To determine which of the given equations models the same quadratic relationship as [tex]\( f(x) = 2x^2 - 12x + 11 \)[/tex], we need to express [tex]\( f(x) \)[/tex] in the standard form [tex]\( a(x-h)^2 + k \)[/tex].
First, start with the given quadratic function:
[tex]\[ f(x) = 2x^2 - 12x + 11 \][/tex]
We will complete the square to rewrite this in vertex form [tex]\( a(x-h)^2 + k \)[/tex].
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the [tex]\( x \)[/tex]-terms:
[tex]\[ f(x) = 2(x^2 - 6x) + 11 \][/tex]
2. To complete the square inside the parentheses, take the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-6\)[/tex]), halve it, then square it:
[tex]\[ \left( \frac{-6}{2} \right)^2 = (-3)^2 = 9 \][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[ f(x) = 2(x^2 - 6x + 9 - 9) + 11 \][/tex]
[tex]\[ f(x) = 2((x - 3)^2 - 9) + 11 \][/tex]
4. Distribute the 2 and simplify:
[tex]\[ f(x) = 2(x - 3)^2 - 18 + 11 \][/tex]
[tex]\[ f(x) = 2(x - 3)^2 - 7 \][/tex]
So, in vertex form, the quadratic function [tex]\( f(x) = 2x^2 - 12x + 11 \)[/tex] can be expressed as:
[tex]\[ f(x) = 2(x - 3)^2 - 7 \][/tex]
Now, comparing this result with the given choices:
A. [tex]\( y = 2(x-6)^2 + 5 \)[/tex]
B. [tex]\( y = 2(x+6)^2 + 2 \)[/tex]
C. [tex]\( y = 2(x+3)^2 - 7 \)[/tex]
D. [tex]\( y = 2(x-3)^2 - 7 \)[/tex]
We observe that option D matches our transformed equation:
[tex]\[ y = 2(x - 3)^2 - 7 \][/tex]
Therefore, the correct answer is
[tex]\[ \boxed{D} \][/tex]
