{"introPage":{"showIntroPage":false,"startButtonText":"Starte Quiz","introduction":""},"progressType":"dots","passPercentage":50,"questions":[{"params":{"media":{"disableImageZooming":false},"text":"

F\u00fclle die L\u00fccken aus.<\/p>\n","overallFeedback":[{"from":0,"to":100}],"showSolutions":"L\u00f6sung anzeigen","tryAgain":"Wiederholen","checkAnswer":"\u00dcberpr\u00fcfen","notFilledOut":"Bitte f\u00fclle alle L\u00fccken aus, um die L\u00f6sung ansehen zu k\u00f6nnen","answerIsCorrect":"':ans' ist korrekt","answerIsWrong":"':ans' ist falsch","answeredCorrectly":"Korrekt beantwortet","answeredIncorrectly":"Falsch beantwortet","solutionLabel":"Korrekte Antwort:","inputLabel":"L\u00fccke @num von @total","inputHasTipLabel":"Tipp verf\u00fcgbar","tipLabel":"Tipp","behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"autoCheck":false,"caseSensitive":true,"showSolutionsRequiresInput":true,"separateLines":false,"confirmCheckDialog":false,"confirmRetryDialog":false,"acceptSpellingErrors":true},"scoreBarLabel":"Du hast :num von :total Punkten erreicht.","confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"questions":["

Eine Funktion f ordnet jeder Zahl x aus dem\/der *Definitionsbereich\/Definitionsmenge* \\( D_f \\) *genau* einen Funktionswert f(x) zu.
\nDie Menge aller Funktionswerte hei\u00dft *Wertemenge\/Wertebereich*<\/p>\n\n

<\/p>\n\n

<\/p>\n"],"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Eingaben werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet.","a11yCheckingModeHeader":"\u00dcberpr\u00fcfen-Modus aktiv"},"library":"H5P.Blanks 1.12","metadata":{"contentType":"Fill in the Blanks","license":"U","title":"Definition Funktion","defaultLanguage":"de"},"subContentId":"c6457ef6-b9c3-4b55-a73c-e7a778466685"},{"params":{"media":{"type":{"params":{"contentName":"Bild","file":{"path":"images\/file-5e5fab3a6cf03.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":376,"height":357}},"library":"H5P.Image 1.1","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"},"subContentId":"20e0b331-9538-48c9-ae20-a9cb5f95249e"},"disableImageZooming":false},"answers":[{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

f(1) = g(1)<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

f(0) = g(0) = 0<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

f(x) > g(x) f\u00fcr x > 1<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

f(x) = 2 hat genau eine L\u00f6sung.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

g(x) = 2 hat genau eine L\u00f6sung.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

Die Funktionen f und g haben den Definitionsbereich \u211d.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

Die Funktionen f und g haben den Wertebereich \u211d.<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":false,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"\u00dcberpr\u00fcfen","showSolutionButton":"L\u00f6sung anzeigen","tryAgainButton":"Wiederholen","tipsLabel":"Hinweis anzeigen","scoreBarLabel":"Du hast :num von :total Punkten erreicht.","tipAvailable":"Hinweis verf\u00fcgbar","feedbackAvailable":"R\u00fcckmeldung verf\u00fcgbar","readFeedback":"R\u00fcckmeldung vorlesen","wrongAnswer":"Falsche Antwort","correctAnswer":"Richtige Antwort","shouldCheck":"H\u00e4tte gew\u00e4hlt werden m\u00fcssen","shouldNotCheck":"H\u00e4tte nicht gew\u00e4hlt werden sollen","noInput":"Bitte antworte, bevor du die L\u00f6sung ansiehst","a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Auswahlen werden als richtig, falsch oder fehlend markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Versuche werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet."},"confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"question":"

Die Skizze zeigt die Graphen der Funktion f (gr\u00fcn) und der Funktion g (rot). Kreuze an, welche Aussagen wahr sind.<\/p>\n"},"library":"H5P.MultiChoice 1.14","metadata":{"contentType":"Multiple Choice","license":"U","title":"Zwei Graphen","defaultLanguage":"de"},"subContentId":"8fdeaaa1-930d-4496-a137-200fcafe0a13"},{"params":{"media":{"type":{"params":{"contentName":"Bild","file":{"path":"images\/file-5ee6184f635fa.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":430,"height":515}},"library":"H5P.Image 1.1","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"},"subContentId":"0ee71635-8275-444c-9b2e-74cc0484b89c"},"disableImageZooming":false},"text":"

Der abgebildete Graph der Funktion f hat die Funktionsgleichung f(x) = a\u00b7(x - b)\u00b3 + c.
\nGib die Werte f\u00fcr a, b und c an.<\/p>\n","overallFeedback":[{"from":0,"to":100}],"showSolutions":"L\u00f6sung anzeigen","tryAgain":"Wiederholen","checkAnswer":"\u00dcberpr\u00fcfen","notFilledOut":"Bitte f\u00fclle alle L\u00fccken aus, um die L\u00f6sung ansehen zu k\u00f6nnen","answerIsCorrect":"':ans' ist korrekt","answerIsWrong":"':ans' ist falsch","answeredCorrectly":"Korrekt beantwortet","answeredIncorrectly":"Falsch beantwortet","solutionLabel":"Korrekte Antwort:","inputLabel":"L\u00fccke @num von @total","inputHasTipLabel":"Tipp verf\u00fcgbar","tipLabel":"Tipp","behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"autoCheck":false,"caseSensitive":true,"showSolutionsRequiresInput":true,"separateLines":false,"confirmCheckDialog":false,"confirmRetryDialog":false,"acceptSpellingErrors":false},"scoreBarLabel":"Du hast :num von :total Punkten erreicht.","confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"questions":["

a = *0,5\/0.5*<\/p>\n\n

b = *1*<\/p>\n\n

c = *-2*<\/p>\n"],"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Eingaben werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet.","a11yCheckingModeHeader":"\u00dcberpr\u00fcfen-Modus aktiv"},"library":"H5P.Blanks 1.12","metadata":{"contentType":"Fill in the Blanks","license":"U","title":"Funktion 3. Grades","defaultLanguage":"de"},"subContentId":"7ed0b0e8-c298-476b-a788-6fb26097d734"},{"params":{"media":{"type":{"params":{"contentName":"Bild","file":{"path":"images\/file-5ee61a18f1307.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":711,"height":563}},"library":"H5P.Image 1.1","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"},"subContentId":"223681a7-d04a-46cc-b531-1d2e1d233ab1"},"disableImageZooming":false},"text":"

Der abgebildete Graph der Funktion f hat die Funktionsgleichung \\( f(x) = \\frac{a}{x-b} + c \\).
\nGib die Werte f\u00fcr a, b und c an.<\/p>\n","overallFeedback":[{"from":0,"to":100}],"showSolutions":"L\u00f6sung anzeigen","tryAgain":"Wiederholen","checkAnswer":"\u00dcberpr\u00fcfen","notFilledOut":"Bitte f\u00fclle alle L\u00fccken aus, um die L\u00f6sung ansehen zu k\u00f6nnen","answerIsCorrect":"':ans' ist korrekt","answerIsWrong":"':ans' ist falsch","answeredCorrectly":"Korrekt beantwortet","answeredIncorrectly":"Falsch beantwortet","solutionLabel":"Korrekte Antwort:","inputLabel":"L\u00fccke @num von @total","inputHasTipLabel":"Tipp verf\u00fcgbar","tipLabel":"Tipp","behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"autoCheck":false,"caseSensitive":true,"showSolutionsRequiresInput":true,"separateLines":false,"confirmCheckDialog":false,"confirmRetryDialog":false,"acceptSpellingErrors":false},"scoreBarLabel":"Du hast :num von :total Punkten erreicht.","confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"questions":["

a = *2*<\/p>\n\n

b = *-1*<\/p>\n\n

c = *1,5\/1.5*<\/p>\n"],"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Eingaben werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet.","a11yCheckingModeHeader":"\u00dcberpr\u00fcfen-Modus aktiv"},"library":"H5P.Blanks 1.12","metadata":{"contentType":"Fill in the Blanks","license":"U","title":"Gebrochen rationale Funktion","defaultLanguage":"de"},"subContentId":"788e912a-4934-4e41-a872-40ffd2a44345"},{"params":{"media":{"disableImageZooming":false},"answers":[{"correct":false,"tipsAndFeedback":{"chosenFeedback":"

f(2) und g(2) sind addiert 0. Es k\u00f6nnte also z.B. f(2) = 1 und g(2) = -1 sein.<\/div>\n","notChosenFeedback":"","tip":""},"text":"

Wenn h(2) = 0 ist, dann ist f(2) = 0 und g(2) = 0.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

Wenn h(2) = 0 ist, dann ist f(2) = -g(2).<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"

Nur das Produkt von f und g muss 0 ergeben. Es k\u00f6nnte also z.B. f(2) = 0 und g(2) = 5 sein.<\/div>\n","tip":"","notChosenFeedback":""},"text":"

Wenn i(2) = 0 ist, dann ist f(2) = 0 und g(2) = 0.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

Wenn i(2) = 0 ist, dann ist f(2) = 0 oder g(2) = 0.<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":false,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"\u00dcberpr\u00fcfen","showSolutionButton":"L\u00f6sung anzeigen","tryAgainButton":"Wiederholen","tipsLabel":"Hinweis anzeigen","scoreBarLabel":"Du hast :num von :total Punkten erreicht.","tipAvailable":"Hinweis verf\u00fcgbar","feedbackAvailable":"R\u00fcckmeldung verf\u00fcgbar","readFeedback":"R\u00fcckmeldung vorlesen","wrongAnswer":"Falsche Antwort","correctAnswer":"Richtige Antwort","shouldCheck":"H\u00e4tte gew\u00e4hlt werden m\u00fcssen","shouldNotCheck":"H\u00e4tte nicht gew\u00e4hlt werden sollen","noInput":"Bitte antworte, bevor du die L\u00f6sung ansiehst","a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Auswahlen werden als richtig, falsch oder fehlend markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Versuche werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet."},"confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"question":"

Gegeben sind die Funktionen f und g und die zusammengesetzten Funktionen h = f + g und i = f \u00b7 g.<\/p>\n\n

Kreuze die wahren Aussagen an.<\/p>\n"},"library":"H5P.MultiChoice 1.14","metadata":{"contentType":"Multiple Choice","license":"U","title":"Zusammengesetzte Funktion","defaultLanguage":"de"},"subContentId":"8ad6dc58-a7d8-43cd-be5b-e66469b0d3a9"},{"params":{"scoreShow":"\u00dcberpr\u00fcfen","tryAgain":"Wiederholen","scoreExplanation":"Richtige Antworten geben +1 Punkte. Bei falschen Antworten gibt es -1 Punkt. Die kleinstm\u00f6gliche Punktzahl ist 0.","question":{"settings":{"size":{"width":620,"height":310}},"task":{"elements":[{"x":4.8333314182207685,"y":48.333359692540306,"width":10.545833375,"height":1.295833375,"dropZones":["0","1","2"],"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

\\( f(x) = x^{3} - 2x + 1 \\)<\/p>\n"},"subContentId":"d75cfc5a-c17f-4da5-949d-9809325afa87","metadata":{"contentType":"Text","license":"U","title":"Unbenannt: Text","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false},{"x":36.392474484627016,"y":48.387096774193544,"width":9.920833375,"height":1.2729172377929663,"dropZones":["0","1","2"],"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

\\( f(x) = x^{5} + 3x \\)<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Unbenannt: Text","defaultLanguage":"de"},"subContentId":"2bdf6629-579e-43a2-823c-b3e690f81060"},"backgroundOpacity":100,"multiple":false},{"x":69.3172004170867,"y":48.387096774193544,"width":9.920833375,"height":1.285421815429685,"dropZones":["0","1","2"],"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

\\( f(x) = x^{4} - 5 \\)<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Unbenannt: Text","defaultLanguage":"de"},"subContentId":"e5d34054-cf6e-493f-81da-981ac4eb89b3"},"backgroundOpacity":100,"multiple":false},{"x":4.836018933971777,"y":74.19354838709677,"width":10.545833375,"height":1.9052109389648413,"dropZones":["0","1","2"],"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

\\( f(x) = {(2x - 1)}^3 \\)<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Unbenannt: Text","defaultLanguage":"de"},"subContentId":"4a0e7cfa-6f1c-4d62-8834-85b40de82a45"},"backgroundOpacity":100,"multiple":false},{"x":38.6881740656502,"y":74.19354838709677,"width":9.295833375,"height":1.9052109389648413,"dropZones":["0","1","2"],"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

\\( f(x) = \\frac{1}{x^{3}-x} \\)<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Unbenannt: Text","defaultLanguage":"de"},"subContentId":"a42006b9-ac58-4f1a-b7fc-621a64f55218"},"backgroundOpacity":100,"multiple":false},{"x":69.3172004170867,"y":74.19354838709677,"width":9.295833375,"height":1.9052109389648413,"dropZones":["0","1","2"],"type":{"library":"H5P.AdvancedText 1.1","params":{"text":"

\\( f(x) = \\frac{1}{x^{2}+1} \\)<\/p>\n"},"metadata":{"contentType":"Text","license":"U","title":"Unbenannt: Text","defaultLanguage":"de"},"subContentId":"be8b4f29-389d-403e-9ab5-34bfde49026d"},"backgroundOpacity":100,"multiple":false}],"dropZones":[{"x":0,"y":6.444683122567813,"width":13.125,"height":5,"correctElements":["2","5"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

achsensymm. zur y-Achse<\/div>\n"},{"x":33.83333437153478,"y":6.430126010584665,"width":14.375,"height":5,"correctElements":["1","4"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

punktsymm. zum Ursprung<\/div>\n"},{"x":70.92960766154798,"y":6.444683122567813,"width":11.25,"height":5,"correctElements":["0","3"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

keine einfache Symm.<\/div>\n"}]}},"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":false,"enableCheckButton":true,"showSolutionsRequiresInput":true,"singlePoint":false,"applyPenalties":true,"enableScoreExplanation":true,"dropZoneHighlighting":"dragging","autoAlignSpacing":2,"enableFullScreen":false,"showScorePoints":true,"showTitle":true},"grabbablePrefix":"Ziehbares Element {num} von {total}.","grabbableSuffix":"Abgelegt in Ablagezone {num}.","dropzonePrefix":"Ablagezone {num} von {total}.","noDropzone":"Keine Ablagezone.","tipLabel":"Hinweis zeigen","tipAvailable":"Hinweis verf\u00fcgbar","correctAnswer":"Richtige Antwort","wrongAnswer":"Falsche Antwort","feedbackHeader":"R\u00fcckmeldung","scoreBarLabel":"Du hast :num von :total Punkten erreicht.","scoreExplanationButtonLabel":"Zeige Erkl\u00e4rung der Punktevergabe","localize":{"fullscreen":"Vollbild","exitFullscreen":"Vollbild beenden"},"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yRetry":"Die Aufgabe wiederholen. Alle Versuche werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet."},"library":"H5P.DragQuestion 1.13","metadata":{"contentType":"Drag and Drop","license":"U","title":"Untersuche die Graphen von f auf Symmetrie und ordne zu.","defaultLanguage":"de"},"subContentId":"4114d0e7-fb28-402f-9843-be8aad79c714"},{"params":{"media":{"disableImageZooming":false},"text":"

Bestimme alle Nullstellen von f.<\/p>\n\n

Beginne mit der kleinsten L\u00f6sung.<\/p>\n","overallFeedback":[{"from":0,"to":100}],"showSolutions":"L\u00f6sung anzeigen","tryAgain":"Wiederholen","checkAnswer":"\u00dcberpr\u00fcfen","notFilledOut":"Bitte f\u00fclle alle L\u00fccken aus, um die L\u00f6sung ansehen zu k\u00f6nnen","answerIsCorrect":"':ans' ist korrekt","answerIsWrong":"':ans' ist falsch","answeredCorrectly":"Korrekt beantwortet","answeredIncorrectly":"Falsch beantwortet","solutionLabel":"Korrekte Antwort:","inputLabel":"L\u00fccke @num von @total","inputHasTipLabel":"Tipp verf\u00fcgbar","tipLabel":"Tipp","behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"autoCheck":false,"caseSensitive":true,"showSolutionsRequiresInput":true,"separateLines":false,"confirmCheckDialog":false,"confirmRetryDialog":false,"acceptSpellingErrors":false},"scoreBarLabel":"Du hast :num von :total Punkten erreicht.","confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"questions":["

a) \\( f(x) = (2x + 4)\u00b7(x - 5) \\) \\(x_1\\) = *-2* \\(x_2\\) = *5*<\/p>\n\n

b) \\( f(x) = x\u00b7(\\frac{1}{3}x - 1) \\) \\(x_1\\) = *0* \\(x_2\\) = *3*<\/p>\n\n

c) \\( f(x) = 3(x^2 + 4)\u00b7(9x^2 - 9) \\) \\(x_1\\) = *-1* \\(x_2\\) = *1*<\/p>\n\n

d) \\( f(x) = (x^3 -1)\u00b7(x^2 - 2x) \\) \\(x_1\\) = *0* \\(x_2\\) = *1* \\(x_3\\) = *2*<\/p>\n\n

e) \\( f(x) = x^3 - 6x^2 + 5x \\) \\(x_1\\) = *0* \\(x_2\\) = *1* \\(x_3\\) = *5*<\/p>\n"],"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Eingaben werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet.","a11yCheckingModeHeader":"\u00dcberpr\u00fcfen-Modus aktiv"},"library":"H5P.Blanks 1.12","metadata":{"contentType":"Fill in the Blanks","license":"U","title":"Nullstellen von f","defaultLanguage":"de"},"subContentId":"6a1a849f-b0c3-4d87-a6b0-3c27f4dbae8a"},{"params":{"text":"

F\u00fclle in die L\u00fccken die passenden \u00c4quivalenzumformungen.<\/p>\n","overallFeedback":[{"from":0,"to":100}],"showSolutions":"L\u00f6sung anzeigen","tryAgain":"Wiederholen","checkAnswer":"\u00dcberpr\u00fcfen","notFilledOut":"Bitte f\u00fclle alle L\u00fccken aus, um die L\u00f6sung ansehen zu k\u00f6nnen","answerIsCorrect":"':ans' ist korrekt","answerIsWrong":"':ans' ist falsch","answeredCorrectly":"Korrekt beantwortet","answeredIncorrectly":"Falsch beantwortet","solutionLabel":"Korrekte Antwort:","inputLabel":"L\u00fccke @num von @total","inputHasTipLabel":"Tipp verf\u00fcgbar","tipLabel":"Tipp","behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"autoCheck":false,"caseSensitive":true,"showSolutionsRequiresInput":true,"separateLines":false,"confirmCheckDialog":false,"confirmRetryDialog":false,"acceptSpellingErrors":false},"scoreBarLabel":"Du hast :num von :total Punkten erreicht.","confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"questions":["

Gegeben sind zwei Funktionen mit den Gleichungen y = -x\u00b2 + 3 und y = -2(x-2)\u00b2 +4.<\/p>\n\n

Gleichsetzungsverfahren zur Bestimmung der Schnittpunkte:<\/p>\n\n

-x\u00b2 + 3 = -2(x-2)\u00b2 +4 | ausmultiplizieren<\/p>\n\n

\u21d4 -x\u00b2 + 3 = *-2*x\u00b2 + *8*x + *-8* +4 | zusammenfassen<\/p>\n\n

\u21d4 x\u00b2 + *-8*x + *+7\/7* = 0 <\/p>\n\n

Mitternachtsformel anwenden ergibt f\u00fcr die beiden L\u00f6sungen<\/p>\n\n

\\( x_{1,2} = \\frac{8 \u00b1 \\sqrt{(-8)^2 - 4\u00b71\u00b77}}{2\u00b71} \\)<\/p>\n\n

und somit<\/p>\n\n

\\(x_1\\) = *1* oder \\(x_2\\) = *7* (beginne mit der kleineren Zahl)<\/p>\n"],"media":{"disableImageZooming":false},"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Eingaben werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet.","a11yCheckingModeHeader":"\u00dcberpr\u00fcfen-Modus aktiv"},"library":"H5P.Blanks 1.12","metadata":{"contentType":"Fill in the Blanks","license":"U","title":"Schnittpunkte bestimmen","defaultLanguage":"de"},"subContentId":"e43f0f7b-81d8-4f85-b18f-96064af82a9d"},{"params":{"scoreShow":"\u00dcberpr\u00fcfen","tryAgain":"Wiederholen","scoreExplanation":"Richtige Antworten geben +1 Punkte. Bei falschen Antworten gibt es -1 Punkt. Die kleinstm\u00f6gliche Punktzahl ist 0.","question":{"settings":{"size":{"width":620,"height":310}},"task":{"elements":[{"x":24.18054806643681,"y":12.889366245135626,"width":5.545833375,"height":7.2350225641891885,"dropZones":["0","1","2","3","4","5"],"type":{"library":"H5P.Image 1.1","params":{"contentName":"Bild","file":{"path":"images\/file-5ee8d0cb7dd0d.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":304,"height":394},"alt":"Graph_-1dopp_2dopp"},"subContentId":"29d5c015-6405-46fb-be4a-1cb40dd9972a","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false},{"x":0,"y":25.778732490271253,"width":5.6738598637104625,"height":6.420835938964841,"dropZones":["0","1","2","3","4","5"],"type":{"library":"H5P.Image 1.1","params":{"contentName":"Bild","file":{"path":"images\/file-5ee8d0f4e4e1c.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":307,"height":345},"alt":"Graph_-1dopp_2einf"},"subContentId":"67c9b3d0-a31c-41ad-9143-86b8c6f4c984","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false},{"x":24.18054806643681,"y":61.22448966439423,"width":6.170833375,"height":6.739015193181818,"dropZones":["0","1","2","3","4","5"],"type":{"library":"H5P.Image 1.1","params":{"contentName":"Bild","file":{"path":"images\/file-5ee8d10706789.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":289,"height":313},"alt":"Graph_-1drei_2ein"},"subContentId":"c15658fa-8bab-4388-bc3f-b02bce0a992a","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false},{"x":83.50349265609512,"y":25.778732490271253,"width":5.685345380455278,"height":5.913542237792966,"dropZones":["0","1","2","3","4","5"],"type":{"library":"H5P.Image 1.1","params":{"contentName":"Bild","file":{"path":"images\/file-5ee8d145549bd.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":289,"height":303},"alt":"Graph_-1ein_2dopp"},"subContentId":"71131ea1-a6a3-400b-bf64-b8ab2ec4fc74","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false},{"x":59.645351897210794,"y":12.889366245135626,"width":5.629296340281517,"height":5.9145874648437475,"dropZones":["0","1","2","3","4","5"],"type":{"library":"H5P.Image 1.1","params":{"contentName":"Bild","file":{"path":"images\/file-5ee8d151b925c.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":289,"height":305},"alt":"Graph_-1ein_2drei"},"subContentId":"c39c700a-00c0-48bd-9428-4027173e1e22","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false},{"x":61.25738843497325,"y":64.12459706954974,"width":5.545833375,"height":6.221509050675675,"dropZones":["0","1","2","3","4","5"],"type":{"library":"H5P.Image 1.1","params":{"contentName":"Bild","file":{"path":"images\/file-5ee8d160992f5.jpg","mime":"image\/jpeg","copyright":{"license":"U"},"width":289,"height":322},"alt":"Graph_-1ein_2ein"},"subContentId":"a9d493ba-7b38-45b3-afde-60e2f1987550","metadata":{"contentType":"Image","license":"U","title":"Unbenannt: Image","defaultLanguage":"de"}},"backgroundOpacity":100,"multiple":false}],"dropZones":[{"x":-3.227696569660287e-6,"y":6.440876073588697,"width":9.375,"height":5.625,"correctElements":["5"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

f(x) = (x+1)(x-2)<\/div>\n"},{"x":38.6881740656502,"y":6.440876073588697,"width":9.375,"height":5.625,"correctElements":["1"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

f(x) = (x+1)\u00b2\u00b7(x-2)<\/div>\n"},{"x":74.15053506300403,"y":6.440876073588697,"width":9.375,"height":5.625,"correctElements":["0"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

f(x) = (x+1)\u00b2\u00b7(x-2)\u00b2<\/div>\n"},{"x":-3.227696569660287e-6,"y":67.74193548387096,"width":9.375,"height":6.23333233691406,"correctElements":["3"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

f(x) = (x+1)\u00b7(x-2)\u00b2<\/div>\n"},{"x":38.6881740656502,"y":67.74193548387096,"width":9.375,"height":6.23333233691406,"correctElements":["2"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

f(x) = (x+1)\u00b3\u00b7(x-2)<\/div>\n"},{"x":75.76343828881049,"y":67.74193548387096,"width":9.375,"height":6.23333233691406,"correctElements":["4"],"showLabel":true,"backgroundOpacity":100,"tipsAndFeedback":{"tip":""},"single":false,"autoAlign":false,"label":"

f(x) = (x+1)\u00b7(x-2)\u00b3<\/div>\n"}]}},"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":false,"enableCheckButton":true,"showSolutionsRequiresInput":true,"singlePoint":false,"applyPenalties":true,"enableScoreExplanation":true,"dropZoneHighlighting":"dragging","autoAlignSpacing":2,"enableFullScreen":false,"showScorePoints":true,"showTitle":true},"grabbablePrefix":"Ziehbares Element {num} von {total}.","grabbableSuffix":"Abgelegt in Ablagezone {num}.","dropzonePrefix":"Ablagezone {num} von {total}.","noDropzone":"Keine Ablagezone.","tipLabel":"Hinweis zeigen","tipAvailable":"Hinweis verf\u00fcgbar","correctAnswer":"Richtige Antwort","wrongAnswer":"Falsche Antwort","feedbackHeader":"R\u00fcckmeldung","scoreBarLabel":"Du hast :num von :total Punkten erreicht.","scoreExplanationButtonLabel":"Zeige Erkl\u00e4rung der Punktevergabe","localize":{"fullscreen":"Vollbild","exitFullscreen":"Vollbild beenden"},"a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Eingaben werden als richtig, falsch oder unbeantwortet markiert.","a11yRetry":"Die Aufgabe wiederholen. Alle Versuche werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet."},"library":"H5P.DragQuestion 1.13","metadata":{"contentType":"Drag and Drop","license":"U","title":"Ordne die Graphen den passenden Funktionsgleichungen zu.","defaultLanguage":"de"},"subContentId":"d2c8c86b-525b-4e72-a783-70ccd12d971c"},{"params":{"media":{"disableImageZooming":false},"answers":[{"correct":false,"tipsAndFeedback":{"chosenFeedback":"

Eine ganzrationale Funktion 3. Grades hat maximal 3 Nullstellen.<\/div>\n","tip":"","notChosenFeedback":""},"text":"

Eine ganzrationale Funktion 3. Grades hat genau 3 Nullstellen.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

Eine ganzrationale Funktion 3. Grades hat mindestens eine Nullstelle.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"

Gegenbeispiel: Die Funktion f mit \\( f(x) = x^{4} + 1 \\) hat keine Nullstellen.<\/div>\n","notChosenFeedback":""},"text":"

Eine ganzrationale Funktion 4. Grades hat mindestens eine Nullstelle.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"

Gegenbeispiel: Die Funktion f mit f(x) = x\u00b2\u00b7(x-1)\u00b7(x+1) ist achsensymmetrisch zum Ursprung, aber hat genau drei Nullstellen (-1, 0 und 1)<\/div>\n","tip":"","notChosenFeedback":""},"text":"

Eine zur y-Achse symmetrische Funktion f hat immer eine gerade Anzahl an verschiedenen Nullstellen.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"chosenFeedback":"","tip":"","notChosenFeedback":""},"text":"

Hat der Funktionsterm einer Funktion f nur x-Potenzen mit geraden Hochzahlen, so ist das Schaubild symmetrisch zur y-Achse.<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":false,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":false,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"\u00dcberpr\u00fcfen","showSolutionButton":"L\u00f6sung anzeigen","tryAgainButton":"Wiederholen","tipsLabel":"Hinweis anzeigen","scoreBarLabel":"Du hast :num von :total Punkten erreicht.","tipAvailable":"Hinweis verf\u00fcgbar","feedbackAvailable":"R\u00fcckmeldung verf\u00fcgbar","readFeedback":"R\u00fcckmeldung vorlesen","wrongAnswer":"Falsche Antwort","correctAnswer":"Richtige Antwort","shouldCheck":"H\u00e4tte gew\u00e4hlt werden m\u00fcssen","shouldNotCheck":"H\u00e4tte nicht gew\u00e4hlt werden sollen","noInput":"Bitte antworte, bevor du die L\u00f6sung ansiehst","a11yCheck":"Die Antworten \u00fcberpr\u00fcfen. Die Auswahlen werden als richtig, falsch oder fehlend markiert.","a11yShowSolution":"Die L\u00f6sung anzeigen. Die richtigen L\u00f6sungen werden in der Aufgabe angezeigt.","a11yRetry":"Die Aufgabe wiederholen. Alle Versuche werden zur\u00fcckgesetzt und die Aufgabe wird erneut gestartet."},"confirmCheck":{"header":"Beenden?","body":"Ganz sicher beenden?","cancelLabel":"Abbrechen","confirmLabel":"Beenden"},"confirmRetry":{"header":"Wiederholen?","body":"Ganz sicher wiederholen?","cancelLabel":"Abbrechen","confirmLabel":"Best\u00e4tigen"},"question":"

Kreuze die richtigen Aussagen an.<\/p>\n"},"library":"H5P.MultiChoice 1.14","metadata":{"contentType":"Multiple Choice","license":"U","title":"Nullstellen ganzrationale Funktionen","defaultLanguage":"de"},"subContentId":"8707af0f-2c1a-4b2d-96c1-9d912c59d803"}],"disableBackwardsNavigation":false,"randomQuestions":true,"endGame":{"showResultPage":true,"showSolutionButton":true,"noResultMessage":"Dein Ergebnis:","message":"Du hast @score Punkte von @total m\u00f6glichen.","overallFeedback":[{"from":0,"to":45,"feedback":"Leider hast du das Thema noch nicht ausreichend verstanden. Du musst dir die Inhalte zum Thema nochmal genauer anschauen."},{"from":46,"to":60,"feedback":"Du hast das Thema in den Grundz\u00fcgen verstanden. Du musst dir einzelne Inhalte zum Thema nochmal genauer anschauen."},{"from":61,"to":75,"feedback":"Gut! Du hast das Thema verstanden. Wenige Inhalte zum Thema musst du dir nochmal genauer anschauen."},{"from":76,"to":90,"feedback":"Prima! Du hast das Thema gut verstanden. Deinem weiteren Lernfortschritt steht nichts im Weg!"},{"from":91,"to":100,"feedback":"Hervorragend! Du hast das Thema sehr gut verstanden und durchdrungen."}],"solutionButtonText":"L\u00f6sung anzeigen","retryButtonText":"Wiederholen","finishButtonText":"Beenden","showAnimations":false,"skippable":false,"skipButtonText":"Skip video","showRetryButton":true},"override":{"checkButton":true},"texts":{"prevButton":"Zur\u00fcck","nextButton":"Weiter","finishButton":"Beenden","textualProgress":"Aktuelle Frage: @current von @total Fragen","jumpToQuestion":"Frage %d von %total","questionLabel":"Frage","readSpeakerProgress":"Frage @current von @total","unansweredText":"Unbeantwortet","answeredText":"Beantwortet","currentQuestionText":"Aktuelle Frage"}}

Funktionen Grundlagen