Question

Use transformations to graph the equation$y=\cos \left(x-\frac{\mathrm{\pi }}{2}\right)-1$.

Short answer

The graph is

Step-by-step solution

Step 1. Given Information 

We are given the equation $y=\cos \left(x-\frac{\mathrm{\pi }}{2}\right)-1$and we need to use transformations to graph it.

Step 2. Using transformations 

From the concept of the vertical and horizontal shift we know, the graph of $g\left(x\right)=f(x+c)+k$will be the graph of $f\left(x\right)$shifted left or right by $c$units depending upon the sign and up or down by $k$units depending upon the sign.

The equation can be written as:

$$\begin{gathered} y=\cos \left(x-\frac{\mathrm{\pi }}{2}\right)-1 \\ \Rightarrow y+1=\cos \left(x-\frac{\mathrm{\pi }}{2}\right) \end{gathered}$$

Let $$\begin{gathered} X\equiv x-\frac{\mathrm{\pi }}{2} \\ Y\equiv y+1 \end{gathered}$$

The lines pass through $X=0,Y=0$.

$$\begin{gathered} \therefore x-\frac{\mathrm{\pi }}{2}=0 \\ \Rightarrow x=\frac{\mathrm{\pi }}{2} \\ y+1=0 \\ \Rightarrow y=-1 \end{gathered}$$

Step 3. Plotting the graph 

The graph of the equation is