Chapter 7, Problem 6CR

To determine

To graph: The equation $y=\cos \left(x-\frac{\pi }{2}\right)-1$ using transformations.

Explanation of Solution

Given information:

The equation, $y=\cos \left(x-\frac{\pi }{2}\right)-1$ .

Graph:

To get the graph of $y=\cos \left(x-\frac{\pi }{2}\right)-1$ ,begin with the graph of the equation $y=\cos x$ .

  

If a function $y=f\left(x\right)$ transforms to $y=f\left(x-a\right),a>1$ then graph of $y=f\left(x\right)$ shifts horizontally ‘ $a$ ’units to the right.

Hence, if a graph of $y=\cos x$ transforms to $y=\cos \left(x-\frac{\pi }{2}\right)$ ,where $\frac{\pi }{2}>1$ it is shifted horizontally ‘ $\frac{\pi }{2}$ ’units to the right.

To obtain the graph of $y=\cos \left(x-\frac{\pi }{2}\right)$, shift the graph of $y=\cos x$ horizontally‘ $\frac{\pi }{2}$ ’units to the right.

  

To obtain the graph of $y=\cos \left(x-\frac{\pi }{2}\right)-1$, shift down the graph of $y=\cos \left(x-\frac{\pi }{2}\right)$ by $1$ unit vertically downward.

  

Interpretation:

The graph of $y=\cos \left(x-\frac{\pi }{2}\right)-1$ is the graph of $y=\cos x$ horizontally shifted‘ $\frac{\pi }{2}$ ’units to the rightand then vertically shifted $1$ unit downward.