Amplitude Period Phase Shift Calculator
Enter the coefficients of a sinusoid y = A sin(Bx + C) + D to instantly read off its amplitude, period, phase shift, vertical shift, and frequency.
One full cycle in units of x
- 1
Amplitude |A|
|2| = 2 - 2
Phase shift −C ÷ B
−1 ÷ 3 = -0.3333Set Bx + C = 0 and solve for x to find where the cycle starts. - 3
Period 2π ÷ |B|
2π ÷ |3| = 2π ÷ 3 = 2.0944
How does this calculator work?
For y = A sin(Bx + C) + D, the amplitude is |A|, the period is 2π/|B|, the phase shift is -C/B (positive means rightward), and the vertical shift is D. Frequency equals |B|/(2π) and the midline is y = D. Enter the four coefficients to read every property at once.
Formula
How this is calculated
A general sinusoid is written y = A·sin(Bx + C) + D. The coefficient A controls the vertical stretch, so the amplitude (the maximum distance from the midline) is |A|. The midline itself is the horizontal line y = D, and D is called the vertical shift because it moves the whole wave up or down.
The coefficient B compresses or stretches the wave horizontally. Since sine repeats every 2π radians of its argument, one full cycle of x spans a period of 2π/|B|. The frequency, the number of cycles per unit of x, is the reciprocal scaled by 2π: |B|/(2π). To find where the basic cycle begins, set the argument Bx + C = 0, giving x = -C/B; this is the phase shift. A positive value shifts the graph to the right and a negative value to the left.
The inputs are treated as the literal coefficients A, B, C, and D, with x and the argument measured in radians. B must be non-zero, otherwise the period and phase shift are undefined (the function would be constant). The graph below samples y over two full periods starting at x = 0 so you can see the shape, amplitude, and offset directly.
Frequently asked questions
The cycle starts where the argument Bx + C equals zero. Solving for x gives x = -C/B, so the horizontal compression by B must be divided out. Only when B = 1 does the phase shift equal -C.
A positive phase shift moves the entire graph to the right along the x-axis by that amount; a negative value moves it to the left.
Yes. A cosine y = A cos(Bx + C) + D has the same amplitude, period, frequency, and vertical shift. Only the starting point of the cycle (phase reference) differs, since cosine leads sine by a quarter period.
Also known as
TG we-Calculate Editorial Team. (2026). Amplitude Period Phase Shift Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/amplitude-period-phase-shift-calculator
TG we-Calculate Editorial Team. "Amplitude Period Phase Shift Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/amplitude-period-phase-shift-calculator.
TG we-Calculate Editorial Team, "Amplitude Period Phase Shift Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/amplitude-period-phase-shift-calculator
@misc{wecalculate_amplitude_period_phase_shift_calculator, title = {Amplitude Period Phase Shift Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/amplitude-period-phase-shift-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?