General Form y = A sin(B(x - C)) + D

Trigonometry · Axiom Academy

LESSON Sine Function Transformations Understanding y = A sin(B(x - C)) + D: How each parameter transforms the basic sine wave The parameter A controls the amplitude—the distance from the midline to the maximum (or minimum) value of the function. It causes a vertical stretch or compression. Amplitude = |A| (absolute value of A) If |A| > 1: wave is vertically stretched (taller) If 0 < |A| < 1: wave is vertically compressed (shorter) If A < 0: wave is reflected across the x-axis The parameter B controls the period—how quickly the wave completes one full cycle. It causes a horizontal stretch or compression. If |B| > 1: wave is horizontally compressed (oscillates faster) If 0 < |B| < 1: wave is horizontally stretched (oscillates slower) Basic sine has B = 1, giving period = 2π 4. Phase Shift: The C Parameter The parameter C controls the phase shift—a horizontal translation of the entire wave. Notice that C appears as (x - C) in the formula. If C > 0: wave shifts right by C units The form (x - C) is standard; (x + C) would shift left 5. Vertical Shift: The D Parameter The parameter D controls the vertical shift—a vertical translation of the entire wave. This moves the midline (equilibrium position) of the wave. If D > 0: wave shifts up by D units Maximum value: D + |A|, Minimum value: D - |A| When all four parameters work together, you can create any sinusoidal function. Understanding each parameter individually allows you to analyze and graph any sine function efficiently.

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