Parabola Calculator

Analyze a quadratic equation, identify the vertex, focus, directrix, and axis of symmetry, and review each step beside the result.

Last updated: March 2026 | By ForgeCalc Engineering

Parabola Solver

Quadratic analysis

Step-by-Step Analysis

1.Standard form: y = 1x^2 + 0x + 0

2.Find h: h = -b / (2a) = -(0) / (2 * 1) = 0

3.Find k: k = f(h) = 0

4.Find p: p = 1 / (4a) = 0.25

5.Focus: (0, 0.25)

6.Directrix: y = -0.25

Parabola Summary

Vertex

(0, 0)

Focus

(0, 0.25)

Directrix

y = -0.25

Axis of Symmetry

x = 0

What Is a Parabola?

A parabola is the graph of a quadratic function. It is symmetric, opens upward or downward based on the sign of a, and has a vertex, focus, and directrix.

How to Analyze a Parabola

Enter the coefficients a, b, and c.
Compute the vertex using -b / (2a).
Evaluate the quadratic at the vertex to get k.
Find p, the focus, the directrix, and the axis of symmetry.

y = ax^2 + bx + c

Worked Example

Example: y = x^2 has vertex at (0, 0).

a = 1, b = 0, c = 0

Frequently Asked Questions

What if a is zero?

A parabola requires a non-zero a value.

Can the parabola open sideways?

This calculator analyzes vertical parabolas only.

What is p?

p is the focal distance, equal to 1 / (4a).

Does this accept decimals?

Yes. Any finite real coefficients are accepted.

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