The problem asks to calculate the average rate of change of the function $f(x)$ from $x = 1$ to $x = 5$. Here's the approach:

Step 1: Identify Points on the Graph

From the graph, we need to find the values of the function $f(x)$ at $x = 1$ and $x = 5$.

• At $x = 1$, the value of $f(1) = 4$.
• At $x = 5$, the value of $f(5) = -2$.

Step 2: Calculate the Average Rate of Change

The average rate of change of the function from $x = 1$ to $x = 5$ is given by the formula:

$\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1}$

Substituting the values:

$\text{Average rate of change} = \frac{-2 - 4}{5 - 1} = \frac{-6}{4} = -1.5$

Answer

The average rate of change from $x = 1$ to $x = 5$ is $-1.5$.

Do you want more details or have any questions?

Here are some related questions to expand your understanding:

1. How do you find the average rate of change for non-linear functions over different intervals?
2. What does a negative average rate of change indicate about a function's behavior?
3. How would the average rate of change differ if calculated over a smaller interval, like from $x = 1$ to $x = 3$?
4. Can the average rate of change be zero, and what would that indicate about the function?
5. How can the average rate of change be interpreted graphically on the function’s plot?

Tip: The average rate of change is essentially the slope of the secant line that passes through two points on the graph of a function.