The image shows a question asking for the average rate of change of a function $f(x)$ from $x = 1$ to $x = 3$ using a given graph.

To calculate the average rate of change of $f(x)$ over the interval $[x_1, x_2]$, we use the formula:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Step-by-step Solution:

1. Identify Points on the Graph:

   • Find $f(1)$ and $f(3)$ from the graph.

2. Calculate Values from the Graph:

   • At $x = 1$, the function value $f(1) = 2$.
   • At $x = 3$, the function value $f(3) = 6$.

3. Apply the Formula: $\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{6 - 2}{2} = \frac{4}{2} = 2$

Therefore, the average rate of change of $f(x)$ from $x = 1$ to $x = 3$ is 2.

Would you like more details, or do you have any questions?

Further Questions to Explore:

1. How do you find the rate of change for a nonlinear function?
2. What does the average rate of change represent in a real-world context?
3. How is the average rate of change different from the instantaneous rate of change?
4. How can you interpret the slope of a line in terms of rate of change?
5. How do you calculate the average rate of change for other intervals?

Tip:

For linear functions, the average rate of change is constant and equal to the slope of the line.