Shifting and Stretching Waves

Trigonometry · Axiom Academy

INTRO Shifting and Stretching Waves Discover how transformations change the sine curve. Watch the wave respond to each adjustment! This is the basic sine wave: y = sin(x) . It oscillates smoothly between -1 and 1. Try adjusting the viewing window to see more of the wave! Adding or subtracting a constant moves the entire wave up or down. Move the slider to shift the wave vertically! ⬅➡ Horizontal Shifts (Phase Shift) Subtracting a value inside the sine function shifts the wave horizontally. This is called a phase shift . Explore how it moves! Multiplying the sine function stretches or compresses it vertically. This changes the amplitude – how tall the wave is! Horizontal Stretch (Period Change) Multiplying x inside the sine function changes how quickly the wave completes its cycle. This affects the period and frequency ! Now combine all transformations! The general form is: y = A·sin(B(x − h)) + k Amplitude (A): Vertical stretch/compression – controls wave height Frequency (B): Horizontal stretch/compression – affects period (2π/B) Phase Shift (h): Horizontal translation – moves wave left/right Vertical Shift (k): Vertical translation – raises/lowers midline Order matters! Apply transformations in this sequence: horizontal stretch → horizontal shift → vertical stretch → vertical shift.

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