The question asks to find the average rate of change for the function $g(x) = 4x^2 - 2x + 1$ on the interval $[-1, 1]$.

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

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

Here, $a = -1$ and $b = 1$.

Step 1: Calculate $g(1)$

$g(1) = 4(1)^2 - 2(1) + 1 = 4 - 2 + 1 = 3$

Step 2: Calculate $g(-1)$

$g(-1) = 4(-1)^2 - 2(-1) + 1 = 4 + 2 + 1 = 7$

Step 3: Apply the formula

$\text{Average Rate of Change} = \frac{g(1) - g(-1)}{1 - (-1)} = \frac{3 - 7}{1 + 1} = \frac{-4}{2} = -2$

So, the average rate of change is $-2$.

Would you like a detailed explanation of any steps?

Here are 5 additional related questions you could explore:

1. How does the average rate of change relate to the derivative?
2. What is the significance of the average rate of change being negative?
3. How would the calculation differ if the interval were $[0, 2]$?
4. Can you verify this result by graphing the function?
5. What would the instantaneous rate of change be at $x = 0$?

Tip: For any quadratic function like $g(x) = 4x^2 - 2x + 1$, the average rate of change over symmetric intervals around the vertex provides insight into the concavity of the graph.