The rate of change of a function between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the average rate of change formula:

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

For the function:

$f(x) = 3x^2 + 9$

Step 1: Evaluate $f(x)$ at the given points

For $x_1 = -3$:

$f(-3) = 3(-3)^2 + 9 = 3(9) + 9 = 27 + 9 = 36$

For $x_2 = 0$:

$f(0) = 3(0)^2 + 9 = 0 + 9 = 9$

Step 2: Compute the average rate of change

$\frac{f(0) - f(-3)}{0 - (-3)} = \frac{9 - 36}{0 + 3} = \frac{-27}{3} = -9$

Final Answer:

The average rate of change from $x = -3$ to $x = 0$ is $-9$.

Would you like a detailed explanation of why this formula works?

Here are five related questions to explore further:

1. What is the difference between average and instantaneous rate of change?
2. How do you find the derivative of $f(x) = 3x^2 + 9$ to get the instantaneous rate of change?
3. How does the rate of change vary for quadratic functions compared to linear ones?
4. Can we use limits to define the instantaneous rate of change at a single point?
5. What are real-world applications of average rate of change in physics or economics?

Tip: The average rate of change of a quadratic function over an interval depends on the slope of the secant line connecting two points.