Question

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

(1 point)
Responses

y = 2x^2+3x−1

y=x^2−2

y = 3(1/2)^x

y = 6x -8

Answer 1

To determine which function has the greatest rate of change over the interval \((-2, 2)\), we will evaluate the average rate of change for each function over the given interval. The average rate of change is calculated using the formula:

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

where \(a = -2\) and \(b = 2\).

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

1. **Function: \(y = 2x^2 + 3x - 1\)**
\[
f(-2) = 2(-2)^2 + 3(-2) - 1 = 2(4) - 6 - 1 = 8 - 6 - 1 = 1
\]
\[
f(2) = 2(2)^2 + 3(2) - 1 = 2(4) + 6 - 1 = 8 + 6 - 1 = 13
\]
\[
\text{Average Rate of Change} = \frac{13 - 1}{2 - (-2)} = \frac{12}{4} = 3
\]

2. **Function: \(y = x^2 - 2\)**
\[
f(-2) = (-2)^2 - 2 = 4 - 2 = 2
\]
\[
f(2) = (2)^2 - 2 = 4 - 2 = 2
\]
\[
\text{Average Rate of Change} = \frac{2 - 2}{2 - (-2)} = \frac{0}{4} = 0
\]

3. **Function: \(y = 3(1/2)^x\)**
\[
f(-2) = 3(1/2)^{-2} = 3 \cdot 4 = 12
\]
\[
f(2) = 3(1/2)^2 = 3 \cdot \frac{1}{4} = \frac{3}{4}
\]
\[
\text{Average Rate of Change} = \frac{\frac{3}{4} - 12}{2 - (-2)} = \frac{\frac{3}{4} - \frac{48}{4}}{4} = \frac{-\frac{45}{4}}{4} = -\frac{45}{16}
\]

4. **Function: \(y = 6x - 8\)**
\[
f(-2) = 6(-2) - 8 = -12 - 8 = -20
\]
\[
f(2) = 6(2) - 8 = 12 - 8 = 4
\]
\[
\text{Average Rate of Change} = \frac{4 - (-20)}{2 - (-2)} = \frac{24}{4} = 6
\]

Now let's summarize the average rates of change we calculated:

- For \(y = 2x^2 + 3x - 1\), the average rate of change is **3**.
- For \(y = x^2 - 2\), the average rate of change is **0**.
- For \(y = 3(1/2)^x\), the average rate of change is **\(-\frac{45}{16}\)** (approximately -2.8125).
- For \(y = 6x - 8\), the average rate of change is **6**.

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

\[
\boxed{y = 6x - 8}
\]