Quadratic Function Plotter (ax² + bx + c)
Plot any quadratic function ax² + bx + c. Get the vertex, discriminant, real roots and a smooth parabola chart. Instant, free.
How to use this tool
- Enter coefficient a (x²), coefficient b (x), constant c, x minimum and x maximum in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your vertex x and the full breakdown beneath it.
A quadratic function f(x) = ax² + bx + c forms a parabola. The vertex is at x = −b/(2a). Real roots exist when the discriminant b²−4ac ≥ 0; they are x = (−b ± √(b²−4ac)) / 2a.
Formula
f(x) = ax2 + bx + c
Vertex: xv = −b ÷ (2a), yv = a xv2 + b xv + c
Discriminant: Δ = b2 − 4ac
Real roots (when Δ ≥ 0): x = (−b ± √Δ) ÷ (2a)
How it works
This tool plots the parabola f(x) = ax² + bx + c by evaluating it at 61 equally spaced points across the requested x range, then analytically computes the vertex coordinates, the discriminant, and any real roots. When the discriminant is negative the parabola does not cross the x-axis and no real roots are reported. The vertex marks the minimum (a > 0) or maximum (a < 0) of the function.
Worked example
- a = 1, b = 0, c = −4. Function: f(x) = x² − 4.
- Vertex: x = −0 ÷ (2×1) = 0; y = 1×0 + 0 − 4 = −4. Vertex at (0, −4).
- Discriminant: 0² − 4×1×(−4) = 16.
- Roots: x = (0 ± √16) ÷ 2 = ±2. So root 1 = −2, root 2 = 2.
Vertex (0, −4); discriminant = 16; roots x = −2 and x = 2.
Common mistakes to avoid
- Entering a=0, which reduces the equation to a line — the quadratic formula requires a non-zero leading coefficient and the chart will be degenerate or error.
- Expecting two real roots when the discriminant is negative (b^2 - 4ac < 0) — the parabola does not cross the x-axis in that case.
- Reading the vertex x-coordinate as a root — the vertex is the turning point of the parabola, not a zero of the function.
Key terms
- Quadratic function
- A polynomial of degree 2 of the form ax² + bx + c (a ≠ 0) whose graph is a parabola opening upward (a > 0) or downward (a < 0).
- Vertex
- The turning point of the parabola — the minimum if a > 0, the maximum if a < 0 — located at x = −b/(2a).
- Discriminant
- The expression b² − 4ac: positive means two distinct real roots, zero means one repeated root, negative means no real roots (complex roots only).
- Real roots
- The x values where f(x) = 0, i.e. where the parabola crosses the x-axis; found using the quadratic formula when the discriminant is non-negative.
- Parabola
- The U-shaped (or inverted-U) curve traced by any quadratic function; it is symmetric about the vertical line through the vertex.
Frequently asked questions
- What does the discriminant tell me?
- If b²−4ac > 0 there are two real roots; = 0 one repeated root; < 0 no real roots (the parabola doesn't cross the x-axis).
- Does a affect the shape?
- Yes. a > 0 opens upward; a < 0 opens downward. Larger |a| makes it narrower.