To estimate the average rate of change of the function on the interval $x = 1$ to $x = 4$, we can use the formula for the average rate of change:

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

Here, $x_1 = 1$ and $x_2 = 4$.

From the graph:

• At $x = 1$, the function value $f(1) \approx 6$.
• At $x = 4$, the function value $f(4) \approx 3$.

Now we can substitute into the formula:

$\text{Average rate of change} = \frac{f(4) - f(1)}{4 - 1} = \frac{3 - 6}{4 - 1} = \frac{-3}{3} = -1$

Thus, the estimated average rate of change from $x = 1$ to $x = 4$ is approximately $-1$.

Do you have any questions or would you like further details?

Here are some related questions to deepen your understanding:

1. How does the average rate of change relate to the slope of a line?
2. What does a negative average rate of change indicate about the function's behavior?
3. How would the average rate of change differ if we selected a different interval?
4. Can the average rate of change be zero? If so, what would that mean?
5. What is the difference between average rate of change and instantaneous rate of change?

Tip: The average rate of change is similar to the slope of a secant line connecting two points on the graph.