{"id":"2fe89ab7-f5b3-4821-9685-1977a4270759","type":"https://serlo.org/editor","variant":"serlo-org","domainOrigin":"serlo.org","version":2,"editorVersion":"0.22.0","dateModified":"2025-02-26T18:25:13.209Z","document":{"plugin":"course","state":{"content":{"plugin":"rows","state":[]},"pages":[{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Ziel dieses Kurses ist es, einen Überblick über das Thema \"Potenzfunktionen mit natürlichen Exponenten\" zu geben."}]},{"type":"p","children":[{"text":" "}]},{"type":"p","children":[{"text":"Inhalte"}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"type":"p","children":[{"text":"Hinführung zum Begriff der Potenzfunktion"}]}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"type":"p","children":[{"text":"Eigenschaften einer Potenzfunktion (Steigungsverhalten, Wertemenge, Symmetrieverhalten)"}]}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"type":"p","children":[{"text":"Graph einer Potenzfunktion"}]}]}]}]}],"id":"36fd74d9-ba55-4c73-8630-7737cbf313ce"}],"id":"7af53c17-ebf3-4d11-9860-dba688515de5"},"id":"51912-course-page","title":"Übersicht"},{"content":{"plugin":"rows","state":[{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Quadratische Funktionen und Parabeln sind dir schon aus früheren Jahrgangsstufen bekannt. 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Oder ganz anders ausschauen? Was passiert für negative "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Werte?"}]},{"type":"p","children":[{"text":"Denke nach, und gehe dann zur nächsten Seite dieses Kurses …"}]}],"id":"c7238fda-eba4-479a-8ea4-7d5eaa8e5653"}],"id":"00abf100-97fe-426b-a27c-f793723e90ba"},"id":"51535-course-page","title":"Potenzfunktion (2|4)"},{"content":{"plugin":"rows","state":[{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Rechts ist zusätzlich zu den Funktionen "},{"type":"math","src":"p(x)=x^2","inline":true,"children":[{"text":"p(x)=x^2"}]},{"text":" und "},{"type":"math","src":"q(x)=x^4","inline":true,"children":[{"text":"q(x)=x^4"}]},{"text":" nun auch die Funktion "},{"type":"math","src":"f(x)=x^3","inline":true,"children":[{"text":"f(x)=x^3"}]},{"text":" in Rot eingezeichnet."}]},{"type":"p","children":[{"text":"Und war deine Vermutung richtig?"}]},{"type":"p","children":[{"text":"Wie du siehst, unterscheidet sich der Graph von "},{"type":"math","src":"x^3","inline":true,"children":[{"text":"x^3"}]},{"text":" vor allem dadurch, dass er in negative Richtung nach "},{"type":"math","src":"-\\infty","inline":true,"children":[{"text":"-\\infty"}]},{"text":" läuft."}]},{"type":"p","children":[{"text":"Für positive "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Werte verläuft der Graph tatsächlich zwischen "},{"type":"math","src":"x^2","inline":true,"children":[{"text":"x^2"}]},{"text":" und "},{"type":"math","src":"x^4","inline":true,"children":[{"text":"x^4"}]},{"text":"."}]},{"type":"h","level":4,"children":[{"text":"Warum läuft "},{"type":"math","src":"x^3","inline":true,"children":[{"text":"x^3"}]},{"text":" ins Negative?"}]},{"type":"p","children":[{"text":"Bei "},{"type":"math","src":"x^3","inline":true,"children":[{"text":"x^3"}]},{"text":" wird der "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Wert dreimal mit sich selbst multipliziert. Bei negativen "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Werten erhält man somit ein negatives Ergebnis."}]},{"type":"p","children":[{"text":"Für "},{"type":"math","src":"-1","inline":true,"children":[{"text":"-1"}]},{"text":" und "},{"type":"math","src":"-2","inline":true,"children":[{"text":"-2"}]},{"text":" gilt zum Beispiel:"}]},{"type":"p","children":[{"text":""},{"type":"math","src":"f(-1)=(-1)^3=(-1)\\cdot(-1) \\cdot (-1) =-1","inline":true,"children":[{"text":"f(-1)=(-1)^3=(-1)\\cdot(-1) \\cdot (-1) =-1"}]},{"text":" "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"f(-2)= (-2)^3=(-2)\\cdot(-2) \\cdot (-2) =-8","inline":true,"children":[{"text":"f(-2)= (-2)^3=(-2)\\cdot(-2) \\cdot (-2) =-8"}]},{"text":" "}]},{"type":"p","children":[{"text":"Ähnlich sieht es bei "},{"type":"math","src":"x^5, x^7, x^9,…","inline":true,"children":[{"text":"x^5, x^7, x^9,…"}]},{"text":" also allgemein bei Potenzen mit ungeradem Exponenten aus."}]},{"type":"p","children":[{"text":"Damit ist der Graph nicht mehr achsensymmetrisch zur "},{"type":"math","src":"y","inline":true,"children":[{"text":"y"}]},{"text":"-Achse, sondern punktsymmetrisch zum Ursprung."}]}],"id":"6f980261-ba4b-46d2-b6aa-3c1055eb8ff0"}],"id":"a37cb89f-84b6-4df6-a3c4-ef0b51050d81"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/legacy/56cb03f14c69d_7f46279f58b92892ba83cb2855e581b5866119c9.png","alt":"x^3","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"f9e1e438-49ae-4c96-a020-4f5f7319bf87"}},"id":"05d54c6d-1788-4f27-a907-f1b91187a576"},"illustrating":true,"width":50},"id":"eb1565b8-44b2-4cca-88db-e48043c8e1b2"}],"id":"351474c4-7eb5-43b6-bda8-1f3f43ae21c6"},"id":"51537-course-page","title":"Potenzfunktion (3|4)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"An dieser Stelle ist es sinnvoll, einen Begriff für Funktionen wie "},{"type":"math","src":"p(x)=x^2, f(x)=x^3, q(x)=x^4, …","inline":true,"children":[{"text":"p(x)=x^2, f(x)=x^3, q(x)=x^4, …"}]},{"text":" einzuführen:"}]}],"id":"1e365d08-78f4-4ebe-be33-80d98976ca7a"},{"plugin":"box","state":{"title":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"251f9e9c-ad54-4649-8ac1-0ccfd2e26eb1"},"content":{"plugin":"rows","state":[{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Eine "},{"text":"Potenzfunktion","strong":true},{"text":" ist eine Funktion der Form "},{"type":"math","src":"f(x)=a\\cdot x^n","inline":true,"children":[{"text":"f(x)=a\\cdot x^n"}]},{"text":" (mit "},{"type":"math","src":"n\\in \\mathbb{N}_0, a\\in \\mathbb{R}","inline":true,"children":[{"text":"n\\in \\mathbb{N}_0, a\\in \\mathbb{R}"}]},{"text":")"}]}],"id":"b2bf84d4-af58-49ad-b8ce-b36c451e3518"}],"id":"464de741-8438-4004-aab9-f50d3bb6f097"}],"id":"2e9de74c-d600-4196-90d2-90eb044cbd31"},"type":"blank","anchorId":"box85286"},"id":"5984a929-387f-4840-94e0-a3e12097edbc"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Form der Potenzfunktion hängt vom "},{"text":"Exponenten","strong":true},{"text":" "},{"type":"math","src":"n","inline":true,"children":[{"text":"n"}]},{"text":" und "},{"text":"Koeffizienten","strong":true},{"text":" "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" ab."}]},{"type":"p","children":[{"text":"Eine "},{"text":"gerade Potenzfunktion","strong":true},{"text":" ist eine Potenzfunktion mit geradem Exponenten "},{"type":"math","src":"(","inline":true,"children":[{"text":"("}]},{"text":"wie "},{"type":"math","src":"1, 3x^2, -2x^4)","inline":true,"children":[{"text":"1, 3x^2, -2x^4)"}]},{"text":"."}]},{"type":"p","children":[{"text":"Eine "},{"text":"ungerade Potenzfunktion","strong":true},{"text":" ist eine Potenzfunktion mit ungeradem Exponenten "},{"type":"math","src":"(","inline":true,"children":[{"text":"("}]},{"text":"wie "},{"type":"math","src":"x, -x^3, 4x^5)","inline":true,"children":[{"text":"x, -x^3, 4x^5)"}]},{"text":"."}]},{"type":"p","children":[{"text":"Den Exponenten in einer Potenzfunktion kann man auch "},{"text":"Grad","strong":true},{"text":" nennen."}]},{"type":"p","children":[{"text":"Dabei kann eine Potenzfunktion auch den Grad "},{"type":"math","src":"0","inline":true,"children":[{"text":"0"}]},{"text":" haben. In diesem Fall handelt es sich um eine konstante Funktion. So ist zum Beispiel "},{"type":"math","src":"f(x)=3=3\\cdot x^0","inline":true,"children":[{"text":"f(x)=3=3\\cdot x^0"}]},{"text":" auch eine Potenzfunktion."}]}],"id":"fc4b4e5f-849b-44f0-a091-cb31d1860078"},{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Rechts findest du die Graphen von "},{"type":"math","src":"1, x, x^2, x^3, …, x^7","inline":true,"children":[{"text":"1, x, x^2, x^3, …, x^7"}]},{"text":"."}]},{"type":"p","children":[{"text":"Beim Untersuchen der Graphen fällt dir bestimmt auf, dass sie sich alle in dem Punkt "},{"type":"math","src":"(1|1)","inline":true,"children":[{"text":"(1|1)"}]},{"text":" schneiden (hier rot eingezeichnet)."}]}],"id":"cbbd0b38-c536-48e5-befb-c3bbe101d876"},{"plugin":"spoiler","state":{"title":"Woran liegt das?","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Setzt man "},{"type":"math","src":"1","inline":true,"children":[{"text":"1"}]},{"text":" zum Beispiel in die Funktion "},{"type":"math","src":"f(x)=x^3","inline":true,"children":[{"text":"f(x)=x^3"}]},{"text":" ein, so erhält man "},{"type":"math","src":"f(1)=(1)^3=1","inline":true,"children":[{"text":"f(1)=(1)^3=1"}]},{"text":". Wie du dir vielleicht denken kannst, erhält man beim Einsetzen des "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Werts "},{"type":"math","src":"1","inline":true,"children":[{"text":"1"}]},{"text":" unabhängig vom Exponenten immer den gleichen Funktionswert. "},{"type":"math","src":"((1)^2=1, (1)^5=1,…)","inline":true,"children":[{"text":"((1)^2=1, (1)^5=1,…)"}]},{"text":""}]}],"id":"3db37b7a-7578-449f-b255-903eb37590a8"}],"id":"1dc84c93-49c4-423d-9e1b-771504fad424"}},"id":"13f79b40-cc0b-4ddc-8b64-f7eac554cd47"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Werden diese Potenzfunktionen mit einem Koeffizienten multipliziert, z. B. "},{"type":"math","src":"2, 2x, 2x^2,…","inline":true,"children":[{"text":"2, 2x, 2x^2,…"}]},{"text":", so verändert sich dieser \"Fixpunkt\" ebenso. Er hätte nun z.B. die Koordinaten "},{"type":"math","src":"(1|2)","inline":true,"children":[{"text":"(1|2)"}]},{"text":"."}]}],"id":"892226f7-2f2b-450a-98c1-d249a365c9b4"}],"id":"9fc050a8-6c64-4a5b-ab48-775119786c88"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/legacy/56cd94ba9f3f2_0ef22c59647a8e5110ee2f4ec5187e2ce75b6c3c.png","alt":"Potenzfunktionen","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"f55dc3d9-226c-4f0a-ba2f-17c49a6e60d2"}},"id":"5e2f450c-a992-43a7-922c-13666f69958e"},"illustrating":true,"width":50},"id":"66577112-510b-46fa-86cc-7468747dafe6"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Auf der nächsten Kursseite kannst du mithilfe von Applets herausfinden, welchen Einfluss der Koeffizient und der Exponent auf den Verlauf des Graphen haben."}]}],"id":"06f0dc12-5229-46b5-a250-5889629d3a46"}],"id":"b8de54f0-2ca9-4dde-b4c7-5a92d76dcf55"},"id":"51868-course-page","title":"Potenzfunktion (4|4)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"h","level":2,"children":[{"text":"Graph einer Potenzfunktion"}]},{"type":"p","children":[{"text":"In den Applets auf den nächsten zwei Seiten siehst du, wie die Veränderung des Exponenten und des Koeffizienten sich auf den Graphen der Potenzfunktion auswirken."}]},{"type":"p","children":[{"text":"Benutze dabei die Schieberegler, um den Koeffizienten "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" und den Exponenten "},{"type":"math","src":"n","inline":true,"children":[{"text":"n"}]},{"text":" zu variieren."}]},{"type":"h","level":4,"children":[{"text":"Potenzfunktion mit geradem Exponenten"}]}],"id":"f223d0a5-848b-4b04-85e4-91c791a80a3e"},{"plugin":"geogebra","state":"2639747","id":"558e012f-871e-4cb7-b125-0ea2753581fd"},{"plugin":"spoiler","state":{"title":"Wie verändert der Exponent n die Form des Graphen?","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Je größer der Exponent, desto flacher wird der Graph in der Umgebung des Koordinatenursprungs. 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