determine the equation of a parabola with focus at (0,4) and directrix y=-4

The equation of a parabola is determined by the position of its focus and its directrix. When the focus and directrix are aligned vertically, the parabola will open either upwards or downwards.

Given:
- Focus: \((0, 4)\)
- Directrix: \(y = -4\)

### Step-by-step process:

1. **General form of the equation**:
   The equation of a parabola with its vertex at \((h, k)\), focus at \((h, k + p)\), and directrix at \(y = k - p\) is:
   \[
   (x - h)^2 = 4p(y - k)
   \]
   where \(p\) is the distance from the vertex to the focus (or the directrix).

2. **Determine the vertex**:
   The vertex of the parabola is halfway between the focus and the directrix. Since the focus is at \(y = 4\) and the directrix is at \(y = -4\), the vertex is at:
   \[
   y = \frac{4 + (-4)}{2} = 0
   \]
   Therefore, the vertex is at \((0, 0)\).

3. **Find the value of \(p\)**:
   The value of \(p\) is the distance from the vertex to the focus, which is \(4\) units (from \(y = 0\) to \(y = 4\)).

4. **Write the equation**:
   Since the vertex is at \((0, 0)\), the equation simplifies to:
   \[
   x^2 = 4py
   \]
   Substituting \(p = 4\):
   \[
   x^2 = 16y
   \]

### Final equation:
The equation of the parabola is:
\[
x^2 = 16y
\]

Would you like any further explanation or details?

Here are 5 related questions for deeper exploration:
1. What is the general form of the equation of a parabola with a horizontal directrix?
2. How do you find the distance between the vertex and the focus for any parabola?
3. How can the directrix and focus be used to derive the equation of an ellipse?
4. What happens to the equation if the directrix is horizontal instead of vertical?
5. Can the same method be used for a parabola with a different focus or directrix?

**Tip**: Always remember that the value of \(p\) gives you the distance between the vertex and the focus (or the directrix) and is key to writing the correct equation.

Determine the equation of a parabola with focus at (0, 4) and directrix y = -4.

Learn how to determine the equation of a parabola with a focus at (0, 4) and a directrix at y = -4. This solution involves key concepts in coordinate geometry and conic sections, including the parabola equation (x - h)^2 = 4p(y - k). Ideal for students in Grades 9-11.

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