Quadratic Function Calculator - Vertex, Zeros, Discriminant

Q: What happens when a = 0?

When a = 0, the function becomes f(x) = bx + c, which is linear, not quadratic. The calculator returns an error because the quadratic formula and vertex formula both divide by a. Use a linear function calculator instead.

Q: How do I convert standard form to vertex form?

Complete the square: from ax2 + bx + c, factor out a from the first two terms, then add and subtract (b/2a)2 inside. Or use the shortcut: p = -b/(2a), q = f(p) = c - b2/(4a). Then write f(x) = a(x - p)2 + q. Example: x2 - 6x + 8 becomes (x - 3)2 - 1.

Q: Can the vertex be below the x-axis?

Yes. When a > 0 (upward parabola) and q 0) and the parabola dips below zero between them. Example: f(x) = x2 - 4x + 3 has vertex (2, -1) and roots at x = 1 and x = 3.

Q: Where are quadratic functions used in real life?

Projectile motion (height = -4.9t2 + v0t + h0), bridge arches (parabolic curves), satellite dish shapes, revenue optimization (R = price x quantity, both linear in price), braking distance (proportional to speed squared), and optics (parabolic mirrors focus light to a point).

Q: How is this different from the Quadratic Equation Calculator?

The Quadratic Equation Calculator solves ax2 + bx + c = 0 and gives you the roots. This Quadratic Function Calculator analyzes the entire function f(x) = ax2 + bx + c - roots, vertex, axis, direction. Use the equation calculator when you just need roots; use this one when you need the full picture for graphing or analysis.

Three coefficients, five answers

Every quadratic function f(x) = ax2 + bx + c is completely defined by three numbers: a, b, and c. From them, this calculator derives everything you need - the discriminant that tells you how many roots exist, the roots themselves, the vertex (the highest or lowest point), the axis of symmetry, and which way the parabola opens.

Quick start

Enter a = 1, b = -6, c = 8. Result: discriminant = 4, roots x1 = 2 and x2 = 4, vertex at (3, -1), axis of symmetry x = 3, parabola opens upward.

The formulas behind the results

Discriminant: delta = b2 - 4ac

Roots: x = (-b +/- sqrt(delta)) / (2a)

Vertex: p = -b/(2a), q = -delta/(4a), so vertex = (p, q)

Axis of symmetry: x = p = -b/(2a)

What the discriminant tells you

Discriminant	Roots	Parabola behavior
delta > 0	Two distinct real roots	Parabola crosses the x-axis at two points
delta = 0	One repeated root	Parabola touches the x-axis at the vertex
delta < 0	No real roots	Parabola floats entirely above or below the x-axis

Worked examples

Two roots: f(x) = x2 - 5x + 6 (a=1, b=-5, c=6). Delta = 25 - 24 = 1 > 0. Roots: x1 = (5-1)/2 = 2, x2 = (5+1)/2 = 3. Vertex: (2.5, -0.25). The parabola dips just below the x-axis between x=2 and x=3.
Factored form: (x - 2)(x - 3)

One root: f(x) = x2 - 4x + 4 (a=1, b=-4, c=4). Delta = 16 - 16 = 0. Root: x0 = 2. Vertex: (2, 0). The parabola sits exactly on the x-axis at its lowest point.
Perfect square: (x - 2)2

No real roots: f(x) = 2x2 + x + 3 (a=2, b=1, c=3). Delta = 1 - 24 = -23 < 0. No x-intercepts. Vertex: (-0.25, 2.88). The entire parabola floats above the x-axis.
Since a > 0 and delta < 0, f(x) > 0 for all x

Downward parabola: f(x) = -x2 + 6x - 5 (a=-1, b=6, c=-5). Delta = 36 - 20 = 16 > 0. Roots: x1 = 1, x2 = 5. Vertex: (3, 4). Maximum value is 4, not minimum - because a < 0.
For a < 0, the vertex is the maximum point

Projectile motion: A ball thrown upward follows h(t) = -4.9t2 + 20t + 1.5 (SI units). Maximum height: vertex q = -(400 - 29.4)/(-19.6) = 21.9 m at t = 20/9.8 = 2.04 s. Hits the ground when h(t) = 0: use the quadratic formula to find t = 4.15 s.
Physics uses quadratic functions constantly

Vertex form vs standard form

Form	Formula	Best for
Standard	f(x) = ax2 + bx + c	Finding roots via quadratic formula, y-intercept (c)
Vertex	f(x) = a(x - p)2 + q	Reading vertex (p, q) directly, graphing
Factored	f(x) = a(x - x1)(x - x2)	Reading roots directly, solving inequalities

FAQ

What happens when a = 0?

When a = 0, the function becomes f(x) = bx + c, which is linear, not quadratic. The calculator returns an error because the quadratic formula and vertex formula both divide by a. Use a linear function calculator instead.

How do I convert standard form to vertex form?

Complete the square: from ax2 + bx + c, factor out a from the first two terms, then add and subtract (b/2a)2 inside. Or use the shortcut: p = -b/(2a), q = f(p) = c - b2/(4a). Then write f(x) = a(x - p)2 + q. Example: x2 - 6x + 8 becomes (x - 3)2 - 1.

Can the vertex be below the x-axis?

Yes. When a > 0 (upward parabola) and q < 0, the vertex sits below the x-axis. This means the function has two real roots (delta > 0) and the parabola dips below zero between them. Example: f(x) = x2 - 4x + 3 has vertex (2, -1) and roots at x = 1 and x = 3.

What does the discriminant mean geometrically?

The discriminant delta = b2 - 4ac measures how far the vertex is from the x-axis, scaled by 4a. When delta > 0, the vertex is on the "other side" of the x-axis from the opening direction - the parabola crosses. When delta = 0, the vertex sits exactly on the x-axis. When delta < 0, the vertex is on the "same side" - the parabola never reaches the x-axis.

Where are quadratic functions used in real life?

Projectile motion (height = -4.9t2 + v0t + h0), bridge arches (parabolic curves), satellite dish shapes, revenue optimization (R = price x quantity, both linear in price), braking distance (proportional to speed squared), and optics (parabolic mirrors focus light to a point).

How is this different from the Quadratic Equation Calculator?

The Quadratic Equation Calculator solves ax2 + bx + c = 0 and gives you the roots. This Quadratic Function Calculator analyzes the entire function f(x) = ax2 + bx + c - roots, vertex, axis, direction. Use the equation calculator when you just need roots; use this one when you need the full picture for graphing or analysis.

Related tools

Quadratic Equation Calculator

Solve ax2 + bx + c = 0 and get the roots with step-by-step working - Open calculator

Standard Deviation Calculator

The variance formula involves squaring deviations - a quadratic operation at its core - Open calculator

Coordinate Calculator

Plot the vertex and roots on a coordinate plane - distance and midpoint between key points - Open calculator

Powers & Roots Calculator

The quadratic formula uses a square root - compute sqrt(delta) independently here - Open calculator

Ready-made calculations

The most searched variants of this calculator. Each link opens it with the value already filled in, ready to calculate.