1. What is a Parabola?

A parabola is a U-shaped curve that is the graph of a quadratic function. It is a conic section produced by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.

2. Vertex and Focus Relationship

The vertex form of a parabola's equation is:

Where:

• \( (h, k) \) — Vertex coordinates
• \( p \) — Distance from vertex to focus (and vertex to directrix)
• Focus is at \( (h, k + p) \)
• Directrix is the line \( y = k - p \)

Note: For horizontal parabolas, the equation becomes \( (y-k)^2 = 4p(x-h) \) with focus at \( (h + p, k) \).

3. Using the Calculator

Instructions: Enter the vertex coordinates (h,k) and the parameter p. The calculator will determine:

• Focus point coordinates
• Directrix equation
• Vertex form equation
• Standard form equation

4. Practical Applications

Applications: Parabolas are used in physics (projectile motion), engineering (satellite dishes, headlights), architecture (arches), and economics (cost/profit curves).

5. Frequently Asked Questions (FAQ)

Q1: What does the parameter p represent?
A: p is the distance from the vertex to the focus (and from the vertex to the directrix). It determines how "wide" or "narrow" the parabola appears.

Q2: How is p related to the standard form coefficient a?
A: They are related by the formula p = 1/(4a), where y = ax² + bx + c is the standard form.

Q3: What if my parabola opens horizontally?
A: Use the form (y-k)² = 4p(x-h). The focus would be at (h + p, k) and directrix at x = h - p.

Q4: Can p be negative?
A: Yes, a negative p value indicates the parabola opens downward (for vertical) or leftward (for horizontal).

Q5: How do I find p if I only have the standard form equation?
A: First complete the square to convert to vertex form, or use p = 1/(4a) where a is the coefficient of the squared term.