Parabola from Points Calculator
Enter three points and get the quadratic equation of the parabola passing through all of them.
Last updated: June 22, 2026 · Reviewed by TG we-Calculate Editorial Team- 1
Discriminant
-1² − 4 × 2 × 1 = -7Positive = two real roots, zero = one, negative = none. - 2
Axis of symmetry
−-1 ÷ (2 × 2) = 0.25 - 3
Vertex y
2 × 0.25² + -1 × 0.25 + 1 = 0.8750
Formula
How this is calculated
A parabola y = ax² + bx + c has three unknown coefficients — a, b and c — so exactly three points are needed to pin it down. Substituting each point gives one linear equation in a, b, c, producing a 3×3 system that the calculator solves directly (the determinant is a Vandermonde determinant in the three x-values).
A unique solution exists only when the three x-values are distinct; if two points share an x-value, or the points are arranged so the system is singular, no single parabola fits and the tool says so. If the solved a comes out as 0 the three points actually lie on a straight line, not a parabola.
From the coefficients the calculator also reports the vertex (the turning point) at x = −b ÷ 2a, the axis of symmetry x = −b ÷ 2a, and the discriminant b² − 4ac, which tells you whether the parabola crosses the x-axis.
Examples
| Input | Result |
|---|---|
| (-1, 4), (0, 1), (2, 7) | y = 2x² − x + 1 |
| (0, 0), (1, 1), (2, 4) | y = x² |
Frequently asked questions
Three, as long as they have distinct x-values. A parabola y = ax² + bx + c has three coefficients, so three independent points determine them.
If two of the points have the same x-value (a vertical pair), no function y = ax² + bx + c can pass through both, so there is no solution.
Also known as
Sources & references
TG we-Calculate Editorial Team. (2026). Parabola from Points Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/parabola-from-points-calculator
TG we-Calculate Editorial Team. "Parabola from Points Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/parabola-from-points-calculator.
TG we-Calculate Editorial Team, "Parabola from Points Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/parabola-from-points-calculator
@misc{wecalculate_parabola_from_points_calculator, title = {Parabola from Points Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/parabola-from-points-calculator}}, year = {2026}, note = {TG we-Calculate} }
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