Parabola Calculator — Vertex, Focus, Directrix & Roots
Enter the coefficients a, b, and c of the standard-form parabola y = ax² + bx + c to instantly find the vertex, focus, directrix, axis of symmetry, all intercepts, and the discriminant — with a live curve plot.
Vertex (1, -4) • Opens Upward
- 1
Vertex x = −b ÷ (2a)
−(-2) ÷ (2 × 1) = 1 - 2
b² ÷ (4a)
-2² ÷ (4 × 1) = 1This is how far the vertex shifts below (or above) the y-intercept. - 3
Vertex y = c − b²/(4a)
-3 − 1 = -4
How does this calculator work?
For y = ax² + bx + c: vertex = (−b/2a, c−b²/4a); focus = (x_v, y_v+1/4a); directrix: y = y_v−1/4a; axis of symmetry: x = −b/2a. X-intercepts exist when b²−4ac ≥ 0. Enter a, b, c to get all properties and a live graph instantly.
Formula
How this is calculated
A parabola in standard form y = ax² + bx + c is completely defined by its three coefficients. The leading coefficient a controls the width and direction: |a| > 1 makes it narrower, |a| < 1 makes it wider; positive a opens upward, negative a opens downward. The vertex — the extreme point of the parabola — sits at x_v = −b/(2a) and y_v = c − b²/(4a).
The vertex form also reveals the geometric optics properties of the parabola: the focus lies exactly 1/(4a) above the vertex (at y_v + 1/4a for an upward parabola), and the directrix is the horizontal line y = y_v − 1/4a. Every point on the parabola is equidistant from the focus and the directrix — this is the geometric definition.
The x-intercepts (real roots) exist when the discriminant b² − 4ac is positive (two roots), zero (one repeated root, the parabola is tangent to the x-axis), or negative (no real roots, entirely above or below the x-axis). The y-intercept is simply c (the value at x = 0). The axis of symmetry is the vertical line x = x_v, which divides the parabola into two mirror halves.
Frequently asked questions
For y = ax² + bx + c, the vertex x-coordinate is −b/(2a). Substitute this back into the equation to get the y-coordinate: y_v = c − b²/(4a). Alternatively, complete the square to rewrite the equation in vertex form y = a(x − h)² + k, where (h, k) is the vertex.
The focus is a special point inside the parabola such that every point on the parabola is equidistant from the focus and the directrix (a horizontal line outside the parabola). For y = ax² + bx + c the focus is at (x_v, y_v + 1/(4a)). This property is used in parabolic satellite dishes and telescopes, where parallel signals reflect off the parabolic surface and converge at the focus.
The discriminant b² − 4ac determines how many x-intercepts (real roots) the parabola has. If it is positive, the parabola crosses the x-axis at two distinct points. If it equals zero, the vertex sits exactly on the x-axis (one repeated root). If it is negative, the parabola does not cross the x-axis at all — all points lie strictly above (a > 0) or below (a < 0) it.
Also known as
TG we-Calculate Editorial Team. (2026). Parabola Calculator — Vertex, Focus, Directrix & Roots [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/parabola-calculator
TG we-Calculate Editorial Team. "Parabola Calculator — Vertex, Focus, Directrix & Roots." TG we-Calculate. 2026. https://we-calculate.com/calculator/parabola-calculator.
TG we-Calculate Editorial Team, "Parabola Calculator — Vertex, Focus, Directrix & Roots," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/parabola-calculator
@misc{wecalculate_parabola_calculator, title = {Parabola Calculator — Vertex, Focus, Directrix & Roots}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/parabola-calculator}}, year = {2026}, note = {TG we-Calculate} }
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