Graphing Variations of the Cotangent Function

• 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 blue graph in Figure 14 shows $y=3\tan \left(4x\right)$ and the green graph shows $y=3\cot \left(4x\right).$

Figure 14