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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.

Given the function g(x) = 4x^2 - 2x + 1, find the average rate of change on the interval [-1, 1]. Simplify your answer. Only write a number as your answer. Do not add any letters, symbols, or space. For a fraction, use /.

Learn how to calculate the average rate of change for the quadratic function g(x) = 4x^2 - 2x + 1 over the interval [-1, 1]. This solution provides a step-by-step guide using the average rate of change formula and explores core algebraic concepts related to quadratic functions. Suitable for high school students in Grades 9-12.

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