To solve the problem of estimating the average rate of change from $x = 4$ to $x = 7$, we can follow these steps:

Step 1: Recall the Formula for Average Rate of Change

The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Where:

• $a = 4$
• $b = 7$

Step 2: Estimate $f(4)$ and $f(7)$

From the graph:

• At $x = 4$, the corresponding $y$-value (or $f(4)$) appears to be approximately $f(4) = 2$.
• At $x = 7$, the corresponding $y$-value (or $f(7)$) is approximately $f(7) = 5$.

Step 3: Apply the Formula

Now, applying the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{f(7) - f(4)}{7 - 4} = \frac{5 - 2}{7 - 4} = \frac{3}{3} = 1$

Conclusion:

The estimated average rate of change from $x = 4$ to $x = 7$ is 1.

Would you like more details or have any further questions?

Here are some related questions to deepen your understanding:

1. What is the significance of the average rate of change in a real-world context?
2. How does the average rate of change relate to the slope of a secant line?
3. Can you estimate the instantaneous rate of change at $x = 5$ from the graph?
4. How would you calculate the average rate of change for a different interval, such as from $x = 1$ to $x = 4$?
5. What does it mean if the average rate of change is negative over an interval?

Tip: When working with graphs, approximate the values carefully by identifying exact points or using interpolation for better accuracy.