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Änderungen von Dokument BPE 3.2 Funktionsgraph

Zuletzt geändert von Holger Engels am 2025/03/31 21:43
Von Version 48.1
bearbeitet von Holger Engels
am 2024/11/15 15:04
Änderungskommentar: Neues Bild Fertig zeichnen.svg hochladen
Auf Version 67.4
bearbeitet von Holger Engels
am 2024/11/25 17:18
Änderungskommentar: Es gibt keinen Kommentar für diese Version

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12 12  Zeichne das Schaubild der Funktion {{formula}}f(x)=-0,5x^4+0,7x^3+2x^2-1{{/formula}} mit Hilfe einer Wertetabelle für {{formula}}-2\leq x\leq 3{{/formula}} in ein geeignetes Koordinatensystem ein.
13 13  {{/aufgabe}}
14 14  
15 -{{aufgabe id="Symmetrie untersuchen" afb="II" kompetenzen="" quelle="Niklas Wunder" cc="by-sa" zeit="10"}}
15 +{{aufgabe id="Punkte" afb="I" kompetenzen="K4" quelle="Stefanie Schmidt" cc="by-sa" zeit="2"}}
16 +Das Schaubild einer Funktion, die punktsymmetrisch zum Ursprung ist, enthält die Punkte {{formula}}P_1(1|-2){{/formula}} und {{formula}}P_2(-3|4){{/formula}}. Nenne drei weitere Punkte, die auf dem Schaubild liegen.
17 +{{/aufgabe}}
18 +
19 +{{aufgabe id="Symmetrie untersuchen" afb="I" kompetenzen="K4,K5" quelle="Niklas Wunder" cc="by-sa" zeit="10"}}
16 16  Untersuche die Graphen der Funktionen auf Symmetrie zum Koordinatenursprung und zur y-Achse.
17 17  (% style="list-style:alphastyle" %)
18 18  1. {{formula}}f(x)=3\,x+1{{/formula}}
... ... @@ -23,8 +23,8 @@
23 23  1. {{formula}}f(x)=x^4\,(x^3-3)\cdot (1-x){{/formula}}
24 24  {{/aufgabe}}
25 25  
26 -{{aufgabe id="Symmetrie Parameter bestimmen" afb="III" kompetenzen="" quelle="Niklas Wunder" cc="by-sa" zeit="8"}}
27 -Bestimme einen Zahlenwert {{formula}}a{{/formula}} so, dass der Graph symmetrisch zum Koordinatenursprung oder zur y- Achse ist.
30 +{{aufgabe id="Symmetrie Parameter bestimmen" afb="II" kompetenzen="K5" quelle="Niklas Wunder" cc="by-sa" zeit="8"}}
31 +Bestimme einen Zahlenwert {{formula}}a{{/formula}} so, dass der Graph symmetrisch zum Koordinatenursprung oder zur y- Achse ist.
28 28  a) {{formula}}f(x)=x+a{{/formula}}
29 29  b) {{formula}}f(x)=(x+1)\cdot (x-a){{/formula}}
30 30  c) {{formula}}f(x)=x\cdot (x+a)^2{{/formula}}
... ... @@ -31,7 +31,15 @@
31 31  d) {{formula}}f(x)=x\cdot (x^2+a){{/formula}}
32 32  {{/aufgabe}}
33 33  
34 -{{aufgabe id="Globalverlauf untersuchen" afb="I" kompetenzen="" quelle="Niklas Wunder, Martin Stern" cc="by-sa" zeit="4"}}
38 +{{aufgabe id="Vergleichsfunktion" afb="I" kompetenzen="K5,K6" quelle="Holger Engels, Martin Rathgeb" cc="by-sa" zeit="6" links="[[KMap>>https://kmap.eu/app/browser/Mathematik/Ganzrationale%20Funktionen/Verlauf#beispiel-----verhalten-im-unendlichen]]"}}
39 +Gegeben ist die Funktion //f// mit {{formula}}f{\left ( x \right )} = \frac{1}{2} x^{3} - 10 x^{2} - 2 x + 1{{/formula}}. Um den globalen Verlauf zu untersuchen, soll die Vergleichsfunktion bestimmt werden. Gehe folgedermaßen vor:
40 +1. Klammere //x// in der höchsten vorkommenden Potenz aus.
41 +1. Du erhältst ein Produkt aus {{formula}}x^3{{/formula}} und einer Summe.
42 +1. Streiche aus der Summe alle Summanden, die für betragsmäßig große //x// vernachlässigbar klein werden.
43 +1. Es bleibt nur ein Summand übrig, die Klammern können aufgelöst werden.
44 +{{/aufgabe}}
45 +
46 +{{aufgabe id="Globalverlauf untersuchen" afb="I" kompetenzen="K5,K6" quelle="Niklas Wunder, Martin Stern" cc="by-sa" zeit="4"}}
35 35  Untersuche das Verhalten der Funktion {{formula}}f{{/formula}} für {{formula}}x\rightarrow\pm \infty{{/formula}}:
36 36  (% style="list-style:alphastyle" %)
37 37  1. {{formula}}f(x)=-x^3{{/formula}}
... ... @@ -40,20 +40,40 @@
40 40  1. {{formula}}f(x)=x\cdot(x+7)\cdot(x-7){{/formula}}
41 41  {{/aufgabe}}
42 42  
43 -{{aufgabe id="Schnittpunkte mit den Koordinatenachsen bestimmen" afb="I" kompetenzen="" quelle="Niklas Wunder, Martin Stern" cc="by-sa" zeit="5"}}
55 +{{aufgabe id="Schnittpunkte mit den Koordinatenachsen" afb="I" kompetenzen="K4,K5" quelle="Niklas Wunder, Martin Stern" cc="by-sa" zeit="5"}}
44 44  Bestimme jeweils die Schnittpunkte mit ihren Vielfachheiten des Graphen der Funktion {{formula}}f{{/formula}} mit den Koordinatenachsen:
45 -(% style="list-style:alphastyle" %)
57 +(% class="abc" %)
46 46  1. {{formula}}f(x)=-2(x-\frac{3}{2}){{/formula}}
47 47  1. {{formula}}f(x)=2\cdot(x-3)^2\cdot(x+2)\cdot(x-2){{/formula}}
48 48  1. {{formula}}f(x)=2\cdot(x-3)^3\cdot(x^2-4){{/formula}}
49 49  {{/aufgabe}}
50 50  
51 -{{aufgabe id="Funktionsgraph mit Nullstellen skizzieren" afb="I" kompetenzen="" quelle="Niklas Wunder, Martin Stern" cc="by-sa" zeit="10"}}
52 -Gib die Nullstellen mit ihrer Vielfachheit an und skizziere anschließend den Graphen in einem geeigneten Intervall. Hinweis: Bei der e) gebe die Stellen mit {{formula}}f(x)=-1{{/formula}} an
63 +{{aufgabe id="Funktionsgraph mit Nullstellen skizzieren" afb="I" kompetenzen="K4,K5" quelle="Niklas Wunder, Martin Stern" cc="by-sa" zeit="10"}}
64 +Gib die Nullstellen mit ihrer Vielfachheit an und skizziere anschließend den Graphen in einem geeigneten Intervall.
53 53  (% style="list-style:alphastyle" %)
54 54  1. {{formula}}f_1(x)=(x-2)^2{{/formula}}
55 55  1. {{formula}}f_2(x)=(x+2)^3{{/formula}}
56 56  1. {{formula}}f_3(x)=(x-2)\cdot(x-3)\cdot x^2{{/formula}}
57 -1. {{formula}}f_4(x)=-\frac{1}{10}(x^2-9)\cdot x^3{{/formula}}
69 +1. {{formula}}f_4(x)=-\frac{1}{10}(x^2-9)\cdot (x-3){{/formula}}
58 58  1. {{formula}}f_5(x) = (x-3)^5{{/formula}}
59 59  {{/aufgabe}}
72 +
73 +{{aufgabe id="Fertig zeichnen" afb="I" kompetenzen="K4" quelle="Stefanie Schmidt" cc="by-sa" zeit="3"}}
74 +Ergänze das Schaubild der Funktion //f// mit {{formula}}f(x)=\frac{1}{11,66}(x^7-8x^5+16x^3){{/formula}} im Intervall {{formula}}[0;2,5]{{/formula}}.
75 +[[image:Fertig zeichnen.svg||width=600]]
76 +{{/aufgabe}}
77 +
78 +{{aufgabe id="Open Middle" afb="I" kompetenzen="K2,K4" quelle="Martina Wagner, Holger Engels" cc="by-sa" zeit="6" tags="problemlösen"}}
79 +Gegeben ist ein Funktionsterm mit Platzhaltern für selbstgewählte Zahlen von -5 bis 5. Jede Zahl darf maximal zweimal verwendet werden.
80 +
81 +{{formula}}f(x)=(x+\square)^\square \cdot (x+\square)^\square \cdot (x+\square)^\square \cdot (x+\square)^\square{{/formula}}
82 +
83 +Ermittle mögliche Zahlen für den Term, sodass das Schaubild folgende Eigenschaften erfüllt.
84 +(% class="abc" %)
85 +1. Symmetrisch zur y-Achse, keine Nullstelle bei //x=0// mit Grad höchstens sechs.
86 +1. Punktsymmetrisch zum Ursprung mit Grad höchstens fünf.
87 +{{/aufgabe}}
88 +
89 +{{lehrende}}K3 wurde bewusst weggelassen .. das kommt in BPE 3.5{{/lehrende}}
90 +
91 +{{seitenreflexion bildungsplan="5" kompetenzen="3" anforderungsbereiche="2" kriterien="3" menge="3"/}}
Fertig zeichnen.svg
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Fertig zeichnen.ggb
Author
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1 +XWiki.holgerengels
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1 +51.6 KB
Inhalt
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Kommentar
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1 -Lösung zu Aufgabe "Schnittpunkte mit den Koordinatenachsen bestimmen" überarbeiten.
1 +Wir haben die Aufgabe "Schnittpunkte mit den Koordinatenachsen" rechnerisch vereinfacht, da das Lösen von Polynomgleichungen erst min BPE 3.4 kommt
XWiki.XWikiComments[2]
Autor
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1 -XWiki.holgerengels
Kommentar
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1 -Eine Aufgabe, die zeigt, warum der Summand mit der höchsten Potenz den Verlauf bestimmt (so etwa [[KMap>>https://kmap.eu/app/browser/Mathematik/Ganzrationale%20Funktionen/Verlauf#beispiel-----verhalten-im-unendlichen]])
Datum
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1 -2024-11-15 11:17:25.685