
Quadratic function

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
For example, a "univariate" (single-variable) quadratic function has the form
in the single variable "x". The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the -axis, as shown at right.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
The bivariate case in terms of variables "x" and "y" has the form
with at least one of "a, b, c" not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).
A quadratic function in three variables "x", "y," and "z" contains exclusively terms "x"2, "y"2, "z"2, "xy", "xz", "yz", "x", "y", "z", and a constant:
with at least one of the coefficients "a, b, c, d, e," or "f" of the second-degree terms being non-zero.
In general there can be an arbitrarily large number of variables, in which case the resulting surface of setting a quadratic function to zero is called a quadric, but the highest degree term must be of degree 2, such as "x"2, "xy", "yz", etc.
Etymology.
The adjective "quadratic" comes from the Latin word "quadrātum" ("square"). A term like is called a square in algebra because it is the area of a "square" with side .
Terminology.
Coefficients.
The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.
Degree.
When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant.
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial.
Variables.
A quadratic polynomial may involve a single variable "x" (the univariate case), or multiple variables such as "x", "y", and "z" (the multivariate case).
The one-variable case.
Any single-variable quadratic polynomial may be written as 
where "x" is the variable, and "a", "b", and "c" represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation formula_5. The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization, completing the square, graphing, Newton's method, or through the use of the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
Bivariate case.
Any quadratic polynomial with two variables may be written as 
where "x" and "y" are the variables and "a", "b", "c", "d", "e", and "f" are the coefficients. Such polynomials are fundamental to the study of conic sections, which are characterized by equating the expression for "f" ("x", "y") to zero. 
Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces. In linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space.
Forms of a univariate quadratic function.
A univariate quadratic function can be expressed in three formats:
The coefficient is the same value in all three forms. To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots and . To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
Graph of the univariate function.
Regardless of the format, the graph of a univariate quadratic function formula_10 is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation formula_11.
The coefficient controls the degree of curvature of the graph; a larger magnitude of gives the graph a more closed (sharply curved) appearance.
The coefficients and together control the location of the axis of symmetry of the parabola (also the -coordinate of the vertex and the "h" parameter in the vertex form) which is at 
The coefficient controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the -axis.
Vertex.
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is . Using the method of completing the square, one can turn the standard form
into
so the vertex, , of the parabola in standard form is
If the quadratic function is in factored form
the average of the two roots, i.e.,
is the -coordinate of the vertex, and hence the vertex is
The vertex is also the maximum point if , or the minimum point if .
The vertical line
that passes through the vertex is also the axis of symmetry of the parabola.
Maximum and minimum points.
Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:
 is a root of if 
resulting in
with the corresponding function value
so again the vertex point coordinates, , can be expressed as
Roots of the univariate function.
Exact roots.
The roots (or "zeros"), and , of the univariate quadratic function
are the values of for which .
When the coefficients , , and , are real or complex, the roots are
Upper bound on the magnitude of the roots.
The modulus of the roots of a quadratic formula_27 can be no greater than formula_28 where formula_29 is the golden ratio formula_30
The square root of a univariate quadratic function.
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.
If formula_31 then the equation formula_32 describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the ordinate of the minimum point of the corresponding parabola formula_33. 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.
If formula_34 then the equation formula_32 describes either a circle or other ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola
formula_33 is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.
Iteration.
To iterate a function formula_37, one applies the function repeatedly, using the output from one iteration as the input to the next.
One cannot always deduce the analytic form of formula_38, which means the "n"th iteration of formula_39. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of formula_39 if the inverse exists.) But there are some analytically tractable cases.
For example, for the iterative equation
one has
where
So by induction,
can be obtained, where formula_46 can be easily computed as
Finally, we have
as the solution.
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<"r"<4 can be solved in certain cases, one of which is chaotic and one of which is not. In the chaotic case "r"=4 the solution is
where the initial condition parameter formula_51 is given by formula_52. For rational formula_51, after a finite number of iterations formula_54 maps into a periodic sequence. But almost all formula_51 are irrational, and, for irrational formula_51, formula_54 never repeats itself – it is non-periodic and exhibits sensitive dependence on initial conditions, so it is said to be chaotic.
The solution of the logistic map when "r"=2 is
formula_58
for formula_59. Since formula_60 for any value of formula_61 other than the unstable fixed point 0, the term formula_62 goes to 0 as "n" goes to infinity, so formula_54 goes to the stable fixed point formula_64
Bivariate (two variable) quadratic function.
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.
Such a function describes a quadratic surface. Setting formula_66 equal to zero describes the intersection of the surface with the plane formula_67, which is a locus of points equivalent to a conic section.
Minimum/maximum.
If formula_68 the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.
If formula_69 the function has a minimum if "A">0, and a maximum if "A"<0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at formula_70 where:
If formula_73 and formula_74 the function has no maximum or minimum; its graph forms a parabolic cylinder.
If formula_73 and formula_76 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.