The question in the image asks for the average rate of change of the function $f(x)$ from $x = 2$ to $x = 5$, based on the graph provided.

To calculate the average rate of change of a function between two points, we use the formula:

$\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Here:

• $x_1 = 2$
• $x_2 = 5$

From the graph, we need to determine the values of $f(2)$ and $f(5)$:

• At $x = 2$, the value of the function $f(2)$ appears to be 4.
• At $x = 5$, the value of the function $f(5)$ appears to be 2.

Now, apply these values to the formula:

$\text{Average rate of change} = \frac{f(5) - f(2)}{5 - 2} = \frac{2 - 4}{5 - 2} = \frac{-2}{3} \approx -0.67$

The correct answer is approximately -0.67, but it seems the provided answer was positive 0.67, which is incorrect since the slope is negative.

Would you like further details or have any additional questions?

Here are 5 related questions to expand on this concept:

1. How is the average rate of change related to the slope of a line?
2. Can the average rate of change be zero, and under what conditions?
3. What is the physical interpretation of the average rate of change in real-life applications?
4. How would you find the instantaneous rate of change at a point from a graph?
5. What happens to the rate of change as the interval between the two points becomes smaller?

Tip: Always check the sign of the rate of change based on the direction of the graph! A downward slope means a negative rate of change.