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

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

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

is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an emptyAbstract algebra Algebra Set theory In mathematics, the empty set is the set with no elements. Notation The standard notation for denoting the empty set, invented by Nicholas Bourbaki, is the symbol , also written as or ∅, and sometimes approximated locus of points.

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

Such a function describes a quadratic surfaceIn mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the. Setting f(x,y) equal to zero describes the intersection of the surface with the plane z=0, which is a locusIn mathematics, a locus (plural loci is a collection of points which share a common property. A locus of points usually forms a continuous figure or figures. For example, the conic sections are defined in terms of loci: # A circle is the locus of points f of points equivalent to a conic sectionIn mathematics, a conic section (or just conic is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their propert.

1 Roots

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

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

2 See also

quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, the distance between two points in three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six var
Matrix representation of conic sectionsIn mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci, tangents, and the relative position of a given point. We can also study conic sections whose axes aren't parallel to our coordina
quadric

Polynomials

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