{"id":"2f486270-d1e2-4c35-b82f-ec5dfb6db79b","type":"https://serlo.org/editor","variant":"serlo-org","domainOrigin":"serlo.org","version":2,"editorVersion":"0.22.0","dateModified":"2025-02-26T18:25:58.996Z","document":{"plugin":"course","state":{"content":{"plugin":"rows","state":[]},"pages":[{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Inhalt des Kurses"}]},{"type":"p","children":[{"text":"Du lernst in diesem Kurs …"}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"was eine gestreckte, gestauchte und gespiegelte Parabel ist."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"was der Öffnungsfaktor einer Parabel ist."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"wie man den Öffnungsfaktor einer vorgegebenen Parabel bestimmen kann."}]}]}]}],"id":"d3513c22-f1a2-492e-b9ba-11947286e2a7"},{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Vorkenntnisse"}]},{"type":"p","children":[{"text":"Bevor du diesen Kurs beginnst, solltest du wissen, …"}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"was eine "},{"type":"a","href":"/1855","children":[{"text":"Normalparabel"}]},{"text":" ist."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"was ein "},{"type":"a","href":"/1843","children":[{"text":"Funktionsgraph"}]},{"text":" ist und wie man diesen in ein "},{"type":"a","href":"/1537","children":[{"text":"Koordinatensystem"}]},{"text":" einzeichnet."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"wie man "},{"type":"a","href":"/2171","children":[{"text":"lineare Gleichungen"}]},{"text":" löst."}]}]}]}],"id":"9c81a38c-d9b3-4e9b-ae28-d853c2109bde"},{"plugin":"text","state":[{"type":"h","level":3,"children":[{"text":"Kursdauer"}]},{"type":"p","children":[{"text":"Der Kurs dauert ca. 45 Minuten."}]}],"id":"ac10548d-7b84-427e-a57a-d209b6bfb524"}],"id":"2eafa3e1-098a-4d6e-951b-fbd6eb714aa5"},"id":"101128-course-page","title":"Übersicht"},{"content":{"plugin":"rows","state":[{"plugin":"image","state":{"src":"https://assets.serlo.org/5aae3fc3dd746_370e1e4cceb60fbdb2271eb1fd5efdd20f549b50.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"2e99541d-3a0d-4b78-846d-e32d44a5e163"}},"id":"0dc5bbd8-4f23-46e1-b9a6-e774b6b9b606"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Rakotzbrücke im Kromlauer Park hat die Form einer quadratischen Funktion, jedoch nicht die einer Normalparabel. Wie könnte der Funktionsterm lauten?"}]}],"id":"fe1356b0-3cbb-4756-a6ac-c25a55fadc1b"},{"plugin":"image","state":{"src":"https://assets.serlo.org/5aae42db145e0_af40177661e041f893e856ffa65fca21264bfe91.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"1507cc17-b85f-457a-8ec4-26a9ca32ba95"}},"id":"9d5fe866-c529-4520-9d02-8e7c7cfaa967"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Marco wirft in der letzten Minute auf den freien Korb. Zum Glück versteht er etwas von Mathematik und wirft die perfekte Parabel!"}]},{"type":"p","children":[{"text":"Welche Form wird diese Parabel beschreiben? Wie könnte ein Funktionsterm lauten?"}]}],"id":"01f2b57f-2249-4417-a35a-552255e29bb6"}],"id":"c3537e35-8bf1-4b99-aa08-ac052bf3738f"},"id":"101132-course-page","title":"Motivation"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Mit einer Normalparabel der Form "},{"type":"math","src":"y =x^2","inline":true,"children":[{"text":"y =x^2"}]},{"text":" kannst du manche Sachverhalte wie beispielsweise die Brücke oder die Flugbahn des Basketballes nicht mit einem Funktionsgraph abbilden. "}]},{"type":"p","children":[{"text":"Du kannst allerdings Parabeln strecken oder stauchen, also \"enger\" oder \"breiter\" als die Normalparabel machen."}]},{"type":"p","children":[{"text":"Zudem kannst du Parabeln an der "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Achse spiegeln, sodass sie nach unten geöffnet ist."}]}],"id":"9e72acf6-5c7d-40a2-b19e-046d76c7e91b"},{"plugin":"serloTable","state":{"rows":[{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Gestreckte Parabel"}]}],"id":"9969c899-ec14-4037-b1ad-9de14428828f"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Normalparabel"}]}],"id":"3d17bf88-94f8-4912-8b81-53f163353bbe"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Gestauchte Parabel"}]}],"id":"c937c819-6e9b-432f-b599-c5c85ea236be"}}]},{"columns":[{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa4cf13993d_2e6248b4d92d6a4ea3be50ce8504860854c9c09c.png","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"dce728a3-0fb7-49d4-b818-6411870e8337"}},"id":"1941aa26-bd10-4928-8b93-4be0b6344658"}},{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa4cfcc65ba_6fa2bf578a4c3d8535fe7f59378ccb045a7dc864.png","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"31485349-7c22-47df-8e88-13780ce51d73"}},"id":"cb9f267b-e5e7-400e-8268-8bc3b87ad6d7"}},{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa4d0a6c625_34d2c64c38b1a3a9b8926e3b1ff81526fb1795ae.png","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"8e7b688e-6a91-436d-bfbb-cb8f415b1ab0"}},"id":"e26334fa-5ada-45b1-8930-3fd4c4277950"}}]}],"tableType":"OnlyColumnHeader"},"id":"0f72398d-0f6b-46cd-a925-80eabd9d91cc"},{"plugin":"rows","state":[{"plugin":"image","state":{"src":"https://assets.serlo.org/5aae46833f906_74d2afaa235b398526e11e7cae5893f3afbb65bf.PNG","alt":"","maxWidth":400,"caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"d8a7e425-36da-42e3-8669-1b7718a2d8d8"}},"id":"39f05d42-2ec1-4365-81db-df79e7e6fd84"},{"plugin":"text","state":[{"type":"p","children":[{"text":"gespiegelte Normalparabel"}]}],"id":"62ceb599-d1d3-4109-a853-2446008c7e0d"}],"id":"15d968b1-bdb3-4cda-9446-79d6558ec90f"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Parabel hat dann die Funktion "},{"type":"math","src":"y = a \\cdot x^2","inline":true,"children":[{"text":"y = a \\cdot x^2"}]},{"text":".\nEs steht also ein Faktor vor dem "},{"type":"math","src":"x^2","inline":true,"children":[{"text":"x^2"}]},{"text":", das den Funktionsgraphen staucht oder streckt.\nDiesen Faktor "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" nennt man "},{"text":"Öffnungsfaktor","strong":true},{"text":"."}]}],"id":"cebdeb7e-f354-49b9-ab2e-4676e47e811a"}],"id":"afd55355-7bf7-4ad9-9c2f-ed578b5315db"},"id":"101136-course-page","title":"Wie kann man Breite und Öffnung einer Parabel verändern?"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Bearbeite die folgende Aufgabe und beobachte, wie sich der Öffnungsfaktor auf den Funktionsgraphen der Parabel auswirkt."}]}],"id":"3bdabf6f-73db-463e-931f-f0c8ac87c249"},{"plugin":"injection","state":"/100922","id":"703f8af8-1d75-4235-b3d3-143b10442040"}],"id":"cd8ab726-b7c5-4fc6-84ef-3170babb866e"},"id":"101142-course-page","title":"Wirkung des Öffnungsfaktors (1/3)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Der Öffnungsfaktor "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" kann auch negativ werden. Bearbeite die folgende Aufgabe anhand des Applets und beobachte die Wirkung von "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" auf den Funktionsgraphen, wenn er negative Werte annimmt!"}]}],"id":"0ac81111-0675-4b7c-9c32-ba0e49239340"},{"plugin":"injection","state":"/100955","id":"6064d7ec-2581-43be-ae46-476230a543b7"}],"id":"61b602a9-2732-44bd-9651-21cea075f8ed"},"id":"101147-course-page","title":"Wirkung des Öffnungsfaktors (2/3)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Hier findest du nochmal eine Zusammenfassung der Ergebnisse aus den vorangegangenen Aufgaben."}]}],"id":"85a84322-0e85-4f8e-ad26-b0a46f02ecd0"},{"plugin":"serloTable","state":{"rows":[{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Öffnungsfaktor 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x^2"}]},{"text":""}]}],"id":"22088753-5e39-47c0-a425-38a9523a87c9"}},"id":"89693ece-1d1e-4e12-8440-d169195ec8f2"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":"Normalparabel"}]}],"id":"1abb22ce-627f-446b-ac93-3b188edac343"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"0 < a < 1","inline":true,"children":[{"text":"0 < a < 1"}]},{"text":""}]}],"id":"c5b6281a-7311-4c00-9a02-bf345a3fcd9c"}},{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa5023aec98_34d2c64c38b1a3a9b8926e3b1ff81526fb1795ae.png","maxWidth":350,"caption":{"plugin":"text","state":[{"type":"p","children":[{"text":"z.B.\n"},{"type":"math","src":"y=0,4x^2","inline":true,"children":[{"text":"y=0,4x^2"}]},{"text":""}]}],"id":"97401b45-2c4f-4036-9521-7baeb7825961"}},"id":"1f777ce2-d307-4c54-abed-9c5e932605fd"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"nach oben geöffnet"}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"gestaucht (breiter als die Normalparabel)"}]}]}]}],"id":"3dcf1844-f28d-4d50-b9c3-62289353a5e4"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"a > 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Normalparabel)"}]}]}]}],"id":"c27b31f3-d9f2-4e26-8d70-b9c5afa71752"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"a=-1","inline":true,"children":[{"text":"a=-1"}]},{"text":""}]}],"id":"6fb4125a-f121-45d9-af6a-b7554bb6bc05"}},{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa516cf18b3_e18ea229e40b356fbc0aafb06d3db751a2f54acf.png","maxWidth":350,"caption":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"y=-1 \\cdot x^2 = -x^2","inline":true,"children":[{"text":"y=-1 \\cdot x^2 = -x^2"}]},{"text":""}]}],"id":"764c5d58-2338-4dde-ab00-7b2295161f68"}},"id":"c2c9f777-f9ba-4ec4-b65d-47df15b4028b"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"nach unten geöffnet"}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"an der "},{"type":"math","src":"x","inline":true,"children":[{"text":"x"}]},{"text":"-Achse gespiegelte Normalparabel"}]}]}]}],"id":"b71dd075-a33a-4258-857c-8aa1a97b00d3"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"-1 < a < 0","inline":true,"children":[{"text":"-1 < a < 0"}]},{"text":""}]}],"id":"9757116c-89e6-4197-b270-3b99c093fbae"}},{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa51c7be8ac_590ad2138a5390cff7ea3be41361a0a59032ef68.png","caption":{"plugin":"text","state":[{"type":"p","children":[{"text":"z.B.\n"},{"type":"math","src":"y=-0,4x^2","inline":true,"children":[{"text":"y=-0,4x^2"}]},{"text":""}]}],"id":"bb39195d-71e7-4966-a6d7-c05f21f8fe20"}},"id":"11edaff1-fe26-4fb2-a7d8-9f94dc15ff06"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"nach unten geöffnet"}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"gestaucht (breiter als die Normalparabel)"}]}]}]}],"id":"58b27263-8288-4b87-9586-8466518b3f4f"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"a< -1","inline":true,"children":[{"text":"a< -1"}]},{"text":""}]}],"id":"8d7baf87-3579-4511-bb10-abd6e2e4c8b3"}},{"content":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa5217c928d_1720b5ac6f944a058e5cd9f1df5df1d0959cc4e9.png","caption":{"plugin":"text","state":[{"type":"p","children":[{"text":"z.B.\n"},{"type":"math","src":"y = -4x^2","inline":true,"children":[{"text":"y = -4x^2"}]},{"text":""}]}],"id":"403a5ef5-64cc-4d89-b819-bc2589287f02"}},"id":"a6cee7bd-f0b2-4f44-b63d-09bfbba9d118"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"nach unten geöffnet"}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"gestreckt (enger als die Normalparabel)"}]}]}]}],"id":"46964d8e-ff81-4d88-900f-4df4db5f4f0f"}}]}],"tableType":"OnlyColumnHeader"},"id":"2258596a-cb94-4cad-8b84-8cefc485b35a"}],"id":"aec166f1-b79e-4791-9511-f90512effcfa"},"id":"101150-course-page","title":"Wirkung des Öffnungsfaktors (3/3)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Du weißt jetzt, wie sich Öffnungsfaktoren auf den Funktionsgraphen einer Parabel auswirken. "}]},{"type":"p","children":[{"text":"Aber wie kannst du den richtigen Öffnungfaktor bestimmen, wenn du einen Funktionsgraphen einer Parabel hast?"}]}],"id":"9b1322cc-694b-44fc-acf2-52ba60a84427"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Beispiel","strong":true},{"text":"\nSchau dir diese Parabel an:"}]}],"id":"0d537766-f834-4a6b-a74e-4f4e5f1d5b86"},{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaa54b33edc6_bec77e5cf2998f84bee3b3773acf89ebd0ac3da8.png","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"2ea22411-d00d-4ed3-bbbd-e2807db30e3d"}},"id":"98dc4ff3-7098-4900-b159-37a140fd3ca8"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Du kannst sofort erkennen, dass diese Parabel nach unten geöffnet ist und gestaucht. Du weißt also, dass der Öffnungsfaktor "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" negativ sein muss. "}]}],"id":"00b4ceae-40bc-4963-954d-67e1ac7530cb"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Bei einer Parabel mit dem Scheitelpunkt "},{"type":"math","src":"(0|0)","inline":true,"children":[{"text":"(0I0)"}]},{"text":" kann man den Öffnungsfaktor im Punkt "},{"type":"math","src":"(1|a)","inline":true,"children":[{"text":"(1Ia)"}]},{"text":" ablesen. Im Beispiel kann man diesen Punkt leider nicht genau ablesen, aber man kann schon erkennen, dass a zwischen "},{"type":"math","src":"-0,2","inline":true,"children":[{"text":"-0,2"}]},{"text":" und "},{"type":"math","src":"-0,3","inline":true,"children":[{"text":"-0,3"}]},{"text":" liegt."}]}],"id":"0d85c549-ae3e-43d1-89e0-102b230c46b4"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Um den Öffnungsfaktor herauszufinden benötigst du einen Punkt, der nicht der Scheitelpunkt ist. Gut ablesen kann man den Punkt "},{"type":"math","src":"(2|-1)","inline":true,"children":[{"text":"(2I-1)"}]},{"text":"."}]}],"id":"40e2dbb4-94c7-48f5-a508-745443e95807"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Koordinaten des Punktes "},{"type":"math","src":"(2|-1) ","inline":true,"children":[{"text":"(2|-1)"}]},{"text":" setzt du in die Funktionsgleichung ein. "}]}],"id":"66df9b34-a1b8-435a-95b3-e463132a69bc"},{"plugin":"text","state":[{"type":"p","children":[{"text":" "},{"type":"math","src":"y=a\\cdot x^2","inline":true,"children":[{"text":"y = ax^2"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\Rightarrow -1 = a \\cdot (-2)^2","inline":true,"children":[{"text":"\\Rightarrow -1 = a \\cdot (-2)^2"}]},{"text":""}]}],"id":"5d766a40-8b02-4be7-840e-dae3874827a2"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Du erhältst also eine lineare Gleichung, die du nach "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" auflösen kannst. "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"-1 = a \\cdot (-2)^2","inline":true,"children":[{"text":"-1 = a \\cdot (-2)^2"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"-1 = a \\cdot 4 \\hspace{2cm} |:4","inline":true,"children":[{"text":"-1 = a \\cdot 4 \\hspace{2cm} |:4"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"-0,25=a","inline":true,"children":[{"text":"- \\dfrac{1}{4} = a"}]},{"text":""}]},{"type":"p","children":[{"text":"Du erhältst also für den Öffnungsfaktor "},{"type":"math","src":"a=-0,25","inline":true,"children":[{"text":"a = - \\dfrac{1}{4}"}]},{"text":"."}]},{"type":"p","children":[{"text":"Als Funktionsgleichung erhältst du "},{"type":"math","src":"y=-0,25x^2","inline":true,"children":[{"text":"y=- \\dfrac{1}{4} x^2"}]},{"text":"."}]}],"id":"6137f8a5-60d8-44e8-91d3-174543627257"}],"id":"4a7648c3-4d28-42cc-9426-d49782832e3e"},"id":"101164-course-page","title":"Öffnungsfaktor bestimmen"},{"content":{"plugin":"rows","state":[{"plugin":"injection","state":"/101202","id":"eec3cb2c-6e50-4092-825b-ecef952b37a5"}],"id":"6b282521-3c3a-434b-bbb4-8f102aa94df2"},"id":"101226-course-page","title":"Aufstellen der Funktionsgleichung"},{"content":{"plugin":"rows","state":[{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Du erinnerst dich sicher an Marco, der den Basketball mit einer perfekten Parabeln in den Korb wirft."}]},{"type":"p","children":[{"text":"Wir können nun die Funktionsgleichung der Flugbahn bestimmen."}]}],"id":"6a319776-9f12-4159-800d-1fa96203cb88"}],"id":"2f555997-5792-4835-b407-95b43fdaf78d"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaec684d1222_af40177661e041f893e856ffa65fca21264bfe91.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"3044b494-3031-451f-9181-4541ab684ac1"}},"id":"774644c5-2902-4983-9732-6ef9c45e58ca"},"illustrating":true,"width":50},"id":"4f1e2753-b58f-42a7-a8bc-d48231621a9d"},{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Wir integrieren dazu das Bild in ein Koordinatensystem…"}]}],"id":"a8a0a962-2fca-419e-9648-670773e419bc"}],"id":"61354659-5827-42dd-9ea5-6bad0634ad3f"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaec6926c0e4_c93a9affd215e35d31c5a1d60b7dd65d53b9aee0.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"5b3cfbd9-e505-4dbf-aedb-d5a1572a9390"}},"id":"56fc628c-3af4-43d3-b4da-2709525b3783"},"illustrating":true,"width":50},"id":"2f674d89-6f93-4b3e-a79a-7630c4a995bb"},{"plugin":"multimedia","state":{"explanation":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"…und lesen die Koordinaten des Basketballes ab."}]},{"type":"p","children":[{"text":"Der Ball hat also die Koordinaten "},{"type":"math","src":"C=(-2,7\\,|-1,8)","inline":true,"children":[{"text":"C=(-2,7\\,|-1,8)"}]},{"text":"."}]},{"type":"p","children":[{"text":"Nehmen wir nun an, dass sich der Scheitelpunkt der Parabel wieder im Ursprung "},{"type":"math","src":"S=(0\\,|\\,0)","inline":true,"children":[{"text":"S=(0\\,|\\,0)"}]},{"text":" befindet."}]}],"id":"de3cb883-19dd-4aa9-bb83-e62fbe3382d3"}],"id":"eb572564-13d1-4dc6-9e53-edf9cfea0cea"},"multimedia":{"plugin":"image","state":{"src":"https://assets.serlo.org/5aaec6a2baf8c_7b3afe6f52bb3052f92f0e2b6d6231e596295c8d.jpg","alt":"","caption":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"bd43f8b3-b08b-444a-af34-960c58c55d7a"}},"id":"bed26dc4-9729-4147-aea7-a94c28feabd9"},"illustrating":true,"width":50},"id":"5bf3323b-28c1-4b92-839b-7e1486e84e83"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Berechne nun den Funktionsterm der Parabel und überprüfe anschließend mit dem Applet."}]}],"id":"e83fe137-e19f-4b6b-bf28-844d4b19ecaa"},{"plugin":"spoiler","state":{"title":"Bestimmen des Funktionsterms","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Stelle den Funktionsterm auf, wie du es auf den vorherigen Seiten geübt hast. "}]},{"type":"p","children":[{"text":"Setze dazu den bereits bekannten Punkt "},{"type":"math","src":"C=(-2,7\\,| \\, -1,8)","inline":true,"children":[{"text":"C=(-2,7\\,| \\, -1,8)"}]},{"text":" in die Funktionsgleichung "},{"type":"math","src":"y=ax^2","inline":true,"children":[{"text":"y=ax^2"}]},{"text":" ein. "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"y = ax^2","inline":true,"children":[{"text":"y = ax^2"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"-1,8 = a \\cdot (-2,7)^2","inline":true,"children":[{"text":"-1,8 = a \\cdot (-2,7)^2"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"-1,8 = a \\cdot 7,29","inline":true,"children":[{"text":"-1,8 = a \\cdot 7,29"}]},{"text":""}]},{"type":"p","children":[{"text":"Löse diese Gleichung nach "},{"type":"math","src":"a","inline":true,"children":[{"text":"a"}]},{"text":" um auf den Öffnungsfaktor zu kommen!"}]},{"type":"p","children":[{"text":""},{"type":"math","src":"-1,8 = a \\cdot 7,29 \\hspace{2cm} | : 7,29","inline":true,"children":[{"text":"-1,8 = a \\cdot 7,29 \\hspace{2cm} | : 7,29"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\dfrac{-1,8}{7,29} = a","inline":true,"children":[{"text":"\\dfrac{-1,8}{7,29} = a"}]},{"text":""}]},{"type":"p","children":[{"text":""},{"type":"math","src":"a \\approx -0,25","inline":true,"children":[{"text":"a \\approx -0,25"}]},{"text":""}]},{"type":"p","children":[{"text":"Du erhältst also den Funktionsterm "},{"type":"math","src":"y = -0,25 x^2","inline":true,"children":[{"text":"y = -0,25 x^2"}]},{"text":" für die Parabel, die den Basketballwurf beschreibt."}]}],"id":"e09c7806-6ed8-4dba-a448-34badb1a5951"}],"id":"728928a1-91f5-4a65-ab00-76ca05e5a601"}},"id":"5ac9b6f2-f66b-42b7-bc1c-e3c3c7479e46"},{"plugin":"geogebra","state":"r3hvpqmw","id":"a86464bd-c39e-405d-8a74-c358cd0c2d65"}],"id":"3a6ef0e0-a092-475f-9dab-f11754d4b65f"},"id":"101562-course-page","title":"Die Flugbahn des Basketballs"}]},"id":"1390aaa4-e072-4daf-9f21-c88ab202e19f"}}

/mathe/101125/stauchen-und-strecken-einer-parabel-oeffnungsfaktor