Overview

Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, is a polynomial function of the form

The graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

of a quadratic function is a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

whose axis of symmetry is parallel to the y-axis.

The expression in the definition of a quadratic function is a polynomial of degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

2
or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.

If the quadratic function is set equal to zero, then the result is a quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

.
Encyclopedia
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, is a polynomial function of the form

The graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

of a quadratic function is a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

whose axis of symmetry is parallel to the y-axis.

The expression in the definition of a quadratic function is a polynomial of degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

2
or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.

If the quadratic function is set equal to zero, then the result is a quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

. The solutions to the equation are called the roots of the equation.

## Origin of word

Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

”). A term like x2 is called a square in algebra because it is the area of a square with side x.

In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and quadrant. Quadratum is the Latin word for square because a square has four sides.

## Roots

The roots (zeros) of the quadratic function

are the values of x for which f(x) = 0.

When the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s a, b, and c, are real or complex, the roots are
where the discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

is defined as

## Forms of a quadratic function

A quadratic function can be expressed in three formats:
• is called the standard form,
• is called the factored form, where and are the roots of the quadratic equation, it is used in logistic map
Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...

• is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively.

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
Completing the square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...

. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

## Graph

Regardless of the format, the graph of a quadratic function is a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

(as shown above).
• If (or is a positive number), the parabola opens upward.
• If (or is a negative number), the parabola opens downward.

The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive a makes the function increase faster and the graph appear more closed.

The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex) which is at .

The coefficient b alone is the declivity of the parabola as y-axis intercepts.

The coefficient c controls the height of the parabola, more specifically, it is the point where the parabola intercept the y-axis.

### Vertex

The vertex of a parabola is the place where it turns, hence, it's also called the turning point. If the quadratic function is in vertex form, the vertex is . By the method of completing the square, one can turn the general form
into

so the vertex of the parabola in the vertex form is

If the quadratic function is in factored form
the average of the two roots, i.e.,

is the x-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
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

:
giving
with the corresponding function value
so again the vertex point coordinates can be expressed as

## The square root of a quadratic function

The square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of a quadratic function gives rise either to an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

or to a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

.If then the equation describes a hyperbola. The axis of the hyperbola is determined by the ordinate
Ordinate
In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa...

of the minimum point of the corresponding parabola .
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 then the equation describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola
is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

locus of points.

## Iteration

Given an , one cannot always deduce the analytic form of , which means the nth iteration of . (The superscript can be extended to negative number referring to the iteration of the inverse of if the inverse exists.) But there is one easier case, in which .

In such case, one has,
where and .
So by induction,
can be obtained, where can be easily computed as.
Finally, we have,
in the case of .

See Topological conjugacy for more detail about such relationship between f and g. And see Complex quadratic polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...

for the chaotic behavior in the general iteration.

## Bivariate (two variable) quadratic function

A bivariate quadratic function is a second-degree polynomial of the form
Such a function describes a quadratic surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

. Setting equal to zero describes the intersection of the surface with the plane , which is a locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

of points equivalent to a conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

.

### Minimum/maximum

If the function has no maximum or minimum, its graph forms an hyperbolic paraboloid
Paraboloid
In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

.

If the function has a minimum if A>0, and a maximum if A<0, its graph forms an elliptic paraboloid.

The minimum or maximum of a bivariate quadratic function is obtained at where:

If and the function has no maximum or minimum, its graph forms a parabolic cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

.

If and the function achieves the maximum/minimum at a line. Similarly, a minimum if A>0 and a maximum if A<0, its graph forms a parabolic cylinder.

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2...

• Matrix representation of conic sections