Question

Which Function has the greatest rate of change in the interval (-2,2)?





(1 point)
Responses

y=x2−2


y = 2x2+3x−1

y = 6x -8
y = 6x -8

y = 3(1/2)x

Answer 1

To determine which function has the greatest rate of change in the interval \((-2, 2)\), we need to calculate the average rate of change for each function over that interval.

The average rate of change of a function \(f(x)\) from \(x = a\) to \(x = b\) is given by the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

### Let's calculate the average rate of change for each function:

1. **Function: \(y = x^2 - 2\)**

- \(f(-2) = (-2)^2 - 2 = 4 - 2 = 2\)
- \(f(2) = (2)^2 - 2 = 4 - 2 = 2\)

\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{2 - 2}{4} = 0
\]

2. **Function: \(y = 2x^2 + 3x - 1\)**

- \(f(-2) = 2(-2)^2 + 3(-2) - 1 = 2(4) - 6 - 1 = 8 - 6 - 1 = 1\)
- \(f(2) = 2(2)^2 + 3(2) - 1 = 2(4) + 6 - 1 = 8 + 6 - 1 = 13\)

\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{13 - 1}{4} = \frac{12}{4} = 3
\]

3. **Function: \(y = 6x - 8\)**

- \(f(-2) = 6(-2) - 8 = -12 - 8 = -20\)
- \(f(2) = 6(2) - 8 = 12 - 8 = 4\)

\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{4 - (-20)}{4} = \frac{4 + 20}{4} = \frac{24}{4} = 6
\]

4. **Function: \(y = 3(1/2)^x\)**

- \(f(-2) = 3(1/2)^{-2} = 3 \cdot (2^2) = 3 \cdot 4 = 12\)
- \(f(2) = 3(1/2)^{2} = 3 \cdot (1/4) = \frac{3}{4}\)

\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{\frac{3}{4} - 12}{4} = \frac{\frac{3 - 48}{4}}{4} = \frac{-\frac{45}{4}}{4} = -\frac{45}{16}
\]

### Summary of Average Rates of Change:
- \(y = x^2 - 2\): 0
- \(y = 2x^2 + 3x - 1\): 3
- \(y = 6x - 8\): 6
- \(y = 3(1/2)^x\): \(-\frac{45}{16} \approx -2.8125\)

### Conclusion
The function with the greatest rate of change in the interval \((-2, 2)\) is:

**\(y = 6x - 8\)** with an average rate of change of **6**.