To find the average rate of change of $f(x)$ from $x = 1$ to $x = 6$, we need to use the formula for the average rate of change:

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

Here, $a = 1$ and $b = 6$. So we need to determine the values of $f(1)$ and $f(6)$ from the graph.

Step 1: Estimate $f(1)$

Looking at the graph at $x = 1$, it appears that $f(1) = -1$.

Step 2: Estimate $f(6)$

At $x = 6$, the graph shows that $f(6) = -6$.

Step 3: Apply the average rate of change formula

Now, plug these values into the formula:

$\text{Average Rate of Change} = \frac{f(6) - f(1)}{6 - 1} = \frac{-6 - (-1)}{6 - 1} = \frac{-6 + 1}{5} = \frac{-5}{5} = -1$

So, the average rate of change is $-1$.

Would you like more details or have any further questions?

Related Questions:

1. How is the average rate of change related to the slope of the secant line?
2. What is the difference between instantaneous rate of change and average rate of change?
3. Can you find the instantaneous rate of change at $x = 1$?
4. How would the average rate of change differ if $x = 1$ was replaced with $x = 0$?
5. What does a negative average rate of change imply about the behavior of the function?

Tip:

The average rate of change of a function between two points is equivalent to the slope of the secant line that connects these points on the graph.