Question

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





(1 point)
Responses

y = 2x2+3x−1
y = 2 x squared plus 3 x minus 1

y=x2−2
y is equal to x squared minus 2

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

y = 3(12)x

Answer 1

To determine which function has the greatest rate of change in the interval \((-2, 2)\), we need to calculate the rate of change for each function. The rate of change of a function over an interval can be determined by the average rate of change from \(x = -2\) to \(x = 2\), which is defined as:

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

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

Let's compute 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:** \(y = 6x - 8\)

\[
f(-2) = 6(-2) - 8 = -12 - 8 = -20
\]
\[
f(2) = 6(2) - 8 = 12 - 8 = 4
\]
\[
\text{Average Rate of Change} = \frac{4 - (-20)}{2 - (-2)} = \frac{24}{4} = 6
\]

4. **Function:** \(y = 3(12)x\) (which simplifies to \(y = 36x\))

\[
f(-2) = 36(-2) = -72
\]
\[
f(2) = 36(2) = 72
\]
\[
\text{Average Rate of Change} = \frac{72 - (-72)}{2 - (-2)} = \frac{144}{4} = 36
\]

Now we compare the average rates of change we calculated:

- \(y = 2x^2 + 3x - 1\): Average Rate of Change = 3
- \(y = x^2 - 2\): Average Rate of Change = 0
- \(y = 6x - 8\): Average Rate of Change = 6
- \(y = 3(12)x\): Average Rate of Change = 36

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

\[
\boxed{y = 3(12)x}
\]