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Quadratic function
 
 
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A quadratic function, in mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, is a polynomial function of the form , where . The graphGraph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs)....
 of a quadratic function is a parabolaParabola

The parabola is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a...
 whose major axis is parallel to the y-axis.

The expression in the definition of a quadratic function is a polynomial of degreeDegree of a polynomial

The degree of a polynomial is the maximum of the degrees of all terms in 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 equationQuadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree....
. The solutions to the equation are called the rootRoot (mathematics)

This article is about the zeroes of a function....
s of the equation or the zeros of the function.

Origin of word


The adjective quadratic comes from the LatinLatin

Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome....
 word quadratum for squareSquare (geometry)

In plane geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides....
. A term like x2 is called a squareSquare (algebra)

In algebra, the square of a number is that number multiplied by itself....
 in algebra because it is the area of a square with side x.

In general, a prefix quadr(i)- indicates the number 44 (number)

This article discusses the number Four....
. Examples are quadrilateralQuadrilateral Summary

In geometry, a quadrilateral is a polygon with four sides and four vertices....
 and quadrantQuadrant

Quadrant may refer to:* A sector equal to one quarter of a circle, or half a semicircle...
. 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 conjugateComplex conjugate

    In mathematics, the complex conjugate...
    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 mapLogistic map

    The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise f...
  • is called the standard form or vertex form.


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 squareFacts About Completing the square

Completing the square is an algebra technique, also used in many types of calculus....
. 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 parabolaParabola Overview

The parabola is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a...
 (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 calculusCalculus

Calculus is a central branch of mathematics, developed from algebra and geometry....
, the advantage of this method is that it works for more general functions.


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

In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the functio...
 points (which depends on , if , it has a minimum point, if , it has a maximum point) we have to first, take its derivativeDerivative

In mathematics, the derivative is defined as the instantaneous rate of change of a function....
:


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 rootSquare root

In mathematics, a square root of a number x is a number whose square is x....
 of a quadratic function gives rise either to an ellipseEllipse

The search term "Elliptical" redirects to this page; for the exercise machine, see Elliptical trainer....
 or to a hyperbolaHyperbola

In mathematics, a hyperbola is a type of conic section defined as the intersection between a right circular conical surface...
.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 emptyEmpty set Summary

In mathematics and more specifically set theory, the empty set is the unique set which contains no elements....
 locus of points.

Bivariate quadratic function

A bivariate quadratic function is a second-degree polynomial of the form
Such a function describes a quadratic surfaceSurface

In mathematics, specifically in topology, a surface is a two-dimensional manifold....
. Setting equal to zero describes the intersection of the surface with the plane , which is a locusLocus (mathematics)

In mathematics, a locus is a collection of points which share a property....
 of points equivalent to a conic sectionConic section

In mathematics, a conic section is a curve that can be formed by intersecting a cone with a plane....
.

Minimum/Maximum


If the function has no maximum or minimum, its graph forms an hyperbolic paraboloidParaboloid

In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation:...
.

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

In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation:...
.

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 formQuadratic form

    In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables....
  • Matrix representation of conic sectionsMatrix representation of conic sections

    In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci...
  • QuadricQuadric

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

    This article on periodic points of complex quadratic mappings describes periodic points of some quadratic polynomial mapping...
  • List of mathematical functionsList of mathematical functions

    In mathematics, several functions or groups of functions are important enough to deserve their own names....


External links