3x2 = -12yC. Given the general equation of parabola, find the following:
a. Standard equation
b. Coordinates of the vertex
c. Coordinates of the focus
d. Equation of axis of symmetry
e. Equation of directrix

To solve the given problem, we need to rewrite the given equation of the parabola into its standard form and then determine the vertex, focus, axis of symmetry, and directrix.

### Step 1: Rewrite the given equation

The given equation is:

\[
3x^2 = -12yC
\]

However, it looks like there might be a slight typo with the 'C'. Assuming you meant to write the equation as:

\[
3x^2 = -12y
\]

Let’s proceed with this corrected equation.

### Step 2: Rewrite into Standard Form

We want to rewrite the equation in a recognizable standard form of a parabola. Dividing both sides by 3:

\[
x^2 = -4y
\]

This matches the standard form of a vertical parabola:

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

where \((h, k)\) is the vertex and \( p \) determines the distance and direction of the focus and directrix from the vertex.

### Step 3: Identify the Standard Form

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

- \( h = 0 \), \( k = 0 \) (vertex at the origin)
- \( 4p = -4 \) so \( p = -1 \)

Thus, the standard equation of the parabola is:

\[
(x - 0)^2 = 4(-1)(y - 0)
\]

or simply:

\[
x^2 = -4y
\]

### Step 4: Determine the Characteristics

**a. Standard Equation:**

\[
x^2 = -4y
\]

**b. Coordinates of the Vertex:**

The vertex is at \((0, 0)\).

**c. Coordinates of the Focus:**

The focus is located at \((0, k + p)\). Substituting \( k = 0 \) and \( p = -1 \):

\[
\text{Focus} = (0, 0 + (-1)) = (0, -1)
\]

**d. Equation of Axis of Symmetry:**

The axis of symmetry is the vertical line passing through the vertex, which is the \(y\)-axis:

\[
x = 0
\]

**e. Equation of the Directrix:**

The directrix is the line given by \( y = k - p \). Substituting \( k = 0 \) and \( p = -1 \):

\[
\text{Directrix} = y = 0 - (-1) = y = 1
\]

### Summary:

1. **Standard Equation:** \(x^2 = -4y\)
2. **Coordinates of the Vertex:** \((0, 0)\)
3. **Coordinates of the Focus:** \((0, -1)\)
4. **Equation of Axis of Symmetry:** \(x = 0\)
5. **Equation of Directrix:** \(y = 1\)

Would you like further details on any of these steps?

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Here are some related questions:

1. How do you determine the direction of a parabola from its equation?
2. What is the role of the parameter \( p \) in the standard form of a parabola?
3. How does the sign of \( p \) affect the location of the focus and directrix?
4. Can the vertex of a parabola be determined directly from its standard form?
5. How do you find the length of the latus rectum of a parabola?

**Tip:** For parabolas, always check the coefficient of the squared term to determine whether it opens up, down, left, or right!

3x2 = -12yC. Given the general equation of parabola, find the following: a. Standard equation b. Coordinates of the vertex c. Coordinates of the focus d. Equation of axis of symmetry e. Equation of directrix

Discover how to find the standard equation, vertex, focus, axis of symmetry, and directrix of the parabola from the equation x^2 = -4y. This guide explains key concepts of conic sections and provides step-by-step solutions for students in Grades 10-12.

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