Phase Shift and Vertical Shift

Pre-Calculus · Axiom Academy

LESSON Phase Shift and Vertical Shift How the C and D in slide a wave sideways and lift its midline. 1. The Phase Shift Slides the Wave In , the number C slides the entire wave horizontally . A positive C moves it C units to the right ; a negative C moves it left. The cleanest way to see it: the point where the parent wave begins its cycle at the origin re-appears at x = C . cycle now starts at x = C (here , a shift right) 2. Factor B Out Before You Read C When there is a coefficient on x inside, the phase shift is not the constant you see. Take . Reading off is wrong. Factor the 2 out first to expose the true shift: The shift is , not the bare constant . Factoring rewrites as , so the wave begins half a unit later than the misread predicts. 3. The Vertical Shift Lifts the Midline Adding D at the end raises (or lowers) the whole curve. The line the wave oscillates around — the midline — moves from y = 0 to y = D . The shape is untouched; it just rides higher. midline moves to y = D ; range becomes The wave now centers on y = D instead of the x -axis. The crest reaches y = D + A (here A = 1 , so D + 1 ). Now build from the parent wave: stretch by A = 2 , halve the period with B = 2 , slide right by , then lift the midline to D = 1 . Watch the four key points land. already factored — shift is , midline D = 1 Start of cycle at sits on the midline y = 1 (ascending); the crest hits y = 3 at and the trough y = -1 at — so the range is .

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