Quadratic function

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In mathematics, a '''quadratic function''' is a polynomial function of the form :<math>f(x)=ax^2+bx+c</math>, 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 :<math> y = \pm \sqrt{a x^2 + b x + c} </math> describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola :<math> y_p = a x^2 + b x + c. </math> 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 :<math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola :<math> y_p = a x^2 + b x + c </math> 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 :<math> f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F. </math> 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 :<math> x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} </math>. For the method of extracting these roots, see quadratic equation. ==See also== * quadratic form * Matrix representation of conic sections * quadricja:二次関数