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

Jun 19, 2015

Period: $\pi$
Amplitude: $6$

#### Explanation:

We start analyzing the argument of sin in $y = 3 \sin 2 \left(x - \frac{\pi}{6}\right)$:
If the function was $y = \sin x$ we would have a period of $2 \pi$ because the sine function makes a complete oscillation (0,1,0,-1,0) in $2 \pi$.
If we sum anything to the argument (for example like the function $y = \sin \left(x + 999 \pi + \sqrt{7}\right)$), 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: $y = 3 \sin 2 x$.

Amplitude
If we consider that the sine function has a range $\left[- 1 , 1\right]$, its amplitude is 2. If the argument varies in all $\mathbb{R}$, the range won't change. For example $y = \sin \left(\sin x + 34 x\right)$ 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 $y = 3 \sin x$ that means a range of $\left[- 3 , 3\right]$ so an amplitude of $3 - \left(- 3\right) = 6$.

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

Verify
As we can see in this graph, the range is [-3,3], and the function is plotted in the interval $\left[0 , \pi\right]$. 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]}