Scaling, stretching and shifting sinusoids

Tags Math

This is a brief and simple [1] explanation of how to adjust the standard sinusoid sin(x) to change its amplitude, frequency and phase shift. More precisely, given the general function:

\[s(x)=A\cdot sin(w\cdot x+\theta)\]

We’ll see how adjusting the parameters A, w and \theta affect the shape of s(x). Each section below covers one of these aspects mathematically, and you can use the demo at the bottom to experiment with the topic visually.

Scaling

Scaling is conceptually the simplest change; we adjust A to increase or decrease the amplitude (maximal height) of s(x). Setting A=2 will make the y value twice as large (in both the positive and negative direction) as the original function.

Stretching

Stretching changes the frequency of sin(x), which is inverse proportional to its period. The baseline function sin(x) has a period of 2\pi, meaning it repeats every 2\pi. In other words, sin(x)=sin(x+2\pi) for any x.

If we set w=2, we get sin(2x). This function repeats itself twice as fast as sin(x), because x is multiplied by 2 before being fed into the sinusoid. If x changes by \pi, the sinusoid’s input changes by 2\pi. Therefore, the period of sin(2x) is \pi, the period of sin(4x) is \frac{\pi}{2} and so on. [2]

More generally, the period of sin(wx) is \frac{2\pi}{w}. Play with the demo below to see this in action, by changing w and observing how the waveform changes.

If we know the period p we want, we can easily calculate the w that gives us this period:

\[p=\frac{2\pi}{w} \implies w=\frac{2\pi}{p}\]

Shifting

The final parameter we discuss is \theta; it’s called the phase of the sinusoid. In the baseline sin(x), \theta=0. The sinusoid is 0 at x=0, achieves its positive peak at x=\frac{\pi}{2}, crosses 0 again at x=\pi, negative peak at x=\frac{3\pi}{2} and returns to its original position at x=2\pi where the repetition begins.

By adding a non-zero \theta, we don’t affect the sinusoid’s amplitude or frequency, but we do shift it right or left along the x axis. For example, suppose we use the function sin(x+\theta) with \theta=\frac{\pi}{2}. Then when x=0, we have sin(\frac{\pi}{2}), so the sinusoid is already at its positive peak; at x=\frac{\pi}{2}, the sinusoid crosses 0 into the negatives, etc. Everything happens earlier (by exactly the value of \theta=\frac{\pi}{2}) than in the baseline sinusoid. In other words, we’ve shifted the function left by \frac{\pi}{2}. Similarly, when \theta is negative, everything happens later, and the function is shifted right.

Putting it all together

We’ve now gone over all the parameters for the function:

\[s(x)=A\cdot sin(w\cdot x+\theta)\]
  • A controls the scaling factor (amplitude).
  • w is the frequency and controls the repetition period
  • \theta controls the phase - how much the sinusoid is shifted left or right

Use the demo below to adjust these parameters and observe their effect on the sinusoid:

Your browser does not support the HTML5 canvas tag.
[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.

For comments, please send me an email.