This free parabola calculator finds the vertex, focus and directrix of any parabola, along with the axis of symmetry, latus rectum, focal length p, intercepts and vertex form, then plots it. Use y = ax² + bx + c, a sideways parabola x = ay² + by + c, or build the equation from a focus and directrix.
Parabola formulas
Worked example: y = x² − 4x + 3
- a = 1, b = −4, c = 3.
- Vertex: h = 4/2 = 2, k = 3 − 16/4 = −1, so V = (2, −1).
- p = 1/(4·1) = 0.25, so the focus is (2, −0.75) and the directrix is y = −1.25.
- x-intercepts from x² − 4x + 3 = 0: x = 1 and x = 3. y-intercept: (0, 3).
What the focus and directrix mean
Every point on a parabola is the same distance from the focus as from the directrix. That reflective property is why satellite dishes, car headlights and solar cookers are parabolic: rays parallel to the axis bounce straight to the focus.
Frequently asked questions
How do I find the vertex of a parabola?
For y = ax² + bx + c, the vertex x-coordinate is −b / 2a. Substitute it back into the equation to get the y-coordinate. For y = x² − 4x + 3 the vertex is (2, −1).
How do I find the focus and directrix?
Compute the focal length p = 1 / 4a. For a vertical parabola the focus is p units above the vertex (h, k + p) and the directrix is the line y = k − p. If a is negative, p is negative and the parabola opens downward.
What is the latus rectum of a parabola?
It is the chord through the focus, parallel to the directrix. Its length is |4p| = |1/a|, which tells you how wide the parabola is at its focus.
Can I find the equation from a focus and directrix?
Yes. Choose Focus & directrix, enter the focus point and the directrix line, and the calculator gives the vertex form, standard form and every other property.
How is a sideways parabola different?
A parabola of the form x = ay² + by + c opens left or right. Its axis of symmetry is horizontal (y = k), its focus is (h + p, k) and its directrix is the vertical line x = h − p.
Related tools
Free educational tool by HawkInc. Results are calculated in your browser and rounded for display; check critical work by hand or with a second method.