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Zusammenfassung

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Seiteneigenschaften
Inhalt
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11 11  Symmetrie
12 12  Stetigkeit
13 13  
14 -{{aufgabe id="Skizzieren" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA"}}
15 -Skizziere die Graphen der Funktionen //f// und //g// mit {{formula}}f(x)=x^3{{/formula}} und {{formula}}g(x) = x^{1/3}{{/formula}} in ein gemeinsames Koordinatensystem, dessen x- und y-Achse jeweils von {{formula}}[-8; +8]{{/formula}} geht.
14 +
15 +{{aufgabe id="Erkunden (Paar von Potenzfunktionen) - Wertetabelle" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
16 +(% style="list-style: alphastyle" %)
17 +1. Ergänze für die Funktionsgleichung {{formula}}f(x)=x^2{{/formula}} folgende Wertetabelle (wo möglich).
18 +((((% class="border" style="width:100%" %)
19 +|={{formula}}x{{/formula}}|-1|| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10|||||||||
20 +|={{formula}}f(x){{/formula}}||-1||||||||||||400|900|1600|2500|3600|4900|6400|8100|10000
21 +)))
22 +1. Ergänze für die Funktionsgleichung {{formula}}g(x)=x^{1/2}{{/formula}} folgende Wertetabelle (wo möglich).
23 +((((% class="border" style="width:100%" %)
24 +|={{formula}}x{{/formula}}|-1||0|1|4|9|16|25|36|49|64|81|100|||||||||
25 +|={{formula}}g(x){{/formula}}||-1||||||||||||20|30|40|50|60|70|80|90|100
26 +)))
27 +1. Erkennst du eine Symmetrie?
28 +1. Sei nun {{formula}}x\in \mathbb{R}^+{{/formula}}. Bestimme
29 +(((
30 +1) {{formula}}g(y){{/formula}} für {{formula}}y=f(x){{/formula}} und
31 +2) {{formula}}f(y){{/formula}} für {{formula}}y=g(x){{/formula}}.
32 +)))
33 +1. Sei nun {{formula}}x\in \mathbb{R}{{/formula}}. Untersuche
34 +(((
35 +1) {{formula}}g(y){{/formula}} für {{formula}}y=f(x){{/formula}} und
36 +2) {{formula}}f(y){{/formula}} für {{formula}}y=g(x){{/formula}}.
37 +)))
16 16  {{/aufgabe}}
17 17  
18 -{{aufgabe id="D und W" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA"}}
19 -Gib jeweils den Definitions- und den Wertebereich an:
40 +{{aufgabe id="Erkunden (eine Potenzfunktion) - Wertetabelle" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
41 +Untersuche die Funktion //f// mit {{formula}}f(x)=\frac{1}{x}{{/formula}} und Definitionsbereich {{formula}}\mathbb{R}^*{{/formula}} im Hinblick auf ihr Randverhalten und ihre Wertemenge. Ergänze dafür zunächst folgende Wertetabellen (wo möglich).
42 +
43 +(% style="list-style: alphastyle" %)
44 +1. (((Randverhalten: Verhalten im Unendlichen
45 +1) Verhalten gegen plus Unendlich ({{formula}}+\infty{{/formula}})
46 +(% class="border" %)
47 +|={{formula}}x{{/formula}}| {{formula}}+1{{/formula}}| {{formula}}+10{{/formula}}| {{formula}}+100{{/formula}}| {{formula}}+1000{{/formula}}| {{formula}}+10^6{{/formula}}| {{formula}}+10^9{{/formula}}| {{formula}}+10^{12}{{/formula}}|
48 +|={{formula}}f(x){{/formula}}||||||||0
49 +
50 +2) Verhalten gegen minus Unendlich ({{formula}}-\infty{{/formula}})
51 +(% class="border" %)
52 +|={{formula}}x{{/formula}}| {{formula}}-1{{/formula}}| {{formula}}-10{{/formula}}| {{formula}}-100{{/formula}}| {{formula}}-1000{{/formula}}| {{formula}}-10^6{{/formula}}| {{formula}}-10^9{{/formula}}|{{formula}}-10^{12}{{/formula}}|
53 +|={{formula}}f(x){{/formula}}||||||||0
54 +)))
55 +1. (((Randverhalten: Verhalten nahe der Definitionslücke ({{formula}}x \approx 0{{/formula}})
56 +1) Verhalten links bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x<0{{/formula}})
57 +(% class="border" %)
58 +|={{formula}}x{{/formula}}| {{formula}}-1{{/formula}}| {{formula}}-0,1{{/formula}}| {{formula}}-0,01{{/formula}}| {{formula}}-0,001{{/formula}}| {{formula}}-10^{-6}{{/formula}}| {{formula}}-10^{-9}{{/formula}}| {{formula}}-10^{-12}{{/formula}}|0
59 +|={{formula}}f(x){{/formula}}||||||||
60 +
61 +2) Verhalten rechts bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x>0{{/formula}})
62 +(% class="border" %)
63 +|={{formula}}x{{/formula}}| {{formula}}+1{{/formula}}| {{formula}}+0,1{{/formula}}| {{formula}}+0,01{{/formula}}| {{formula}}+0,001{{/formula}}| {{formula}}+10^{-6}{{/formula}}| {{formula}}+10^{-9}{{/formula}}| {{formula}}+10^{-12}{{/formula}}|0
64 +|={{formula}}f(x){{/formula}}||||||||
65 +)))
66 +1. Erkennst du eine Symmetrie?
67 +1. Beschreibe das Randverhalten der Funktion und nenne ihre Wertemenge.
68 +1. Bestimme {{formula}}g(y){{/formula}} für {{formula}}y=g(x){{/formula}} und {{formula}}x\in \mathbb{R}^*{{/formula}}.
69 +{{/aufgabe}}
70 +
71 +{{aufgabe id="Erkunden - Graph und Asymptoten (gerader Parameter)" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
72 +Gib zu den Funktionsgleichungen {{formula}}f(x)=x^2{{/formula}}, {{formula}}g(x)=x^{1/2}{{/formula}} und {{formula}}h(x)=x^{-2}{{/formula}} jeweils den maximalen Definitionsbereich mit zugehörigem Wertebereich an und skizziere die Graphen der Funktionen ggf. mit ihren Asymptoten in ein gemeinsames Koordinatensystem, dessen x-Achse von {{formula}}[-3; +3]{{/formula}} geht. - Erkennst du bei einem Graphen bzw. zwischen zwei Graphen eine Symmetrie?
73 +
74 +{{lehrende}}
75 +Diese Aufgabe folgt gleich noch in anderem Layout; das bessere Layout soll sich für diese und die (nach-)folgende Aufgabe durchsetzen.
76 +{{/lehrende}}
77 +{{/aufgabe}}
78 +
79 +{{aufgabe id="Erkunden - Graph und Asymptoten (gerader Parameter)" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
80 +Gegeben sind drei Funktionsgleichungen {{formula}}f(x)=x^2{{/formula}}, {{formula}}g(x)=x^{1/2}{{/formula}} und {{formula}}h(x)=x^{-2}{{/formula}}.
81 +(% style="list-style: alphastyle" %)
82 +1. Gib jeweils den maximalen Definitionsbereich mit zugehörigem Wertebereich an.
83 +1. Skizziere jeweils den Graphen der Funktion ggf. mit Asymptoten; benutze dafür ein gemeinsames Koordinatensystem, dessen x-Achse von {{formula}}[-3; +3]{{/formula}} geht.
84 +1. Erkennst du bei einem Graphen bzw. zwischen zwei Graphen eine Symmetrie?
85 +{{/aufgabe}}
86 +
87 +{{aufgabe id="Erkunden - Graph und Asymptoten (ungerader Parameter)" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
88 +Gib zu den Funktionsgleichungen {{formula}}f(x)=x^3{{/formula}}, {{formula}}g(x)=x^{1/3}{{/formula}} und {{formula}}h(x)=x^{-3}{{/formula}} jeweils den maximalen Definitionsbereich mit zugehörigem Wertebereich an und skizziere die Graphen der Funktionen ggf. mit ihren Asymptoten in ein gemeinsames Koordinatensystem, dessen x- und y-Achse jeweils von {{formula}}[-8; +8]{{/formula}} geht. - Erkennst du bei einem Graphen bzw. zwischen zwei Graphen eine Symmetrie?
89 +{{/aufgabe}}
90 +
91 +{{aufgabe id="D und W" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
92 +Gib jeweils den maximalen Definitionsbereich mit zugehörigem Wertebereich an und skizziere die Graphen der Funktionen ggf. mit ihren Asymptoten:
93 +
94 +(% style="list-style: alphastyle" %)
20 20  1. {{formula}}f(x)=\frac{1}{x-2}+1{{/formula}}
21 21  1. {{formula}}g(x)=\sqrt{x+2}-1{{/formula}}
22 22  {{/aufgabe}}
23 23  
24 -{{aufgabe id="Eigenschaften" afb="I" kompetenzen="K1, K5" quelle="??" cc="BY-SA"}}
25 -Bestimme zu den unten genannten Funktionen den (1) Globalverlauf, die (2) Symmetrie, den (3) Definitions- und den (4) Wertebereich und gegebenenfalls (5) waagerechte und senkrechte Asymptoten.
99 +{{aufgabe id="Eigenschaften" afb="I" kompetenzen="K1, K5" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
100 +Gegeben ist die Funktionsgleichung {{formula}}f(x) = \frac{-3}{x-2}+4{{/formula}}.
26 26  
27 27  (% style="list-style: alphastyle" %)
28 -1. Das Schaubild der Funktion g ist eine Parabel vierter Ordnung mit dem Scheitel {{formula}}S(-2| 3){{/formula}}, die um den Streckungsfaktor {{formula}}\frac{1}{2}{{/formula}} in y-Richtung gestreckt wurde.
29 -1. Die Funktion h ist eine Potenzfunktion mit {{formula}}h(x) = \frac{-3}{x-2}+4{{/formula}}
103 +1. Gib für die Funktion //f// den maximalen Definitionsbereich mit zugehörigem Wertebereich und den Globalverlauf an.
104 +1. Nenne für den Graphen von //f// die waagerechte Asymptote und die senkrechte Asymptote.
105 +1. Zeige durch Rechnung, dass der Graph der Funktion weder symmetrisch zum Ursprung noch symmetrisch zur y-Achse ist.
30 30  {{/aufgabe}}
31 31  
32 32  {{aufgabe id="Venn - Eigenschaften" afb="II" kompetenzen="K2, K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="8" tags="problemlösen"}}
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46 46  **Zusatzaufgabe:** Finde möglichst einfache/ komplexe Lösungen.
47 47  {{/aufgabe}}
48 48  
49 -{{aufgabe id="Stetigkeit" afb="II" kompetenzen="" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5"}}
125 +{{aufgabe id="Stetigkeit - Anschaulische Einführung (Gegenlese)" afb="II" kompetenzen="" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5"}}
50 50  Sascha behauptet, die Funktion //f// mit {{formula}}f(x) = \frac{1}{x}{{/formula}} sei auf ihrem maximalen Definitionsbereich nicht stetig, weil man ihren Graphen nicht ohne Absetzen zeichnen kann. Nimm dazu Stellung!
51 51  {{/aufgabe}}
52 52  
129 +{{aufgabe id="Stetigkeitsbetrachtungen" afb="II" kompetenzen="" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5"}}
130 +Beurteile für jedes Schaubild, ob der Graph zu einer (zusammengesetzten) Funktion gehören kann und ob diese im dargestellten Bereich stetig ist!
131 +[[image:Stetigkeit ee.svg||style="margin: 8px"]] [[image:Stetigkeit ie.svg||style="margin: 8px"]] [[image:Stetigkeit ei.svg||style="margin: 8px"]] [[image:Stetigkeit ii.svg||style="margin: 8px"]]
132 +[[image:Stetigkeit lee.svg||style="margin: 8px"]] [[image:Stetigkeit o.svg||style="margin: 8px"]] (% style="display: inline-block" %) Hinweis:
133 +⬤ schließt den Punkt ein
134 +⭘ schließt ihn aus
135 +{{/aufgabe}}
53 53  
137 +{{aufgabe id="Umkehrung" afb="III" kompetenzen="K1, K2, K5" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5" niveau=p}}
138 +Sascha behauptet, die Funktion //f// mit {{formula}}f(x) = \frac{1}{x^2}{{/formula}} sei auf ihrem maximalen Definitionsbereich ihre eigene Umkehrfunktion. Nimm dazu Stellung!
139 +{{/aufgabe}}
54 54  
Stetigkeit lie.svg
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