Quadratic function
The graph of a real single-variable quadratic function is a parabola.
In general the zeros of such a quadratic function describe a conic section (a circle or other ellipse, a parabola, or a hyperbola) in the
A quadratic function can have an arbitrarily large number of variables.
The adjective quadratic comes from the Latin word quadrātum ("square").
The coefficients of a quadratic 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).
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial.
Any single-variable quadratic polynomial may be written as where x is the variable, and a, b, and c represent the coefficients.
The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula.
Such polynomials are fundamental to the study of conic sections, as the implicit equation of a conic section is obtained by equating to zero a quadratic polynomial, and the zeros of a quadratic function form a (possibly degenerate) conic section.
Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces or hypersurfaces.
A univariate quadratic function can be expressed in three formats:[2] The coefficient a is the same value in all three forms.
The coefficients b and a together control the location of the axis of symmetry of the parabola (also the x-coordinate of the vertex and the h parameter in the vertex form) which is at The coefficient c controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the y-axis.
The vertical line that passes through the vertex is also the axis of symmetry of the parabola.
Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative: x is a root of f '(x) if f '(x) = 0 resulting in with the corresponding function value so again the vertex point coordinates, (h, k), can be expressed as The roots (or zeros), r1 and r2, of the univariate quadratic function are the values of x for which f(x) = 0.
[4] The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola.
The directions of the axes of the hyperbola are determined by the ordinate of the minimum point of the corresponding parabola
If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.
, one applies the function repeatedly, using the output from one iteration as the input to the next.
(The superscript can be extended to negative numbers, referring to the iteration of the inverse of
See Topological conjugacy for more detail about the relationship between f and g. And see Complex quadratic polynomial for the chaotic behavior in the general iteration.
The logistic map with parameter 2 In the chaotic case r=4 the solution is where the initial condition parameter never repeats itself – it is non-periodic and exhibits sensitive dependence on initial conditions, so it is said to be chaotic. A bivariate quadratic function is a second-degree polynomial of the form where A, B, C, D, and E are fixed coefficients and F is the constant term. equal to zero describes the intersection of the surface with the plane which is a locus of points equivalent to a conic section. the function has no maximum or minimum; its graph forms a hyperbolic paraboloid. the function has a minimum if both A > 0 and B > 0, and a maximum if both A < 0 and B < 0; its graph forms an elliptic paraboloid. the function has no maximum or minimum; its graph forms a parabolic cylinder. the function achieves the maximum/minimum at a line—a minimum if A>0 and a maximum if A<0; its graph forms a parabolic cylinder.
