About the Parabola Calculator
This parabola calculator takes a quadratic equation in standard form — y = ax² + bx + c for a parabola that opens up or down, or x = ay² + by + c for one that opens left or right — and finds all of its geometric features: the vertex, the focus, the directrix, the axis of symmetry, the focal length p, the length of the latus rectum, the direction it opens and the vertex form of the equation.
For upward and downward parabolas it also returns the y-intercept and any real x-intercepts (roots), and plots the curve so you can see the vertex and the axis of symmetry. It is designed for algebra and precalculus students working with conic sections, and for anyone designing a parabolic reflector, dish or arch who needs to know where the focus sits.
If you already have vertex form y = a(x − h)² + k, expand it (b = −2ah, c = ah² + k) or simply enter a, then −2ah and ah² + k.
How to use the parabola calculator
- 1Choose whether your equation is y = … (opens up/down) or x = … (opens left/right).
- 2Enter the coefficients a, b and c; use 0 for any missing term.
- 3Read the vertex form, vertex, focus and directrix.
- 4Check the axis of symmetry and intercepts, and use the graph to visualise the curve.
Formula and method
Completing the square turns y = ax² + bx + c into vertex form y = a(x − h)² + k with h = −b/(2a) and k = c − b²/(4a). Comparing with the focus–directrix form (x − h)² = 4p(y − k) shows that 4p = 1/a, so the focal length is p = 1/(4a). The focus sits p units from the vertex along the axis of symmetry x = h and the directrix is the line p units on the opposite side; when a is negative p is negative and the parabola opens downward.
For x = ay² + by + c the roles of x and y swap: the vertex is (c − b²/(4a), −b/(2a)), the focus is (h + p, k), the directrix is x = h − p and the axis of symmetry is horizontal. The latus rectum — the chord through the focus parallel to the directrix — has length |4p| = |1/a|.
- a, b, c
- Coefficients of the quadratic
- (h, k)
- Vertex
- p
- Signed focal length (vertex to focus)
Worked examples
y = x² − 4x + 3
h = 4 ÷ 2 = 2 and k = 3 − 16/4 = −1, so the vertex is (2, −1). With p = 1/4 the focus is (2, −0.75) and the directrix is y = −1.25. The roots are x = 1 and x = 3.
Downward parabola y = −0.5x² + 2x + 1
h = −2 ÷ (−1) = 2 and k = 1 − 4/(−2) = 3. Since a < 0, p = −0.5: the focus is below the vertex at (2, 2.5) and the directrix is above it at y = 3.5.
Sideways parabola x = 0.25y²
This is y² = 4x, so 4p = 4 and p = 1. The focus is at (1, 0), the directrix is x = −1 and the latus rectum is 4 units long.
Frequently asked questions
How do you find the focus of a parabola?+
Find the vertex (h, k), then compute p = 1/(4a). For y = ax² + bx + c the focus is (h, k + p); for x = ay² + by + c it is (h + p, k).
How do you find the directrix of a parabola?+
The directrix is the line p units from the vertex on the opposite side to the focus: y = k − p for an up/down parabola, x = h − p for a left/right one.
What is the vertex formula?+
For y = ax² + bx + c the vertex x-coordinate is −b/(2a). Substitute it back in, or use k = c − b²/(4a), to get the y-coordinate.
Which way does a parabola open?+
For y = ax² + bx + c it opens up when a > 0 and down when a < 0. For x = ay² + by + c it opens right when a > 0 and left when a < 0.
Why does a satellite dish use a parabola?+
Every ray parallel to the axis reflects off a parabola through the focus. Placing the receiver at the focus collects the whole signal — the same property is used in headlights and solar cookers.