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im neuen Projektwiki (projekte.zum.de).Achsensymmetrie zur y- Achse: Unterschied zwischen den Versionen
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− | | | + | |<popup name="Hilfe zu GeoGebra"> |
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− | <popup name="Hilfe zu GeoGebra"> | + | |
*Einen Punkt kannst du, wie im letzten Kapitel, über das Symbol "Neuer Punkt" in der Werkzeugleiste direkt an die jeweilige Stelle im Applet setzen. | *Einen Punkt kannst du, wie im letzten Kapitel, über das Symbol "Neuer Punkt" in der Werkzeugleiste direkt an die jeweilige Stelle im Applet setzen. | ||
*Es gibt aber auch die Möglichkeit Objekte direkt zu spiegeln: | *Es gibt aber auch die Möglichkeit Objekte direkt zu spiegeln: | ||
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|width="0,5%"| | |width="0,5%"| | ||
− | | valign="top"| < | + | | valign="top"|<big>Spiegle die Punkte '''<span style="color: #008B00 ">A</span>''', '''<span style="color: #008B00 ">B</span>''', '''<span style="color: #008B00 ">C</span>''', '''<span style="color: #008B00 ">D</span>''' und '''<span style="color: #008B00 ">E</span>''' an der '''<span style="color: #551A8B ">y- Achse</span>'''.<br /> |
<br /> | <br /> | ||
+ | Vergleiche die Koordinaten der gespiegelten Punkte mit denen der ursprünglichen Punkte.<br /> | ||
+ | Was fällt dir auf?<br /> | ||
<br /> | <br /> | ||
− | + | Verbinde die Punkte zu einem Funktionsgraphen.<br /> | |
− | + | Welche Funktion wird hier abgebildet?</big><br /> | |
− | + | <br /> | |
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− | </ | + | |
|} | |} | ||
+ | <br /> | ||
+ | <iframe src="http://LearningApps.org/watch?v=pbvmnw5o3" style="border:0px;width:100%;height:585px" webkitallowfullscreen="true" mozallowfullscreen="true"></iframe> | ||
+ | <br /> | ||
</td></tr></table></center> | </td></tr></table></center> | ||
</div> | </div> | ||
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<tr><td width="800px" valign="top"> | <tr><td width="800px" valign="top"> | ||
=== <big>Allgemein</big> === | === <big>Allgemein</big> === | ||
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<div class="lueckentext-quiz"> | <div class="lueckentext-quiz"> | ||
Ist der Graph einer Funktion f '''achsensymmetrisch zur y- Achse''',<br /> | Ist der Graph einer Funktion f '''achsensymmetrisch zur y- Achse''',<br /> | ||
so besitzen '''gleich weit vom Ursprung entfernte''' x- Werte immer den '''gleichen Funktionswert'''.<br /> | so besitzen '''gleich weit vom Ursprung entfernte''' x- Werte immer den '''gleichen Funktionswert'''.<br /> | ||
− | Es gilt also: f (x) = f (-x)<br /> | + | Es gilt also: f (x) = f (-x).<br /> |
<br /> | <br /> | ||
Man kann aber auch vom '''Funktionsterm''' auf den '''Verlauf des Graphen''' schließen:<br /> | Man kann aber auch vom '''Funktionsterm''' auf den '''Verlauf des Graphen''' schließen:<br /> | ||
− | Gilt für eine Funktion f mit der '''Definitionsmenge D<sub>f</sub>''' für alle x ∈ D<sub>f</sub><br /> | + | Gilt für eine Funktion f mit der '''Definitionsmenge D<sub>f</sub>''' für alle x ∈ D<sub>f</sub>:<br /> |
f (x) = f (-x),<br /> | f (x) = f (-x),<br /> | ||
dann verläuft der Graph von f '''achsensymmetrisch zur y- Achse'''. | dann verläuft der Graph von f '''achsensymmetrisch zur y- Achse'''. | ||
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Stelle die Parameter '''<span style="color: red">a</span>''', '''<span style="color: #00BFFF ">b</span>''', '''<span style="color: #76EE00 ">c</span>''', '''<span style="color: orange">d</span>''', '''<span style="color: #EE00EE">e</span>''' so ein, dass '''f''' '''<span style="color: #551A8B">achsensymmetrisch zur y- Achse</span>''' ist.<br /></big> | Stelle die Parameter '''<span style="color: red">a</span>''', '''<span style="color: #00BFFF ">b</span>''', '''<span style="color: #76EE00 ">c</span>''', '''<span style="color: orange">d</span>''', '''<span style="color: #EE00EE">e</span>''' so ein, dass '''f''' '''<span style="color: #551A8B">achsensymmetrisch zur y- Achse</span>''' ist.<br /></big> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
Zeile 93: | Zeile 86: | ||
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|} | |} | ||
Zeile 103: | Zeile 96: | ||
<br /> | <br /> | ||
''Antwort:''<br /> | ''Antwort:''<br /> | ||
− | Für Achsensymmetrie zur y- Achse muss immer '''h(x) = h(-x)''' für alle möglichen | + | Für Achsensymmetrie zur y- Achse muss immer '''h(x) = h(-x)''' für alle möglichen x- Werte einer Funktion h gegeben sein.<br /> |
Gibt es <span style="color: red">nur gerade Exponenten</span>, wird jedes negative Vorzeichen vor einem x- Wert aufgehoben: <br /> | Gibt es <span style="color: red">nur gerade Exponenten</span>, wird jedes negative Vorzeichen vor einem x- Wert aufgehoben: <br /> | ||
Z. B.: '''<span style="color: #00CD00 ">h: x -> x<sup>12</sup> - 4x<sup>8</sup> - 1</span>'''<br /> | Z. B.: '''<span style="color: #00CD00 ">h: x -> x<sup>12</sup> - 4x<sup>8</sup> - 1</span>'''<br /> | ||
Zeile 119: | Zeile 112: | ||
= x<sup>12</sup> '''<span style="color: #912CEE ">+</span>''' 4x<sup>9</sup> - 1<br /> | = x<sup>12</sup> '''<span style="color: #912CEE ">+</span>''' 4x<sup>9</sup> - 1<br /> | ||
'''<span style="color: #912CEE">≠ k(x)</span>''' | '''<span style="color: #912CEE">≠ k(x)</span>''' | ||
+ | </td></tr></table></center> | ||
+ | </div> | ||
+ | |||
+ | <div style="padding:1px;background:#B452CD;border:0px groove;"> | ||
+ | |||
+ | |||
+ | <center><table border="0" width="800px" cellpadding=5 cellspacing=15> | ||
+ | <tr><td width="800px" valign="top"> | ||
+ | === <big>Übung</big> === | ||
<br /> | <br /> | ||
<br /> | <br /> |
Version vom 30. Juni 2013, 14:19 Uhr
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AllgemeinIst der Graph einer Funktion f achsensymmetrisch zur y- Achse,
Kennst du weitere Beispiele für achsensymmetrische Funktionen; Wie muss der Graph einer solchen Funktion aussehen?
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Übung
Manipulationen an Funktionen |