Quadratic function

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In mathematics, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree. For example, 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:

f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz +j,

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

x^2 - x - 2\!

A univariate (single-variable) quadratic function has the form[1]

f(x)=ax^2+bx+c,\quad a \ne 0

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

 f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\!

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).

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

Etymology[edit]

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.

In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and quadrant. Quadratum is the Latin word for square because a square has four sides.

Terminology[edit]

Coefficients[edit]

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[edit]

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[edit]

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[edit]

Any single-variable quadratic polynomial may be written as

ax^2 + bx + c,\,\!

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 ax^2 + bx + c = 0. 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[edit]

Any quadratic polynomial with two variables may be written as

 f(x,y) = a x^2 + b y^2 + cxy + dx+ e y + f, \,\!

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. 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[edit]

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

  • f(x) = a x^2 + b x + c \,\! is called the standard form,
  • f(x) = a(x - x_1)(x - x_2)\,\! is called the factored form, where x1 and x2 are the roots of the quadratic function and the solutions of the corresponding quadratic equation.
  • f(x) = a(x - h)^2 + k \,\! is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively.

To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots x1 and x2. 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[edit]

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

Regardless of the format, the graph of a univariate quadratic function f(x)=ax2+bx+c is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation y = ax2+bx+c.

  • If a > 0, (or is a positive number), the parabola opens upward.
  • If a < 0, (or is a negative number), the parabola opens downward.

The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex, greater positive a values makes the function increase faster and the graph appears more closed.

The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex) which is at x = -\frac{b}{2a}.

The coefficient b alone is the declivity of the parabola as y-axis intercepts.

The coefficient c controls the height of the parabola, more specifically, it is the point where the parabola intercept the y-axis.

Vertex[edit]

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). By the method of completing the square, one can turn the standard form

f(x) = a x^2 + b x + c \,\!

into

 f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} ,

so the vertex of the parabola in standard form is

 \left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right).

If the quadratic function is in factored form

f(x) = a(x - r_1)(x - r_2) \,\!

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

\frac{r_1 + r_2}{2} \,\!

is the x-coordinate of the vertex, and hence the vertex is

 \left(\frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right)\right).\!

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

The vertical line

 x=h=-\frac{b}{2a}

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

Maximum and minimum points[edit]

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

f(x)=ax^2+bx+c \quad \Rightarrow \quad f'(x)=2ax+b \,\!,

giving

x=-\frac{b}{2a}

with the corresponding function value

f(x) = a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c = -\frac{(b^2-4ac)}{4a} = -\frac{\Delta}{4a} \,\!,

so again the vertex point coordinates can be expressed as

 \left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right).

Roots of the univariate function[edit]

Further information: Quadratic equation

The roots (zeros) of the univariate quadratic function

f(x) = ax^2+bx+c\,

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

When the coefficients a, b, and c, are real or complex, the roots are

x=\frac{-b \pm \sqrt{\Delta}}{2 a},

where the discriminant is defined as

\Delta = b^2 - 4 a c \, .

The square root of a univariate quadratic function[edit]

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

Iteration[edit]

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

For example, for the iterative equation

f(x)=a(x-x_0)^2+x_0

one has

f(x)=a(x-x_0)^2+x_0=h^{(-1)}(g(h(x))),\,\!

where

g(x)=ax^2\,\! and h(x)=x-x_0.\,\!

So by induction,

f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!

can be obtained, where g^{(n)}(x) can be easily computed as

g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.\,\!

Finally, we have

f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}+x_0\,\!

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

 x_{n+1} = r x_n (1-x_n), \quad  0\leq x_0<1

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

x_{n}=\sin^{2}(2^{n} \theta \pi)

where the initial condition parameter \theta is given by \theta = \tfrac{1}{\pi}\sin^{-1}(x_0^{1/2}). For rational \theta, after a finite number of iterations x_n maps into a periodic sequence. But almost all \theta are irrational, and, for irrational \theta, x_n 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

x_n = \frac{1}{2} - \frac{1}{2}(1-2x_0)^{2^{n}}

for x_0 \in [0,1). 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)^{2^{n}} goes to 0 as n goes to infinity, so x_n goes to the stable fixed point \tfrac{1}{2}.

Bivariate (two variable) quadratic function[edit]

Further information: Quadric and Quadratic form

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. Setting f(x,y)\,\! equal to zero describes the intersection of the surface with the plane z=0\,\!, which is a locus of points equivalent to a conic section.

Minimum/maximum[edit]

If  4AB-E^2 <0 \, the function has no maximum or minimum, its graph forms an hyperbolic paraboloid.

If  4AB-E^2 >0 \, 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  (x_m, y_m) \, where:

x_m = -\frac{2BC-DE}{4AB-E^2},
y_m = -\frac{2AD-CE}{4AB-E^2}.

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.

If  4AB- E^2 =0 \, and  DE-2CB=2AD-CE =0 \, 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[edit]

References[edit]

  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

External links[edit]