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use the distance formula to write an equation of the parabola for each

To write the equation of a parabola using the Distance Formula, you need to know the relationship between the vertex, focus, and directrix.

### General Concept:
For a parabola with a vertical axis of symmetry:
- **Vertex**: \((h, k)\)
- **Focus**: \((h, k + p)\)
- **Directrix**: \(y = k - p\)

The equation is given by:
\[
(x - h)^2 = 4p(y - k)
\]
Where \(p\) is the distance from the vertex to the focus (or from the vertex to the directrix).

Given that the vertex is at \((0, 0)\) for all the provided problems, the equations simplify as:

\[
x^2 = 4py
\]

Now, let's find the equation for each given problem.

### Problem 4:
- **Vertex**: \((0, 0)\)
- **Directrix**: \(y = 4\)

**Step 1**: Determine the focus. Since the directrix is above the vertex, the parabola opens downwards. The focus is equidistant from the vertex on the opposite side:
- Focus: \((0, -4)\)

**Step 2**: Calculate \(p\):
- \(p = -4\)

**Equation**:
\[
x^2 = -16y
\]

### Problem 5:
- **Vertex**: \((0, 0)\)
- **Focus**: \((0, -1)\)

**Step 1**: Since the focus is below the vertex, the parabola opens downwards.

**Step 2**: Calculate \(p\):
- \(p = -1\)

**Equation**:
\[
x^2 = -4y
\]

### Problem 6:
- **Vertex**: \((0, 0)\)
- **Directrix**: \(y = 2\)

**Step 1**: Determine the focus. Since the directrix is above the vertex, the parabola opens downwards. The focus is equidistant from the vertex on the opposite side:
- Focus: \((0, -2)\)

**Step 2**: Calculate \(p\):
- \(p = -2\)

**Equation**:
\[
x^2 = -8y
\]

### Do you have any more questions or need further details?

### Related Questions:
1. What is the significance of the value of \(p\) in the equation of a parabola?
2. How would the equation change if the vertex is not at the origin?
3. Can you derive the equation of a parabola with a horizontal axis of symmetry?
4. How do the directrix and focus affect the shape of the parabola?
5. What happens to the parabola if the directrix is moved further from the vertex?

### Tip:
Remember that the sign of \(p\) determines the direction the parabola opens: if \(p\) is positive, it opens upwards (or to the right for horizontal parabolas); if \(p\) is negative, it opens downwards (or to the left for horizontal parabolas).

Learn how to write equations of parabolas using the Distance Formula. Understand the relationship between the vertex, focus, and directrix for parabolas with a vertical axis of symmetry. Detailed examples and step-by-step solutions provided.

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