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Input three point coordinates to quickly find the parabola's standard form, vertex coordinates, and axis of symmetry.
Enter coefficients or three points to view the parabola's analysis.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
This solver works with a vertical quadratic, written y = ax² + bx + c. You can enter the coefficients directly or provide three points on the curve. In three-point mode, the x-coordinates must be distinct so the points determine one quadratic function. A zero value of a is not a parabola, and three collinear points do not define a non-degenerate quadratic curve.
The result includes the equation, vertex, axis of symmetry, opening direction, focus, directrix, y-intercept, discriminant, and x-intercepts when real. The vertex is the turning point. For a parabola described by a vertical quadratic, the axis is the vertical line x = −b/(2a); the sign of a determines whether it opens up or down. OpenStax summarizes the standard focus and directrix relationships for parabolas.
y = ax² + bx + c: vertex = (−b/(2a), c − b²/(4a)); focus = (h, k + 1/(4a)); directrix = y = k − 1/(4a), where (h, k) is the vertex.Math reference: OpenStax Algebra and Trigonometry 2e, section 12.3, describes the vertical and horizontal standard forms and their vertex, focus, and directrix. Checked September 28, 2026.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Select coefficients when the equation is already in the form y = ax² + bx + c. Select three points when you have three coordinates sampled from a vertical parabola.
In coefficient mode, enter a, b, and c; a must be nonzero. In point mode, enter all three x and y values and ensure no two x-coordinates are equal.
Check the equation and vertex first, then use the axis and opening direction to interpret the graph. The focus and directrix are also outputs, not fields to enter.
Review the displayed algebra and plot. Copy or share the values only after confirming that the entered points or coefficients match the problem.
Use cases
See how the tool fits into real work and everyday tasks.
Enter given coefficients to compare the vertex, intercepts, and opening direction with a hand calculation or graph sketch.
Use three known coordinates to calculate the exact quadratic passing through them, then verify that each supplied point appears on the plotted curve.
For a simple vertical parabolic model, examine the vertex and intercepts as a quick mathematical check before using the equation elsewhere.
Q&A
Find concise answers to common questions and confusing cases.
This mode solves for one function of the form y = ax² + bx + c. Repeated x-values do not supply three independent inputs for that fit.
No. The available inputs are the three coefficients or three points. Focus and directrix are calculated outputs.
The graph does not cross the x-axis at a real coordinate. The solver still reports the equation and other features; its root result is shown as a complex pair.
Notes
Review scope, result limitations, and important precautions before use.
This solver handles vertical parabolas representable as a function of x. It does not fit noisy data statistically, accept focus/directrix as inputs, or solve sideways parabolas of the form x = ay² + by + c. Three points produce an exact interpolating curve; if measurements contain noise or more than three observations, an exact three-point solution may not describe the overall trend. Decimal output is formatted for readability, so retain full precision if your next calculation requires it.
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