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| − | Bei ganzrationalen Funktionen ist der '''Grad der Funktion''', sowie das '''Vorzeichen des Leitkoeffizienten''' ausschlaggebend, zu welchem der vier charakteristischen Verläufe die Funktion gehört:<br /> | + | __NOTOC__ |
| | + | <div style="padding:1px;background:#66CD00;border:0px groove;"> |
| | + | |
| | + | |
| | + | <center><table border="0" width="850px" cellpadding=5 cellspacing=15> |
| | + | <tr><td width="800px" valign="top"> |
| | + | <big>Der charakteristische Verlauf von ganzrationalen Funktionen ist von zwei Faktoren abhängig. Worauf es genau ankommt, kannst du im folgenden Applet mit den Schiebereglern untersuchen.<br /> |
| | + | Welche '''vier Fälle''' unterscheidet man?</big> |
| | + | <br /> |
| | + | <center> |
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| | + | </center> |
| | + | <br /> |
| | + | <popup name="Lösung"> |
| | + | Bei ganzrationalen Funktionen ist der '''Grad der Funktion''', sowie das '''Vorzeichen des Leitkoeffizienten''' dafür ausschlaggebend, zu welchem der vier charakteristischen Verläufe die Funktion gehört:<br /> |
| | + | <br /> |
| | | | |
| | {| | | {| |
| − | | valign="top"|<big><center>'''<span style="color: #00BFFF ">"von links oben nach rechts oben"</span>'''</center><br /> | + | | valign="top"|<big><center>'''<span style="color:#00BFFF">"von links oben nach rechts oben"</span>'''</center></big><br /> |
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| + | [[Datei:Oben-oben.png|300px]] |
| | |width="1%"| | | |width="1%"| |
| − | | valign="top"|<big><center>'''<span style="color: #9ACD32 ">"von links unten nach rechts oben"</span>'''</center><br /> | + | | valign="top"|<big><center>'''<span style="color:#9ACD32">"von links unten nach rechts oben"</span>'''</center></big><br /> |
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| + | [[Datei:Unten-oben.png|300px]] |
| | |- | | |- |
| | | <br> | | | <br> |
| | |- | | |- |
| − | | valign="top"|<big><center>'''<span style="color: #0000CD ">"von links unten nach rechts unten"</span>'''</center><br /> | + | | valign="top"|<big><center>'''<span style="color:#0000CD">"von links unten nach rechts unten"</span>'''</center></big><br /> |
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| + | [[Datei:Unten-unten.png|300px]] |
| | |width="1%"| | | |width="1%"| |
| − | | valign="top"|<big><center>'''<span style="color: #008B00 ">"von links oben nach rechts unten"</span>'''</center><br /> | + | | valign="top"|<big><center>'''<span style="color:#008B00">"von links oben nach rechts unten"</span>'''</center></big><br /> |
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| | + | |} |
| | + | |
| | + | </popup> |
| | + | </td></tr></table></center> |
| | + | </div> |
| | + | |
| | + | <div style="padding:1px;background:#66CD00;border:0px groove;"> |
| | + | |
| | + | |
| | + | <center><table border="0" width="850px" cellpadding=5 cellspacing=15> |
| | + | <tr><td width="800px" valign="top"> |
| | + | === <big>Übung === |
| | + | Teste dein Wissen mit dieser Aufgabe aus dem Känguru-Wettbewerb 2013:</big><br /> |
| | + | <br /> |
| | + | <iframe src="http://LearningApps.org/watch?v=pfiz98zzc" style="border:0px;width:100%;height:1160px" webkitallowfullscreen="true" mozallowfullscreen="true"></iframe> |
| | + | <br /> |
| | + | |
| | + | <popup name="Lösung"> |
| | + | W(x) = (a - x) (b - x)<sup>2</sup><br /> |
| | + | <br /> |
| | + | Die Funktion hat eine einfache Nullstelle bei a (der Graph schneidet die x- Achse) und eine doppelte Nullstelle bei b (der Graph berührt die x- Achse).<br /> |
| | + | -> mögliche Lösungen: A, B, C, D<br /> |
| | + | <br /> |
| | + | Da a < b ist, muss die einfache Nullstelle vor der doppelten liegen.<br /> |
| | + | -> mögliche Lösungen: A, D<br /> |
| | + | <br /> |
| | + | Ausmultiplizieren des Funktionsterms ergibt W(x) = - x<sup>3</sup> +...<br /> |
| | + | W(x) hat den Leitkoeffizienten: -1.<br /> |
| | + | Damit verläuft die Funktion von links oben nach rechts unten.<br /> |
| | + | -> Lösung: A |
| | + | |
| | + | </popup> |
| | + | |
| | + | </td></tr></table></center> |
| | + | </div> |
| | + | <br /> |
| | + | <br /> |
| | + | |
| | + | {| |
| | + | {{Vorlage:Lesepfad Ende |
| | + | |Link zurück=[[Manipulationen an Funktionen/Grenzwerte im Unendlichen|Zurück zu den Grenzwerten im Unendlichen]] |
| | + | |Link vor= |
| | + | |Text Copyright=<colorize>Manipulationen an Funktionen</colorize> |
| | + | }} |
| | |} | | |} |
