To solve the problem of finding the average rate of change of the function $f(x)$ on the interval $[-2, 3]$, we use the following formula for the average rate of change:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

where $a = -2$ and $b = 3$, and $f(a)$ and $f(b)$ are the function values at $x = -2$ and $x = 3$, respectively.

From the graph:

• At $x = -2$, the function value $f(-2)$ appears to be $3$.
• At $x = 3$, the function value $f(3)$ appears to be $6$.

Now, we substitute these values into the formula:

$\text{Average rate of change} = \frac{f(3) - f(-2)}{3 - (-2)} = \frac{6 - 3}{3 + 2} = \frac{3}{5} = 0.6$

Thus, the average rate of change of the function $f(x)$ on the interval $[-2, 3]$ is $0.6$.

Would you like further details or have any questions about this solution?

Here are 5 follow-up questions related to this concept:

1. How does the average rate of change relate to the slope of the secant line between two points on a graph?
2. What would the average rate of change be if the interval were changed to $[0, 3]$?
3. How do instantaneous rate of change and average rate of change differ?
4. Can the average rate of change be negative, and what would that indicate about the function's behavior?
5. How does the average rate of change help in understanding the overall trend of a function over an interval?

Tip: The average rate of change is essentially the slope of a straight line connecting two points on a curve, so visualizing this on a graph can help in understanding how the function behaves between those points.