Achsensymmetrie zur y-Achse

In diesem Artikel erkläre ich dir die Achsensymmetrie zur y-Achse. Als erstes veranschauliche ich sie graphisch, dann zeige ich dir einen kleinen Trick mit dem du sofort erkennen kannst, ob eine ganzrationale Funktion achsensymmetrisch ist und als letztes zeige ich dir, wie man mathematisch die Achsensymmetrie nachweist.

Graphische Erklärung

achsensymmetrischer Graph

Dieser Graph ist achsensymmetrisch zur y-Achse. Warum?

achsensymmetrischer Graph

Wie du siehst, dient die y-Achse als eine Spiegelachse. Wenn also rechts und links dieser Achse der Funktionsgraph exakt gleich aussieht, dann gehört dieser Graph zu einer achsensymmetrischen Funktion. Hierbei sagt man, dass die Funktion spiegelsymmetrisch zur y-Achse ist.

Achsensymmetrie zur y-Achse bei ganzrationalen Funktion

Bei einer ganzrationalen Funktion genügt ein einziger Blick auf die Exponenten (Hochzahlen) um zu entscheiden, ob die vorliegende Funktion achsensymmetrisch ist, denn es gilt:

Die Funktion

f(x) = x4 + 5x2 -1

ist achsensymmstrisch, denn die Exponenten (4 und 2) sind ausschließlich gerade Zahlen!

Die Funktion

f(x) = x4 + 2x3 + 5x2 -1

ist NICHT achsensymmetrisch, da sie einen ungeraden Exponenten (3) hat!

Mathematischer Nachweis für Symmetrie zur y-Achse:

f(x) = f(-x)

Natürlich gibt es einen mathematischen Nachweis für achsensymmetrische Funktionen.

Hierzu bildet man f(-x) und vergleicht dieses Ergebnis mit der Ausgangsfunktion. Stimmen die gebildete Funktion und die Ausgangsfunktion überein, dann ist die vorliegende Funktion symmetrisch zur y-Achse, andernfalls nicht.

Doch wie wird f(-x) gebildet?

Hierzu ersetzt man jedes vorkommdende x in der Ausgangsfunktion durch -x. Dieser Ausdruck wird dann vereinfacht. Beachte bitte, dass beim Vereinfachen die Potenzen Vorrang haben. Wenn man also 2(-x)4 vereinfachen soll, dann liegt das erste Augenmerk auf (-x)4. (-x)4 steht für (-x)•(-x)•(-x)•(-x) und dies ergibt x4. Das Ganze mit zwei multipliziert ist 2x4

Beispiele:

f(x) = 3x4 + 5x2 + 2

f(-x) = 3(-x)4 + 5(-x)2 + 2 = 3x4 + 5x2 + 2

Da hier f(x) = f(-x) gilt, ist die vorliegende Funktion symmetrisch zur y-Achse!

f(x) = 4x4 – 6x3 + 7x2 – 3x -1

f(-x) = 4(-x)4 – 6(-x)3 + 7(-x)2 – 3(-x) -1

= 4x4 + 6x3 + 7x2 + 3x -1

Bei diesem Beispiel stimmen f(x) und f(-x) nicht überein. Diese Funktion ist also nicht symmetrisch zur y-Achse!

Video auf Youtube anschauen: >>>Hier klicken<<<

 

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