✏️ Wertetabellen zuordnen

Ordne die folgenden Wertetabellen den entsprechenden Wachstumsmodellen (lineares Wachstum, quadratisches Wachstum, exponentielle Zunahme und exponentielle Abnahme) zu.

$$\begin{array}{c|c} x & f(x) \\ \hline 0 & 2 \\ 2 & 4 \\ 4 & 8 \\ 6 & 18 \\ 8 & 32 \\ 10 & 50 \\ \end{array}$$

Lösung

Quadratisches Wachstum, da

$$\begin{array}{c|c} x & f(x) \\ \hline 0 & 0 \\ \footnotesize{2^2=4} & \footnotesize{\frac{1}{2}\cdot 4=2} \\ 2 & 2 \\ \footnotesize{4^2=16} & \footnotesize{\frac{1}{2}\cdot 16 = 8} \\ 4 & 8 \\ \footnotesize{6^2=36} & \footnotesize{\frac{1}{2}\cdot 36 = 18} \\ 6 & 18 \\ \end{array}$$

$$\begin{array}{c|c} x & g(x) \\ \hline 0 & 3 \\ 2 & 7 \\ 4 & 11 \\ 6 & 15 \\ 8 & 19 \\ 10 & 23 \\ \end{array}$$

Lösung

lineares Wachstum, da

$$\begin{array}{c|c} x & g(x) \\ \hline 0 & 3 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow +4} \\ 2 & 7 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow +4} \\ 4 & 11 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow +4} \\ 6 & 15 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow +4} \\ 8 & 19 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow +4} \\ 10 & 23 \\ \end{array}$$

$$\begin{array}{c|c} x & h(x) \\ \hline 0 & 3 \\ 2 & 6,75 \\ 4 & 15,188 \\ 6 & 34,172 \\ 8 & 76,887 \\ 10 & 172,995 \\ \end{array}$$

Lösung

exponentielle Zunahme, da

$$\begin{array}{c|c} x & h(x) \\ \hline 0 & 3 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 2,25} \\ 2 & 6,75 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 2,25} \\ 4 & 15,188 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 2,25} \\ 6 & 34,172 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 2,25} \\ 8 & 76,887 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 2,25} \\ 10 & 172,995 \\ \end{array}$$

$$\begin{array}{c|c} x & i(x) \\ \hline 0 & 3 \\ 2 & 2,43 \\ 4 & 1,968 \\ 6 & 1,594 \\ 8 & 1,291 \\ 10 & 1,046 \\ \end{array}$$

Lösung

exponentielle Abnahme, da

$$\begin{array}{c|c} x & i(x) \\ \hline 0 & 3 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 0,81} \\ 2 & 2,43 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 0,81} \\ 4 & 1,968 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 0,81} \\ 6 & 1,594 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 0,81} \\ 8 & 1,291 \\ \footnotesize{\downarrow +2} & \footnotesize{\downarrow \cdot 0,81} \\ 10 & 1,046 \\ \end{array}$$