What does this calculator find from a parabola equation?

The vertex, axis of symmetry, focus, and directrix from y = ax² + bx + c.

How is the vertex calculated?

x-coordinate = −b ÷ (2a); plug back in for the y-coordinate.

What does the sign of "a" tell you?

Positive opens upward; negative opens downward.

Parabola Calculator - CalculatorLabs.net

Parabola Calculator

A parabola is the smooth U-shaped curve you get from a quadratic equation, y = ax² + bx + c — the same shape traced by a thrown ball, a satellite dish, or a suspension bridge cable.

Coefficient a

Coefficient b

Coefficient c

Instant Result

Vertex X -

Vertex Y -

Focus X -

Focus Y -

Directrix (y =) -

What Is a Parabola Calculator?

A parabola is the smooth U-shaped curve you get from a quadratic equation, y = ax² + bx + c — the same shape traced by a thrown ball, a satellite dish, or a suspension bridge cable. This calculator finds the vertex (the curve's turning point), the focus (a special point used in optics and antenna design), and the directrix (a reference line) from the equation's coefficients.

How to Use This Parabola Calculator

Write your parabola in the form y = ax² + bx + c and identify the coefficients a, b, and c.
Enter all three coefficients into the fields above (a cannot be zero).
Click Calculate to get the vertex, focus, and directrix.

Formula

Vertex: h = −b / (2a), k = c − b² / (4a)
Focus: (h, k + p), where p = 1 / (4a)
Directrix: y = k − p

Worked Example

For the parabola y = 2x² − 4x + 1: a=2, b=−4, c=1. Vertex h = 4/4 = 1, k = 1 − 16/8 = −1, so vertex is (1, −1). p = 1/8 = 0.125. Focus = (1, −0.875). Directrix: y = −1.125.