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Zuletzt geändert von Simone Hochrein am 2026/02/03 14:25
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1 (%class=abc%)
2 1. ((({{formula}} a=4, \quad P(2,5 | 14), \quad f(x)=c \cdot a^{x} {{/formula}}
3
4 {{formula}}f(x)=c \cdot 4^x {{/formula}}
5
6 {{formula}}
7 \begin{aligned}
8 f(2,5) &= 4 \\
9 4 &= c \cdot 4^{2,5} && | : 4^{2,5} \\
10 c &= \frac{4}{4^{2,5}} = 0,125 \\
11 \end{aligned}
12 {{/formula}}
13
14 {{formula}}\Rightarrow f(x)= 0,125 \cdot 4^{x} {{/formula}}
15 )))
16 1. ((({{formula}} c=6, \quad P(-1 | 9), \quad f(x)=c \cdot a^{x} {{/formula}}
17
18 {{formula}}f(x)=6 \cdot a^x {{/formula}}
19
20 {{formula}}
21 \begin{aligned}
22 f(-1) &= 9 \\
23 9 &= 6 \cdot a^{-1} \\
24 9 &= \frac{6}{a} && | \cdot a \quad | : 9 \\
25 a &= \frac{6}{9} = \frac{2}{3} \\
26 \end{aligned}
27 {{/formula}}
28
29 {{formula}}\Rightarrow f(x)= 6 \cdot \left(\frac{2}{3}\right)^{x} {{/formula}})))
30 1. ((({{formula}} A(0 | -2), \quad B(2 | -4,5), \quad f(x)=c \cdot a^{x} {{/formula}}
31
32 {{formula}}
33 \begin{aligned}
34 f(0) &= -2 \Rightarrow -2 = c \cdot a^{0} \Rightarrow c = -2 \\
35 f(x) &= -2 \cdot a^{x} \\
36 f(2) &= -4,5 \\
37 -4,5 &= -2 \cdot a^{2} && | : (-2) \\
38 2,25 &= a^{2} && | \sqrt{\phantom{x}} \\
39 a &= 1,5 \\
40 \end{aligned}
41 {{/formula}}
42
43 {{formula}}\Rightarrow f(x)= -2 \cdot 1,5^{x} {{/formula}})))
44 1. ((({{formula}} A(1 | 1,5), \quad B(2 | 4,5), \quad f(x)=c \cdot a^{x} {{/formula}}
45
46 {{formula}}
47 \begin{aligned}
48 1,5 &= c \cdot a && (1) \\
49 4,5 &= c \cdot a^{2} && (2) \\
50 \text{aus (1): } c &= \frac{1,5}{a} && (3) \\
51 \text{(3) in (2): } 4,5 &= \frac{1,5}{a} \cdot a^{2} \\
52 4,5 &= 1,5 \cdot a && | : 1,5 \\
53 a &= 3 && (4) \\
54 \text{(4) in (3): } c &= \frac{1,5}{3} = 0,5 \\
55 \end{aligned}
56 {{/formula}}
57
58 {{formula}}\Rightarrow f(x)= 0,5 \cdot 3^{x} {{/formula}})))
59 1. ((({{formula}} A(2 | 1), \quad B(5 | 27), \quad f(x)=c \cdot a^{x} {{/formula}}
60
61 {{formula}}
62 \begin{aligned}
63 1 &= c \cdot a^{2} && (1) \\
64 27 &= c \cdot a^{5} && (2) \\
65 \text{aus (1): } c &= \frac{1}{a^{2}} && (3) \\
66 \text{(3) in (2): } 27 &= \frac{1}{a^{2}} \cdot a^{5} \\
67 27 &= a^{3} && | \sqrt[3]{\phantom{x}} \\
68 a &= 3 && (4) \\
69 \text{(4) in (3): } c &= \frac{1}{3^{2}} = \frac{1}{9} \\
70 \end{aligned}
71 {{/formula}}
72
73 {{formula}}\Rightarrow f(x)= \frac{1}{9} \cdot 3^{x} {{/formula}})))
74 1. ((({{formula}} A(2 | 16), \quad B(-2 | \frac{81}{16}), \quad f(x)=c \cdot a^{x} {{/formula}}
75
76 {{formula}}
77 \begin{aligned}
78 16 &= c \cdot a^{2} && (1) \\
79 \frac{81}{16} &= c \cdot a^{-2} && (2) \\
80 \text{aus (1): } c &= \frac{16}{a^{2}} && (3) \\
81 \text{(3) in (2): } \frac{81}{16} &= \frac{16}{a^{2}} \cdot a^{-2} \\
82 \frac{81}{16} &= \frac{16}{a^{2}} \cdot \frac{1}{a^{2}} \\
83 \frac{81}{16} &= \frac{16}{a^{4}} && | \text{ Kehrwert} \\
84 \frac{16}{81} &= \frac{a^{4}}{16} && | \cdot 16 \\
85 a^{4} &= \frac{256}{81} && | \sqrt[4]{\phantom{x}} \\
86 a &= \frac{4}{3} && (4) \\
87 \text{(4) in (3): } c &= \frac{16}{(4/3)^{2}} = 9 \\
88 \end{aligned}
89 {{/formula}}
90
91 {{formula}}\Rightarrow f(x)= 9 \cdot \left(\frac{4}{3}\right)^{x} {{/formula}})))