Änderungen von Dokument BPE 2.1 Funktionstypen und deren Eigenschaften

Zuletzt geändert von Holger Engels am 2025/03/31 21:42

Von 64.1 nach 63.1 Von 187.1 nach 186.1

Von Version 186.1

bearbeitet von Holger Engels
am 2024/10/15 10:35

Änderungskommentar: Neues Bild venn.svg hochladen

Auf Version 64.1

bearbeitet von Martin Rathgeb
am 2024/10/14 15:55

Änderungskommentar: Es gibt keinen Kommentar für diese Version

Rohform
Im Layout

Zusammenfassung

Seiteneigenschaften (2 geändert, 0 hinzugefügt, 0 gelöscht)
Anhänge (1 geändert, 0 hinzugefügt, 1 gelöscht)
- Funktionsskizze.png
- venn.svg

Details

Seiteneigenschaften

Dokument-Autor

@@ -1,1 +1,1 @@
--XWiki.holgerengels
++XWiki.martinrathgeb

Inhalt

@@ -11,80 +11,11 @@
  Symmetrie
  Stetigkeit
--
--{{aufgabe id="Erkunden (Paar von Potenzfunktionen) - Wertetabelle" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
--(% style="list-style: alphastyle" %)
--1. Ergänze für die Funktionsgleichung {{formula}}f(x)=x^2{{/formula}} folgende Wertetabelle (wo möglich).
--((((% class="border" style="width:100%" %)
--|={{formula}}x{{/formula}}|-1|| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9| 10|||||||||
--|={{formula}}f(x){{/formula}}||-1||||||||||||400|900|1600|2500|3600|4900|6400|8100|10000
--)))
--1. Ergänze für die Funktionsgleichung {{formula}}g(x)=x^{1/2}{{/formula}} folgende Wertetabelle (wo möglich).
--((((% class="border" style="width:100%" %)
--|={{formula}}x{{/formula}}|-1||0|1|4|9|16|25|36|49|64|81|100|||||||||
--|={{formula}}g(x){{/formula}}||-1||||||||||||20|30|40|50|60|70|80|90|100
--)))
--1. Erkennst du eine Symmetrie?
--1. Sei nun {{formula}}x\in \mathbb{R}^+{{/formula}}. Bestimme
--(((
--1) {{formula}}g(y){{/formula}} für {{formula}}y=f(x){{/formula}} und
--2) {{formula}}f(y){{/formula}} für {{formula}}y=g(x){{/formula}}.
--)))
--1. Sei nun {{formula}}x\in \mathbb{R}{{/formula}}. Untersuche
--(((
--1) {{formula}}g(y){{/formula}} für {{formula}}y=f(x){{/formula}} und
--2) {{formula}}f(y){{/formula}} für {{formula}}y=g(x){{/formula}}.
--)))
--{{/aufgabe}}
--
--{{aufgabe id="Erkunden (eine Potenzfunktion) - Wertetabelle" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
--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).
--
--(% style="list-style: alphastyle" %)
--1. (((Randverhalten: Verhalten im Unendlichen
--1) Verhalten gegen plus Unendlich ({{formula}}+\infty{{/formula}})
--(% class="border" %)
--|={{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}}|
--|={{formula}}f(x){{/formula}}||||||||0
--
--2) Verhalten gegen minus Unendlich ({{formula}}-\infty{{/formula}})
--(% class="border" %)
--|={{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}}|
--|={{formula}}f(x){{/formula}}||||||||0
--)))
--1. (((Randverhalten: Verhalten nahe der Definitionslücke ({{formula}}x \approx 0{{/formula}})
--1) Verhalten links bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x<0{{/formula}})
--(% class="border" %)
--|={{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
--|={{formula}}f(x){{/formula}}||||||||
--
--2) Verhalten rechts bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x>0{{/formula}})
--(% class="border" %)
--|={{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
--|={{formula}}f(x){{/formula}}||||||||
--)))
--1. Erkennst du eine Symmetrie?
--1. Beschreibe das Randverhalten der Funktion und nenne ihre Wertemenge.
--1. Bestimme {{formula}}g(y){{/formula}} für {{formula}}y=g(x){{/formula}} und {{formula}}x\in \mathbb{R}^*{{/formula}}.
--{{/aufgabe}}
--
--{{aufgabe id="Erkunden - Graph und Asymptoten (gerader Parameter)" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
++{{aufgabe id="Erkunden - Gerader Parameter" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
  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?
--
--{{lehrende}}
--Diese Aufgabe folgt gleich noch in anderem Layout; das bessere Layout soll sich für diese und die (nach-)folgende Aufgabe durchsetzen.
--{{/lehrende}}
  {{/aufgabe}}
--{{aufgabe id="Erkunden - Graph und Asymptoten (gerader Parameter)" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
--Gegeben sind drei Funktionsgleichungen {{formula}}f(x)=x^2{{/formula}}, {{formula}}g(x)=x^{1/2}{{/formula}} und {{formula}}h(x)=x^{-2}{{/formula}}.
--(% style="list-style: alphastyle" %)
--1. Gib jeweils den maximalen Definitionsbereich mit zugehörigem Wertebereich an.
--1. Skizziere jeweils den Graphen der Funktion ggf. mit Asymptoten; benutze dafür ein gemeinsames Koordinatensystem, dessen x-Achse von {{formula}}[-3; +3]{{/formula}} geht.
--1. Erkennst du bei einem Graphen bzw. zwischen zwei Graphen eine Symmetrie?
--{{/aufgabe}}
--
--{{aufgabe id="Erkunden - Graph und Asymptoten (ungerader Parameter)" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
++{{aufgabe id="Erkunden - Ungerader Parameter" afb="I" kompetenzen="" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
  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?
  {{/aufgabe}}
@@ -96,13 +96,12 @@
 . {{formula}}g(x)=\sqrt{x+2}-1{{/formula}}
  {{/aufgabe}}
--{{aufgabe id="Eigenschaften" afb="I" kompetenzen="K1, K5" quelle="Holger Engels, Martin Rathgeb" cc="BY-SA"}}
--Gegeben ist die Funktionsgleichung {{formula}}f(x) = \frac{-3}{x-2}+4{{/formula}}.
++{{aufgabe id="Eigenschaften" afb="I" kompetenzen="K1, K5" quelle="??" cc="BY-SA"}}
++Bestimme zu den unten genannten Funktionen (1) den maximalen Definitionsbereich mit (2) zugehörigem Wertebereich, (3) den Globalverlauf, (4) die Symmetrie und gegebenenfalls (5) waagerechte und senkrechte Asymptoten.
  (% style="list-style: alphastyle" %)
--1. Gib für die Funktion //f// den maximalen Definitionsbereich mit zugehörigem Wertebereich und den Globalverlauf an.
--1. Nenne für den Graphen von //f// die waagerechte Asymptote und die senkrechte Asymptote.
--1. Zeige durch Rechnung, dass der Graph der Funktion weder symmetrisch zum Ursprung noch symmetrisch zur y-Achse ist.
++1. Das Schaubild der Funktion g ist eine Parabel vierter Ordnung mit dem Scheitel {{formula}}S(-2|  3){{/formula}}, die mit dem Faktor {{formula}}\frac{1}{2}{{/formula}} in y-Richtung gestreckt wurde.
++1. Die Funktion h ist eine transformierte Potenzfunktion mit {{formula}}h(x) = \frac{-3}{x-2}+4{{/formula}}.
  {{/aufgabe}}
  {{aufgabe id="Venn - Eigenschaften" afb="II" kompetenzen="K2, K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="8"  tags="problemlösen"}}
@@ -122,19 +122,15 @@
  **Zusatzaufgabe:** Finde möglichst einfache/ komplexe Lösungen.
  {{/aufgabe}}
--{{aufgabe id="Stetigkeit - Anschaulische Einführung (Gegenlese)" afb="II" kompetenzen="" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5"}}
++{{aufgabe id="Stetigkeit" afb="II" kompetenzen="" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5"}}
  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!
  {{/aufgabe}}
  {{aufgabe id="Stetigkeitsbetrachtungen" afb="II" kompetenzen="" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5"}}
--Beurteile für jedes Schaubild, ob der Graph zu einer (zusammengesetzten) Funktion gehören kann und ob diese im dargestellten Bereich stetig ist!
++Beurteile für jedes Schaubild, ob der Graph zu einer (zusammengesetzten) Funktion gehören kann und ob diese im dargestellten stetig sind!
  [[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"]]
--[[image:Stetigkeit lee.svg||style="margin: 8px"]] [[image:Stetigkeit o.svg||style="margin: 8px"]] (% style="display: inline-block" %) Hinweis:
++[[image:Stetigkeit lee.svg||style="margin: 8px"]] [[image:Stetigkeit lie.svg||style="margin: 8px"]] [[image:Stetigkeit lei.svg||style="margin: 8px"]] [[image:Stetigkeit lii.svg||style="margin: 8px"]]  [[image:Stetigkeit o.svg||style="margin: 8px"]] (% style="display: inline-block" %)(((Hinweis:
  ⬤ schließt den Punkt ein
--⭘ schließt ihn aus
++⭘ schließt ihn aus)))
  {{/aufgabe}}
--{{aufgabe id="Umkehrung" afb="III" kompetenzen="K1, K2, K5" quelle="Martin Rathgeb, Holger Engels" cc="BY-SA" zeit="5" niveau=p}}
--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!
--{{/aufgabe}}
--

Funktionsskizze.png

Author

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1		-XWiki.fujan

Größe

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1		-17.9 KB

Inhalt

venn.svg

Größe

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--34.4 KB
++35.1 KB

Inhalt

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Änderungen von Dokument BPE 2.1 Funktionstypen und deren Eigenschaften

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