In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quadratic polynomial is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
of degree two in one or more variables. A quadratic function is the
polynomial function defined by a quadratic polynomial. Before 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 example, a
univariate (single-variable) quadratic function has the form
:
where is its variable. The
graph of a univariate quadratic function is a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
, a
curve that has an
axis of symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. parallel to the -axis.
If a quadratic function is
equated with zero, then the result is a
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
. The solutions of a quadratic equation are the
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
s of the corresponding quadratic function.
The
bivariate case in terms of variables and has the form
:
with at least one of 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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
(a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
or other
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
, a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
, or a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
).
A quadratic function in three variables , , and contains exclusively terms , , , , and a constant:
:
where at least one of the
coefficients 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
The adjective ''quadratic'' comes from the
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
word ''
quadrātum'' ("
square"). A term raised to the second power like is called a
square in algebra because it is the area of a ''square'' with side .
Terminology
Coefficients
The
coefficients of a quadric function are often taken to be
real or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, but they may be taken in any
ring, in which case the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
and the
codomain are this ring (see
polynomial evaluation).
Degree
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
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being 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
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
.
Variables
A quadratic polynomial may involve a single
variable ''x'' (the univariate case), or multiple variables such as ''x'', ''y'', and ''z'' (the multivariate case).
The one-variable case
Any single-variable quadratic polynomial may be written as
:
where ''x'' is the variable, and ''a'', ''b'', and ''c'' represent the
coefficients. Such polynomials often arise in a
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
The solutions to this equation are called the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
and can be expressed in terms of the coefficients as the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
. Each quadratic polynomial has an associated quadratic function, whose
graph is a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
.
Bivariate and multivariate cases
Any quadratic polynomial with two variables may be written as
:
where and are the variables and are the coefficients, and one of , and is nonzero. Such polynomials are fundamental to the study of
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s, as the
implicit equation of a conic section is obtained by equating to zero a quadratic polynomial, and the
zeros of a quadratic function form a (possibly degenerate) conic section.
Similarly, quadratic polynomials with three or more variables correspond to
quadric surfaces or
hypersurfaces.
Quadratic polynomials that have only terms of degree two are called
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s.
Forms of a univariate quadratic function
A univariate 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 function and the solutions of the corresponding quadratic equation.
*
is called the vertex form, where and are the and coordinates of the vertex, respectively.
The coefficient is the same value in all three forms. To convert the standard form to factored form, one needs only the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
to determine the two roots and . 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.
Graph of the univariate function
Regardless of the format, the graph of a univariate quadratic function
is a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
(as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation
.
* If , the parabola opens upwards.
* If , the parabola opens downwards.
The coefficient controls the degree of curvature of the graph; a larger magnitude of gives the graph a more closed (sharply curved) appearance.
The coefficients and together control the location of the axis of symmetry of the parabola (also the -coordinate of the vertex and the ''h'' parameter in the vertex form) which is at
:
The coefficient controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the -axis.
Vertex
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is . Using the method of completing the square, one can turn the standard form
:
into
:
so the vertex, , of the parabola in standard form is
:
If the quadratic function is in factored form
:
the average of the two roots, i.e.,
:
is the -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, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the vertex point, being a
maximum or minimum of the function, can be obtained by finding the roots of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
:
:
is a root of if
resulting in
:
with the corresponding function value
:
so again the vertex point coordinates, , can be expressed as
:
Roots of the univariate function
Exact roots
The
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
(or ''zeros''), and , of the univariate quadratic function
:
are the values of for which .
When the
coefficients , , and , are
real or
complex, the roots are
:
:
Upper bound on the magnitude of the roots
The
modulus of the roots of a quadratic
can be no greater than
where
is the
golden ratio [Lord, Nick, "Golden bounds for the roots of quadratic equations", ''Mathematical Gazette'' 91, November 2007, 549.]
The square root of a univariate quadratic function
The
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of a univariate quadratic function gives rise to one of the four conic sections,
almost always
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
either to an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
or to a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
.
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 the
ordinate of the
minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
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 the
maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
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 may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
locus of points.
Iteration
To
iterate 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 the ''n''
th 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 analytically
tractable cases.
For example, for the iterative equation
:
one has
:
where
:
and
So by induction,
:
can be obtained, where
can be easily computed as
:
Finally, we have
:
as the solution.
See
Topological conjugacy for more detail about the relationship between ''f'' and ''g''. And see
Complex quadratic polynomial for the chaotic behavior in the general iteration.
The
logistic map
:
with parameter 2<''r''<4 can be solved in certain cases, one of which is
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
and one of which is not. In the chaotic case ''r''=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 exhibits
sensitive dependence on initial conditions, so it is said to be chaotic.
The solution of the logistic map when ''r''=2 is
for
. Since
(1-2x_0)\in (-1,1) for any value of
x_0 other than the unstable fixed point 0, the term
(1-2x_0)^ goes to 0 as ''n'' goes to infinity, so
x_n goes to the stable fixed point
\tfrac.
Bivariate (two variable) quadratic function
A bivariate quadratic function is a second-degree polynomial of the form
:
f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F,
where ''A, B, C, D'', and ''E'' are fixed
coefficients and ''F'' is the constant term.
Such a function describes a quadratic Surface (mathematics)">surface. Setting
f(x,y) equal to zero describes the intersection of the surface with the plane
z=0, which is a locus (mathematics)">locus of points equivalent to a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
.
Minimum/maximum
If
4AB-E^2 <0 , the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.
If
4AB-E^2 >0 , the function has a minimum if both and , and a maximum if both and ; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at
(x_m, y_m) , where:
:
x_m = -\frac,
:
y_m = -\frac.
If
4AB- E^2 =0 and
DE-2CB=2AD-CE \ne 0 , the function has no maximum or minimum; its graph forms a parabolic
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an ...
.
If
4AB- E^2 =0 and
DE-2CB=2AD-CE =0 , the function achieves the maximum/minimum at a line—a minimum if ''A''>0 and a maximum if ''A''<0; its graph forms a parabolic cylinder.
See also
*
Quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
*
Quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
*
Matrix representation of conic sections In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relatio ...
*
Quadric
*
Periodic points of complex quadratic mappings
*
List of mathematical functions
References
*Algebra 1, Glencoe,
*Algebra 2, Saxon,
External links
*
{{DEFAULTSORT:Quadratic Function
Polynomial functions
Parabolas