How to use this calculator

For a quadratic written as y = ax^2 + bx + c, the vertex is its turning point. The sign of a determines whether it opens upward or downward. This calculator also reports roots, the focus, and the directrix using the standard coordinate-plane model.

The method

h = -b/(2a); k = ah^2 + bh + c; vertex = (h,k)

Enter the values in the labeled fields, check the units or selected mode, and choose Calculate. The result includes the applicable working and a breakdown or visual when useful. Change an input and calculate again to update the result.

WORKED EXAMPLE

Coefficient a: 1 Coefficient b: -4 Constant c: 3

Parabola vertex: (2, -1)

y = x^2 - 4x + 3

Scope and limitations

Nonzero a is required. Coefficients and results use floating-point arithmetic. The graph shows a finite neighborhood around the vertex rather than the entire unbounded parabola.

Common questions

What happens when a is zero?

The expression becomes linear or constant and is not a parabola, so the calculator asks you to use a nonzero value.

Why are some roots complex?

A negative discriminant means the parabola does not meet the real x-axis, so its two algebraic roots are complex.

References & further reading

External references provide background, not an endorsement of this implementation. See our methodology for numerical conventions and limitations.