Find the vertex and focus on the parabola x square + 12y = 0

The given equation is:

\[
x^2 + 12y = 0
\]

First, we rearrange the equation into standard form by isolating \( y \):

\[
x^2 = -12y
\]

This is a parabola that opens downward since the coefficient of \( y \) is negative. The standard form of a parabola that opens vertically is:

\[
x^2 = 4p y
\]

Here, comparing \( x^2 = -12y \) with \( x^2 = 4p y \), we can see that \( 4p = -12 \). Solving for \( p \):

\[
p = \frac{-12}{4} = -3
\]

### Vertex:
The vertex of the parabola is at the origin, \((0, 0)\), since the equation is in the form \( x^2 = 4p(y - k) \) and there is no shift in either the \(x\)- or \(y\)-direction.

### Focus:
The focus of a parabola is located at a distance \(p\) from the vertex along the axis of symmetry. Since \( p = -3 \), the focus is 3 units below the vertex (because \( p \) is negative and the parabola opens downward).

Thus, the focus is at:

\[
(0, -3)
\]

### Summary:
- **Vertex:** \( (0, 0) \)
- **Focus:** \( (0, -3) \)

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the directrix of the given parabola?
2. How do you find the axis of symmetry of a parabola?
3. What would be the equation if the parabola opened to the right?
4. How do you determine the direction in which a parabola opens?
5. How does the value of \( p \) affect the shape of the parabola?

**Tip:** The focus and directrix of a parabola always lie on opposite sides of the vertex.

Learn how to find the vertex and focus of the parabola x^2 + 12y = 0 with a detailed solution. This problem explores key concepts of parabolas, including the vertex at (0, 0) and the focus at (0, -3), suitable for students in Grades 9-12.

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