Beispiel Nr: 06
$\text{Aufstellen von Funktionsgleichungen}\\$
$\textbf{Aufgabe:}$ eine ganzrationale Funktion 3.Grades, mit den Nullstellen bei $x_1=-1$ und $x_2=4$ und dem Hochpunkt $H(2/2)$. \\
$\textbf{Rechnung:}\\ \\ \text{Funktion} \\ f\left(x\right)=a\cdot x^3+b\cdot x^2+c\cdot x+d\\ f'\left(x\right)=3a\cdot x^2+2b\cdot x+c\\ f''\left(x\right)=6a\cdot x+2b\\ \text{Gegeben:}\\ f\left(2\right)=2 \qquad a\cdot 2^3+b\cdot 2^2+c\cdot 2+d=2 \\ f'\left(2\right)=0 \qquad 3a\cdot 2^2+2b\cdot 2+c=0 \\ f\left(-1\right)=0 \qquad a\cdot (-1)^3+b\cdot (-1)^2+c\cdot (-1)+d=0 \\ f\left(4\right)=0 \qquad a\cdot 4^3+b\cdot 4^2+c\cdot 4+d=0 \\\small \begin{array}{l} 8a+4b+2c+d=2 \\ 12a+4b+c=0 \\ -a+b -c+d=0 \\ 64a+16b+4c+d=0 \\ \\ \end{array} \qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 4 & 2 & 1 & 2 \\ 12 & 4 & 1 & 0 & 0 \\ -1 & 1 & -1 & 1 & 0 \\ 64 & 16 & 4 & 1 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{12}{8}\\z2s1=12-8\cdot \frac{12}{8}=0 \\ z2s2=4-4\cdot \frac{12}{8}=-2 \\ z2s3=1-2\cdot \frac{12}{8}=-2 \\ z2s4=0-1\cdot \frac{12}{8}=-1\frac{1}{2} \\ z2s5=0-2\cdot \frac{12}{8}=-3 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 4 & 2 & 1 & 2 \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ -1 & 1 & -1 & 1 & 0 \\ 64 & 16 & 4 & 1 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{-1}{8}\\z3s1=-1-8\cdot \frac{-1}{8}=0 \\ z3s2=1-4\cdot \frac{-1}{8}=1\frac{1}{2} \\ z3s3=-1-2\cdot \frac{-1}{8}=-\frac{3}{4} \\ z3s4=1-1\cdot \frac{-1}{8}=1\frac{1}{8} \\ z3s5=0-2\cdot \frac{-1}{8}=\frac{1}{4} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 4 & 2 & 1 & 2 \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ 0 & 1\frac{1}{2} & -\frac{3}{4} & 1\frac{1}{8} & \frac{1}{4} \\ 64 & 16 & 4 & 1 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}1\cdot \frac{64}{8}\\z4s1=64-8\cdot \frac{64}{8}=0 \\ z4s2=16-4\cdot \frac{64}{8}=-16 \\ z4s3=4-2\cdot \frac{64}{8}=-12 \\ z4s4=1-1\cdot \frac{64}{8}=-7 \\ z4s5=0-2\cdot \frac{64}{8}=-16 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 4 & 2 & 1 & 2 \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ 0 & 1\frac{1}{2} & -\frac{3}{4} & 1\frac{1}{8} & \frac{1}{4} \\ 0 & -16 & -12 & -7 & -16 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{4}{-2}\\z1s2=4-(-2)\cdot \frac{4}{-2}=0 \\ z1s3=2-(-2)\cdot \frac{4}{-2}=-2 \\ z1s4=1-(-1\frac{1}{2})\cdot \frac{4}{-2}=-2 \\ z1s5=2-(-3)\cdot \frac{4}{-2}=-4 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & -2 & -2 & -4 \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ 0 & 1\frac{1}{2} & -\frac{3}{4} & 1\frac{1}{8} & \frac{1}{4} \\ 0 & -16 & -12 & -7 & -16 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{1\frac{1}{2}}{-2}\\z3s2=1\frac{1}{2}-(-2)\cdot \frac{1\frac{1}{2}}{-2}=0 \\ z3s3=-\frac{3}{4}-(-2)\cdot \frac{1\frac{1}{2}}{-2}=-2\frac{1}{4} \\ z3s4=1\frac{1}{8}-(-1\frac{1}{2})\cdot \frac{1\frac{1}{2}}{-2}=0 \\ z3s5=\frac{1}{4}-(-3)\cdot \frac{1\frac{1}{2}}{-2}=-2 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & -2 & -2 & -4 \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & -16 & -12 & -7 & -16 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}2\cdot \frac{-16}{-2}\\z4s2=-16-(-2)\cdot \frac{-16}{-2}=0 \\ z4s3=-12-(-2)\cdot \frac{-16}{-2}=4 \\ z4s4=-7-(-1\frac{1}{2})\cdot \frac{-16}{-2}=5 \\ z4s5=-16-(-3)\cdot \frac{-16}{-2}=8 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & -2 & -2 & -4 \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & 0 & 4 & 5 & 8 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-2}{-2\frac{1}{4}}\\z1s3=-2-(-2\frac{1}{4})\cdot \frac{-2}{-2\frac{1}{4}}=0 \\ z1s4=-2-0\cdot \frac{-2}{-2\frac{1}{4}}=-2 \\ z1s5=-4-(-2)\cdot \frac{-2}{-2\frac{1}{4}}=-2\frac{2}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & 0 & -2 & -2\frac{2}{9} \\ 0 & -2 & -2 & -1\frac{1}{2} & -3 \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & 0 & 4 & 5 & 8 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-2}{-2\frac{1}{4}}\\z2s3=-2-(-2\frac{1}{4})\cdot \frac{-2}{-2\frac{1}{4}}=0 \\ z2s4=-1\frac{1}{2}-0\cdot \frac{-2}{-2\frac{1}{4}}=-1\frac{1}{2} \\ z2s5=-3-(-2)\cdot \frac{-2}{-2\frac{1}{4}}=-1\frac{2}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & 0 & -2 & -2\frac{2}{9} \\ 0 & -2 & 0 & -1\frac{1}{2} & -1\frac{2}{9} \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & 0 & 4 & 5 & 8 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}3\cdot \frac{4}{-2\frac{1}{4}}\\z4s3=4-(-2\frac{1}{4})\cdot \frac{4}{-2\frac{1}{4}}=0 \\ z4s4=5-0\cdot \frac{4}{-2\frac{1}{4}}=5 \\ z4s5=8-(-2)\cdot \frac{4}{-2\frac{1}{4}}=4\frac{4}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & 0 & -2 & -2\frac{2}{9} \\ 0 & -2 & 0 & -1\frac{1}{2} & -1\frac{2}{9} \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & 0 & 0 & 5 & 4\frac{4}{9} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}4\cdot \frac{-2}{5}\\z1s4=-2-5\cdot \frac{-2}{5}=0 \\ z1s5=-2\frac{2}{9}-4\frac{4}{9}\cdot \frac{-2}{5}=-\frac{4}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & 0 & 0 & -\frac{4}{9} \\ 0 & -2 & 0 & -1\frac{1}{2} & -1\frac{2}{9} \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & 0 & 0 & 5 & 4\frac{4}{9} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}4\cdot \frac{-1\frac{1}{2}}{5}\\z2s4=-1\frac{1}{2}-5\cdot \frac{-1\frac{1}{2}}{5}=0 \\ z2s5=-1\frac{2}{9}-4\frac{4}{9}\cdot \frac{-1\frac{1}{2}}{5}=\frac{1}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline8 & 0 & 0 & 0 & -\frac{4}{9} \\ 0 & -2 & 0 & 0 & \frac{1}{9} \\ 0 & 0 & -2\frac{1}{4} & 0 & -2 \\ 0 & 0 & 0 & 5 & 4\frac{4}{9} \\ \end{array} \\ \\ a=\frac{-\frac{4}{9}}{8}=-\frac{1}{18}\\b=\frac{\frac{1}{9}}{-2}=-\frac{1}{18}\\c=\frac{-2}{-2\frac{1}{4}}=\frac{8}{9}\\d=\frac{4\frac{4}{9}}{5}=\frac{8}{9}\\L=\{-\frac{1}{18}/-\frac{1}{18}/\frac{8}{9}/\frac{8}{9}\} \\ \text{Funktion} \\ f\left(x\right)=-\frac{1}{18}x^3-\frac{1}{18}x^2+\frac{8}{9}x+\frac{8}{9}$