8.2 Graphs of the other trigonometric functions  (Page 6/9)

• Step 1. The given function is already written in the general form, $y=A\csc \left(Bx\right).$
• Step 2. $\left|A\right|=\left|-3\right|=3,$ so the stretching factor is 3.
• Step 3. $B=4,$ so $P=\frac{2\pi }{4}=\frac{\pi }{2}.$ The period is $\frac{\pi }{2}$ units.
• Step 4. Sketch the graph of the function $g(x)=\mathrm{-3}\sin (4x).$
• Step 5. Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function .
• Steps 6–7. Sketch three asymptotes at $x=0,x=\frac{\pi }{4},$ and $x=\frac{\pi }{2}.$ We can use two reference points, the local maximum at $\left(\frac{\pi }{8},\mathrm{-3}\right)$ and the local minimum at $\left(\frac{3\pi }{8},3\right).$ [link] shows the graph.

  

• Step 1. Express the function given in the form $y=2\csc \left(\frac{\pi }{2}x\right)+1.$
• Step 2. Identify the stretching/compressing factor, $\left|A\right|=2.$
• Step 3. The period is $\frac{2\pi }{\left|B\right|}=\frac{2\pi }{\frac{\pi }{2}}=\frac{2\pi }{1}\cdot \frac{2}{\pi }=4.$
• Step 4. The phase shift is $\frac{0}{\frac{\pi }{2}}=0.$
• Step 5. Draw the graph of $y=A\csc (Bx)$ but shift it up $D=1.$
• Step 6. Sketch the vertical asymptotes, which occur at $x=0,x=2,x=4.$

The graph for this function is shown in [link] .