Quadratic function

Last updated

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.

Contents

A quadratic polynomial with two real roots (crossings of the x axis) and hence no complex roots. Some other quadratic polynomials have their minimum above the x axis, in which case there are no real roots and two complex roots. Polynomialdeg2.svg
A quadratic polynomial with two real roots (crossings of the x axis) and hence no complex roots. Some other quadratic polynomials have their minimum above the x axis, in which case there are no real roots and two complex roots.

For example, a univariate (single-variable) quadratic function has the form [1]

in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right.

If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.

The bivariate case in terms of variables x and y has the form

with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).

A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:

with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.

In general there can be an arbitrarily large number of variables, in which case the resulting surface of setting a quadratic function to zero is called a quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.

Etymology

The adjective quadratic comes from the Latin word quadrātum ("square"). A term like x2 is called a square in algebra because it is the area of a square with side x.

Terminology

Coefficients

The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.

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". 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.

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. In elementary algebra, such polynomials often arise in the form of a quadratic equation . The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization, completing the square, graphing, Newton's method, or through the use of the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.

Bivariate case

Any quadratic polynomial with two variables may be written as

where x and y are the variables and a, b, c, d, e, and f are the coefficients. Such polynomials are fundamental to the study of conic sections, which are characterized by equating the expression for f (x, y) to zero. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces. In linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space.

Forms of a univariate quadratic function

A univariate quadratic function can be expressed in three formats: [2]

The coefficient a is the same value in all three forms. To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots r1 and r2. 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

f
(
x
)
=
a
x
2
|
a
=
{
0.1
,
0.3
,
1
,
3
}
{\displaystyle f(x)=ax^{2}|_{a=\{0.1,0.3,1,3\}}\!} Function ax^2.svg
f
(
x
)
=
x
2
+
b
x
|
b
=
{
1
,
2
,
3
,
4
}
{\displaystyle f(x)=x^{2}+bx|_{b=\{1,2,3,4\}}\!} Function x^2+bx.svg
f
(
x
)
=
x
2
+
b
x
|
b
=
{
-
1
,
-
2
,
-
3
,
-
4
}
{\displaystyle f(x)=x^{2}+bx|_{b=\{-1,-2,-3,-4\}}\!} Function x^2-bx.svg

Regardless of the format, the graph of a univariate quadratic function is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation .

The coefficient a controls the degree of curvature of the graph; a larger magnitude of a gives the graph a more closed (sharply curved) appearance.

The coefficients b and a together control the location of the axis of symmetry of the parabola (also the x-coordinate of the vertex and the h parameter in the vertex form) which is at

The coefficient c controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the y-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 (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 the x-coordinate of the vertex, and hence the vertex (h, k) is

The vertex is also the maximum point if a< 0, or the minimum point if a> 0.

The vertical line

that passes through the vertex is also the axis of symmetry of the parabola.

Maximum and minimum points

Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:

x is a root of f '(x) if f '(x) = 0 resulting in

with the corresponding function value

so again the vertex point coordinates, (h, k), can be expressed as

Roots of the univariate function

Graph of y = ax + bx + c, where a and the discriminant b - 4ac are positive, with
Roots and y-intercept in red
Vertex and axis of symmetry in blue
Focus and directrix in pink Quadratic function graph key values.svg
Graph of y = ax + bx + c, where a and the discriminant b − 4ac are positive, with
  • Roots and y-intercept in red
  • Vertex and axis of symmetry in blue
  • Focus and directrix in pink
Visualisation of the complex roots of y = ax + bx + c: the parabola is rotated 180deg about its vertex (orange). Its x-intercepts are rotated 90deg around their mid-point, and the Cartesian plane is interpreted as the complex plane (green). Quadratic function graph complex roots.svg
Visualisation of the complex roots of y = ax + bx + c: the parabola is rotated 180° about its vertex (orange). Its x-intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex plane (green).

Exact roots

The roots (or zeros), r1 and r2, of the univariate quadratic function

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

When the coefficients a, b, and c, 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 [4] [ importance? ]

The square root of a univariate quadratic function

The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola.

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 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 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 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 nth 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 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 for any value of other than the unstable fixed point 0, the term goes to 0 as n goes to infinity, so goes to the stable fixed point

Bivariate (two variable) quadratic function

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

where A, B, C, D, and E are fixed coefficients and F is the constant term. Such a function describes a quadratic surface. Setting equal to zero describes the intersection of the surface with the plane , which is a locus of points equivalent to a conic section.

Minimum/maximum

If the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.

If the function has a minimum if A>0, and a maximum if A<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 parabolic cylinder.

If and 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

Related Research Articles

Asymptote In geometry, limit of the tangent at a point that tends to infinity

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.

In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.

Linear equation Equation that does not involve powers or products of variables

In mathematics, a linear equation is an equation that may be put in the form

Polynomial In mathematics, sum of products of variables, power of variables, and coefficients

In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 1.

Parabola Plane curve: conic section

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

In algebra, a quadratic equation is any equation that can be rearranged in standard form as

Tangent In mathematics, straight line touching a plane curve without crossing it

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f ) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and determines various properties of the roots. It is generally defined as a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. It is often denoted by the symbol .

Quadratic formula solution of the quadratic equation

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, completing the square, graphing and others.

Cubic equation Polynomial equation of degree 3

In algebra, a cubic equation in one variable is an equation of the form

Cubic function Polynomial function of degree 3

In mathematics, a cubic function is a function of the form

Quintic function Polynomial function of degree 5

In algebra, a quintic function is a function of the form

Completing the square Method for solving quadratic equations

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form

Quartic function Polynomial function of degree four

In algebra, a quartic function is a function of the form

In algebra, a monic polynomial is a single-variable polynomial in which the leading coefficient is equal to 1. Therefore, a monic polynomial has the form

Linear differential equation Differential equations that are linear with respect to the unknown function and its derivatives

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

In mathematics, an algebraic equation or polynomial equation is an equation of the form

In mathematics, a variable is a symbol which functions as a placeholder for varying expression or quantities, and is often used to represent an arbitrary element of a set. In addition to numbers, variables are commonly used to represent vectors, matrices and functions.

Conic section Curve obtained by intersecting a cone and a plane

In mathematics, a conic section 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 special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

References

  1. "Quadratic Equation -- from Wolfram MathWorld" . Retrieved January 6, 2013.
  2. Hughes-Hallett, Deborah; Connally, Eric; McCallum, William G. (2007), College Algebra, John Wiley & Sons Inc., p. 205, ISBN   9780471271758 , Search result
  3. "Complex Roots Made Visible – Math Fun Facts" . Retrieved 1 October 2016.
  4. Lord, Nick, "Golden bounds for the roots of quadratic equations", Mathematical Gazette 91, November 2007, 549.