Quadratic function

Template:Short description In mathematics, a quadratic function of a single variable is a function of the form<ref name="wolfram">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

where Template:Tmath is its variable, and Template:Tmath, Template:Tmath, and Template:Tmath are coefficients. The expression Template:Tmath, especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished and the terms quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic.

File:Polynomialdeg2.svg

A quadratic polynomial with two real roots (crossings of the Template:Mvar axis).

The graph of a real single-variable quadratic function is a parabola. If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula.

A quadratic polynomial or quadratic function can involve more than one variable. For example, a two-variable quadratic function of variables Template:Tmath and Template:Tmath has the form

with at least one of Template:Tmath, Template:Tmath, and Template:Tmath not equal to zero. In general the zeros of such a quadratic function describe a conic section (a circle or other ellipse, a parabola, or a hyperbola) in the Template:Tmath–Template:Tmath plane. 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.

EtymologyEdit

The adjective quadratic comes from the Latin word quadrātum ("square"). A term raised to the second power like Template:Tmath is called a square in algebra because it is the area of a square with side Template:Tmath.

TerminologyEdit

CoefficientsEdit

The coefficients of a quadratic function are often taken to be real or complex numbers, but they may be taken in any ring, in which case the domain and the codomain are this ring (see polynomial evaluation).

DegreeEdit

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 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.

VariablesEdit

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 caseEdit

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 <math>ax^2 + bx + c = 0.</math> The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.

Bivariate and multivariate casesEdit

Any quadratic polynomial with two variables may be written as

where Template:Math and Template:Math are the variables and Template:Math are the coefficients, and one of Template:Mvar, Template:Mvar and Template:Mvar is nonzero. Such polynomials are fundamental to the study of conic sections, 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 forms.

Forms of a univariate quadratic functionEdit

A univariate quadratic function can be expressed in three formats:<ref>Template:Cite book</ref>

<math>f(x) = a x^2 + b x + c</math> is called the standard form,
<math>f(x) = a(x - r_1)(x - r_2)</math> is called the factored form, where Template:Math and Template:Math are the roots of the quadratic function and the solutions of the corresponding quadratic equation.
<math>f(x) = a(x - h)^2 + k</math> is called the vertex form, where Template:Math and Template:Math are the Template:Math and Template:Math coordinates of the vertex, respectively.

The coefficient Template:Math 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 Template:Math and Template:Math. 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 functionEdit

File:Function ax^2.svg

_{a=\{0.1,0.3,1,3\}}</math>

File:Function x^2+bx.svg

_{b=\{1,2,3,4\}}</math>

File:Function x^2-bx.svg

_{b=\{-1,-2,-3,-4\}}</math>

Regardless of the format, the graph of a univariate quadratic function <math>f(x) = ax^2 + bx + c</math> is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation <math>y = ax^2 + bx + c</math>.

If Template:Math, the parabola opens upwards.
If Template:Math, the parabola opens downwards.

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

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

The coefficient Template:Math controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the Template:Math-axis.

VertexEdit

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 Template:Math. Using the method of completing the square, one can turn the standard form

into

<math>\begin{align}

 f(x) &= a x^2 + b x + c \\
      &= a (x - h)^2 + k \\
      &= a\left(x - \frac{-b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right),\\

\end{align}</math> so the vertex, Template:Math, of the parabola in standard form is

<math> \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right). </math>Template:Citation needed

If the quadratic function is in factored form

the average of the two roots, i.e.,

is the Template:Math-coordinate of the vertex, and hence the vertex Template:Math is

<math> \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).</math>

The vertex is also the maximum point if Template:Math, or the minimum point if Template:Math.

The vertical line

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

Maximum and minimum pointsEdit

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

<math>f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b</math>

Template:Math is a root of Template:Math if Template:Math resulting in

with the corresponding function value

<math>f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = c-\frac{b^2}{4a},</math>

so again the vertex point coordinates, Template:Math, can be expressed as

<math> \left (-\frac {b}{2a}, c-\frac {b^2}{4a} \right). </math>

Roots of the univariate functionEdit

Template:Quadratic equation graph key points.svg Template:Quadratic function graph complex roots.svg Template:Further

Exact rootsEdit

The roots (or zeros), Template:Math and Template:Math, of the univariate quadratic function

<math>\begin{align}

f(x) &= ax^2+bx+c \\
 &= a(x-r_1)(x-r_2), \\

\end{align}</math>

are the values of Template:Math for which Template:Math.

When the coefficients Template:Math, Template:Math, and Template:Math, are real or complex, the roots are

Upper bound on the magnitude of the rootsEdit

The modulus of the roots of a quadratic <math>ax^2+bx+c</math> can be no greater than <math>\frac{\max(|a|, |b|, |c|)}{|a|}\times \phi, </math> where <math>\phi</math> is the golden ratio <math>\frac{1+\sqrt{5}}{2}.</math><ref>Template:Cite journal</ref>

The square root of a univariate quadratic functionEdit

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 <math>a>0,</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> 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 <math> y_p = a x^2 + b x + c .</math> 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 <math>a<0,</math> then the equation <math> y = \pm \sqrt{a x^2 + b x + c} </math> describes either a circle or other ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola <math> y_p = a x^2 + b x + c</math> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

IterationEdit

To iterate a function <math>f(x)=ax^2+bx+c</math>, 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 <math>f^{(n)}(x)</math>, which means the n^th iteration of <math>f(x)</math>. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of <math>f(x)</math> if the inverse exists.) But there are some analytically tractable cases.

For example, for the iterative equation

one has

where

So by induction,

can be obtained, where <math>g^{(n)}(x)</math> 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

<math>x_{n}=\sin^{2}(2^{n} \theta \pi)</math>

where the initial condition parameter <math>\theta</math> is given by <math>\theta = \tfrac{1}{\pi}\sin^{-1}(x_0^{1/2})</math>. For rational <math>\theta</math>, after a finite number of iterations <math>x_n</math> maps into a periodic sequence. But almost all <math>\theta</math> are irrational, and, for irrational <math>\theta</math>, <math>x_n</math> 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 <math>x_0 \in [0,1)</math>. Since <math>(1-2x_0)\in (-1,1)</math> for any value of <math>x_0</math> other than the unstable fixed point 0, the term <math>(1-2x_0)^{2^{n}}</math> goes to 0 as n goes to infinity, so <math>x_n</math> goes to the stable fixed point <math>\tfrac{1}{2}.</math>

Bivariate (two variable) quadratic functionEdit

Template:Further 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 <math>f(x,y)</math> equal to zero describes the intersection of the surface with the plane <math>z=0,</math> which is a locus of points equivalent to a conic section.

Minimum/maximumEdit

If <math> 4AB-E^2 <0 ,</math> the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.

If <math> 4AB-E^2 >0 ,</math> the function has a minimum if both Template:Nowrap and Template:Nowrap, and a maximum if both Template:Nowrap and Template:Nowrap; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at <math> (x_m, y_m) ,</math> where:

If <math> 4AB- E^2 =0 </math> and <math> DE-2CB=2AD-CE \ne 0 ,</math> the function has no maximum or minimum; its graph forms a parabolic cylinder.

If <math> 4AB- E^2 =0 </math> and <math> DE-2CB=2AD-CE =0 ,</math> 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.

ReferencesEdit

Template:Reflist

Template:Polynomials Template:Authority control