To calculate the average rate of change between $x = 1$ and $x = 4$, we can use the formula for the average rate of change of a function $f(x)$ over the interval $[a, b]$:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Here, $a = 1$ and $b = 4$. From the graph:

• $f(1) \approx 7$
• $f(4) \approx 10$

Now, applying these values into the formula:

$\text{Average rate of change} = \frac{10 - 7}{4 - 1} = \frac{3}{3} = 1.0$

So, the average rate of change from $x = 1$ to $x = 4$ is approximately 1.0.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the average rate of change compare to the instantaneous rate of change?
2. What is the significance of a positive average rate of change?
3. Can you determine where the slope of the curve is steepest between $x = 1$ and $x = 4$?
4. How would the result change if we used different points, such as $x = 2$ and $x = 5$?
5. What real-world phenomena could this graph represent?

Tip: The average rate of change over an interval gives the slope of the secant line connecting two points, providing a general idea of the behavior of the function over that interval.