• $f(5) = 9$.

$\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{9 - 1}{3} = \frac{8}{3}.$

Ordering from Least to Greatest:

1. B: $-4$ (Smallest)
2. A: $-\frac{9}{4} = -2.25$
3. C: $-\frac{1}{2} = -0.5$
4. D: $\frac{8}{3} = 2.67$ (Greatest)

Final Order: $B, A, C, D$.

Would you like more details about the computations or graph analysis?

Here are some follow-up questions:

1. How is the average rate of change related to the slope of a line between two points on a graph?
2. Can the average rate of change be positive, and what does it signify in a real-world context?
3. How do you interpret intervals where the average rate of change is zero?
4. What does a steep negative average rate of change (like in B) tell us about the function's behavior?
5. How would these calculations change if the graph were a straight line?

Tip: Always mark the key points on the graph and write out the formula before performing calculations for clarity.