To determine the average rate of change of the function $f(x)$ from $x = 2$ to $x = 4$, we use the following formula for the average rate of change:

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

where $x_1 = 2$ and $x_2 = 4$.

From the graph:

• At $x = 2$, the corresponding value of $f(2)$ is approximately $4$.
• At $x = 4$, the corresponding value of $f(4)$ is approximately $-2$.

Now, we substitute these values into the formula:

$\text{Average rate of change} = \frac{f(4) - f(2)}{4 - 2} = \frac{-2 - 4}{4 - 2} = \frac{-6}{2} = -3$

Thus, the average rate of change of $f(x)$ from $x = 2$ to $x = 4$ is $-3$.

Would you like further clarification on this?

Here are five related questions you could explore:

1. What is the geometric interpretation of the average rate of change on the graph?
2. How would the average rate of change differ if the interval was from $x = 1$ to $x = 3$?
3. Can the average rate of change be interpreted as the slope of the secant line between two points on a curve?
4. How does the average rate of change compare to the instantaneous rate of change?
5. What happens to the average rate of change if $f(x)$ is a linear function?

Tip: The average rate of change between two points on a graph is the same as the slope of the straight line (secant line) connecting those two points.