To calculate the average rate of change of the function $f(x)$ from $x = 2$ to $x = 5$, we will use the formula:

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

Steps:

1. Read the $y$-values $f(2)$ and $f(5)$ from the graph.
2. Plug the values into the formula and simplify.

Let me analyze the graph closely:

• At $x = 2$, it appears that $f(2) = 4$.
• At $x = 5$, it appears that $f(5) = 0$.

Calculation:

$\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{0 - 4}{5 - 2} = \frac{-4}{3}.$

Final Answer:

The average rate of change of $f(x)$ from $x = 2$ to $x = 5$ is: $\boxed{-\frac{4}{3}}$

Would you like further explanation or details?

Related Questions:

1. What does the average rate of change represent in this context?
2. How does the average rate of change relate to the slope of a secant line?
3. Can you determine the equation of the secant line through these two points?
4. How would the average rate of change differ if the interval was changed to $x = 1$ to $x = 5$?
5. What are the units (if applicable) of the rate of change in this problem?

Tip:

When calculating the average rate of change, it helps to carefully observe the graph for exact coordinates or use equations if given. Always simplify the final result!