Let’s find the average rate of change of $f(x) = 6x^2 + 4$ on the interval $[-1, 3]$.

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

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

Here:
$a = -1$ and $b = 3$.

Step 1: Calculate $f(-1)$
$f(-1) = 6(-1)^2 + 4 = 6(1) + 4 = 6 + 4 = 10$

Step 2: Calculate $f(3)$
$f(3) = 6(3)^2 + 4 = 6(9) + 4 = 54 + 4 = 58$

Step 3: Apply the formula
$\text{Average rate of change} = \frac{58 - 10}{3 - (-1)} = \frac{48}{3 + 1} = \frac{48}{4} = 12$

Final Answer:
The average rate of change of $f(x) = 6x^2 + 4$ on the interval $[-1, 3]$ is 12.

Would you like me to explain any step in more detail or explore related ideas?

Here are some related questions:

1. How is the average rate of change related to the slope of a secant line?
2. What’s the difference between average rate of change and instantaneous rate of change?
3. How do we interpret the average rate of change of a quadratic function geometrically?
4. Can the average rate of change be negative? What would that mean?
5. How do we find the average rate of change over non-integer intervals?

Tip:
For quadratic functions like this, the average rate of change depends on the shape of the parabola and the interval chosen — so choosing different intervals gives different rates of change!