To find the average rate of change of the function $f(x)$ from $x = 1$ to $x = 2$, we use the formula:

$\text{Average rate of change} = \frac{f(2) - f(1)}{2 - 1}$

From the graph:

• $f(1) = 1$
• $f(2) = 3$

Now, substituting these values into the formula:

$\text{Average rate of change} = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2$

Thus, the average rate of change from $x = 1$ to $x = 2$ is 2.

Would you like a more detailed explanation or have any questions? Here are some related questions to explore:

1. What is the average rate of change of $f$ from $x = 0$ to $x = 1$?
2. What is the value of $f(3)$ based on the graph?
3. How does the average rate of change from $x = 4$ to $x = 5$ compare to that from $x = 1$ to $x = 2$?
4. Can the average rate of change be negative? If so, where does it happen on this graph?
5. What does the average rate of change represent geometrically on the graph?

Tip: The average rate of change of a function over an interval is simply the slope of the secant line that passes through the points on the graph corresponding to the endpoints of the interval.