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Änderungen von Dokument BPE 4.2 Transformationen

Zuletzt geändert von Dirk Tebbe am 2025/06/26 14:10
Von Version 74.1
bearbeitet von Martin Rathgeb
am 2025/04/25 14:17
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 39.1
bearbeitet von Holger Engels
am 2025/02/25 16:21
Änderungskommentar: Neues Bild exp f.svg hochladen

Zusammenfassung

Details

Seiteneigenschaften
Dokument-Autor
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1 -XWiki.martinrathgeb
1 +XWiki.holgerengels
Inhalt
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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="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:
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"]]
22 22  (% class="abc" %)
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.
17 +1. Berechne die Parameter //a// und //c//.
18 +1. Nenne Gemeinsamkeiten bzw. Unterschiede der beiden Graphen und ihrer Funktionsterme. Begründe.
25 25  {{/aufgabe}}
26 26  
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.
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.
33 33  {{/aufgabe}}
34 34  
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}}.
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.
37 37  (% class="abc" %)
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.
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
40 40  {{/aufgabe}}
41 41  
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.
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. \\
48 48  {{/aufgabe}}
49 49  
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.
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.
52 52  (% class="abc" %)
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}}.
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}}
55 55  {{/aufgabe}}
56 56  
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.
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}}
61 61  {{/aufgabe}}
62 62  
63 -{{aufgabe id="Nullstelle" afb="II" kompetenzen="K4,K5,K6" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="8"}}
59 +{{aufgabe id="Transformationen und mehr" afb="III" kompetenzen="K6,K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="10"}}
60 +
64 64  Gegeben ist die Funktion {{formula}}f{{/formula}} mit {{formula}}f(x)=3e^{2x}-4{{/formula}}.
65 65  (% class="abc" %)
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.
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.
68 68  {{/aufgabe}}
69 69  
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"/}}
69 +{{seitenreflexion bildungsplan="5" kompetenzen="5" anforderungsbereiche="5" kriterien="5" menge="3"/}}
Transformationen aus Schaubild.ggb
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1 -XWiki.holgerengels
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Transformationen aus Schaubild.svg
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