To solve this problem, we need to calculate the average rate of change of the function $f(x) = 3x^2 - 4$ over the interval $[1, 3]$.

The formula for the average rate of change of a function $f(x)$ over the interval $[a, b]$ is:

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

Step-by-step solution:

1. Identify the values:

   • $a = 1$
   • $b = 3$
   • The function is $f(x) = 3x^2 - 4$.

2. Find $f(3)$: $f(3) = 3(3)^2 - 4 = 3(9) - 4 = 27 - 4 = 23$

3. Find $f(1)$: $f(1) = 3(1)^2 - 4 = 3(1) - 4 = 3 - 4 = -1$

4. Apply the formula: $\text{Average rate of change} = \frac{f(3) - f(1)}{3 - 1} = \frac{23 - (-1)}{3 - 1} = \frac{23 + 1}{2} = \frac{24}{2} = 12$

Thus, the average rate of change of the function $f(x) = 3x^2 - 4$ over the interval $[1, 3]$ is 12.

Would you like further explanation or details on any step? Here are some related questions:

1. How do you interpret the average rate of change geometrically?
2. How would the average rate of change differ for a linear function?
3. Can the average rate of change ever be negative? If so, under what conditions?
4. How does the average rate of change relate to the derivative?
5. What happens if the interval changes? How would the rate of change differ over $[2, 4]$?

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