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To find the focus and directrix of the parabola defined by the equation:

\[
x^2 = -36y
\]

### Step 1: Identify the Parabola's Standard Form

This equation resembles the standard form of a vertically oriented parabola:
\[
x^2 = 4py
\]
where \( p \) is the distance from the vertex to the focus (or the directrix).

### Step 2: Rewrite the Equation to Match the Standard Form

For \( x^2 = -36y \), we can see that:
\[
4p = -36
\]
To solve for \( p \), divide both sides by 4:
\[
p = -9
\]

### Step 3: Determine the Vertex, Focus, and Directrix

Since the equation is in the form \( x^2 = -36y \), the parabola opens downward. For this type of parabola:

1. The **vertex** is at the origin, \((0, 0)\).
2. The **focus** is located at a distance \( p = -9 \) from the vertex, so it is at:
   \[
   (0, -9)
   \]
3. The **directrix** is a horizontal line \( p = 9 \) units above the vertex, at:
   \[
   y = 9
   \]

### Final Answer

- **Focus**: \((0, -9)\)
- **Directrix**: \(y = 9\)

Would you like a detailed explanation of the graphing process or have any questions?

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### Related Questions

1. How do you find the focus and directrix of a parabola that opens sideways?
2. What changes in the focus and directrix if the parabola opens upward instead?
3. How do you determine the orientation of a parabola based on its equation?
4. What is the significance of the value \( p \) in the equation \( x^2 = 4py \)?
5. How would you graph this parabola step by step?

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### Tip
Remember that the sign of \( p \) determines the direction in which the parabola opens.

Find the focus and directrix of the parabola with the given equation: x^2 = -36y.

Explore how to find the focus and directrix of the parabola defined by x^2 = -36y. This solution covers conic sections concepts, including identifying focus and directrix positions and the standard form of a vertical parabola. Ideal for high school students studying Grades 10-12.

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