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To find the focus and directrix of the parabola represented by the equation \( x^2 = -24y \), we can analyze its form and properties.

### Step 1: Identify the Parabola’s Structure
The given equation \( x^2 = -24y \) resembles the standard form of a vertical parabola:
\[
x^2 = 4py
\]
where \( p \) is the distance from the vertex to the focus (if \( p \) is negative, the parabola opens downward).

### Step 2: Find \( p \)
From \( x^2 = -24y \), we can compare to \( x^2 = 4py \) and see that \( 4p = -24 \). Solving for \( p \),
\[
p = -6
\]

### Step 3: Identify the Focus and Directrix
- **Vertex**: Since the equation is in the form \( x^2 = -24y \), the vertex is at the origin \( (0, 0) \).
- **Focus**: The focus is located \( p = -6 \) units below the vertex, so the focus is at \( (0, -6) \).
- **Directrix**: The directrix is a horizontal line \( p = 6 \) units above the vertex, so the directrix is \( y = 6 \).

### Step 4: Determine the Graph
Since \( x^2 = -24y \) represents a parabola that opens downward (due to the negative \( p \) value), the correct graph would be one that shows a downward-facing parabola.

Based on the options provided:
- **Option D** correctly shows a downward-facing parabola.

### Answer Summary
- **Focus**: \( (0, -6) \)
- **Directrix**: \( y = 6 \)
- **Correct Graph**: Option D

Would you like further details on any part of this solution?

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### Related Questions
1. How do we convert any quadratic equation to vertex form?
2. What are the general steps to determine the focus and directrix of a parabola?
3. How would the equation change if the parabola opened upward instead?
4. What if the parabola’s vertex is not at the origin? How would that affect the calculation?
5. How can we verify the correctness of the chosen graph mathematically?

### Tip
Always remember that for a parabola, the sign of \( p \) determines the direction: positive for upward or rightward, and negative for downward or leftward.

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

Learn how to find the focus and directrix of the parabola represented by the equation x^2 = -24y and graph the result. This solution includes a step-by-step explanation using the standard form of a parabola and properties of parabolas. Suitable for high school students.

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