To determine the function that reveals the vertex of the parabola given by [tex]\(y = x^2 + 6x + 3\)[/tex], we need to complete the square. Here is the detailed, step-by-step solution:
1. Move the constant term to the other side of the equation:
[tex]\[
y - 3 = x^2 + 6x
\][/tex]
2. Take half of the coefficient of [tex]\( x \)[/tex], square it, and add it to both sides:
[tex]\[
\left(\frac{6}{2}\right)^2 = 3^2 = 9
\][/tex]
Adding 9 to both sides:
[tex]\[
y - 3 + 9 = x^2 + 6x + 9
\][/tex]
Simplify the left side:
[tex]\[
y + 6 = (x + 3)^2
\][/tex]
3. Isolate [tex]\( y \)[/tex]:
[tex]\[
y = (x + 3)^2 - 6
\][/tex]
The function [tex]\( y = (x + 3)^2 - 6 \)[/tex] reveals the vertex of the parabola. Therefore, the correct answer is:
D. [tex]\( y = (x + 3)^2 - 6 \)[/tex]
