Parabola Calculator

Find the vertex, focus, directrix and real roots of y = ax² + bx + c.

Result and method

Supported equation

Enter a vertical parabola in the form y = ax² + bx + c. Horizontal and rotated parabolas are not supported. Coefficients must have magnitude at most 1e6, and |a| must be at least 0.000001.

Method

Complete the square: y = a(x−h)² + k, where h = −b/(2a) and k = c−b²/(4a). The focus is (h, k+1/(4a)); the directrix is y = k−1/(4a). Real roots exist when b²−4ac ≥ 0.

Worked example

For y = x²−4x+3 = (x−2)²−1, the vertex is (2,−1), focus (2,−0.75), directrix y = −1.25, and roots 1 and 3.

Why is a = 0 not allowed?

That produces a line, not a parabola. Numeric output uses floating-point arithmetic; near-zero discriminants may be sensitive to rounding.