https://file.aiquickdraw.com/mathbot/file/1728937435051y8wbm7sv.jpg

The equation given is \( x^2 + 12y = 0 \). To find the vertex, focus, and directrix, we first need to rewrite the equation into the standard form of a parabola.

### Step 1: Rewrite the equation
Starting with:
\[
x^2 + 12y = 0
\]
Solve for \( y \):
\[
y = -\frac{1}{12}x^2
\]
This is the equation of a parabola that opens downward (since the coefficient of \(x^2\) is negative), with its vertex at the origin \((0,0)\).

### Step 2: Identify the standard form
The general form of a vertical parabola is:
\[
(y - k) = \frac{1}{4p}(x - h)^2
\]
For our equation:
\[
y = -\frac{1}{12}x^2
\]
we can see that \( h = 0 \), \( k = 0 \), and \( \frac{1}{4p} = -\frac{1}{12} \). Solving for \( p \):
\[
4p = 12 \quad \Rightarrow \quad p = 3
\]
Thus, the parabola has a vertex at \((0, 0)\) and a focal length \( p = 3 \).

### Step 3: Find the focus and directrix
For a parabola opening downward:
- The focus is located \( p \) units below the vertex, i.e., at \( (0, -3) \).
- The directrix is a horizontal line \( p \) units above the vertex, i.e., \( y = 3 \).

### Final Answer:
- **Vertex**: \( (0, 0) \)
- **Focus**: \( (0, -3) \)
- **Directrix**: \( y = 3 \)

Would you like more details or have any questions?

### Related Questions:
1. How do you determine the direction in which a parabola opens?
2. What are the properties of a parabola when \( x^2 \) is negative versus positive?
3. How do you calculate the focal distance \( p \) in a parabola?
4. What is the significance of the directrix in the geometry of a parabola?
5. Can this method be applied to non-vertical parabolas?

### Tip:
When solving parabolic equations, converting them into standard form helps in identifying key features such as the vertex, focus, and directrix directly from the equation.

Find the Vertex, Focus, and Directrix for x^2 + 12y = 0

Step-by-step solution for finding the vertex, focus, and directrix of the parabola x^2 + 12y = 0. Learn how to convert the equation into standard form, identify the vertex, and calculate the focal point and directrix for parabolas that open downward. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation