Polynomial Functions

A function $f$ : $x \mapsto f (x)$ , whose function term $f (x)$ is a Polynomial, is called an integral function or polynomial function.

Thus a " $n$ -th" integer function is a function of the form.

$f (x) = a_{n} \cdot x^{n} + a_{n - 1} \cdot x^{n - 1} + \dots + a_{2} \cdot x^{2} + a_{1} \cdot x + a_{0}$

The numbers $a_{n}, a_{n - 1}, \dots, a_{2}, a_{1}, a_{0}$ are known as $Coefficients$ .
These numbers $n, n - 1, \dots$ are known as $Exponents$ .
The largest occuring exponent (here: $n$ ) determines the degree of the polynomial function.
The coefficient before the largest occurring exponent is called the leading coefficient(here: $a_{n}$ ).

Example

$f (x) = - 2 x^{3} - \frac{1}{2} x + 4$

The coefficients are $a_{3} = - 2$ ; $a_{1} = - \frac{1}{2}$ and $a_{0} = 4$ .
The exponents occurring are $n = 3$ und $n - 2 = 1$
$f (x)$ has the degree $3$ and the conducting coefficients $- 2$ .

The Zero

An integer function of the nth degree has at most n zero points.

For polynomial functions up to degree 2, solution formulas exist such as the Square solution formula.

At higher degrees, the Polynomial division, simplify a Polynomial if you already know a zero (e.g by guessing)
For 3rd and 4th degree polynomial functions, there are formulas (which are not commonly used in school and are complicated). For higher degrees, one cannot form a general formula for the zeros.

Limiting Values

If x is allowed to plus or minus infinitely, the limit value of the polynomial function is always plus or minus infinity $\lim_{x \to \pm \infty}$ of the polynomial function is always plus or minus infinity. The sign of the two limit values is the same for integer functions of even degree, and different for odd degree. The sign of the parameter with the highest power (called a in the table) determines the sign of the limit values.

Beispiele

Im Folgenden werden die Grenzwerte der Funktionen

$f (x) = a x^{3} + x^{2} - 2 x + 0,5$ und

$g (x) = a x^{4} - 2 x^{3} + x$

für jeweils $a > 0$ und $a < 0$ betrachtet.

Ungerader Grad

$f (x) = a x^{3} + x^{2} - 2 x + 0,5$

$a = 3$

$\lim_{x \to - \infty} = - \infty$

$\lim_{x \to \infty} = \infty$

$a = - 3$

$\lim_{x \to - \infty} = \infty$

$\lim_{x \to \infty} = - \infty$

Gerader Grad

$g (x) = a x^{4} - 2 x^{3} + x$

$a = 1$

$\lim_{x \to - \infty} = \infty$

$\lim_{x \to \infty} = \infty$

$a = - 1$

$\lim_{x \to - \infty} = - \infty$

$\lim_{x \to \infty} = - \infty$

Spezielle Polynomfunktionen

Im Folgenden werden spezielle Polynomfunktionen vorgestellt:

Konstante Funktionen (Grad 0)

Graph der Abbildung $f (x) = 5$

Die Konstante Funktion ordnet jedem $x$ dasselbe $c$ zu.

$f (x) = c$

Der Graph der konstanten Funktion ist eine Parallele zur $x$ -Achse, die die $y$ -Achse auf der Höhe $c$ schneidet.

Lineare Funktionen (Grad 1)

Graph der Funktion $f (x) = 2 x + 4$

Lineare Funktionen sind ganzrationale Funktionen ersten Grades. Sie haben die Form $f (x) = m x + t$

Lineare Funktion Allgemeine Geradengleichung

Quadratische Funktionen (Grad 2)

Graph der Funktion $f (x) = x^{2} - 3 x + 2$

Quadratische Funktionen sind Polynomfunktionen vom Grad 2. Sie haben die Form $f (x) = a x^{2} + b x + c$

Beispiele und Nicht-Beispiele

$f (x) = 2 x^{4} - \frac{1}{2} x + 1$
$g (x) = \sqrt{2} \cdot x^{2} - π \cdot x^{7}$
$h (x) = - x^{2} - 5 x + 1$
$i (x) = 2 x + 3$
$k (x) = - 2,3$

$f (x) = \sin (x) + 1$
$g (x) = e^{2 x}$
$h (x) = 1 - \ln (x)$
$i (x) = \sqrt{x - 1}$
$k (x) = x + \frac{1}{x}$
$l (x) = \frac{x^{2} - x + 1}{x^{3} + 3}$

Extrema

Abbildung: Graph einer Polynomfunktion 5-ten Grades

Um die Extrema einer Polynomfunktion $f (x)$ $n$ -ten Grades zu bestimmen, berechnet man zunächst die Ableitung $f^{'} (x)$ und bestimmt davon die Nullstellen. $f^{'} (x)$ ist eine Polynomfunktion $(n - 1)$ -ten Grades. Diese hat maximal $(n - 1)$ Nullstellen.

Also folgt:

Eine Polynomfunktion $n$ -ten Grades hat höchstens $(n - 1)$ Extrema.

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