Reshaping a sinusoid with three parameters

The general form of a sinusoid is often written with three adjustable parameters: scaling, frequency, and phase. Each one independently controls a different aspect of the waveform. This walkthrough covers what each parameter does and how to compute the values you need for a desired period.

Amplitude scaling

The simplest adjustment is scaling. Multiplying the function by a constant changes its amplitude — the maximum height of the wave in both positive and negative directions. Setting the scaling factor to 2, for example, makes the output twice as large everywhere compared to the baseline function.

Frequency and period

Frequency and period are inversely related. The standard sinusoid has a period of x, meaning it completes one full cycle and then repeats. If you multiply the input variable by a factor like 2, the function runs through its cycle twice as fast, because the input to the sinusoid doubles for any given change in the independent variable.

Generalizing, a sinusoid with input multiplied by the frequency parameter w repeats every 2π / ω units. Conversely, if you know the period you need, you can solve for the required frequency parameter directly.

Phase shifting

The third parameter, \theta, is the phase. It does not change the shape or frequency of the wave; it only shifts it along the horizontal axis. With a positive phase, the entire waveform is moved left — events like the positive peak and zero crossings occur earlier than in the baseline function. A negative phase shifts the wave to the right, making everything happen later.

Summary of parameters

In the general sinusoid equation, each parameter has a distinct role:

  • A controls the amplitude scaling.
  • w sets the frequency, which determines the repetition period.
  • \theta is the phase, shifting the wave left or right.
[1]The math level of this post is high-school, at best. My main goal here is to test how to integrate interactive demos into my blog posts.
[2]This can be a bit counter-intuitive at first; we scale w by 2, but the period scales by half. Why? The reason is that w affects the sinusoid’s domain, while the period is a property of its range. Therefore, an inverse relation is reasonable, once we put more thought into it. In fact, w is often called the angular frequency of the sinusoid, and frequency is inverse proportional to the period.