Question

graph each function for ( 0 leq x leq 2 pi ). find the period of each function.

1. ( y = sin \frac { 1 } { 6 } x )
2. ( y = sin 4 x )

Explanation:

Step1: Recall the period formula for sine function

The general form of a sine function is \(y = A\sin(Bx - C)+D\), and its period \(T=\frac{2\pi}{|B|}\).

Step2: Identify \(B\) for \(y = \sin\frac{1}{6}x\)

For the function \(y=\sin\frac{1}{6}x\), \(B = \frac{1}{6}\).

Step3: Calculate the period

Using the formula \(T=\frac{2\pi}{|B|}\), substitute \(B=\frac{1}{6}\). Then \(T=\frac{2\pi}{\frac{1}{6}}= 12\pi\).

Step4: Identify \(B\) for \(y=\sin4x\)

For the function \(y = \sin4x\), \(B = 4\).

Step5: Calculate the period

Using the formula \(T=\frac{2\pi}{|B|}\), substitute \(B = 4\). Then \(T=\frac{2\pi}{4}=\frac{\pi}{2}\).

Answer:

The period of \(y=\sin\frac{1}{6}x\) is \(12\pi\), and the period of \(y=\sin4x\) is \(\frac{\pi}{2}\).