Question

How do you find the amplitude, period, and phase shift for a sine function?

Answer 1

If you have `Asin(omega x + phi)`, then:

• `A` is the amplitude;
• `(2pi)/omega` is the period;
• `phi` is the shift.

Explanation:

The most general case of a sine function is

`Asin(omega x + phi)`. Let's see what they do.

• Since the sine function is bounded between `-1` and `1`, if we multiply the function by `A` the result will be bounded between `-A` and `A`. So, the amplitude is "decided" by the outter coefficient.

• The `omega` coefficient affects the period: we know that "normally" `x` runs a full period from `0` to `2pi`. Multipling `x` by a factor `w` means to change the speed of the forementioned run. For example, if you change your variable from `x` to `2x`, your variable will be running at twice the speed, and so you will go from `0` to `2pi` as `x` goes from `0` to `pi`.
  On the other hand, if you change from `x` to `x/2`, you will go at half the speed, and so `x/2` goes from `0` to `2pi` (i.e, runs a full period) as `x` goes from `0` to `4pi`.
  So, we see that the period is `(2pi)/omega`

• Finally, the term `phi` is the shift, it simply means that instead of starting from zero, you start from `phi`: take for example `sin(x)` and `sin(x+pi/3)`. When `x` is zero, `sin(x+pi/3)` is "already" at `pi/3`, representing what I said before: the function starts from `phi` at not from zero.