Quadratic polynomial

f(x) = x² - x - 2

In mathematics, a quadratic function is a polynomial function of the form

f(x) = ax² + bx + c,

where a is nonzero. It takes its name from the Latin quadratus for square, because quadratic functions arise in the calculation of areas of squares. In the case where the domain and codomain are R (the real numbers), the graph of such a function is a parabola.

If the quadratic function is set to be equal to zero, then the result is a quadratic equation.

The square root of a quadratic function gives rise either to an ellipse or to a hyperbola. If a>0 then the equation

describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola

y_p = ax² + bx + c.

If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is positive, then the hyperbola's axis is vertical.

If a<0 then the equation

describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola

y_p = ax² + bx + c

is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

A bivariate quadratic function is a second-degree polynomial of the form

f(x,y) = Ax² + By² + Cx + Dy + Exy + F.

Such a function describes a quadratic surface. Setting f(x,y) equal to zero describes the intersection of the surface with the plane z=0, which is a locus of points equivalent to a conic section.

Roots

The roots, or solutions to the quadratic function, for variable x, are

.

For the method of extracting these roots, see quadratic equation.

See also

• quadratic form
• Matrix representation of conic sections
• quadric