Änderungen von Dokument BPE 2.1 Funktionstypen und deren Eigenschaften

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

Von 14.2 nach 14.1 Von 48.2 nach 48.1

Von Version 48.1

bearbeitet von Martin Rathgeb
am 2024/10/14 14:35

Änderungskommentar: Es gibt keinen Kommentar für diese Version

Auf Version 14.2

bearbeitet von Holger Engels
am 2023/11/10 11:08

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Zusammenfassung

Seiteneigenschaften (2 geändert, 0 hinzugefügt, 0 gelöscht)
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Seiteneigenschaften

Dokument-Autor

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--XWiki.martinrathgeb
++XWiki.holgerengels

Inhalt

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  [[Kompetenzen.K1]] [[Kompetenzen.K4]] Ich kann die Eigenschaften von Potenzfunktionen ausgehend von den Funktionsgraphen erläutern
  [[Kompetenzen.K1]] Ich kann den Stetigkeitsbegriff anschaulich anhand der Graphen von Potenzfunktionen erläutern
--Verhalten +/- oo
--Verhalten nahe Definitionslücke
--Asymptoten
--Symmetrie
--Stetigkeit
++{{aufgabe id="Eigenschaften" afb="I" Kompetenzen="K1, K5" quelle="??" cc="BY-SA"}}
++Bestimmen Sie zu den unten genannten Funktionen den (1) Globalverlauf, die (2) Symmetrie, den (3) Definitions- und den (4) Wertebereich und gegebenenfalls (5) waagerechte und senkrechte Asymptoten.
--{{aufgabe id="Skizzieren" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA"}}
--Skizziere die Graphen der Funktionen //f// und //g// mit {{formula}}f(x)=x^3{{/formula}} und {{formula}}g(x) = x^{1/3}{{/formula}} in ein gemeinsames Koordinatensystem, dessen x- und y-Achse jeweils von {{formula}}[-8; +8]{{/formula}} geht.
++a) Das Schaubild der Funktion g ist eine Parabel vierter Ordnung mit dem Scheitel {{formula}}S(-2|  3){{/formula}}, die um den Streckungsfaktor {{formula}}\frac{1}{2} {{/formula}} in y-Richtung gestreckt wurde.
++b) Die Funktion h ist eine Potenzfunktion mit {{formula}}h(x) = \frac{-3}{(x-2)} +4 {{/formula}}
  {{/aufgabe}}
--{{aufgabe id="Skizzieren" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA"}}
--Gib zum beschriebenen Funktionsterm jeweils die Funktionsgleichung, den maximalen Definitions- und den maximalen Wertebereich an und skizziere die  Graphen der Funktionen in ein gemeinsames Koordinatensystem, dessen x- und y-Achse jeweils von {{formula}}[0; 8]{{/formula}} geht.
--1. {{formula}}x^n{{/formula}} mit {{formula}}n=2,3{{/formula}}
--1. {{formula}}x^{-n}{{/formula}} mit {{formula}}n=2,3{{/formula}}
--1. {{formula}}x^{1/n}{{/formula}} mit {{formula}}n=2,3{{/formula}}
--{{/aufgabe}}
--{{aufgabe id="D und W" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA"}}
--Gib jeweils den Definitions- und den Wertebereich an:
--1. {{formula}}f(x)=\frac{1}{x-2}+1{{/formula}}
--1. {{formula}}g(x)=\sqrt{x+2}-1{{/formula}}
--{{/aufgabe}}
--{{aufgabe id="Eigenschaften" afb="I" kompetenzen="K1, K5" quelle="??" cc="BY-SA"}}
--Bestimme zu den unten genannten Funktionen den (1) Globalverlauf, die (2) Symmetrie, den (3) Definitions- und den (4) Wertebereich und gegebenenfalls (5) waagerechte und senkrechte Asymptoten.
--(% style="list-style: alphastyle" %)
--1. Das Schaubild der Funktion g ist eine Parabel vierter Ordnung mit dem Scheitel {{formula}}S(-2|  3){{/formula}}, die um den Streckungsfaktor {{formula}}\frac{1}{2}{{/formula}} in y-Richtung gestreckt wurde.
--1. Die Funktion h ist eine 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"}}
--[[image:venn.svg|| width="500" style="float: left"]]
--Gib für jedes Feld **A** .. **H** eine passende Funktion {{formula}}f(x)=a\cdot x^n{{/formula}} an. Sollte ein Feld nicht gefüllt werden können, begründe bitte, warum es nicht geht.
--
--(% style="width: calc(100% - 500px); min-width: 300px" %)
--|= A |
--|= B |
--|= C |
--|= D |
--|= E |
--|= F |
--|= G |
--|= H |
--
--**Zusatzaufgabe:** Finde möglichst einfache/ komplexe Lösungen.
--{{/aufgabe}}
--
--{{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 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 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"]] (((Hinweis:
--⬤ schließt den Punkt ein
--⭘ schließt ihn aus)))
--{{/aufgabe}}
--

Stetigkeit ee.svg

Author

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1		-XWiki.holgerengels

Größe

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1		-17.7 KB

Inhalt

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Stetigkeit ei.svg

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Stetigkeit ie.svg

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Stetigkeit ii.svg

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Stetigkeit lie.svg

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Stetigkeit lii.svg

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Stetigkeit o.svg

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venn.svg

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