Parabola Calculator (vertex, focus, directrix)

Understanding the parabola calculator — concept and principle

A parabola is the U-shaped curve of a quadratic function y = ax² + bx + c. This calculator finds its key features: the vertex (the turning point), the focus and the directrix, the axis of symmetry and the y-intercept. These appear in algebra and analytic geometry, in projectile motion, and in the design of satellite dishes, car headlights and solar cookers, which rely on the focus.

How it is calculated

• Vertex: x_v = −b ÷ (2a), y_v = c − b² ÷ (4a).
• Focal length: p = 1 ÷ (4a), the signed distance from the vertex to the focus.
• Focus: (x_v, y_v + p). Directrix: y = y_v − p.
• Axis of symmetry: x = x_v. y-intercept: c.

a must not be 0, otherwise the equation is a straight line.

The principle

Completing the square rewrites the equation as y = a(x − x_v)² + y_v. The squared term is never negative, so when a > 0 the smallest y is y_v at x = x_v (and when a < 0 that is the largest y). That is why the vertex has these coordinates and why the curve is symmetric about the vertical line through it.

Geometrically, a parabola is the set of points equally distant from a fixed point (the focus) and a fixed line (the directrix). With the vertex at the origin, a focus at (0, p) and a directrix y = −p, this condition simplifies to x² = 4py, that is, y = x² ÷ (4p). Matching with y = ax² gives p = 1/(4a). This equal-distance property is what makes a parabolic mirror send rays parallel to the axis through the focus.

Worked example

1. Inputs: a = 1, b = −4, c = 3, so y = x² − 4x + 3.
2. x_v = 4 ÷ 2 = 2; y_v = 3 − 16 ÷ 4 = −1. Vertex (2, -1).
3. p = 1 ÷ 4 = 0.25. Focus (2, −1 + 0.25) = (2, -0.75).
4. Directrix: y = −1 − 0.25 = -1.25.
5. Axis of symmetry x = 2, y-intercept 3. In vertex form: y = (x − 2)² − 1.

How to read the result

With a > 0 the parabola opens upward and the vertex is a minimum; with a < 0 it opens downward and the vertex is a maximum. A larger |a| gives a narrower curve and a focus closer to the vertex. The vertex value tells you the optimum: here the smallest possible y is −1. Because y_v is negative and the curve opens up, it crosses the x-axis twice, at x = 1 and x = 3.

Common mistakes

• Forgetting the minus sign in −b ÷ (2a).
• Placing the focus on the wrong side when a is negative; then p is negative and the focus lies below the vertex.
• Assuming the focus is at distance a from the vertex; it is 1/(4a).

Frequently asked questions

How do I find the x-intercepts?

Solve ax² + bx + c = 0 with the quadratic formula. For the example, x = 1 and x = 3.

Where should the receiver of a dish be placed?

At the focus. For a dish with depth d and diameter D, the focal length is D² ÷ (16d).

What is the latus rectum of a parabola?

The chord through the focus parallel to the directrix; its length is |1/a|, here 1.

Sources: Apollonius of Perga, Conics, Books I–III · J. Stewart, L. Redlin and S. Watson, Precalculus: Mathematics for Calculus, Cengage, chapter on conic sections