graph x² = –24y. Identify the vertex, focus, equation of directrix, and end points of the latus rectum.

The given equation is \( x^2 = -24y \). This is the equation of a parabola. To identify the vertex, focus, equation of the directrix, and the endpoints of the latus rectum, we proceed as follows:

### 1. **Standard Form and Vertex**
The standard form of a parabola that opens downward (or upward) is:

\[
(x - h)^2 = 4p(y - k)
\]

Here, \( (h, k) \) is the vertex of the parabola, and \( p \) is the distance from the vertex to the focus (or the directrix).

Comparing \( x^2 = -24y \) with the standard form \( (x - h)^2 = 4p(y - k) \), we have:

- \( h = 0 \), \( k = 0 \) (since there is no horizontal or vertical shift),
- \( 4p = -24 \), so \( p = -6 \).

Thus, the **vertex** of the parabola is at \( (0, 0) \).

### 2. **Focus**
The focus lies \( p \) units away from the vertex along the axis of symmetry. Since \( p = -6 \), the focus is 6 units below the vertex along the y-axis:

\[
\text{Focus} = (0, 0 + (-6)) = (0, -6)
\]

### 3. **Equation of the Directrix**
The directrix is a horizontal line \( p \) units away from the vertex in the opposite direction of the focus. Since \( p = -6 \):

\[
\text{Equation of the directrix} = y = 0 - (-6) = y = 6
\]

### 4. **Endpoints of the Latus Rectum**
The latus rectum is a line segment that passes through the focus and is perpendicular to the axis of symmetry. Its length is \( |4p| \). Here \( |4p| = 24 \), so the distance from the focus to each endpoint of the latus rectum is \( 12 \) units horizontally (since it is centered at the focus).

The endpoints of the latus rectum are:

\[
\text{Endpoints} = \left( 0 \pm \frac{|4p|}{2}, -6 \right) = (12, -6) \text{ and } (-12, -6)
\]

### Summary:
- **Vertex**: \( (0, 0) \)
- **Focus**: \( (0, -6) \)
- **Equation of the Directrix**: \( y = 6 \)
- **Endpoints of the Latus Rectum**: \( (12, -6) \) and \( (-12, -6) \)

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would the graph change if the equation were \( x^2 = 24y \) instead?
2. What is the general method to convert a quadratic equation into standard form?
3. How do you find the focus of a parabola given its equation in vertex form?
4. What is the significance of the latus rectum in the geometry of a parabola?
5. How would you find the equation of a parabola given its focus and directrix?

**Tip:** When identifying the focus and directrix of a parabola, always remember that they are equidistant from the vertex, but in opposite directions along the axis of symmetry.

Understand the parabola graph of x² = –24y, including finding the vertex, focus, equation of the directrix, and endpoints of the latus rectum. Detailed step-by-step explanation with formulas and examples. Suitable for Grades 11-12.

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