Plot y = ax² + bx + c with vertex, roots, axis of symmetry, and SVG graph.
Coefficients
Presets
Graph
Properties
Formulas used
Quadratic formula
The roots of ax² + bx + c = 0 are given by the quadratic formula:
x = (−b ± √(b² − 4ac)) / (2a)
The expression b² − 4ac under the square root is the discriminant. Its sign determines how many real x-intercepts the parabola has.
Vertex form
Any parabola y = ax² + bx + c can be rewritten in vertex form as y = a(x − h)² + k, where:
h = −b / (2a) and k = c − b² / (4a)
The point (h, k) is the vertex — the minimum if a > 0 (opens up) or the maximum if a < 0 (opens down). The axis of symmetry is the vertical line x = h.
Discriminant meaning
D = b² − 4ac
D > 0 — two distinct real roots; the parabola crosses the x-axis at two points.
D = 0 — one repeated real root; the parabola touches the x-axis at the vertex.
D < 0 — no real roots; the parabola lies entirely above (if a > 0) or below (if a < 0) the x-axis.
FAQ
How do I find the vertex of a parabola?
Use h = −b / (2a) for the x-coordinate, then substitute back into the equation to get k. Equivalently, k = c − b² / (4a). The vertex (h, k) is the turning point — the lowest point if the parabola opens up, the highest if it opens down.
What does the discriminant tell me?
The discriminant D = b² − 4ac counts the real x-intercepts without solving. Positive D means two crossings, zero means one touch at the vertex, negative means the parabola never meets the x-axis. It is the quickest way to check if a quadratic equation has real solutions.
Vertex form vs standard form — which should I use?
Standard form y = ax² + bx + c is useful for finding the y-intercept (just read off c) and for plugging into the quadratic formula. Vertex form y = a(x − h)² + k immediately shows the vertex and the axis of symmetry, which makes graphing and transformations straightforward.
Why must a be non-zero?
If a = 0 the ax² term disappears and the equation reduces to y = bx + c — a straight line, not a parabola. A parabola needs the quadratic term, so a ≠ 0 is required. This plotter shows a warning if you enter a = 0.
How does this plotter auto-scale the graph?
The viewport is centred on the vertex and stretched to include the roots (if they exist) and the y-intercept, with padding on every side. This keeps the key points on-screen for any combination of a, b, c — no manual zooming needed.
Enter the coefficients a, b, c of a quadratic y = ax² + bx + c and the tool instantly computes the vertex (h, k) with h = −b/(2a) and k = c − b²/(4a), the axis of symmetry x = h, the discriminant D = b² − 4ac, the x-intercepts from the quadratic formula, and the y-intercept (0, c). An auto-scaling SVG graph plots the curve with labelled markers for every key point. Example: for y = x² − 4 the vertex is (0, −4), roots are ±2, y-intercept is (0, −4), discriminant is 16 and the parabola opens up. For y = −2x² + 4x + 1 the vertex (1, 3) is a maximum since a < 0. Presets, a copy-formula button and formula notes are built in.