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Concept
Fonction quadratique
Résumé
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".
For example, a univariate (single-variable) quadratic function has the form
where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis.
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros of the corresponding quadratic function.
The bivariate case in terms of variables x and y has the form
with at least one of a, b, c not equal to zero. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), a conic section (a circle or other ellipse, a parabola, or a hyperbola).
A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:
where at least one of the coefficients a, b, c, d, e, f of the second-degree terms is not zero.
A quadratic function can have an arbitrarily large number of variables. The set of its zero form a quadric, which is a surface in the case of three variables and a hypersurface in general case.
The adjective quadratic comes from the Latin word quadrātum ("square"). A term raised to the second power like x2 is called a square in algebra because it is the area of a square with side x.
The coefficients of a quadric function are often taken to be real or complex numbers, but they may be taken in any ring, in which case the domain and the codomain are this ring (see polynomial evaluation).
Source officielle
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