$\begin{array}{l} \\ \text{ Aufstellen von Funktionsgleichungen}\\ \\ \textbf{Aufgabe:}\\eine quadratische Funktion, deren Graph P(-2/1) die Steigung m=2 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(2\right)=3 \qquad a\cdot 2^2+b\cdot 2+c=3 \\ f'\left(1\right)=-\frac{1}{2} \qquad 2a\cdot 1+b=(-\frac{1}{2}) \\\small \begin{array}{l} 4a-2b+c=1 \\ 4a+2b+c=3 \\ 2a+b=-\frac{1}{2} \\ \\ \end{array} \qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & -2 & 1 & 1 \\ 4 & 2 & 1 & 3 \\ 2 & 1 & 0 & -\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}2=\text{Zeile}2\text{-Zeile}1\cdot \frac{4}{4}\\z2s1=4-4\cdot \frac{4}{4}=0 \\ z2s2=2-(-2)\cdot \frac{4}{4}=4 \\ z2s3=1-1\cdot \frac{4}{4}=0 \\ z2s4=3-1\cdot \frac{4}{4}=2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & -2 & 1 & 1 \\ 0 & 4 & 0 & 2 \\ 2 & 1 & 0 & -\frac{1}{2} \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}1\cdot \frac{2}{4}\\z3s1=2-4\cdot \frac{2}{4}=0 \\ z3s2=1-(-2)\cdot \frac{2}{4}=2 \\ z3s3=0-1\cdot \frac{2}{4}=-\frac{1}{2} \\ z3s4=-\frac{1}{2}-1\cdot \frac{2}{4}=-1 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & -2 & 1 & 1 \\ 0 & 4 & 0 & 2 \\ 0 & 2 & -\frac{1}{2} & -1 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}2\cdot \frac{-2}{4}\\z1s2=-2-4\cdot \frac{-2}{4}=0 \\ z1s3=1-0\cdot \frac{-2}{4}=1 \\ z1s4=1-2\cdot \frac{-2}{4}=2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 1 & 2 \\ 0 & 4 & 0 & 2 \\ 0 & 2 & -\frac{1}{2} & -1 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}3=\text{Zeile}3\text{-Zeile}2\cdot \frac{2}{4}\\z3s2=2-4\cdot \frac{2}{4}=0 \\ z3s3=-\frac{1}{2}-0\cdot \frac{2}{4}=-\frac{1}{2} \\ z3s4=-1-2\cdot \frac{2}{4}=-2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 1 & 2 \\ 0 & 4 & 0 & 2 \\ 0 & 0 & -\frac{1}{2} & -2 \\ \end{array} \\ \\ \small \begin{array}{l}\text{Zeile}1=\text{Zeile}1\text{-Zeile}3\cdot \frac{1}{-\frac{1}{2}}\\z1s3=1-(-\frac{1}{2})\cdot \frac{1}{-\frac{1}{2}}=0 \\ z1s4=2-(-2)\cdot \frac{1}{-\frac{1}{2}}=-2 \\ \end{array}\qquad \small \begin{array}{ccc|cc } a & b & c & & \\ \hline4 & 0 & 0 & -2 \\ 0 & 4 & 0 & 2 \\ 0 & 0 & -\frac{1}{2} & -2 \\ \end{array} \\ \\ a=\frac{-2}{4}=-\frac{1}{2}\\b=\frac{2}{4}=\frac{1}{2}\\c=\frac{-2}{-\frac{1}{2}}=4\\L=\{-\frac{1}{2}/\frac{1}{2}/4\} \\ \text{Funktion} \\ f\left(x\right)=-\frac{1}{2}x^2+\frac{1}{2}x+4 \end{array}$