Question
Find the average rate of change of \( g(x)=-3 x+5 \) between the paint \( (-1,8) \) and \( (3,-4) \)
Ask by Salazar Guerrero. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The average rate of change is -3.
Solution
To find the **average rate of change** of the function \( g(x) = -3x + 5 \) between the points \((-1, 8)\) and \((3, -4)\), you can use the following formula:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]
where:
- \( a = -1 \) and \( g(a) = 8 \)
- \( b = 3 \) and \( g(b) = -4 \)
**Step-by-Step Calculation:**
1. **Identify the values:**
- \( g(-1) = 8 \)
- \( g(3) = -4 \)
2. **Apply the formula:**
\[
\text{Average Rate of Change} = \frac{g(3) - g(-1)}{3 - (-1)} = \frac{-4 - 8}{3 + 1} = \frac{-12}{4} = -3
\]
**Conclusion:**
The average rate of change of the function \( g(x) = -3x + 5 \) between the points \((-1, 8)\) and \((3, -4)\) is **\(-3\)**.
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Extra Insights
To find the average rate of change of the function \( g(x) = -3x + 5 \) between the points \( (-1, 8) \) and \( (3, -4) \), we first need to compute the function values at these \( x \)-coordinates. At \( x = -1 \): \[ g(-1) = -3(-1) + 5 = 3 + 5 = 8 \] At \( x = 3 \): \[ g(3) = -3(3) + 5 = -9 + 5 = -4 \] Now, we can use the formula for the average rate of change: \[ \text{Average rate of change} = \frac{g(3) - g(-1)}{3 - (-1)} = \frac{-4 - 8}{3 + 1} = \frac{-12}{4} = -3 \] So, the average rate of change of \( g(x) \) between these two points is \( -3 \).