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. Parabola Equations

The standard forms of parabola equations are:

Where:

• \( a \) — Determines the parabola's width and direction (up/down)
• \( h, k \) — Vertex coordinates
• \( p \) — Distance from vertex to focus

3. Key Features of a Parabola

Vertex: The highest or lowest point of the parabola.
Focus: A fixed point inside the parabola.
Directrix: A line perpendicular to the axis of symmetry.
Axis of Symmetry: The vertical line that divides the parabola into two mirror images.
Roots: Points where the parabola intersects the x-axis (if any).

4. Using the Calculator

Instructions: Enter the coefficients a, b, and c of your quadratic equation in the form y = ax² + bx + c. The calculator will compute and display all key features of the parabola.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola opens horizontally?
A: This calculator handles vertical parabolas (y as a function of x). For horizontal parabolas, you would use x = ay² + by + c.

Q2: What does it mean if there are no real roots?
A: If the discriminant (b² - 4ac) is negative, the parabola doesn't intersect the x-axis.

Q3: How does coefficient 'a' affect the parabola?
A: Larger |a| makes the parabola narrower. Positive a opens upward, negative a opens downward.

Q4: What is the relationship between focus and directrix?
A: Every point on the parabola is equidistant to the focus and the directrix.

Q5: Can I use this for vertex form equations?
A: Yes, just expand your vertex form to standard form to find coefficients a, b, and c.