What is the phase shift?

How it works

For y = A sin(Bx − C) + D or y = A cos(Bx − C) + D, with A and B not 0:

• Amplitude = |A|.
• Period P = 2π ÷ |B|, and the frequency is 1 ÷ P = |B| ÷ 2π (cycles per unit of x).
• Phase shift h = C ÷ B, because A sin(Bx − C) = A sin(B(x − C/B)). A positive h moves the graph right; a negative h moves it left; 0 means no horizontal shift.
• Vertical shift D: the midline is y = D, the maximum is D + |A| and the minimum is D − |A|.

If your function has Bx + C, type −C: sin(x + π/6) has C = −π/6. The chart draws y from h − P/4 to h + 7P/4 and marks x = h.

What you can type

• A and D: a whole number, a decimal, a fraction or a mixed number (3, −0.5, 1/3, 1 1/2), read exactly. Commas and exponents are not accepted.
• B and C: a number (2, −0.5, 1,000; commas only between groups of three digits) with an optional whole number below it (3/4), or a multiple of π: π or pi, optionally with a number in front (2π, 0.5pi, 2*π) and a whole number below it (π/4, −3π/2). One pair of outer brackets is allowed, and spaces around π, / and * are ignored. Up to 30 characters. B and C are read exactly as a fraction r times π⁰ or π¹.
• A = 0 (a flat line) and B = 0 (no period) have no answer, and so does text in B or C that is not one of the forms above.

Output format

• Exact forms. The phase shift is r × π^k with k = 1 (C has π and B does not), 0 (both or neither have π) or −1 (B has π and C does not); the period is r × π^k with k = 1 when B has no π and 0 when it has. Each is written with r in lowest terms and the true minus sign: k = 0 as an integer or a fraction (8, −2, 3/2); k = 1 as nπ/d (π, 2π, π/4, −3π/2; 1 is not written, nor /1); k = −1 as n/π or n/(dπ) (1/π, 3/(4π)).
• The headline is the size of the phase shift followed by "to the right" or "to the left", or "0 (no horizontal shift)".
• The decimals (phase shift, amplitude, period, frequency, vertical shift, maximum, minimum) show to 6 significant figures. π is a 64-bit float; the amplitude, vertical shift, maximum and minimum are exact from the typed values, rounded once.

Worked examples by hand

y = 3 sin(2x) + 1. A = 3, B = 2, C = 0, D = 1: amplitude 3, period 2π ÷ 2 = π, phase shift 0, midline y = 1.

f(x) = 3 sin((π/4)x − π/4). B = π/4 and C = π/4: phase shift (π/4) ÷ (π/4) = 1, to the right; period 2π ÷ (π/4) = 8; maximum 3, minimum −3.

y = −2 cos((π/2)x + π) + 3. B = π/2 and C = −π: phase shift −π ÷ (π/2) = −2, 2 to the left; period 2π ÷ (π/2) = 4; amplitude 2; maximum 3 + 2 = 5, minimum 3 − 2 = 1.

f(x) = sin(x + π/6) − 2. B = 1, C = −π/6: phase shift −π/6 = −0.523599, π/6 to the left; period 2π; the graph is 2 units down.

f(x) = sin((π/6)x). B = π/6: period 2π ÷ (π/6) = 12; frequency 1/12 = 0.0833333.

y = 0.5 cos(4x − 1). B = 4, C = 1: phase shift 1/4 = 0.25, to the right; period 2π ÷ 4 = π/2 = 1.5708.

y = 2 sin(πx − 1). B = π, C = 1: phase shift 1 ÷ π = 1/π = 0.31831, to the right; period 2π ÷ π = 2; frequency 0.5.