Calculate phase shift, amplitude, period, and vertical shift for y = A·sin(Bx + C) + D. Visualize the function with an interactive SVG graph and key-point tables.
Every sinusoidal function can be written in the standard form y = A·sin(Bx + C) + D, where four parameters control the shape and position of the wave. The amplitude |A| determines the height, the period 2π/|B| determines how wide each cycle is, the phase shift -C/B moves the wave left or right, and the vertical shift D moves the midline up or down.
Phase shift is especially important in physics and engineering: in AC circuits, the relative phase between voltage and current determines power delivery. In signal processing, phase alignment is critical for interference and superposition. In music, phase differences produce constructive and destructive interference patterns.
This calculator takes the four parameters A, B, C, D and any of sin, cos, or tan, then computes all wave properties: amplitude, period, frequency, phase shift (in radians and degrees), vertical shift, range, and midline. A live SVG graph shows the wave with phase shift and midline markers, and a key-points table shows the y-values at evenly spaced x-positions across the first period.
Graphing sinusoidal functions by hand requires computing multiple parameters, plotting key points, and drawing smooth curves — all of which are error-prone and time-consuming. This calculator performs all the analysis instantly and provides a publication-quality SVG graph.
It is especially valuable for precalculus and trigonometry students who need to understand how each parameter transforms the parent function, and for physics or engineering students analyzing waveforms, AC circuits, or signal processing.
Phase shift = -C/B. Amplitude = |A|. Period = 2π/|B| for sin/cos, π/|B| for tan. Frequency = 1/period. Midline: y = D. Range: [D - |A|, D + |A|] for sin/cos, all reals for tan.
Result: Amplitude = 3, Period ≈ 3.14, Phase shift ≈ -0.39 rad (−22.5°), Midline = -1
For y = 3·sin(2x + π/4) − 1: amplitude |3| = 3, period = 2π/2 = π ≈ 3.14, phase shift = −(π/4)/2 = −π/8 ≈ −0.393 rad (shift left 22.5°), vertical shift = −1.
The equation y = A·sin(Bx + C) + D is not just a math exercise — it models countless real phenomena. Sound waves, electromagnetic radiation, ocean tides, cardiac rhythms, and seasonal temperature patterns all follow sinusoidal models. The four parameters map directly to physical properties: A → loudness (sound) or intensity (light), B → pitch (sound) or color (light), -C/B → timing offset, and D → baseline level.
For example, the daily temperature in a city can be modeled as T(t) = A·sin(2π/365 · (t - φ)) + D, where A is half the annual range, φ is the day of peak temperature, and D is the annual average. The phase shift gives the lag between the solstice and the hottest day — typically about 3-4 weeks.
In signal processing, relative phase is everything. Two signals of the same frequency can add constructively (in phase, φ = 0°) or destructively (out of phase, φ = 180°). This principle underlies noise-canceling headphones, radio antenna arrays, and Fourier analysis. The Discrete Fourier Transform decomposes any signal into sinusoidal components, each with its own amplitude and phase — the phase spectrum carries as much information as the amplitude spectrum.
Euler's formula e^(ix) = cos(x) + i·sin(x) provides the deepest view of phase: a complex exponential rotates in the complex plane, and the phase determines the starting angle. Engineers use the phasor representation V = V₀·e^(iφ) to simplify AC circuit analysis, converting differential equations into algebraic ones. The phase shift between voltage and current phasors determines the power factor of the circuit.
C is the phase parameter in the equation y = A·sin(Bx + C) + D. The actual horizontal shift is -C/B, not just C or -C. The division by B accounts for the frequency coefficient.
For a sine function, find where the wave crosses the midline going upward — that x-value is the phase shift. For cosine, find the first maximum. The distance from x = 0 to that point is the phase shift.
Yes. A negative A reflects the wave across the midline (y = D). The amplitude is always |A| (positive), but the shape is inverted — peaks become troughs and vice versa.
Phase shift has the same units as x. If x is in radians (standard math), phase shift is in radians. If x represents time, phase shift is in the same time units. The calculator shows both radians and degrees.
In AC circuits, the voltage V(t) = V₀·sin(ωt) and current I(t) = I₀·sin(ωt + φ) may have a phase difference φ. This difference determines real vs. reactive power: P = V₀I₀·cos(φ)/2.
Yes. A negative B reflects the function horizontally (since sin(-x) = -sin(x)). The period formula uses |B|, so the period is always positive. The phase shift -C/B reverses direction when B changes sign.