To calculate the average rate of change of the function $f(x)$ from $x = -1$ to $x = 3$, we can use the formula:

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

Where:

• $a = -1$ and $b = 3$,
• $f(a)$ is the value of the function at $x = -1$,
• $f(b)$ is the value of the function at $x = 3$.

Step 1: Find $f(-1)$

From the graph:

• When $x = -1$, the corresponding value of $f(x)$ appears to be $f(-1) = 2$.

Step 2: Find $f(3)$

From the graph:

• When $x = 3$, the corresponding value of $f(x)$ appears to be $f(3) = 1$.

Step 3: Apply the formula

Now, substituting the values into the formula:

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

Thus, the average rate of change of $f(x)$ from $x = -1$ to $x = 3$ is $-\frac{1}{4}$.

Would you like further clarification or details on any of the steps? Here are five related questions to deepen your understanding:

1. How is the average rate of change related to the slope of a line?
2. What does a negative average rate of change indicate about the graph's behavior?
3. How do you interpret the rate of change in terms of real-world applications?
4. Can the average rate of change be calculated over different intervals on the same graph?
5. How would the answer change if we selected different points, like $x = -2$ to $x = 1$?

Tip: The average rate of change between two points on a graph is the same as the slope of the secant line connecting those two points.