To find the average rate of change of the function $f(x)$ over the interval $2 \leq x \leq 8$, we can use the formula for the average rate of change:

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

Where:

• $a = 2$
• $b = 8$
• $f(a)$ and $f(b)$ are the values of the function $f(x)$ at $x = a$ and $x = b$, respectively.

Step 1: Find $f(2)$ and $f(8)$ from the graph.

• From the graph, when $x = 2$, $f(2) = 10$.
• When $x = 8$, $f(8) = 50$.

Step 2: Apply the formula.

$\text{Average Rate of Change} = \frac{f(8) - f(2)}{8 - 2} = \frac{50 - 10}{8 - 2} = \frac{40}{6} \approx 6.67$

So, the average rate of change of the function $f(x)$ over the interval $2 \leq x \leq 8$ is approximately $6.67$.

Would you like more details on this? Here are 5 related questions you can explore:

1. How is the average rate of change related to the slope of a secant line?
2. How can you calculate the average rate of change for a different interval on the graph?
3. What is the difference between instantaneous rate of change and average rate of change?
4. How would the average rate of change differ for negative intervals or different function types?
5. How can the concept of average rate of change be applied in real-world problems?

Tip: The average rate of change between two points on a curve can be thought of as the slope of the line connecting those points (the secant line).