Answered

Select the correct answer.

Which equation models the same quadratic relationship as function [tex]\( f \)[/tex]?

[tex]\[ f(x) = 2x^2 - 12x + 11 \][/tex]

A. [tex]\( y = 2(x + 3)^2 - 7 \)[/tex]

B. [tex]\( y = 2(x + 6)^2 + 2 \)[/tex]

C. [tex]\( y = 2(x - 6)^2 + 5 \)[/tex]

D. [tex]\( y = 2(x - 3)^2 - 7 \)[/tex]



Answer :

GinnyAnswer
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]

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