To find the average rate of change for the function $f(x)$ from $x = 4$ to $x = 7$, we need to use the formula:

$\text{Average Rate of Change} = \frac{f(7) - f(4)}{7 - 4}$

Looking at the graph:

• At $x = 4$, the function value $f(4)$ appears to be approximately 1.
• At $x = 7$, the function value $f(7)$ appears to be approximately 5.

Now applying these values in the formula:

$\text{Average Rate of Change} = \frac{f(7) - f(4)}{7 - 4} = \frac{5 - 1}{7 - 4} = \frac{4}{3}$

So, the average rate of change from $x = 4$ to $x = 7$ is approximately $\frac{4}{3}$ or $1.33$.

Would you like a more detailed explanation or further assistance?

Here are 5 related questions to extend your understanding:

1. What is the difference between instantaneous rate of change and average rate of change?
2. How can you estimate the instantaneous rate of change at a particular point from a graph?
3. What does the sign (positive or negative) of the average rate of change tell you about the function’s behavior over an interval?
4. How does the shape of the graph between $x = 4$ and $x = 7$ influence the average rate of change?
5. How can we apply the concept of the average rate of change in real-world contexts, such as velocity or economics?

Tip: The average rate of change between two points on a graph is the slope of the line connecting those two points. This is often called the "secant line."