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A quadratic function, in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, is a polynomial function of the form , where . 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 coordinate system, together with Cartesian axes, etc....
 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 major axis is parallel to the y-axis.

The expression in the definition of a quadratic function is a polynomial of degree
Degree of a polynomial

When a polynomial is expressed as a sum or difference of term s , the exponent of the term with the highest exponent is the degree of the polynomial....
 2 or a 2nd degree polynomial, because the highest exponent of is 2.

If the quadratic function is set equal to zero, then the result is a quadratic equation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
.






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A quadratic function, in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, is a polynomial function of the form , where . 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 coordinate system, together with Cartesian axes, etc....
 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 major axis is parallel to the y-axis.

The expression in the definition of a quadratic function is a polynomial of degree
Degree of a polynomial

When a polynomial is expressed as a sum or difference of term s , the exponent of the term with the highest exponent is the degree of the polynomial....
 2 or a 2nd degree polynomial, because the highest exponent of is 2.

If the quadratic function is set equal to zero, then the result is a quadratic equation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
. The solutions to the equation are called the root
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
s of the equation or the zeros of the function.

Origin of word


The adjective quadratic comes from the Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 word quadratum for square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
. A term like x2 is called a square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 in algebra because it is the area of a square with side x.

In general, a prefix quadr(i)- indicates the number 4
4 (number)

This article discusses the number Four. For the year 4 AD, see 4. For other uses of 4, see 4 4 is a number, numeral, and glyph....
. Examples are quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
 and quadrant
Quadrant

Quadrant may refer to:* One of the four sections of the Cartesian coordinate system#Two-dimensional coordinate system* Quadrant , a measuring instrument capable of measuring angles up to 90°...
. Quadratum is the Latin word for square because a square has four sides.

Roots


The two roots of the quadratic equation , where are

This formula is called the quadratic formula.

  • Let '
  • If ', then there are two distinct roots since is a positive real number.
  • If , then the two roots are equal, since is zero.
  • If , then the two roots are complex conjugate
    Complex conjugate

    In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number...
    s, since is imaginary.


By letting and or vice versa, one can factor as .

Forms of a quadratic function

A quadratic function can be expressed in three formats:
  • is called the general form or polynomial 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 Quadratic function, often cited as an archetypal example of how complex, chaos theory behaviour can arise from very simple non-linear dynamical equations....
  • is called the standard form or vertex form where and are the x and y coordinates of the vertex.


To convert the general form to factored form, one needs only the quadratic formula to determine the two roots and . To convert the general form to standard 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 formto the formThe expression inside the parenthesis is of the form x − constant....
. To convert the factored form (or standard form) to general 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 , the parabola opens upward.
  • If , 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).

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

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

x–intercepts


The x-intercepts of the graph are the same as the roots of the quadratic function (see above).

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 standard form, the vertex is . By the method of completing the square, one can turn the general form: to



so the vertex of the parabola in the general form will be



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


The maximum or minimum of the function is always obtained at the vertex, the following method is another derivation of the same fact using calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
, the advantage of this method is that it works for more general functions.


Taking as sample quadratic equation, to find its maximum or minimum
Maxima and minima

In mathematics, maxima and minima, known collectively as extrema, are the largest value or smallest value , that a function takes in a point either within a given neighbourhood or on the function domain in its entirety ....
 points (which depends on , if , it has a minimum point, if , it has a maximum point) we have to first, take its 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 a quantity is changing at a given point....
:


Then, we find the roots of :




So, is the value of . Now, to find the value, we substitute on :




Thus, the maximum or minimum point coordinates are:




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 mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 or to a hyperbola
Hyperbola

In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
.If then the equation describes a hyperbola. The axis of the hyperbola is determined by the ordinate 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 members. 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.

Bivariate 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 E3....
. Setting equal to zero describes the intersection of the surface with the plane , which is a locus
Locus (mathematics)

In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve....
 of points equivalent to a conic section
Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
.

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
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....
.

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.

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.

See also


  • Quadratic form
    Quadratic form

    In mathematics, a quadratic form is a homogeneous polynomial of Degree_ two in a number of variables. For example,is a quadratic form in the variables x and y....
  • Quadratic polynomial
    Quadratic polynomial

    In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z....
  • Matrix representation of conic sections
    Matrix representation of conic sections

    In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis of rotation, vertex , Focus , tangents, and the relative position of a given point....
  • Quadric
    Quadric

    In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
  • Periodic points of complex quadratic mappings
    Periodic points of complex quadratic mappings

    This article describes periodic points of some Complex quadratic polynomial. This theory is applied in relation with the theories of Fatou set and Julia sets....
  • List of mathematical functions
    List of mathematical functions

    In mathematics, several function s or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain these functions in more detail....


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