Let's solve the problem using the uploaded graph.

The average rate of change of a function $f(x)$ from $x = a$ to $x = b$ is calculated using the formula:

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

From the problem:

• $a = 0$
• $b = 3$

Step 1: Identify $f(0)$ and $f(3)$ from the graph.

From the graph:

• $f(0) = 1$ (The point where $x = 0$ on the graph has $y = 1$).
• $f(3) = 16$ (The point where $x = 3$ on the graph has $y = 16$).

Step 2: Apply the formula.

$\text{Average Rate of Change} = \frac{f(3) - f(0)}{3 - 0} = \frac{16 - 1}{3 - 0} = \frac{15}{3} = 5.$

Final Answer:

The average rate of change is $\boxed{5}$.

Would you like me to explain further or provide a detailed breakdown?

Here are 5 related questions to expand your understanding:

1. What is the geometric interpretation of the average rate of change on a graph?
2. How does the average rate of change differ from the instantaneous rate of change?
3. If $f(x) = x^2$, what would be the average rate of change from $x = 0$ to $x = 4$?
4. How does the formula for slope relate to the formula for the average rate of change?
5. What happens to the average rate of change when the interval is very small?

Tip: The average rate of change gives the slope of the secant line connecting two points on the graph of a function.