Quadratic function

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

Related Topics:
Mathematics - Polynomial - Function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:f(x)=ax^2+bx+c,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
Latin - Square - Domain - Codomain - Real number - Graph - 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

Related Topics:
Square root - Ellipse - Hyperbola

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: y = pm sqrt{a x^2 + b x + c}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola

Related Topics:
Ordinate - Minimum

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: y_p = a x^2 + b x + c.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: y = pm sqrt{a x^2 + b x + c}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: y_p = a x^2 + b x + c

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
Surface - Locus - Conic section

~ ~ ~ ~ ~ ~ ~ ~ ~ ~