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.