Question

How do you find the amplitude and period for `y=3sin2 (x-pi/6)`?

Answer 1

Period: `pi`
Amplitude: `6`

Explanation:

We start analyzing the argument of sin in `y=3sin2 (x-pi/6)`:
If the function was `y=sinx` we would have a period of `2pi` because the sine function makes a complete oscillation (0,1,0,-1,0) in `2pi`.
If we sum anything to the argument (for example like the function `y=sin(x+999pi+sqrt(7))`), we make a horizontal translation, so we won't change neither the period or the amplitude.
So we can consider a similar function with the same amplitude and period that is easier to study: `y=3sin2x`.

Amplitude
If we consider that the sine function has a range `[-1,1]`, its amplitude is 2. If the argument varies in all `RR`, the range won't change. For example `y=sin(sinx+34x)` would have always the same amplitude (2) and range ([-1,1]).
What really matters in amplitude is the vertical dilation/compression factor outside of sin(x): in this case we would have `y=3sinx` that means a range of `[-3,3]` so an amplitude of `3-(-3)=6`.

Period
If we consider the sine function, it has a period of `2pi`. The cohefficient inside the argument is a horizontal dilation/compression of the function. We could find the value of the period deviding `2pi` by this cohefficient. For example in this case we should have `(2pi)/2=pi`.

Verify
As we can see in this graph, the range is [-3,3], and the function is plotted in the interval `[0,pi]`. Zooming out, we can see that it's periodic and it's obviously translated.

graph{3sin(2(x-pi/6)) [0, 3.14, -3, 3]}