Why does a positive φ shift the graph left instead of right?

Because sin(x + φ) reaches any given function value at a smaller x than sin(x) does — the function arrives at the same phase earlier. This is consistent with the general rule for y = f(x + a).

How does ω relate to the period?

The standard sine function sin(x) has period 2π. When the argument becomes ωx, one full cycle completes in T = 2π/ω. A larger ω means a shorter period and more waveforms per unit length.

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Parameters

Amplitude (A)1.0

Angular Frequency (ω)1.0

Phase Shift (φ)0.00π

Vertical Offset (k)0.0

Show baseline y = sin(x)

Trigonometric Function Transformations

Adjust A, ω, φ, and k in real time to observe vertical scaling, horizontal compression, phase shifting, and vertical displacement of a sine curve — and build precise intuition for each parameter.

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Amplitude (A)

Controls vertical scaling. |A| determines the distance between maximum and minimum values; a negative A reflects the curve across the x-axis.

Angular Frequency (ω)

Controls horizontal scaling. Period T = 2π/ω — higher ω compresses the waveform into a shorter interval.

Phase Shift (φ)

Controls horizontal displacement. The actual shift distance is φ/ω; positive φ shifts the graph left, negative φ shifts it right.

Vertical Offset (k)

Translates the entire curve along the y-axis. The axis of symmetry moves from y = 0 to y = k.

How Do Parameters Affect the Graph?

The general sinusoidal function y = Asin(ωx + φ) + k has four independent parameters, each controlling a distinct transformation: vertical scaling, horizontal scaling, horizontal translation, and vertical translation.

Use this simulator to adjust each parameter individually and observe its effect on the sine curve in real time. Toggle the baseline y = sin(x) overlay to see exactly what has changed.

Recommended reading order for exams: determine A from the max–min range, find ω from the distance between consecutive peaks, locate φ using a known reference point (zero crossing or extremum), and identify k from the axis of symmetry.