$\begin{array}{l} \\ \text{ Aufstellen von Funktionsgleichungen}\\ \\ \textbf{Aufgabe:}\\ eine quadratische Funktion, deren Graph durch durch die Punkte P(-2/1),Q(1/1),T(2/-3) geht.\\ \\ \textbf{Rechnung:}\\ \\ \text{Funktion} \\ f\left(x\right)=a\cdot x^2+b\cdot x+c\\ f'\left(x\right)=2a\cdot x+b\\ f''\left(x\right)=2a\\ \text{Gegeben:}\\ f\left(-2\right)=1 \qquad a\cdot (-2)^2+b\cdot (-2)+c=1 \\ f\left(1\right)=1 \qquad a\cdot 1^2+b\cdot 1+c=1 \\ f\left(2\right)=(-3) \qquad a\cdot 2^2+b\cdot 2+c=(-3) \\\small \begin{array}{l} 4a-2b+c=1 \\ a+b+c=1 \\ 4a+2b+c=-3 \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & -2 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 4 & 2 & 1 & -3 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{1}{4}\\z2s1=1-4\cdot \frac{1}{4}=0 \\ z2s2=1-(-2)\cdot \frac{1}{4}=1\frac{1}{2} \\ z2s3=1-1\cdot \frac{1}{4}=\frac{3}{4} \\ z2s4=1-1\cdot \frac{1}{4}=\frac{3}{4} \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & -2 & 1 & 1 \\ 0 & 1\frac{1}{2} & \frac{3}{4} & \frac{3}{4} \\ 4 & 2 & 1 & -3 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{4}{4}\\z3s1=4-4\cdot \frac{4}{4}=0 \\ z3s2=2-(-2)\cdot \frac{4}{4}=4 \\ z3s3=1-1\cdot \frac{4}{4}=0 \\ z3s4=-3-1\cdot \frac{4}{4}=-4 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & -2 & 1 & 1 \\ 0 & 1\frac{1}{2} & \frac{3}{4} & \frac{3}{4} \\ 0 & 4 & 0 & -4 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{-2}{1\frac{1}{2}}\\z1s2=-2-1\frac{1}{2}\cdot \frac{-2}{1\frac{1}{2}}=0 \\ z1s3=1-\frac{3}{4}\cdot \frac{-2}{1\frac{1}{2}}=2 \\ z1s4=1-\frac{3}{4}\cdot \frac{-2}{1\frac{1}{2}}=2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 2 & 2 \\ 0 & 1\frac{1}{2} & \frac{3}{4} & \frac{3}{4} \\ 0 & 4 & 0 & -4 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{4}{1\frac{1}{2}}\\z3s2=4-1\frac{1}{2}\cdot \frac{4}{1\frac{1}{2}}=0 \\ z3s3=0-\frac{3}{4}\cdot \frac{4}{1\frac{1}{2}}=-2 \\ z3s4=-4-\frac{3}{4}\cdot \frac{4}{1\frac{1}{2}}=-6 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 2 & 2 \\ 0 & 1\frac{1}{2} & \frac{3}{4} & \frac{3}{4} \\ 0 & 0 & -2 & -6 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{2}{-2}\\z1s3=2-(-2)\cdot \frac{2}{-2}=0 \\ z1s4=2-(-6)\cdot \frac{2}{-2}=-4 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 0 & -4 \\ 0 & 1\frac{1}{2} & \frac{3}{4} & \frac{3}{4} \\ 0 & 0 & -2 & -6 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}3\cdot \frac{\frac{3}{4}}{-2}\\z2s3=\frac{3}{4}-(-2)\cdot \frac{\frac{3}{4}}{-2}=0 \\ z2s4=\frac{3}{4}-(-6)\cdot \frac{\frac{3}{4}}{-2}=-1\frac{1}{2} \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 0 & -4 \\ 0 & 1\frac{1}{2} & 0 & -1\frac{1}{2} \\ 0 & 0 & -2 & -6 \\ \end{array} \\ \\ a=\frac{-4}{4}=-1\\b=\frac{-1\frac{1}{2}}{1\frac{1}{2}}=-1\\c=\frac{-6}{-2}=3\\L=\{-1/-1/3\} \\ \text{Funktion} \\ f\left(x\right)=-1x^2-1x+3 \end{array}$