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Änderungen von Dokument BPE 4.2 Transformationen

Zuletzt geändert von Dirk Tebbe am 2025/06/26 14:10
Von Version 39.1
bearbeitet von Holger Engels
am 2025/02/25 16:21
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Auf Version 74.1
bearbeitet von Martin Rathgeb
am 2025/04/25 14:17
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Zusammenfassung

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1 -XWiki.holgerengels
1 +XWiki.martinrathgeb
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10 10  [[KMap Interaktiv erkunden>>https://kmap.eu/app/browser/Mathematik/Exponentialfunktionen/Verschieben%2C%20Strecken%2C%20Spiegeln]]
11 11  {{/lernende}}
12 12  
13 -{{aufgabe id="Aussagen" afb="I" kompetenzen="K5" quelle="Holger Engels" cc="BY-SA" zeit="4"}}
14 -Gegeben sind die Schaubilder //K,,f,,// und //K,,g,,// und die Funktionsterme {{formula}}f(x)=a\cdot2^x{{/formula}} und {{formula}}g(x)=2^{x-c}{{/formula}}.
15 -[[image:exp f.svg||style="margin:8px;width:400px"]] [[image:exp g.svg||style="margin:8px;width:400px"]]
13 +{{aufgabe id="Funktionsterm aus Transformationen" afb="II" kompetenzen="K4,K5" quelle="Martina Wagner" cc="BY-SA" zeit="15"}}
14 +Gegeben ist die Exponentialfunktion //f// mit der Funktionsgleichung {{formula}}f(x)=2^x{{/formula}}. Die folgenden Graphen {{formula}}K_{g_1}{{/formula}}, {{formula}}K_{g_2}{{/formula}} und {{formula}}K_{g_3}{{/formula}} entstehen jeweils aus dem Graphen {{formula}}K_f{{/formula}} durch Transformationen:
15 +(((
16 +(% class="noborder slim" style="margin-left: 2em;" %)
17 +|{{formula}}K_{g_1}{{/formula}} :|//Streckung// mit dem Faktor {{formula}}-\frac{1}{2}{{/formula}} in y-Richtung, //Verschiebung// um {{formula}}-5{{/formula}} in y-Richtung
18 +|{{formula}}K_{g_2}{{/formula}} :|//Spiegelung// an der y-Achse, //Streckung// mit dem Faktor {{formula}}1{,}5{{/formula}} in y-Richtung, //Verschiebung// um {{formula}}1{{/formula}} in y-Richtung
19 +|{{formula}}K_{g_3}{{/formula}} :|//Streckung// mit dem Faktor {{formula}}0{,}5{{/formula}} in x-Richtung, //Verschiebung// um {{formula}}-2{{/formula}} in y-Richtung
20 +)))
21 +Bearbeite zu jedem der drei Fälle folgende Teilaufgaben:
16 16  (% class="abc" %)
17 -1. Berechne die Parameter //a// und //c//.
18 -1. Nenne Gemeinsamkeiten bzw. Unterschiede der beiden Graphen und ihrer Funktionsterme. Begründe.
23 +1. Gib die Funktionsgleichungen von //g//₁, //g//₂ und //g//₃ an.
24 +1. Skizziere die Graphen {{formula}}K_{g_1}{{/formula}}, {{formula}}K_{g_2}{{/formula}} und {{formula}}K_{g_3}{{/formula}} im Vergleich zu {{formula}}K_f{{/formula}} in einem gemeinsamen Koordinatensystem.
19 19  {{/aufgabe}}
20 20  
21 -{{aufgabe id="Aufstellen eines Funktionstermes" afb="II" kompetenzen="K2,K4,K5" quelle="[[IQB e.V.>>https://www.iqb.hu-berlin.de/abitur/pools2022/abitur/pools2022/mathematik/grundlegend/2022_M_grundlege_20.pdf]]" niveau="g" tags="iqb" cc="by"}}
22 -[[image:Graphexponentialfunktion.PNG||width="180" style="float: right"]](% class="abc" %)
23 -1. Die Abbildung zeigt den Graphen der Funktion {{formula}}f: x \mapsto a \cdot b^x{{/formula}} mit {{formula}} a,b \in \mathbb{R}^+{{/formula}}. Bestimme passende Werte von {{formula}}a{{/formula}} und {{formula}}b{{/formula}}.
24 -1. Der Graph der in {{formula}}\mathbb{R}{{/formula}} definierten Funktion {{formula}}g: x \mapsto 3^x{{/formula}} wird um 2 in negative x-Richtung verschoben. Zeige, dass der dadurch entstandene Graph durch eine Streckung des Graphen von {{formula}}g{{/formula}} in y-Richtung erzeugt werden kann.
27 +{{aufgabe id="Transformationen aus Schaubild" afb="I" kompetenzen="K4,K5,K6" quelle="Martina Wagner" cc="BY-SA" zeit="5"}}
28 +Gegeben ist der Graph der Funktion //g// mit {{formula}}g(x)=a\cdot2^{\pm x}+d{{/formula}}.
29 +[[Abbildung 1>>image:Transformationen aus Schaubild.svg||style="float:right;width:350px;margin-left:8px"]]
30 +(% class="abc" %)
31 +1. Beschreibe, durch welche Transformationen der Graph von //g// aus dem Graphen der Funktion //f// mit {{formula}}f(x)=2^x{{/formula}} hervorgeht.
32 +1. Gib die Funktionsgleichung von //g// an.
25 25  {{/aufgabe}}
26 26  
27 -{{aufgabe id="Term und Skizze" afb="I" kompetenzen="K5" quelle="Martina Wagner" cc="BY-SA" zeit="4"}}
28 -Der Graph der Funktion //f// mit {{formula}}f(x)=2^x{{/formula}} wird durch mehrere Transformationen verändert. Stelle den zugehörigen Funktionsterm auf und skizziere den neuen Graphen.
35 +{{aufgabe id="Skizzieren" afb="II" kompetenzen="K4,K5" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="15"}}
36 +Gegeben sind die Funktionen //f//, //g//, //h// und //i// mit {{formula}}f(x)=e^x-2{{/formula}}, {{formula}}g(x)=-e^x+2{{/formula}}, {{formula}}h(x)=e^{-x-2}{{/formula}} und {{formula}}i(x)=-e^{-x}+1{{/formula}}.
29 29  (% class="abc" %)
30 -1. Verschiebung in y-Richtung um 3
31 -1. Streckung in y-Richtung mit dem Faktor {{formula}}-\frac{1}{2}{{/formula}} und Verschiebung in y-Richtung um -5
32 -1. Spiegelung an der y-Achse; Streckung in y-Richtung mit dem Faktor 1,5; Verschiebung in y-Richtung um 1
33 -1. Streckung in x-Richtung mit dem Faktor 0,5 und Verschiebung in y-Richtung um -2
38 +1. Skizziere die Graphen der Funktionen in ein gemeinsames Schaubild.
39 +1. Beschreibe Gemeinsamkeiten und Unterschiede zwischen den dargestellten Graphen hinsichtlich ihrer Lage, Symmetrie und Verschiebung.
34 34  {{/aufgabe}}
35 35  
36 -{{aufgabe id="Transformationen aus Schaubild" afb="I" kompetenzen="K5" quelle="Martina Wagner" cc="BY-SA" zeit="4"}}
37 -[[Abbildung 1>>image:Exp-Funktion.png||style="float:right;width:250px"]]Gegeben ist der untenstehende Graph der Funktion //f// mit {{formula}}f(x)=a\cdot2^{\pm x}+d{{/formula}}. Beschreibe durch welche Transformationen der Graph von //f// aus dem Graphen der Funktion //g// mit {{formula}}g(x)=2^x{{/formula}} hervorgeht, und stelle den zugehörigen Funktionsterm auf. \\
42 +{{aufgabe id="Analogie 1" afb="II" kompetenzen="K1,K4,K5" quelle="Elke Hallmann" cc="BY-SA" zeit="5"}}
43 +Gegeben sind die Funktionen {{formula}}f{{/formula}} und {{formula}}g{{/formula}} mit {{formula}}f(x)=a\cdot2^x{{/formula}} und {{formula}}g(x)=2^{x-c}{{/formula}} sowie ihre Graphen {{formula}}K_f{{/formula}} und {{formula}}K_g{{/formula}}.
44 +[[image:exp f.svg||style="margin:8px;width:360px"]] [[image:exp g.svg||style="margin:8px;width:360px"]]
45 +(% class="abc" %)
46 +1. Bestimme die Parameter //a// und //c//.
47 +1. Gib Gemeinsamkeiten bzw. Unterschiede der beiden Graphen und ihrer Funktionsterme an. Begründe deine Beobachtung.
38 38  {{/aufgabe}}
39 39  
40 -{{aufgabe id="Transformationen aus Funktionsterm" afb="II" kompetenzen="K6,K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="6"}}
41 -
42 -Skizziere das Schaubild von {{formula}} g(x) {{/formula}} und beschreibe wie {{formula}}K_g {{/formula}} aus dem Graphen von {{formula}} f {{/formula}} mit {{formula}} f(x)=e^x {{/formula}} entsteht.
50 +{{aufgabe id="Analogie 2" afb="II" kompetenzen="K5" quelle="Holger Engels" cc="BY-SA" zeit="5"}}
51 +Gegeben ist die Funktion {{formula}}f{{/formula}} mit {{formula}}f(x)=2^x{{/formula}}. Der Graph der Funktion {{formula}}g{{/formula}} entsteht aus dem Graphen der Funktion {{formula}}f{{/formula}} durch Streckung mit Faktor //1/2// in x-Richtung.
43 43  (% class="abc" %)
44 -1. {{formula}} g(x)=e^x-2 {{/formula}}
45 -1. {{formula}} g(x)=e^{3x}+2,5 {{/formula}}
46 -1. {{formula}} g(x)=-1,5e^x {{/formula}}
47 -1. {{formula}} g(x)=e^{-0,5x}+1 {{/formula}}
53 +1. Bestimme den Funktionsterm von {{formula}}g{{/formula}}.
54 +1. Ermittle einen weiteren Funktionsterm {{formula}}h{{/formula}} des Graphens {{formula}}K_g{{/formula}} in der Form {{formula}}h(x)=q^x{{/formula}}.
48 48  {{/aufgabe}}
49 49  
50 -{{aufgabe id="Asymtoten bestimmen" afb="II" kompetenzen="K6,K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="8"}}
51 -Skizziere jeweils das Schaubild der Funktion und bestimme die Gleichung der Asymptoten.
52 -(% class="abc" %)
53 -1. {{formula}} f(x)=e^x-1,5 {{/formula}}
54 -1. {{formula}} g(x)=e^{-x}+\pi {{/formula}}
55 -1. {{formula}} h(x)=3^{-x}+6^{-x} {{/formula}}
56 -1. {{formula}} i(x)=(\frac{2}{3})^x{{/formula}}
57 +{{aufgabe id="Aufstellen eines Funktionstermes" afb="II" kompetenzen="K4,K5" quelle="[[IQB e.V.>>https://www.iqb.hu-berlin.de/abitur/pools2022/abitur/pools2022/mathematik/grundlegend/2022_M_grundlege_20.pdf]]" niveau="g" tags="iqb" cc="by" zeit="8"}}
58 +[[image:Graphexponentialfunktion.PNG||width="180" style="float: right"]](% class="abc" %)
59 +1. Die Abbildung zeigt den Graphen der Funktion {{formula}}f: x \mapsto a \cdot b^x{{/formula}} mit {{formula}} a,b \in \mathbb{R}^+{{/formula}}. Bestimme passende Werte von {{formula}}a{{/formula}} und {{formula}}b{{/formula}}.
60 +1. Der Graph der in {{formula}}\mathbb{R}{{/formula}} definierten Funktion {{formula}}g: x \mapsto 3^x{{/formula}} wird um 2 in negative x-Richtung verschoben. Zeige, dass der dadurch entstandene Graph durch eine Streckung des Graphen von {{formula}}g{{/formula}} in y-Richtung erzeugt werden kann.
57 57  {{/aufgabe}}
58 58  
59 -{{aufgabe id="Transformationen und mehr" afb="III" kompetenzen="K6,K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="10"}}
60 -
63 +{{aufgabe id="Nullstelle" afb="II" kompetenzen="K4,K5,K6" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="8"}}
61 61  Gegeben ist die Funktion {{formula}}f{{/formula}} mit {{formula}}f(x)=3e^{2x}-4{{/formula}}.
62 62  (% class="abc" %)
63 -1. Beschreibe den Verlauf von dem Schaubild {{formula}}K_f{{/formula}}.
64 -1. Wie entsteht {{formula}}K_f{{/formula}} aus dem Schaubild der Funktion {{formula}}g{{/formula}} mit {{formula}}g(x)=e^x{{/formula}}?
65 -1. Zeige: Für {{formula}}x<-1{{/formula}} hat jeder Punkt {{formula}}P\in K_f{{/formula}} einen Abstand von höchstens 4 und mindestens 3,5 LE.
66 -1. Zeige, dass die Nullstelle von {{formula}}f{{/formula}} zwischen 0,1 und 0,2 liegt.
66 +1. Begründe, dass die Funktion {{formula}}f{{/formula}} eine Nullstelle haben muss.
67 +1. Zeige, dass die Nullstelle von {{formula}}f{{/formula}} im Intervall {{formula}}[0,1; 0,2]{{/formula}} liegt.
67 67  {{/aufgabe}}
68 68  
69 -{{seitenreflexion bildungsplan="5" kompetenzen="5" anforderungsbereiche="5" kriterien="5" menge="3"/}}
70 +{{aufgabe id="Umkehraufgabe" afb="II" kompetenzen="K2,K5" quelle="Holger Engels" cc="BY-SA" zeit="8"}}
71 +Das Schaubild der Funktion {{formula}}g{{/formula}} mit {{formula}}g(x)=2^{x+4}{{/formula}} ist aus dem Schaubild der Funktion {{formula}}f{{/formula}} entstanden, indem dieses zunächst um zwei nach links verschoben und dann horizontal mit Faktor //2// gestreckt wurde. Bestimme den Funktionsterm der Ausgangsfunktion {{formula}}f{{/formula}}.
72 +{{/aufgabe}}
73 +
74 +{{lehrende}}
75 +K3 wird in [[BPE 4.6>>BPE_4_6]] behandelt
76 +AFB III kann mit dem Thema kaum erreicht werden.
77 +{{/lehrende}}
78 +
79 +{{seitenreflexion bildungsplan="4" kompetenzen="4" anforderungsbereiche="5" kriterien="5" menge="5"/}}
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Transformationen aus Schaubild.ggb
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