Inmathematics, aquadratic function of a singlevariable is afunction of the form[1]
where is its variable, and,, and arecoefficients. Theexpression, especially when treated as anobject in itself rather than as a function, is aquadratic polynomial, apolynomial of degree two. Inelementary mathematics a polynomial and its associatedpolynomial function are rarely distinguished and the termsquadratic function andquadratic polynomial are nearly synonymous and often abbreviated asquadratic.
Thegraph of areal single-variable quadratic function is aparabola. If a quadratic function isequated with zero, then the result is aquadratic equation. The solutions of a quadratic equation are thezeros (orroots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by thequadratic formula.
A quadratic polynomial or quadratic function can involve more than one variable. For example, a two-variable quadratic function of variables and has the form
with at least one of,, and not equal to zero. In general the zeros of such a quadratic function describe aconic section (acircle or otherellipse, aparabola, or ahyperbola) in the– plane. A quadratic function can have an arbitrarily large number of variables. The set of its zero form aquadric, which is asurface in the case of three variables and ahypersurface in general case.
The adjectivequadratic comes from theLatin wordquadrātum ("square"). A term raised to the second power like is called asquare in algebra because it is the area of asquare with side.
Thecoefficients of a quadratic function are often taken to bereal orcomplex numbers, but they may be taken in anyring, in which case thedomain and thecodomain are this ring (seepolynomial evaluation).
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 thelargest degree of a non-zero term of the polynomial, more typically "order" refers to thelowest degree of a non-zero term of apower series.
A quadratic polynomial may involve a singlevariablex (theunivariate case), or multiple variables such asx,y, andz (the multivariate case).
Any single-variable quadratic polynomial may be written as
wherex is the variable, anda,b, andc represent thecoefficients. Such polynomials often arise in aquadratic equation The solutions to this equation are called theroots and can be expressed in terms of the coefficients as thequadratic formula. Each quadratic polynomial has an associated quadratic function, whosegraph is aparabola.
Any quadratic polynomial with two variables may be written as
wherex andy are the variables anda,b,c,d,e,f are the coefficients, and one ofa,b andc is nonzero. Such polynomials are fundamental to the study ofconic sections, as theimplicit equation of a conic section is obtained by equating to zero a quadratic polynomial, and thezeros of a quadratic function form a (possibly degenerate) conic section.
Similarly, quadratic polynomials with three or more variables correspond toquadric surfaces orhypersurfaces.
Quadratic polynomials that have only terms of degree two are calledquadratic forms.
A univariate quadratic function can be expressed in three formats:[2]
The coefficienta is the same value in all three forms. To convert thestandard form tofactored form, one needs only thequadratic formula to determine the two rootsr1 andr2. To convert thestandard form tovertex form, one needs a process calledcompleting the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
Regardless of the format, the graph of a univariate quadratic function is aparabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation.
The coefficienta controls the degree of curvature of the graph; a larger magnitude ofa gives the graph a more closed (sharply curved) appearance.
The coefficientsb anda together control the location of the axis of symmetry of the parabola (also thex-coordinate of the vertex and theh parameter in the vertex form) which is at
The coefficientc controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts they-axis.
Thevertex of a parabola is the place where it turns; hence, it is also called theturning point. If the quadratic function is in vertex form, the vertex is(h,k). Using the method of completing the square, one can turn the standard form
into
so the vertex,(h,k), of the parabola in standard form is
If the quadratic function is in factored form
the average of the two roots, i.e.,
is thex-coordinate of the vertex, and hence the vertex(h,k) is
The vertex is also the maximum point ifa < 0, or the minimum point ifa > 0.
The vertical line
that passes through the vertex is also theaxis of symmetry of the parabola.
Usingcalculus, the vertex point, being amaximum or minimum of the function, can be obtained by finding the roots of thederivative:
x is a root off '(x) iff '(x) = 0resulting in
with the corresponding function value
so again the vertex point coordinates,(h,k), can be expressed as
Theroots (orzeros),r1 andr2, of the univariate quadratic function
are the values ofx for whichf(x) = 0.
When thecoefficientsa,b, andc, arereal orcomplex, the roots are
Themodulus of the roots of a quadratic can be no greater than where is thegolden ratio[5]
Thesquare root of a univariate quadratic function gives rise to one of the four conic sections,almost always either to anellipse or to ahyperbola.
If then the equation describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by theordinate of theminimum point of the corresponding parabola 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 then the equation describes either a circle or other ellipse or nothing at all. If the ordinate of themaximum point of the corresponding parabola is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes anempty locus of points.
Toiterate a function, 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, which means thenth iteration of. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of if the inverse exists.) But there are some analyticallytractable cases.
For example, for the iterative equation
one has
where
So by induction,
can be obtained, where can be easily computed as
Finally, we have
as the solution.
SeeTopological conjugacy for more detail about the relationship betweenf andg. And seeComplex quadratic polynomial for the chaotic behavior in the general iteration.
Thelogistic map
with parameter 2<r<4 can be solved in certain cases, one of which ischaotic and one of which is not. In the chaotic caser=4 the solution is
where the initial condition parameter is given by. For rational, after a finite number of iterations maps into a periodic sequence. But almost all are irrational, and, for irrational, never repeats itself – it is non-periodic and exhibitssensitive dependence on initial conditions, so it is said to be chaotic.
The solution of the logistic map whenr=2 is
for. Since for any value of other than the unstable fixed point 0, the term goes to 0 asn goes to infinity, so goes to the stable fixed point
Abivariate quadratic function is a second-degree polynomial of the form
whereA, B, C, D, andE are fixedcoefficients andF is theconstant term.Such a function describes a quadraticsurface. Setting equal to zero describes the intersection of the surface with the plane which is alocus of points equivalent to aconic section.
If the function has no maximum or minimum; its graph forms a hyperbolicparaboloid.
If the function has a minimum if bothA > 0 andB > 0, and a maximum if bothA < 0 andB < 0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at where:
If and the function has no maximum or minimum; its graph forms a paraboliccylinder.
If and the function achieves the maximum/minimum at a line—a minimum ifA>0 and a maximum ifA<0; its graph forms a parabolic cylinder.
