6.2 Graphs of the other trigonometric functions  (Page 7/9)

• Step 1. Expressing the function in the form $f\left(x\right)=A\cot \left(Bx\right)$ gives $f\left(x\right)=3\cot \left(4x\right).$
• Step 2. The stretching factor is $\left|A\right|=3.$
• Step 3. The period is $P=\frac{\pi }{4}.$
• Step 4. Sketch the graph of $y=3\tan (4x).$
• Step 5. Plot two reference points. Two such points are $\left(\frac{\pi }{16},3\right)$ and $\left(\frac{3\pi }{16},\mathrm{-3}\right).$
• Step 6. Use the reciprocal relationship to draw $y=3\cot (4x).$
• Step 7. Sketch the asymptotes, $x=0,x=\frac{\pi }{4}.$

The orange graph in [link] shows $y=3\tan \left(4x\right)$ and the blue graph shows $y=3\cot \left(4x\right).$

• Step 1. The function is already written in the general form $f\left(x\right)=A\cot \left(Bx-C\right)+D.$
• Step 2. $A=4,$ so the stretching factor is 4.
• Step 3. $B=\frac{\pi }{8},$ so the period is $P=\frac{\pi }{\left|B\right|}=\frac{\pi }{\frac{\pi }{8}}=8.$
• Step 4. $C=\frac{\pi }{2},$ so the phase shift is $\frac{C}{B}=\frac{\frac{\pi }{2}}{\frac{\pi }{8}}=4.$
• Step 5. We draw $f\left(x\right)=4\tan \left(\frac{\pi }{8}x-\frac{\pi }{2}\right)-2.$
• Step 6-7. Three points we can use to guide the graph are $(6,2),(8,-2),$ and $(10,-6).$ We use the reciprocal relationship of tangent and cotangent to draw $f\left(x\right)=4\cot \left(\frac{\pi }{8}x-\frac{\pi }{2}\right)-2.$
• Step 8. The vertical asymptotes are $x=4$ and $x=12.$

The graph is shown in [link] .