To determine the correct function that represents a parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and a [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex], we need to check each of the given functions against the specified conditions.
Condition 1: Zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex]
A parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex] can be written in the factored form as:
[tex]\[ f(x) = a(x + 2)(x - 4) \][/tex]
Condition 2: [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex]
This means that when [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -16 \)[/tex]. Using this information, we can test each given function.
### Option A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
1. Check the zeros:
[tex]\[
f(-2) = 2(-2)^2 + 4(-2) - 16 = 8 - 8 - 16 = -16 \quad (\text{Not } 0)
\][/tex]
[tex]\[
f(4) = 2(4)^2 + 4(4) - 16 = 32 + 16 - 16 = 32 \quad (\text{Not } 0)
\][/tex]
Option A fails at checking the zeros.
### Option B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
1. Check the zeros:
[tex]\[
f(-2) = (-2)^2 - 2(-2) - 8 = 4 + 4 - 8 = 0
\][/tex]
[tex]\[
f(4) = 4^2 - 2(4) - 8 = 16 - 8 - 8 = 0
\][/tex]
Both zeros are correct.
2. Check the [tex]\( y \)[/tex]-intercept:
[tex]\[
f(0) = 0^2 - 2(0) - 8 = -8 \quad (\text{Not } -16)
\][/tex]
Option B seems promising but does not match the [tex]\( y \)[/tex]-intercept.
### Option C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
1. Check the zeros:
[tex]\[
f(-2) = (-2)^2 + 2(-2) - 8 = 4 - 4 - 8 = -8 \quad (\text{Not } 0)
\][/tex]
[tex]\[
f(4) = 4^2 + 2(4) - 8 = 16 + 8 - 8 = 16 \quad (\text{Not } 0)
\][/tex]
Option C fails at checking the zeros.
### Option D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
1. Check the zeros:
[tex]\[
f(-2) = 2(-2)^2 - 4(-2) - 16 = 8 + 8 - 16 = 0
\][/tex]
[tex]\[
f(4) = 2(4)^2 - 4(4) - 16 = 32 - 16 - 16 = 0
\][/tex]
Both zeros are correct.
2. Check the [tex]\( y \)[/tex]-intercept:
[tex]\[
f(0) = 2(0)^2 - 4(0) - 16 = -16
\][/tex]
[tex]\( y \)[/tex]-intercept is correct.
### Final Decision:
Based on the given conditions and the evaluations, Option D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex] is the function that represents a parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and a [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex].
