Getting the general form:

only put the solution in getting focus(with the formula) and get the opening of it

Vertex (0,1), Focus (0,-1)

The general form of a parabola with a vertical axis is:

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

Where:
- \((h, k)\) is the vertex,
- \(p\) is the distance from the vertex to the focus.

Given:
- Vertex: \((0, 1)\),
- Focus: \((0, -1)\).

### Step 1: Find \(p\)
The distance from the vertex to the focus is 2 units downward, so \(p = -2\).

### Step 2: Apply the formula
Using \(h = 0\), \(k = 1\), and \(p = -2\), the equation becomes:

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

Simplifying:

\[
x^2 = -8(y - 1)
\]

This is the general form of the parabola.

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Would you like further details on this or any step? Here are some related questions:

1. How do you find the equation of a parabola given other properties?
2. What does the focus of a parabola represent?
3. Can you convert this to standard form?
4. How do you determine whether a parabola opens up or down?
5. What is the relationship between the directrix and the focus?

**Tip**: The sign of \(p\) determines whether the parabola opens upwards (positive \(p\)) or downwards (negative \(p\)).

Getting the general form: Vertex (0,1), Focus (0,-1).

Discover how to derive the general form of a parabola with vertex (0,1) and focus (0,-1). This step-by-step solution explains the formula for a parabola with a vertical axis and covers key concepts in conic sections, suitable for students in Grades 10-12.

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