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To solve this problem, we start by examining the given equation:

\[
x^2 = -24y
\]

This equation is in the form \( x^2 = 4py \), where \( p \) is a constant that represents the distance between the vertex and both the focus and the directrix of the parabola. This form tells us that the parabola opens downward, as the \( y \)-term has a negative coefficient.

### Step 1: Identify \( p \)
To rewrite \( x^2 = -24y \) in the form \( x^2 = 4py \), we need to match coefficients:

\[
4p = -24
\]

Solving for \( p \):

\[
p = -6
\]

### Step 2: Determine the Focus and Directrix
Since the vertex of the parabola is at the origin \((0, 0)\) and the parabola opens downward, we use \( p = -6 \) to locate the focus and the directrix:

- **Focus**: The focus is located \( p \) units away from the vertex in the direction the parabola opens. Thus, the focus is at \((0, -6)\).
- **Directrix**: The directrix is a line \( p \) units in the opposite direction of the focus. So, the directrix is the horizontal line \( y = 6\).

### Answer
The focus of the parabola is:

\[
(0, -6)
\]

Would you like a further breakdown of how to graph this parabola or have any additional questions?

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Here are five related questions to deepen your understanding:

1. How would the focus and directrix change if the equation were \( x^2 = 24y \)?
2. What is the focus if the equation is \( y^2 = 8x \)?
3. How do you find the vertex, focus, and directrix for parabolas that open to the left or right?
4. Can you explain how the value of \( p \) affects the width and orientation of a parabola?
5. What is the general form of a parabola that opens horizontally versus vertically?

**Tip:** For parabolas, knowing the value of \( p \) helps quickly locate both the focus and directrix, as these points are always symmetrically positioned around the vertex.

Find the focus and directrix of the parabola with the given equation: x^2 = -24y.

Learn how to find the focus and directrix of a parabola with the equation x^2 = -24y. This solution covers key concepts in understanding the geometry of parabolas and calculating focus and directrix points.

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