What is "quadratic function"

n. (context mathematics English) Any function whose value is the solution of a quadratic polynomial

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x, y, z, xy, xz, yz, x, y, z, and a constant:

f(x, y, z) = ax + by + cz + dxy + exz + fyz + gx + hy + iz + j, 

with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.

[[Image:Polynomialdeg2.svg|thumb|right|

x − x − 2

|frame|A quadratic polynomial with two real roots (crossings of the x axis) and hence no complex roots. Some other quadratic polynomials have their minimum above the x axis, in which case there are no real roots and two complex roots.]]

A univariate (single-variable) quadratic function has the form

f(x) = ax + bx + c,  a ≠ 0
in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the -axis, as shown at right.

If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.

The bivariate case in terms of variables x and y has the form

f(x, y) = ax + by + cxy + dx + ey + f 

with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).

In general there can be an arbitrarily large number of variables, in which case the resulting surface is called a quadric, but the highest degree term must be of degree 2, such as x, xy, yz, etc.