To determine the average rate of change of $f(x)$ from $x = 1$ to $x = 3$, we need to use the formula for the average rate of change:

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

where:

• $x_1 = 1$
• $x_2 = 3$

Now, let’s find the values of $f(1)$ and $f(3)$ from the graph.

1. Locate $f(1)$: From the graph, $f(1) = 4$.
2. Locate $f(3)$: From the graph, $f(3) = 1$.

Now, substitute these values into the formula:

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

So, the average rate of change of $f(x)$ from $x = 1$ to $x = 3$ is $-\frac{3}{2}$.

Would you like further details on this process or have additional questions?

Here are five related questions to further your understanding:

1. What does a negative average rate of change indicate about a function over an interval?
2. How would the average rate of change be different if $f(1)$ and $f(3)$ were closer in value?
3. Can you find the average rate of change from $x = 2$ to $x = 4$ for this function?
4. How does the average rate of change relate to the slope of a secant line on a graph?
5. If $f(x)$ was linear, how would the average rate of change vary across different intervals?

Tip: The average rate of change formula is similar to finding the slope of a line connecting two points on a graph. This can help you interpret the function's behavior between these points.