Geometrie & Funktionen · Common Core High School Algebra I (Quadratic Functions & Graphs), AP Precalculus, GCSE Higher Mathematics (Parabolas & Turning Points).
Parabel & Quadratische Funktionen
Quadratische Funktionen & Parabel-Parameter: Scheitelpunktform, Achsenverschiebung und Nullstellen
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Geometrie-Demofenster● 60 FPS
Experimentelle Steuerungsparameter
Echtzeit-Herleitung & Kennzahlen
Scheitel (h, k)
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Stationary turning point
Minimum im Intervall
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Global minimum within [p, q]
Maximum im Intervall
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Global maximum within [p, q]
Diskriminante Δ
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Determines number of real roots
Echtzeit-Formelherleitung
Grundformel:
--Einsetzungsschritte:
--Ergebnis: --
Typische Beispielaufgaben & Prüfungsfallen
Classic Moving-Axis Problem: Min of y = x² - 2tx + 1 on [-1, 2]
Aufgabe: Given f(x) = x² - 2tx + 1 for x ∈ [-1, 2], find the minimum value function g(t) in terms of parameter t.
Analysestrategie: The parabola opens upward with symmetry axis x = t. For closed interval [-1, 2], classify into 3 cases: (1) t < -1: increasing, min at f(-1); (2) -1 ≤ t ≤ 2: vertex inside, min at f(t) = 1 - t²; (3) t > 2: decreasing, min at f(2).
Schrittweise Herleitung:
- Step 1: Complete the square: f(x) = (x - t)² + 1 - t², axis at x = t.
- Step 2: Case 1 (t < -1): Axis lies to left; f(x) strictly increases on [-1, 2]; min is f(-1) = 2t + 2.
- Step 3: Case 2 (-1 ≤ t ≤ 2): Axis inside interval; min is vertex value f(t) = 1 - t².
- Step 4: Case 3 (t > 2): Axis lies to right; f(x) strictly decreases on [-1, 2]; min is f(2) = 5 - 4t.
- Step 5: Synthesize piecewise definition of g(t).
⚠️ Häufige Fehlvorstellung: When maximizing an upward-opening parabola, the maximum only occurs at the farther boundary; the critical threshold is the interval midpoint (p+q)/2.
Arbeitsblatt für den Unterricht & Hausaufgaben
Q1. What are the axis of symmetry and vertex coordinates for y = 2(x - 3)² + 5?
Answer Key: Axis of symmetry x = 3, vertex (3, 5)
Q2. Find maximum and minimum of y = x² - 4x + 3 on [0, 5].
Answer Key: Min = -1 (at x=2), Max = 8 (at x=5)
Häufige Fragen & typische Missverständnisse
What is the rationale behind 'add left, subtract right; add up, subtract down' for function transformations?
Replacing x with (x - h) shifts right because each x must be h units larger to produce the same function input; vertical shift simply adds k directly to the output y.