Question
Finding the average rate of change of a function Find the average rate of change of \( g(x)=3 x^{2}-2 x \) from \( x=-4 \) to \( x=-2 \). Simplify your answer as much as possible. \( \square \)
Ask by Campos Higgins. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The average rate of change of \( g(x) = 3x^{2} - 2x \) from \( x = -4 \) to \( x = -2 \) is \(-20\).
Solution
To find the **average rate of change** of the function \( g(x) = 3x^{2} - 2x \) from \( x = -4 \) to \( x = -2 \), we use the formula:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
where \( a = -4 \) and \( b = -2 \).
1. **Calculate \( g(-4) \):**
\[
g(-4) = 3(-4)^2 - 2(-4) = 3(16) + 8 = 48 + 8 = 56
\]
2. **Calculate \( g(-2) \):**
\[
g(-2) = 3(-2)^2 - 2(-2) = 3(4) + 4 = 12 + 4 = 16
\]
3. **Compute the average rate of change:**
\[
\text{Average Rate of Change} = \frac{16 - 56}{-2 - (-4)} = \frac{-40}{2} = -20
\]
So, the average rate of change of \( g(x) \) from \( x = -4 \) to \( x = -2 \) is:
\[
\boxed{-20}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the average rate of change of the function \( g(x)=3 x^{2}-2 x \) from \( x=-4 \) to \( x=-2 \), you first need to evaluate the function at both points. Calculate \( g(-4) \): \[ g(-4) = 3(-4)^2 - 2(-4) = 3(16) + 8 = 48 + 8 = 56 \] Now calculate \( g(-2) \): \[ g(-2) = 3(-2)^2 - 2(-2) = 3(4) + 4 = 12 + 4 = 16 \] Now, use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{g(-2) - g(-4)}{-2 - (-4)} = \frac{16 - 56}{-2 + 4} = \frac{-40}{2} = -20 \] So, the average rate of change of \( g(x) \) from \( x=-4 \) to \( x=-2 \) is \( -20 \).
