BYTETOOLS

Parabola Calculator

Analyze any parabola from standard or vertex form: vertex, focus, directrix, axis of symmetry, intercepts and latus rectum — with a live graph.

(2, 3)
Vertex
(2, 3.25)
Focus
y = 2.75
Directrix
1
Latus rectum

Graph

All properties

  • Equation: y = 1x² − 4x + 7
  • Vertex: (2, 3)
  • Axis of symmetry: x = 2
  • Focus: (2, 3.25)
  • Directrix: y = 2.75
  • Latus rectum: 1
  • Opens: upward
  • y-intercept: (0, 7)
  • x-intercepts: none (discriminant < 0)

What is the Parabola Calculator?

The ByteTools Parabola Calculator takes a quadratic in standard form y = ax² + bx + c or vertex form y = a(x − h)² + k and returns everything about the parabola: vertex, axis of symmetry, focus, directrix, latus rectum length, x-intercepts and y-intercept, plus whether it opens upward or downward.

  • Accepts standard and vertex form input
  • Vertex, focus, directrix, axis of symmetry and latus rectum
  • x-intercepts (real or none) and y-intercept
  • Canvas graph with self-drawn axes, grid, and marked focus/directrix
  • PNG export of the graph
  • 100% private — computed and drawn in your browser

How to use the Parabola Calculator

  1. 1

    Choose standard form y = ax² + bx + c or vertex form y = a(x − h)² + k.

  2. 2

    Enter the coefficients (a must be non-zero).

  3. 3

    Read the vertex, focus, directrix, axis of symmetry, latus rectum and intercepts.

  4. 4

    Inspect the live graph with the vertex, focus and directrix marked.

  5. 5

    Download the graph as a PNG or copy the properties.

About the Parabola Calculator

The ByteTools Parabola Calculator takes a quadratic in standard form y = ax² + bx + c or vertex form y = a(x − h)² + k and returns everything about the parabola: vertex, axis of symmetry, focus, directrix, latus rectum length, x-intercepts and y-intercept, plus whether it opens upward or downward.

The tool also draws the parabola on a canvas with axes, gridlines and tick labels rendered point by point, marking the vertex, focus and directrix so you can see how the algebra maps onto the picture. The graph is exportable as a PNG for worksheets and notes.

All computation and drawing happens 100% locally in your browser — nothing is uploaded. It is built for precalculus and conic-sections students, teachers preparing materials, and anyone who wants parabola properties without plotting software.

Frequently asked questions

How do you find the focus of a parabola?

Write the parabola as y = a(x − h)² + k, then the focal distance is p = 1 / (4a). The focus sits at (h, k + p) — inside the curve — and the directrix is the horizontal line y = k − p. For y = x², the focus is (0, 0.25) and the directrix is y = −0.25.

What is the directrix of a parabola?

The directrix is a fixed line such that every point on the parabola is equally distant from the focus and from that line. That equal-distance property is the geometric definition of a parabola. For an up/down opening parabola the directrix is horizontal, at y = k − 1/(4a).

What is the latus rectum?

It is the chord through the focus parallel to the directrix, and its length is |1/a| (equivalently |4p|). It tells you how wide the parabola is at the focus level — a handy guide for sketching: the parabola passes 1/(2a) either side of the focus.

How do you convert standard form to vertex form?

Complete the square: h = −b / 2a and k = c − b² / 4a. For y = x² − 4x + 7 that gives h = 2, k = 3, so y = (x − 2)² + 3. This calculator does the conversion automatically and shows both forms.

Why does my parabola have no x-intercepts?

The parabola crosses the x-axis only when ax² + bx + c = 0 has real solutions, which requires the discriminant b² − 4ac to be non-negative. When it is negative, the whole curve sits above or below the axis, and the calculator reports "no real x-intercepts" while still showing everything else.

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