Beispiel Nr: 07
$\text{Aufstellen von Funktionsgleichungen}\\$
$\textbf{Aufgabe:}$ eine ganzrationale Funktion 3.Grades, mit der Wendtangente $y=2x+1=0$ an der Stelle $x_2=1$ und dem Schnittpunkt mit der x-Achse $x=-2$. \\
$\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(1\right)=3 \qquad a\cdot 1^3+b\cdot 1^2+c\cdot 1+d=3 \\ f'\left(1\right)=2 \qquad 3a\cdot 1^2+2b\cdot 1+c=2 \\ f'\left(-2\right)=0 \qquad 3a\cdot (-2)^2+2b\cdot (-2)+c=0 \\ f''\left(1\right)=0 \qquad 6a\cdot 1+2b=0 \\\small \begin{array}{l} a+b+c+d=3 \\ 3a+2b+c=2 \\ 12a-4b+c=0 \\ 6a+2b=0 \\ \\ \end{array} \qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 1 & 1 & 1 & 3 \\ 3 & 2 & 1 & 0 & 2 \\ 12 & -4 & 1 & 0 & 0 \\ 6 & 2 & 0 & 0 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{3}{1}\\z2s1=3-1\cdot \frac{3}{1}=0 \\ z2s2=2-1\cdot \frac{3}{1}=-1 \\ z2s3=1-1\cdot \frac{3}{1}=-2 \\ z2s4=0-1\cdot \frac{3}{1}=-3 \\ z2s5=2-3\cdot \frac{3}{1}=-7 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 1 & 1 & 1 & 3 \\ 0 & -1 & -2 & -3 & -7 \\ 12 & -4 & 1 & 0 & 0 \\ 6 & 2 & 0 & 0 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{12}{1}\\z3s1=12-1\cdot \frac{12}{1}=0 \\ z3s2=-4-1\cdot \frac{12}{1}=-16 \\ z3s3=1-1\cdot \frac{12}{1}=-11 \\ z3s4=0-1\cdot \frac{12}{1}=-12 \\ z3s5=0-3\cdot \frac{12}{1}=-36 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 1 & 1 & 1 & 3 \\ 0 & -1 & -2 & -3 & -7 \\ 0 & -16 & -11 & -12 & -36 \\ 6 & 2 & 0 & 0 & 0 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}1\cdot \frac{6}{1}\\z4s1=6-1\cdot \frac{6}{1}=0 \\ z4s2=2-1\cdot \frac{6}{1}=-4 \\ z4s3=0-1\cdot \frac{6}{1}=-6 \\ z4s4=0-1\cdot \frac{6}{1}=-6 \\ z4s5=0-3\cdot \frac{6}{1}=-18 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 1 & 1 & 1 & 3 \\ 0 & -1 & -2 & -3 & -7 \\ 0 & -16 & -11 & -12 & -36 \\ 0 & -4 & -6 & -6 & -18 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{1}{-1}\\z1s2=1-(-1)\cdot \frac{1}{-1}=0 \\ z1s3=1-(-2)\cdot \frac{1}{-1}=-1 \\ z1s4=1-(-3)\cdot \frac{1}{-1}=-2 \\ z1s5=3-(-7)\cdot \frac{1}{-1}=-4 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & -1 & -2 & -4 \\ 0 & -1 & -2 & -3 & -7 \\ 0 & -16 & -11 & -12 & -36 \\ 0 & -4 & -6 & -6 & -18 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{-16}{-1}\\z3s2=-16-(-1)\cdot \frac{-16}{-1}=0 \\ z3s3=-11-(-2)\cdot \frac{-16}{-1}=21 \\ z3s4=-12-(-3)\cdot \frac{-16}{-1}=36 \\ z3s5=-36-(-7)\cdot \frac{-16}{-1}=76 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & -1 & -2 & -4 \\ 0 & -1 & -2 & -3 & -7 \\ 0 & 0 & 21 & 36 & 76 \\ 0 & -4 & -6 & -6 & -18 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}2\cdot \frac{-4}{-1}\\z4s2=-4-(-1)\cdot \frac{-4}{-1}=0 \\ z4s3=-6-(-2)\cdot \frac{-4}{-1}=2 \\ z4s4=-6-(-3)\cdot \frac{-4}{-1}=6 \\ z4s5=-18-(-7)\cdot \frac{-4}{-1}=10 \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & -1 & -2 & -4 \\ 0 & -1 & -2 & -3 & -7 \\ 0 & 0 & 21 & 36 & 76 \\ 0 & 0 & 2 & 6 & 10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{-1}{21}\\z1s3=-1-21\cdot \frac{-1}{21}=0 \\ z1s4=-2-36\cdot \frac{-1}{21}=-\frac{2}{7} \\ z1s5=-4-76\cdot \frac{-1}{21}=-\frac{8}{21} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & 0 & -\frac{2}{7} & -\frac{8}{21} \\ 0 & -1 & -2 & -3 & -7 \\ 0 & 0 & 21 & 36 & 76 \\ 0 & 0 & 2 & 6 & 10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{-2}{21}\\z2s3=-2-21\cdot \frac{-2}{21}=0 \\ z2s4=-3-36\cdot \frac{-2}{21}=\frac{3}{7} \\ z2s5=-7-76\cdot \frac{-2}{21}=\frac{5}{21} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & 0 & -\frac{2}{7} & -\frac{8}{21} \\ 0 & -1 & 0 & \frac{3}{7} & \frac{5}{21} \\ 0 & 0 & 21 & 36 & 76 \\ 0 & 0 & 2 & 6 & 10 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}4=\text{Zeile}4\text{-Zeile}3\cdot \frac{2}{21}\\z4s3=2-21\cdot \frac{2}{21}=0 \\ z4s4=6-36\cdot \frac{2}{21}=2\frac{4}{7} \\ z4s5=10-76\cdot \frac{2}{21}=2\frac{16}{21} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & 0 & -\frac{2}{7} & -\frac{8}{21} \\ 0 & -1 & 0 & \frac{3}{7} & \frac{5}{21} \\ 0 & 0 & 21 & 36 & 76 \\ 0 & 0 & 0 & 2\frac{4}{7} & 2\frac{16}{21} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}4\cdot \frac{-\frac{2}{7}}{2\frac{4}{7}}\\z1s4=-\frac{2}{7}-2\frac{4}{7}\cdot \frac{-\frac{2}{7}}{2\frac{4}{7}}=0 \\ z1s5=-\frac{8}{21}-2\frac{16}{21}\cdot \frac{-\frac{2}{7}}{2\frac{4}{7}}=-\frac{2}{27} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & 0 & 0 & -\frac{2}{27} \\ 0 & -1 & 0 & \frac{3}{7} & \frac{5}{21} \\ 0 & 0 & 21 & 36 & 76 \\ 0 & 0 & 0 & 2\frac{4}{7} & 2\frac{16}{21} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}4\cdot \frac{\frac{3}{7}}{2\frac{4}{7}}\\z2s4=\frac{3}{7}-2\frac{4}{7}\cdot \frac{\frac{3}{7}}{2\frac{4}{7}}=0 \\ z2s5=\frac{5}{21}-2\frac{16}{21}\cdot \frac{\frac{3}{7}}{2\frac{4}{7}}=-\frac{2}{9} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & 0 & 0 & -\frac{2}{27} \\ 0 & -1 & 0 & 0 & -\frac{2}{9} \\ 0 & 0 & 21 & 36 & 76 \\ 0 & 0 & 0 & 2\frac{4}{7} & 2\frac{16}{21} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}4\cdot \frac{36}{2\frac{4}{7}}\\z3s4=36-2\frac{4}{7}\cdot \frac{36}{2\frac{4}{7}}=0 \\ z3s5=76-2\frac{16}{21}\cdot \frac{36}{2\frac{4}{7}}=37\frac{1}{3} \\ \end{array}\qquad \small \begin{array}{cccc|cc } a & b & c & d & & \\ \hline1 & 0 & 0 & 0 & -\frac{2}{27} \\ 0 & -1 & 0 & 0 & -\frac{2}{9} \\ 0 & 0 & 21 & 0 & 37\frac{1}{3} \\ 0 & 0 & 0 & 2\frac{4}{7} & 2\frac{16}{21} \\ \end{array} \\ \\ a=\frac{-\frac{2}{27}}{1}=-\frac{2}{27}\\b=\frac{-\frac{2}{9}}{-1}=\frac{2}{9}\\c=\frac{37\frac{1}{3}}{21}=1\frac{7}{9}\\d=\frac{2\frac{16}{21}}{2\frac{4}{7}}=1\frac{2}{27}\\L=\{-\frac{2}{27}/\frac{2}{9}/1\frac{7}{9}/1\frac{2}{27}\} \\ \text{Funktion} \\ f\left(x\right)=-\frac{2}{27}x^3+\frac{2}{9}x^2+1\frac{7}{9}x+1\frac{2}{27}$