To find the average rate of change of the function $f(x)$ over the interval $2 \leq x \leq 6$, we can use the formula for the average rate of change between two points:

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

Here:

• $x_1 = 2$ and $f(x_1) = 54$
• $x_2 = 6$ and $f(x_2) = 42$

Substituting these values:

$\text{Average Rate of Change} = \frac{42 - 54}{6 - 2} = \frac{-12}{4} = -3$

So, the average rate of change of the function over the interval $2 \leq x \leq 6$ is -3.

Would you like further explanation on this solution?

Here are some additional questions for further exploration:

1. What does the average rate of change tell us about the behavior of the function on this interval?
2. How would the answer change if we used a different interval, say $0 \leq x \leq 8$?
3. Is the average rate of change the same for each interval given in the table? Why or why not?
4. How could you interpret the negative sign in the average rate of change in a real-world context?
5. What is the average rate of change over the interval $4 \leq x \leq 8$?

Tip: Remember that the average rate of change between two points on a function is the slope of the line connecting those points on the graph of the function.