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Änderungen von Dokument BPE 3.1 Eigenschaften und Formen

Zuletzt geändert von Holger Engels am 2025/04/09 13:57
Von Version 91.1
bearbeitet von Niklas Wunder
am 2024/12/17 17:42
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 113.1
bearbeitet von Holger Engels
am 2024/12/18 13:40
Änderungskommentar: Neues Bild 1-fach.svg hochladen

Zusammenfassung

Details

Seiteneigenschaften
Dokument-Autor
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1 -XWiki.niklaswunder
1 +XWiki.holgerengels
Inhalt
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2 2  
3 3  [[Kompetenzen.K5]] Ich kenne die allgemeine Form der Polynomfunktion
4 4  [[Kompetenzen.K4]] Ich kenne die Produktform der Polynomfunktion
5 -[[Kompetenzen.K3]] [[Kompetenzen.K4]] Ich kann die für den Anwendungsfall geeignete Darstellungsform wählen
6 -[[Kompetenzen.K1]] [[Kompetenzen.K4]] Ich kann die Wahl der Darstellungsform im Anwendungskontext begründen
5 +[[Kompetenzen.K4]] Ich kenne die Scheitelform der quadratischen Funktion [[→ BPE 2.2>>BPE_2_2]]
6 +[[Kompetenzen.K4]] Ich kann Polynomfunktionen mithilfe unterschiedlicher Darstellungsformen beschreiben
7 +[[Kompetenzen.K1]] [[Kompetenzen.K6]] Ich kann die Wahl der Form im mathematischen Kontext begründen
8 +[[Kompetenzen.K1]] [[Kompetenzen.K3]] [[Kompetenzen.K6]] Ich kann die Wahl der Form im anwendungsorientierten Kontext begründen [[→ BPE 3.5>>BPE_3_5]]
7 7  
10 +Wiederholen (qF): Darstellungsformen von quadratischen Funktionen (SF, PF, HF)
11 +Wiederholen (qF): Eingehen auf verschiedene Eigenschaften (Vorteile, Nachteile) der DF
12 +Kennen: algebraische DF von PF, HF von Polynomfunktionen
13 +Input: Vorgegebene Schaubilder vergleichen (Gemeinsamkeiten, Unterschiede)
14 +"Beschreiben": Form 'fühlen' (Globalverhalten, Lokalverhalten); vgl. Buchstaben-Formen (N, W) Nulltellentypen (einfach vs mehrfach (gerade vs ungerade))
15 +
8 8  {{lernende}}
9 9  [[Nullstellen und Vielfachheiten interaktiv>>https://kmap.eu/app/browser/Mathematik/Ganzrationale%20Funktionen/Produktform#erkunden]]
10 10  {{/lernende}}
11 11  
12 -{{aufgabe id="Schaubilder zuordnen Teil 1" afb="II" kompetenzen="K3,K4" quelle="Niklas Wunder, Katharina Schneider" zeit="5"}}
13 -Ordne die Funktionsterme den 5 Schaubildern zu. Begründe deine Wahl.
20 +{{aufgabe id="Arithmagon Quadratische Formen" afb="I" kompetenzen="K4" quelle="Martina Wagner" cc="by-sa" zeit="5"}}
21 +[[image:Arithmagon Quadratische Formen.svg||width=500 style=float:left]]
22 +{{/aufgabe}}
23 +
24 +{{aufgabe id="Schaubilder zuordnen Teil 1" afb="II" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" cc="by-sa" zeit="5"}}
25 +[[image:Polynome_zuordnen-Grad_drei.svg||width=500 style=float:right]]Ordne die Funktionsterme den 5 Schaubildern zu. Begründe deine Wahl.
14 14  (% style="list-style: alphastyle" %)
15 15  1. {{formula}}f_1(x)=x^3{{/formula}}
16 16  1. {{formula}}f_2(x)=-x^2\cdot(x-3){{/formula}}
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17 17  1. {{formula}}f_3(x)=0{,}5\,x^3{{/formula}}
18 18  1. {{formula}}f_4(x)=0{,}5\,x^3+2\,x^2-3{{/formula}}
19 19  1. {{formula}}f_5(x)=-x^3-2\,x^2+2{{/formula}}
20 -
21 -[[Abbildung 1>>image:geogebra_polynome_dritten_Grades.png||width=640 height=402]]
22 -
23 23  {{/aufgabe}}
24 24  
25 -{{aufgabe id="Schaubilder zuordnen Teil 2" afb="II" kompetenzen="K3,K4" quelle="Niklas Wunder, Katharina Schneider" zeit="5"}}
26 -Ordne die Funktionsterme den 5 Schaubildern zu. Begründe deine Wahl.
34 +{{aufgabe id="Schaubilder zuordnen Teil 2" afb="II" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" cc="by-sa" zeit="5"}}
35 +[[image:Polynome_zuordnen-Grad_vier.svg||width=500 style="float:right"]]Ordne die Funktionsterme den 5 Schaubildern zu. Begründe deine Wahl.
27 27  (% style="list-style: alphastyle" %)
28 28  1. {{formula}}f_1(x)=-0{,}25\,x^4{{/formula}}
29 -1. {{formula}}f_2(x)=-0{,}5\,x^4-1{,}5\,x^3-1{,}5\,x^2-1{{/formula}}
38 +1. {{formula}}f_2(x)=-0{,}5\,x^4-1{,}5\,x^3-1{,}5\,x^2+1{{/formula}}
30 30  1. {{formula}}f_3(x)=-x^4{{/formula}}
31 31  1. {{formula}}f_4(x)=-x^4-x^3+2x^2+2{{/formula}}
32 32  1. {{formula}}f_5(x)=-0{,}3\cdot (x+2)^2\cdot(x-2)^2+4{{/formula}}
33 -
34 -[[Abbildung 1>>image:Polynome_zuordnen-Grad_vier.png||width=640 height=402]]
35 -
36 36  {{/aufgabe}}
37 37  
38 38  {{aufgabe id="Produktform" afb="I" kompetenzen="K4" quelle="Juliane Maier" cc="BY-SA" zeit="10"}}
39 39  Bestimme zu den abbgebildeten Funktionsgraphen eine mögliche Funktionsgleichung in Produktform.
40 -
41 -[[Abbildung 1>>image:Graphen Produktform.png||width=640 height=402]]
42 -
46 +[[image:Graphen Produktform.png||width=600]]
43 43  {{/aufgabe}}
44 44  
45 -{{aufgabe id="Skizzieren" afb="I" kompetenzen="K4" quelle="Juliane Maier" cc="BY-SA"}}
46 -Gegeben ist die Funktion {{formula}}f{{/formula}} mit {{formula}}D=\mathbb{R}{{/formula}}. Skizziere den Funktionsgraphen.
47 -(% style="list-style: alphastyle" %)
48 -1. {{formula}}f(x)=(x-2)^3{{/formula}}
49 -1. {{formula}}f(x)=x^4-x^2{{/formula}}
50 -{{/aufgabe}}
51 -
52 -{{aufgabe id="Immer, manchmal, nie" afb="III" kompetenzen="K1,K5" quelle="Niklas Wunder, Katharina Schneider" zeit="12"}}
49 +{{aufgabe id="Immer, manchmal, nie" afb="III" kompetenzen="K1,K5" quelle="Niklas Wunder, Katharina Schneider" cc="by-sa" zeit="12"}}
53 53  Beurteile, ob die folgenden Aussagen immer, nie oder manchmal unter bestimmten Bedingungen zutreffen. Begründe deine Entscheidung.
54 54  (% style="list-style: alphastyle" %)
55 55  1. Der Graph von {{formula}}f{{/formula}} mit {{formula}}f(x)=-3\cdot x^n {{/formula}} verläuft für ein gerades n von links unten nach rechts unten.
56 -1. Der Graph einer Polynomfunktion mit einem ungeraden Grad hat mindestens eine Nullstelle.
53 +1. Der Graph einer Polynomfunktion mit einem ungeraden Grad hat mindestens eine Nullstelle.
57 57  1. Der Graph einer zum Ursprung symmetrischen Funktion geht durch den Punkt (1|1).
58 58  1. Es gibt mindestens eine Funktion 5.Grades, die keine Nullstelle besitzt.
59 59  1. Der Graph einer achsensymmetrischen Funktion hat mindestens eine Nullstelle.
60 -1. Durch die beiden Punkte P(-2|1) und Q(2|2) verläuft kein Graph einer Funktion vierten Grades.
57 +1. Durch die beiden Punkte P(-2|1) und Q(2|2) verläuft kein Graph einer Funktion vierten Grades.
58 +{{/aufgabe}}
61 61  
62 - {{/aufgabe}}
60 +{{aufgabe id="Vieta" afb="II" kompetenzen="K2,K5" quelle="Martin Rathgeb, Martina Wagner" cc="BY-SA" zeit="10"}}
61 +Ermittle die fehlenden Zahlen bzw. Terme.
62 +(% class="abc" %)
63 +1. {{formula}}x^2+\square x + \square=(x-5)(x+7){{/formula}}
64 +1. {{formula}}x^2+\square x - 12=(x-4)(x-\square){{/formula}}
65 +1. {{formula}}x^2-12 x + \square=(x-4)(x-\square){{/formula}}
66 +1. {{formula}}x^2+\square x + \square=(x-a)(x-b){{/formula}}
67 +{{/aufgabe}}
63 63  
64 -
65 -{{aufgabe id="Darstellungsformen umwandeln" afb="II" kompetenzen="K3,K4" quelle="Niklas Wunder, Katharina Schneider" zeit="15"}}
69 +{{aufgabe id="Darstellungsformen umwandeln" afb="II" kompetenzen="K5" quelle="Niklas Wunder, Katharina Schneider" cc="by-sa" zeit="15"}}
66 66  Wandle in die entsprechend andere Darstellungsform um (Hauptform bzw. Produktform).
67 67  (% style="list-style: alphastyle" %)
68 68  1. {{formula}}f(x)=-\frac{1}{16}\cdot (x-2)^2\cdot (x-8){{/formula}}
69 69  1. {{formula}}f(x)=(x-3)\cdot (x^2+3x+9){{/formula}}
70 -1. {{formula}}f(x)=3\,x^3-33\,x^2+96\,x-84{{/formula}}
71 - Hinweis: Die Funktion f besitzt nur die beiden Nullstellen {{formula}} x_1 =1 {{/formula}} und {{formula}} x_2 =7 {{/formula}}.
74 +1. {{formula}}f(x)=3\,x^3-33\,x^2+96\,x-84{{/formula}}
75 +Hinweis: Die Funktion //f// besitzt nur die beiden Nullstellen {{formula}} x_1 =1 {{/formula}} und {{formula}} x_2 =7 {{/formula}}.
72 72  1. {{formula}}f(x)=-2\,x^4+18\,x^2+8\,x-24{{/formula}}
73 -Hinweis: Die Funktion f besitzt nur die Nullstellen {{formula}} x_1 =-2, x_2=1 {{/formula}} und {{formula}} x_3 =3 {{/formula}}.
77 +Hinweis: Die Funktion //f// besitzt nur die Nullstellen {{formula}} x_1 =-2, x_2=1 {{/formula}} und {{formula}} x_3 =3 {{/formula}}.
74 74  {{/aufgabe}}
75 75  
76 -{{aufgabe id="Parabelmaschine" afb="II" kompetenzen="K2, K5" tags="problemlösen" quelle="Simon Oswald" cc="BY-SA" zeit="20"}}
80 +{{aufgabe id="Parabelmaschine" afb="II" kompetenzen="K2, K5" tags="problemlösen" quelle="Simon Oswald" cc="BY-SA" cc="by-sa" zeit="20"}}
77 77  [[image:Parabelmaschine.PNG||width="240" style="float: right"]]
78 78  Denke dir zwei Zahlen, eine positiv, eine negativ.
79 79  Wenn du diese Zahlen quadrierst, erhältst du zwei Punkte auf der Normalparabel.
80 80  
81 -Ermitteln Sie, wo die Verbindungslinie dieser zwei Punkte die y-Achse schneidet!
85 +Ermittle, wo die Verbindungslinie dieser zwei Punkte die y-Achse schneidet!
82 82  
83 83  {{lehrende}}
84 84  **Variante :** Offene Aufgabe für den Unterricht & für die Klassenarbeit
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98 98  {{/lehrende}}
99 99  {{/aufgabe}}
100 100  
101 -{{lehrende}}
102 -[[Polynomfunktionsgraphen begreifen]]
103 -{{/lehrende}}
104 -
105 -{{aufgabe id="Parameter bestimmen" afb="III" kompetenzen="K4" quelle="Katharina Schneider,Niklas Wunder" cc="BY-SA"}}
105 +{{aufgabe id="Parameter bestimmen" afb="III" kompetenzen="K4,K5" quelle="Katharina Schneider,Niklas Wunder" cc="BY-SA"}}
106 106  Gegeben sind die Funktionsterme der Funktionen {{formula}}f,g,h,k{{/formula}} sowie Punkte, durch die das Schaubild der jeweiligen Funktion verläuft. Bestimme die fehlenden Parameter für jede Funktion.
107 107  (% style="list-style: alphastyle" %)
108 108  1. {{formula}}f(x)=a\cdot (x-3)\cdot (x-5)^2{{/formula}} mit {{formula}} P(5|20) {{/formula}}
... ... @@ -111,4 +111,9 @@
111 111  1. {{formula}} k(x)= a\cdot(x-b)^3-7 {{/formula}} mit {{formula}} P(2|-7) {{/formula}} und {{formula}} Q(0|-5) {{/formula}}
112 112  {{/aufgabe}}
113 113  
114 -{{seitenreflexion bildungsplan="1" kompetenzen="2" anforderungsbereiche="3" kriterien="1" menge="0"/}}
114 +{{lehrende}}
115 +[[Polynomfunktionsgraphen begreifen]]
116 +K3 soll hier nicht bedient werden .. das kommt in BPE 3.5
117 +{{/lehrende}}
118 +
119 +{{seitenreflexion bildungsplan="4" kompetenzen="4" anforderungsbereiche="5" kriterien="4" menge="5"/}}
Polynome_zuordnen-Grad_drei.ggb
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