Chapter 6, Problem 25CT

To determine

To graph: The function $y=2\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$

Explanation of Solution

Given information:

The function $y=2\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$

Graph:

To get the graph of $y=2\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$ perform the transformations as follows:

Step 1: Begin with the graph of the base function $y=\sin x$

  

Step 2: Shift the graph horizontally to the right by $\frac{\pi }{6}$ units gives $y=\sin \left(x-\frac{\pi }{6}\right)$

  

Step 3: Stretch the graph horizontally by $\frac{1}{3}$ units gives $y=\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$ .

  

Step 4: Vertically, stretch the graph by factor $2$ units gives $y=2\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$ .

The graph of the function $y=2\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$ is as follows:

  

Interpretation:

The graph represents function of $y=2\sin \left(\frac{x}{3}-\frac{\pi }{6}\right)$