Quadratic function

A quadratic function, in mathematics, is a polynomial function of the form

$f(x)=ax^2%2Bbx%2Bc,\quad a \ne 0.$

The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis.

The expression $ax^2%2Bbx%2Bc$ in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.

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

1 Origin of word
2 Roots
3 Forms of a quadratic function
4 Graph
- 4.1 Vertex
  - 4.1.1 Maximum and minimum points
5 The square root of a quadratic function
6 Iteration
7 Bivariate (two variable) quadratic function
- 7.1 Minimum/maximum
8 See also
9 References
10 External links

Origin of word

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.

Roots

Further information: Quadratic equation

The roots (zeros) of the quadratic function

$f(x) = ax^2%2Bbx%2Bc\,$

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

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

Forms of a quadratic function

A quadratic function can be expressed in three formats:^[1]

$f(x) = a x^2 %2B b x %2B c \,\!$ is called the standard form,
$f(x) = a(x - x_1)(x - x_2)\,\!$ is called the factored form, where $x_1$ and $x_2$ are the roots of the quadratic equation, it is used in logistic map
$f(x) = a(x - h)^2 %2B k \,\!$ 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 $x_1$ and $x_2$ . 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

Regardless of the format, the graph of a quadratic function is a parabola (as shown above).

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, bigger positive a makes the function increase faster and the graph appear 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

The vertex of a parabola is the place where it turns, hence, it's 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 general form

$f(x) = a x^2 %2B b x %2B c \,\!$

into

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

so the vertex of the parabola in the vertex 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.,

$\frac{r_1 %2B r_2}{2} \,\!$

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

$\left(\frac{r_1 %2B r_2}{2}, f\left(\frac{r_1 %2B r_2}{2}\right)\right).\!$

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

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

$f(x)=ax^2%2Bbx%2Bc \quad \Rightarrow \quad f'(x)=2ax%2Bb \,\!,$

giving

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

with the corresponding function value

$f(x) = a \left (-\frac{b}{2a} \right)^2%2Bb \left (-\frac{b}{2a} \right)%2Bc = -\frac{(b^2-4ac)}{4a} = -\frac{\Delta}{4a} \,\!,$

so again the vertex point coordinates can be expressed as

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

The square root of a quadratic function

The square root of a quadratic function gives rise either to an ellipse or to a hyperbola.If $a>0\,\!$ then the equation $y = \pm \sqrt{a x^2 %2B b x %2B c}$ describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola $y_p = a x^2 %2B b x %2B c \,\!$ .
If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is positive, then the hyperbola's axis is vertical.
If $a<0\,\!$ then the equation $y = \pm \sqrt{a x^2 %2B b x %2B c}$ describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola $y_p = a x^2 %2B b x %2B c \,\!$ is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

Iteration

Given an $f(x)=ax^2%2Bbx%2Bc$ , 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 number referring to the iteration of the inverse of $f(x)$ if the inverse exists.) But there is one easier case, in which $f(x)=a(x-x_0)^2%2Bx_0$ .

In such case, one has

$f(x)=a(x-x_0)^2%2Bx_0=h^{(-1)}(g(h(x)))\,\!$ ,

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

$f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}%2Bx_0\,\!$ ,

in the case of $f(x)=a(x-x_0)^2%2Bx_0$ .

See Topological conjugacy for more detail about such relationship between f and g. And see Complex quadratic polynomial for the chaotic behavior in the general iteration.

Bivariate (two variable) quadratic function

Further information: Quadric and Quadratic form

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

$f(x,y) = A x^2 %2B B y^2 %2B C x %2B D y %2B E x y %2B F \,\!$

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

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.

The minimum or maximum of a bivariate quadratic function is obtained 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.

References

^ Hughes-Hallett, Deborah; Connally, Eric; McCallum, William G. (2007), College Algebra, John Wiley & Sons Inc, p. 205, ISBN 0471271756, 9780471271758, http://books.google.be/books?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enBE306BE306&q=%22three+different+forms+for+a+quadratic+expression+are%22 , Search result

External links

Weisstein, Eric W., "Quadratic" from MathWorld.

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