Quadratic function

In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".

For the zeros of a quadratic function, see Quadratic equation and Quadratic formula.

For example, a univariate (single-variable) quadratic function has the form^[1]

where $x$ is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the $y$ -axis.

If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros of the corresponding quadratic 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. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), 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:

where at least one of the coefficients $a, b, c, d, e, f$ of the second-degree terms is not zero.

A quadratic function can have an arbitrarily large number of variables. The set of its zero form a quadric, which is a surface in the case of three variables and a hypersurface in general case.

Etymology[edit]

The adjective quadratic comes from the Latin word quadrātum ("square"). A term raised to the second power like $x 2$ is called a square in algebra because it is the area of a square with side $x$ .

Terminology[edit]

Coefficients[edit]

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).

Degree[edit]

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. However, where the "degree of a polynomial" refers to the largest degree of a non-zero term of the polynomial, more typically "order" refers to the lowest degree of a non-zero term of a power series.

Variables[edit]

A quadratic polynomial may involve a single variable x (the univariate case), or multiple variables such as x, y, and z (the multivariate case).

$f(x)=ax^{2}+bx+c$ is called the standard form,

$f(x)=a(x-r_{1})(x-r_{2})$ is called the factored form, where $r 1$ and $r 2$ are the roots of the quadratic function and the solutions of the corresponding quadratic equation.

$f(x)=a(x-h)^{2}+k$ is called the vertex form, where $h$ and $k$ are the $x$ and $y$ coordinates of the vertex, respectively.

A univariate quadratic function can be expressed in three formats:^[2]

The coefficient $a$ 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 $r 1$ and $r 2$ . 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.

If $a > 0$ , the parabola opens upwards.

If $a < 0$ , the parabola opens downwards.

The square root of a univariate quadratic function[edit]

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 $a>0,$ then the equation $y=\pm {\sqrt {ax^{2}+bx+c}}$ 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 $y_{p}=ax^{2}+bx+c.$ 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 $a<0,$ then the equation $y=\pm {\sqrt {ax^{2}+bx+c}}$ describes either a circle or other ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola $y_{p}=ax^{2}+bx+c$ is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.