Chapter 8, Problem 5CR

To determine

To graph: The function $y=-2\cos \left(2x-\pi \right)$

Explanation of Solution

Given information:

The function $y=-2\cos \left(2x-\pi \right)$

Graph:

To get the graph of $y=-2\cos \left(2x-\pi \right)$ perform the transformations as follows:

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

  

Step 2: Shift the graph of $y=\cos x$ by $\frac{\pi }{2}$ units to the right to get the graph of $y=\cos \left(x-\frac{\pi }{2}\right)$

The graph of $y=\cos \left(x-\frac{\pi }{2}\right)$ is

  

Step 3: To get the graph of $y=\cos \left(2x-\pi \right)$, stretch the graph of $y=\cos \left(x-\frac{\pi }{2}\right)$ horizontally by $\frac{1}{2}$ units .

  

Step 4: Vertically, stretch the graph by factor $2$ units to get the graph of $y=2\cos \left(2x-\pi \right)$

The graph of $y=2\cos \left(2x-\pi \right)$ is

  

Step 5: To get the graph of $y=-2\cos \left(2x-\pi \right)$, reflect the graph of $y=2\cos \left(2x-\pi \right)$ about $x-$ axis.

The graph of $y=-2\cos \left(2x-\pi \right)$ is

  

Interpretation:

The graph represents function of $y=-2\cos \left(2x-\pi \right)$