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

1 Answer
Jun 19, 2015

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]}