The given equation is \((x)^2 = -8y\).

This is a quadratic equation that can be analyzed further.

### Step-by-Step Solution:

1. **Standard Form of the Equation**:
   \[
   x^2 = -8y
   \]
   This is already in standard form for a parabola that opens downward.

2. **Identifying the Type of Conic Section**:
   - Since the equation is in the form \(x^2 = 4py\), it represents a parabola.
   - Here, \(4p = -8\), so \(p = -2\).

3. **Properties of the Parabola**:
   - **Vertex**: The vertex of this parabola is at the origin \((0, 0)\).
   - **Direction**: The parabola opens downward because the coefficient of \(y\) is negative.
   - **Focus**: The focus of the parabola is at \((0, -2)\).
   - **Directrix**: The directrix is the line \(y = 2\).

### Conclusion:
The equation \((x)^2 = -8y\) represents a parabola that opens downward with its vertex at the origin \((0, 0)\), a focus at \((0, -2)\), and a directrix at \(y = 2\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you derive the equation of a parabola from its vertex and focus?
2. What is the general form of a parabola's equation in vertex form?
3. How do you find the axis of symmetry for a given parabola?
4. How would the equation change if the parabola opened upward instead?
5. What are the applications of parabolas in real-world problems?

**Tip**: Remember that the sign of the coefficient determines the direction in which the parabola opens (positive for upwards, negative for downwards).

Explore the quadratic equation (x)² = -8y and its solution. Understand the properties of parabolas, including vertex, focus, and directrix. Suitable for students in Grades 9-12.

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