To rewrite the quadratic function [tex]\( y = 2x^2 - 4x + 12 \)[/tex] in vertex form, we can follow these steps:
1. Identify the coefficient of [tex]\( x^2 \)[/tex]: The coefficient of [tex]\( x^2 \)[/tex] in the given quadratic equation is 2.
2. Complete the square: To complete the square, we need to focus on the quadratic and linear terms [tex]\( 2x^2 - 4x \)[/tex].
a. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the quadratic and linear terms:
[tex]\[
y = 2(x^2 - 2x) + 12
\][/tex]
b. Complete the square inside the parentheses. To complete the square, take the coefficient of [tex]\( x \)[/tex] (which is -2), divide it by 2, and then square it:
[tex]\[
\left(\frac{-2}{2}\right)^2 = 1
\][/tex]
c. Add and subtract this square inside the parentheses:
[tex]\[
y = 2(x^2 - 2x + 1 - 1) + 12
\][/tex]
[tex]\[
y = 2((x - 1)^2 - 1) + 12
\][/tex]
d. Distribute the 2 and combine like terms:
[tex]\[
y = 2(x - 1)^2 - 2 + 12
\][/tex]
[tex]\[
y = 2(x - 1)^2 + 10
\][/tex]
3. Write the equation in vertex form: The quadratic equation is now in vertex form:
[tex]\[
y = 2(x - 1)^2 + 10
\][/tex]
The vertex form of the given quadratic function is:
[tex]\[
y = 2(x - 1)^2 + 10
\][/tex]
Therefore, among the given options, the correct form is:
[tex]\[
y = 2(x - 1)^2 + 10
\][/tex]
