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Änderungen von Dokument BPE 4.2 Transformationen

Zuletzt geändert von Martin Rathgeb am 2025/04/22 13:52
Von Version 52.1
bearbeitet von Holger Engels
am 2025/03/03 20:57
Änderungskommentar: Neues Bild Transformationen aus Schaubild.svg hochladen
Auf Version 31.1
bearbeitet von Niklas Wunder
am 2024/12/18 13:50
Änderungskommentar: Es gibt keinen Kommentar für diese Version

Zusammenfassung

Details

Seiteneigenschaften
Dokument-Autor
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1 -XWiki.holgerengels
1 +XWiki.niklaswunder
Inhalt
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1 1  {{seiteninhalt/}}
2 2  
3 +{{lernende}}
4 +[[GeoGebra-Buch>>https://www.geogebra.org/m/khnsgz5a#material/c7yGDeph]]
5 +[[KMap Interaktive Elemente>>https://kmap.eu/app/browser/Mathematik/Exponentialfunktionen/Verschieben%2C%20Strecken%2C%20Spiegeln]]
6 +{{/lernende}}
7 +
3 3  [[Kompetenzen.K6]] [[Kompetenzen.K4]] Ich kann beschreiben, durch welche Kette von Transformationen ein gegebener Funktionsterm aus dem der Standard Exponentialfunktion hervorgegangen ist
4 4  [[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
12 +(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}}
14 +{{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"}}
15 +[[image:Graphexponentialfunktion.PNG||width="180" style="float: right"]]
16 +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}}.
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
18 +
19 +2. 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.
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}}
22 +{{aufgabe id="Term und Skizze" afb="I" kompetenzen="K5" quelle="Martina Wagner" cc="BY-SA" zeit="4"}}
23 +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.
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}}
25 +a) Verschiebung in y-Richtung um 3
33 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.
27 +b) Streckung in y-Richtung mit dem Faktor {{formula}}-\frac{1}{2}{{/formula}} und Verschiebung in y-Richtung um -5
28 +
29 +c) Spiegelung an der y-Achse; Streckung in y-Richtung mit dem Faktor 1,5; Verschiebung in y-Richtung um 1
30 +
31 +d) 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 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}}.
34 +{{aufgabe id="Transformationen aus Schaubild" afb="I" kompetenzen="K5" quelle="Martina Wagner" cc="BY-SA" zeit="4"}}
35 +[[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. \\
47 47  {{/aufgabe}}
48 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.
38 +{{aufgabe id="Transformationen aus Funktionsterm" afb="II" kompetenzen="K6,K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="6"}}
39 +
40 +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.
41 +
42 +a) {{formula}} g(x)=e^x-2 {{/formula}}
43 +
44 +b) {{formula}} g(x)=e^{3x}+2,5 {{/formula}}
45 +
46 +c) {{formula}} g(x)=-1,5e^x {{/formula}}
47 +
48 +d) {{formula}} g(x)=e^{-0,5x}+1 {{/formula}}
49 +
50 +
53 53  {{/aufgabe}}
54 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.
53 +{{aufgabe id="Asymtoten bestimmen" afb="II" kompetenzen="K6,K4" quelle="Niklas Wunder, Katharina Schneider" cc="BY-SA" zeit="8"}}
54 +
55 +Skizziere jeweils das Schaubild der Funktion und bestimme die Gleichung der Asymptoten.
56 +
57 +a) {{formula}} f(x)=e^x-1,5 {{/formula}}
58 +
59 +b) {{formula}} g(x)=e^{-x}+\pi {{/formula}}
60 +
61 +c) {{formula}} h(x)=3^{-x}+6^{-x} {{/formula}}
62 +
63 +d) {{formula}} i(x)=(\frac{2}{3})^x{{/formula}}
62 62  {{/aufgabe}}
63 63  
64 64  {{seitenreflexion bildungsplan="5" kompetenzen="5" anforderungsbereiche="5" kriterien="5" menge="3"/}}
Transformationen aus Schaubild.svg
Author
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1 -XWiki.holgerengels
Größe
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1 -41.8 KB
Inhalt
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