To determine which equation models the same quadratic relationship as the function [tex]\( f(x) = 2x^2 - 12x + 11 \)[/tex], we need to compare the given function with each of the possible answers in the expanded form.
Let's explore each option one-by-one:
### Option A: [tex]\( y = 2(x + 3)^2 - 7 \)[/tex]
First, we need to expand [tex]\( 2(x + 3)^2 - 7 \)[/tex]:
[tex]\[
2(x + 3)^2 - 7 = 2(x^2 + 6x + 9) - 7
\][/tex]
[tex]\[
= 2x^2 + 12x + 18 - 7
\][/tex]
[tex]\[
= 2x^2 + 12x + 11
\][/tex]
### Option B: [tex]\( y = 2(x + 6)^2 + 2 \)[/tex]
Next, let's expand [tex]\( 2(x + 6)^2 + 2 \)[/tex]:
[tex]\[
2(x + 6)^2 + 2 = 2(x^2 + 12x + 36) + 2
\][/tex]
[tex]\[
= 2x^2 + 24x + 72 + 2
\][/tex]
[tex]\[
= 2x^2 + 24x + 74
\][/tex]
### Option C: [tex]\( y = 2(x - 6)^2 + 5 \)[/tex]
Now, let's expand [tex]\( 2(x - 6)^2 + 5 \)[/tex]:
[tex]\[
2(x - 6)^2 + 5 = 2(x^2 - 12x + 36) + 5
\][/tex]
[tex]\[
= 2x^2 - 24x + 72 + 5
\][/tex]
[tex]\[
= 2x^2 - 24x + 77
\][/tex]
### Option D: [tex]\( y = 2(x - 3)^2 - 7 \)[/tex]
Finally, let's expand [tex]\( 2(x - 3)^2 - 7 \)[/tex]:
[tex]\[
2(x - 3)^2 - 7 = 2(x^2 - 6x + 9) - 7
\][/tex]
[tex]\[
= 2x^2 - 12x + 18 - 7
\][/tex]
[tex]\[
= 2x^2 - 12x + 11
\][/tex]
Upon expanding each equation, we find that Option D produces the same quadratic equation as [tex]\( f(x) = 2x^2 - 12x + 11 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
