Amplitude, Period and Phase Shift Calculator

How it works

y = A sin(Bx − C) + D or A cos(Bx − C) + D · amplitude = |A| · period = 2π/|B| (360°/|B|) · phase shift = C/B (positive → right) · vertical shift = D, midline y = D · max = D + |A|, min = D − |A| · frequency = 1/period

Each constant in a sinusoid controls one transformation of the basic sine or cosine wave. |A| stretches the wave vertically, B squeezes it horizontally so one cycle takes 2π/|B| instead of 2π, C/B slides it left or right, and D lifts the midline. Factor B out of the argument first: A sin(Bx − C) = A sin(B(x − C/B)), so the shift is C/B, not C (OpenStax Precalculus §6.1). The five key points split one cycle into quarters for sketching the graph.

Worked example

y = 3 sin(2x − 1) − 1 in radians: amplitude |3| = 3; period 2π/2 = π ≈ 3.141593; phase shift 1/2 = 0.5 to the right; vertical shift −1, so the midline is y = −1; maximum −1 + 3 = 2 and minimum −1 − 3 = −4; frequency 1/π ≈ 0.31831 cycles per unit. One cycle starts at x = 0.5 and ends at x = 3.641593.

Sources: OpenStax Precalculus 2e — 6.1 Graphs of the Sine and Cosine Functions · Khan Academy — Midline, amplitude and period review

Frequently asked questions