Parabola Calculator

The Parabola Calculator analyzes a quadratic curve of the form y = ax² + bx + c — its vertex, axis of symmetry, opening direction, x- and y-intercepts, and its focus and directrix — and plots the curve so you can see the shape, not just the numbers. Enter the three coefficients and it returns a full breakdown table, a diagram, and a step-by-step derivation of every value.

Unlike a tool that only solves for x, this one treats the equation as a curve: the vertex and axis of symmetry describe where it sits, the opening direction and focus/directrix describe its shape, and the roots (when real) are just one row among several. It's built for algebra and pre-calculus students graphing parabolas by hand, teachers preparing worked examples, and anyone who needs a parabola's key features without doing the algebra themselves.

How to use it

1. Enter the coefficient a (the number multiplying x²). It must not be zero — with a = 0 the equation is a line, not a parabola.
2. Enter the coefficient b (the number multiplying x).
3. Enter the coefficient c (the constant term).
4. Read the Diagram for a plot of the curve, the Analysis table for every computed value, and Step-by-Step Solution for the full derivation.

Results update automatically as you type. For example, entering a = 1, b = -4, c = 3 (for y = x² − 4x + 3) returns a vertex at (2, -1), axis of symmetry x = 2, x-intercepts at (1, 0) and (3, 0), and a curve that opens upward.

What do the vertex, focus, and directrix mean?

The vertex (h, k) is the parabola's turning point — its minimum when the curve opens upward, its maximum when it opens downward:

h = −b / (2a)
k = c − b² / (4a)

The axis of symmetry is the vertical line x = h the curve is mirrored across. The focus and directrix are the two things that define a parabola geometrically — every point on the curve is equidistant from the focus (a point) and the directrix (a line):

Focus:      (h, k + 1/(4a))
Directrix:  y = k − 1/(4a)

A larger |a| pulls the focus closer to the vertex (a narrower curve); a smaller |a| pushes it farther away (a wider, flatter curve).

How are the x-intercepts (roots) found?

The x-intercepts are where y = 0, found via the discriminant Δ = b² − 4ac, same as the quadratic formula:

• Δ > 0 — two real x-intercepts, one on each side of the vertex.
• Δ = 0 — the vertex itself touches the x-axis (one repeated root).
• Δ < 0 — the curve never crosses the x-axis; the table reports "None" rather than a pair of complex numbers, since a complex root has no point on the plotted diagram. To see those complex roots explicitly, use the Quadratic Formula Solver.

Why can't a be zero?

If a = 0, the x² term disappears and y = bx + c is a straight line — it has no vertex, no axis of symmetry, and no focus/directrix to compute. The calculator flags a = 0 as invalid input rather than silently treating it as a line.

Privacy

This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation, and the diagram, happen locally on your device.

Related tools

To solve the same equation for x via the discriminant alone (including complex roots), see the Quadratic Formula Solver. To convert it to vertex form by completing the square instead, see the Completing the Square Calculator.