Quadratic function

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$f(x) = x^2 - x - 2\,\!$

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

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

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

The expression $a x 2 + b x + c$ 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 x–intercepts
- 4.2 Vertex
5 The square root of a quadratic function
6 Iteration
7 Bivariate quadratic function
- 7.1 Minimum/maximum
8 See also
9 External links

[ Origin of word

The adjective quadratic comes from the Latin word quadratum for 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:

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

are the values of x for which f(x) = 0. When the coefficients a, b, and c, are real or complex, then the roots are given by the quadratic formula:

$\frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}.$

[ Forms of a quadratic function

A quadratic function can be expressed in three formats:

$f(x) = a x^2 + b x + c \,\!$ is called the standard form,

$f(x) = a(x - r_1)(x - r_2)\,\!$ is called the factored form, where $r 1$ and $r 2$ are the roots of the quadratic equation, it is used in logistic map
$f(x) = a(x - h)^2 + k \,\!$ is called the vertex form where $h$ and $k$ are the x and y coordinates of the vertex.

To convert the general form to factored form, one needs only the quadratic formula to determine the two roots $r 1$ and $r 2$ . To convert the general form to standard form, one needs a process called completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute the factors.

[ Graph

$f(x) = ax^2 ,\!$ $a=\{0.1,0.3,1,3\}\!$

$f(x) = x^2 + bx,\!$ $b=\{1,2,3,4\}\!$

$f(x) = x^2 + bx,\!$ $b=\{-1,-2,-3,-4\}\!$

Regardless of the format, the graph of a 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 = -b/2a.

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

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

[ x–intercepts

Inspection of the factored form shows that the x-intercepts of the graph are given by the roots of the quadratic function.

[ 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 standard form, the vertex is $(h, k)\,\!$ . By the method of completing the square, one can turn the general form: $f(x) = a x^2 + b x + c \,\!$ to

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

so the vertex of the parabola in the general form will be

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

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

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

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

The maximum or minimum of the function is always obtained at the vertex, the following method is another derivation of the same fact using calculus, the advantage of this method is that it works for more general functions.

Taking $f(x) = ax^2 + bx + c \,\!$ as sample quadratic equation, to find its maximum or minimum points (which depends on $a \,\!$ , if $a > 0 \,\!$ , it has a minimum point, if $a < 0\,\!$ , it has a maximum point) we have to first, take its derivative:

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

Then, we find the roots of $f'(x)\,\!$ :

$2ax+b=0 \Rightarrow \,\!$ $2ax=-b \Rightarrow\,\!$ $x=-\frac{b}{2a}$

So, $-\frac{b} {2a}$ is the $x\,\!$ value of $f(x)\,\!$ . Now, to find the $y\,\!$ value, we substitute $x = -\frac{b} {2a}$ on $f(x)\,\!$ :

$y=a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c\Rightarrow y= \frac{ab^2}{4a^2} - \frac{b^2}{2a} + c \Rightarrow y= \frac{b^2}{4a} - \frac{b^2}{2a} + c \Rightarrow$

$y= \frac{b^2 - 2b^2 + 4ac}{4a} \Rightarrow y= \frac{-b^2+4ac}{4a} \Rightarrow y= -\frac{(b^2-4ac)}{4a} \Rightarrow y= -\frac{\Delta}{4a}$

Thus, the maximum or minimum point coordinates are:

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

[ 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 $y = \pm \sqrt{a x^2 + b x + c}$ describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola $y_p = a x^2 + b x + c \,\!$ .
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 $y = \pm \sqrt{a x^2 + b x + c}$ describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola $y_p = a x^2 + b x + c \,\!$ is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

[ Iteration

Given an $f (x) = a x 2 + b x + c$ , 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 $f (x) = a (x - x 0) 2 + x 0$ .

In such case, 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

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

in the case of $f (x) = a (x - x 0) 2 + x 0$ .

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

[ Bivariate quadratic function

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

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

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 thee maximum/minimum at a line. Similarly, a minimum if A>0 and a maximum if A<0, its graph forms a parabolic cylinder.