Quadratic Equations

Graph, explore, and understand parabolas with an AI tutor

A quadratic equation has the standard form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Its graph is a parabola — a symmetric U-shaped curve that opens upward when a > 0 and downward when a < 0.

The vertex is the highest or lowest point on the parabola, located at x = −b / (2a). The discriminant Δ = b² − 4ac tells you the number of real roots: two roots when Δ > 0, one repeated root when Δ = 0, and no real roots when Δ < 0.

Use the sliders below to change a, b, and c and watch how the parabola transforms. Ask the AI anything — try "What happens when a is negative?" or "Find the vertex."

Related Topics

Slope of a LineSystems of EquationsFactoring & Roots

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What is the quadratic formula?
The quadratic formula solves ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / 2a. It works for any quadratic equation, even when factoring is difficult.
How do I find the vertex of a parabola?
For the standard form y = ax² + bx + c, the vertex is at x = −b/(2a). Plug that x back into the equation to get y. In vertex form y = a(x − h)² + k, the vertex is simply (h, k).
What does the discriminant tell you?
The discriminant Δ = b² − 4ac reveals the nature of the roots. If Δ > 0, the parabola crosses the x-axis at two points. If Δ = 0, it touches the x-axis at exactly one point (the vertex). If Δ < 0, the parabola doesn't cross the x-axis — the equation has no real solutions.
What is vertex form and why is it useful?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. It makes graphing easy: you can read the vertex directly, and a controls the width and direction of the parabola.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.