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We are tasked with finding the vertex of the quadratic function \( y = x^2 + 6x + 12 \).

The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the vertex formula:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

For the given equation \( y = x^2 + 6x + 12 \), we have:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 12 \)

### Step 1: Find the x-coordinate of the vertex

\[
x_{\text{vertex}} = \frac{-6}{2(1)} = \frac{-6}{2} = -3
\]

### Step 2: Find the y-coordinate of the vertex

Substitute \( x = -3 \) into the equation:

\[
y = (-3)^2 + 6(-3) + 12 = 9 - 18 + 12 = 3
\]

### Conclusion:
The vertex is \( (-3, 3) \).

Therefore, the correct answer is:
\[
\boxed{(-3, 3)}
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you derive the vertex formula from the general quadratic form?
2. What is the axis of symmetry for a quadratic function, and how does it relate to the vertex?
3. How would you complete the square for \( y = x^2 + 6x + 12 \) to find the vertex?
4. What impact does the value of \( a \) have on the shape and direction of the parabola?
5. How can you find the vertex if the quadratic equation is given in factored form?

**Tip:** The vertex represents the highest or lowest point of a parabola, depending on whether the parabola opens upwards or downwards.

Which coordinate pair is the vertex of y = x^2 + 6x + 12?

Explore how to find the vertex of the quadratic equation y = x^2 + 6x + 12 using the vertex formula. This algebra problem demonstrates key concepts in quadratic equations, including identifying the vertex point. Suitable for students in Grades 9-12.

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