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Änderungen von Dokument BPE 4.2 Transformationen

Zuletzt geändert von Martin Rathgeb am 2025/04/25 16:25
Von Version 54.1
bearbeitet von Holger Engels
am 2025/03/03 20:57
Änderungskommentar: Löschung des Bildes Exp-Funktion.png
Auf Version 12.1
bearbeitet von kickoff kickoff
am 2023/10/09 15:32
Änderungskommentar: Neues Bild Exp-Funktion.png hochladen

Zusammenfassung

Details

Seiteneigenschaften
Dokument-Autor
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1 -XWiki.holgerengels
1 +XWiki.kickoff
Inhalt
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1 -{{seiteninhalt/}}
1 +{{box cssClass="floatinginfobox" title="**Contents**"}}
2 +{{toc start=2 depth=2 /}}
3 +{{/box}}
2 2  
3 -[[Kompetenzen.K6]] [[Kompetenzen.K4]] Ich kann beschreiben, durch welche Kette von Transformationen ein gegebener Funktionsterm aus dem der Standard Exponentialfunktion hervorgegangen ist
4 -[[Kompetenzen.K6]] [[Kompetenzen.K4]] Ich kann beschreiben, durch welche Kette von Transformationen ein gegebenes Schaubild aus dem der Standard Exponentialfunktion hervorgegangen ist
5 +[[Kompetenzen.K6]], [[Kompetenzen.K4]] Ich kann beschreiben, durch welche Kette von Transformationen ein gegebener Funktionsterm aus dem der Standard Exponentialfunktion hervorgegangen ist
6 +[[Kompetenzen.K6]], [[Kompetenzen.K4]] Ich kann beschreiben, durch welche Kette von Transformationen ein gegebenes Schaubild aus dem der Standard Exponentialfunktion hervorgegangen ist
5 5  [[Kompetenzen.K4]] Ich kann den Funktionsterm zu einer verbal gegebenen Transformation angeben
6 6  [[Kompetenzen.K4]] Ich kann den Funktionsterm zu einer grafisch gegebenen Transformation angeben
9 +(Im grundlegenden Anforderungsniveau wird horizontal nur entweder veschoben oder gestreckt. Im erhöhten Anforderungsniveau werden auch Kombinationen dieser beiden Transformationen betrachtet)
7 7  
8 -{{lernende}}
9 -[[GeoGebra-Buch>>https://www.geogebra.org/m/khnsgz5a#material/c7yGDeph]]
10 -[[KMap Interaktiv erkunden>>https://kmap.eu/app/browser/Mathematik/Exponentialfunktionen/Verschieben%2C%20Strecken%2C%20Spiegeln]]
11 -{{/lernende}}
11 +{{aufgabe afb="I" kompetenzen="K5" quelle="Martina Wagner" cc="BY-SA" zeit="4"}}
12 +Der Graph der Funktion //f// mit {{formula}}f(x)=2^x{{/formula}} wird durch mehrere Transformationen vedrändert. Stelle den zugehörigen Funktionsterm auf und skizziere den neuen Graphen.
12 12  
13 -{{aufgabe id="Funktionsterm aus Transformationen" afb="I" kompetenzen="K4,K5" quelle="Martina Wagner" cc="BY-SA" zeit="12"}}
14 -Der Graph der Funktion //f// mit {{formula}}f(x)=2^x{{/formula}} wird jeweils durch eine oder mehrere Transformationen verändert. Stelle den zugehörigen Funktionsterm auf und skizziere den neuen Graphen.
15 -(% class="abc" %)
16 -1. Streckung in y-Richtung mit dem Faktor {{formula}}-\frac{1}{2}{{/formula}} und Verschiebung in y-Richtung um -5
17 -1. Spiegelung an der y-Achse; Streckung in y-Richtung mit dem Faktor 1,5; Verschiebung in y-Richtung um 1
18 -1. Streckung in x-Richtung mit dem Faktor 0,5 und Verschiebung in y-Richtung um -2
14 +a) Verschiebung in y-Richtung um 3
15 +
16 +b) Streckung in y-Richtung mit dem Faktor {{formula}}-\frac{1}{2}{{/formula}} und Verschiebung in y-Richtung um -5
17 +
18 +c) Spiegelung an der y-Achse; Streckung in y-Richtung mit dem Faktor 1,5; Verschiebung in y-Richtung um 1
19 +
20 +d) Streckung in x-Richtung mit dem Faktor 0,5 und Verschiebung in y-Richtung um -2
19 19  {{/aufgabe}}
20 20  
21 -{{aufgabe id="Transformationen aus Schaubild" afb="I" kompetenzen="K4,K5,K6" quelle="Martina Wagner" cc="BY-SA" zeit="5"}}
22 -[[Abbildung 1>>image:Exp-Funktion.png||style="float:right;width:250px"]]Gegeben ist der 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.
23 -{{/aufgabe}}
24 24  
25 -{{aufgabe id="Skizzieren" afb="I" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="12"}}
26 -Skizziere jeweils das Schaubild.
27 -(% class="abc" %)
28 -1. {{formula}} f(x)=e^x-2 {{/formula}}
29 -1. {{formula}} g(x)=-e^{3x}+2 {{/formula}}
30 -1. {{formula}} h(x)=2e^{-x} {{/formula}}
31 -1. {{formula}} i(x)=-e^{-0,5x}+1 {{/formula}}
32 -{{/aufgabe}}
33 -
34 -{{aufgabe id="Analogie 1" afb="II" kompetenzen="K1,K4,K5" quelle="Elke Hallmann" cc="BY-SA" zeit="5"}}
35 -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}}.
36 -[[image:exp f.svg||style="margin:8px;width:360px"]] [[image:exp g.svg||style="margin:8px;width:360px"]]
37 -(% class="abc" %)
38 -1. Bestimme die Parameter //a// und //c//.
39 -1. Gib Gemeinsamkeiten bzw. Unterschiede der beiden Graphen und ihrer Funktionsterme an. Begründe deine Beobachtung.
40 -{{/aufgabe}}
41 -
42 -{{aufgabe id="Analogie 2" afb="II" kompetenzen="K5" quelle="Holger Engels" cc="BY-SA" zeit="5"}}
43 -Die Gleichung der Funktion {{formula}}f{{/formula}} lautet {{formula}}f(x)=2^x{{/formula}}. Die Funktion {{formula}}g{{/formula}} entsteht aus {{formula}}f{{/formula}} durch horizontale Streckung um den Faktor //1/2//.
44 -(% class="abc" %)
45 -1. Bestimme den Funktionsterm von {{formula}}g{{/formula}}?
46 -1. Ermittle einen weiteren Funktionsterm {{formula}}h{{/formula}} des Graphens {{formula}}K_g{{/formula}} in der Form {{formula}}h(x)=q^x{{/formula}}.
47 -{{/aufgabe}}
48 -
49 -{{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"}}
50 -[[image:Graphexponentialfunktion.PNG||width="180" style="float: right"]](% class="abc" %)
51 -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}}.
52 -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.
53 -{{/aufgabe}}
54 -
55 -{{aufgabe id="Transformationen und mehr" afb="II" kompetenzen="K4,K5,K6" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="8"}}
56 -Gegeben ist die Funktion {{formula}}f{{/formula}} mit {{formula}}f(x)=3e^{2x}-4{{/formula}}.
57 -(% class="abc" %)
58 -1. Beschreibe den Verlauf des Graphen {{formula}}K_f{{/formula}}.
59 -1. Wie entsteht {{formula}}K_f{{/formula}} aus dem Schaubild der Funktion {{formula}}g{{/formula}} mit {{formula}}g(x)=e^x{{/formula}}?
60 -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 von der x-Achse.
61 -1. Zeige, dass die Nullstelle von {{formula}}f{{/formula}} im Intervall {{formula}}[0,1; 0,2]{{/formula}} liegt.
62 -{{/aufgabe}}
63 -
64 -{{seitenreflexion bildungsplan="5" kompetenzen="5" anforderungsbereiche="5" kriterien="5" menge="3"/}}
Graphexponentialfunktion.PNG
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Transformationen aus Schaubild.ggb
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Transformationen aus Schaubild.svg
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