Sketch the Graph of f(x) = cot(πx/2)
Question:
Sketch the graph of the following function :
\[ f(x)=\cot\frac{\pi x}{2} \]
Solution:
We know that
\[ \cot\theta=\frac{\cos\theta}{\sin\theta} \]
Therefore
\[ f(x)=\cot\frac{\pi x}{2} \]
The cotangent graph decreases from \(+\infty\) to \(-\infty\) between consecutive asymptotes.
Whenever
\[ \sin\frac{\pi x}{2}=0 \]
the function becomes undefined.
Thus vertical asymptotes occur at
\[ \frac{\pi x}{2}=n\pi \Rightarrow x=2n \]
Important properties:
- Period \(=\dfrac{\pi}{\pi/2}=2\)
- Vertical asymptotes at \(x=0,\pm2,\pm4,\dots\)
- The graph decreases continuously in each interval
Now calculate some important points:
\[ \begin{aligned} x=\frac12 &\Rightarrow y=\cot\frac{\pi}{4}=1\\[8pt] x=1 &\Rightarrow y=\cot\frac{\pi}{2}=0\\[8pt] x=\frac32 &\Rightarrow y=\cot\frac{3\pi}{4}=-1 \end{aligned} \]
Thus the graph passes through the points
\[ \left(\frac12,1\right),\quad (1,0),\quad \left(\frac32,-1\right) \]
The pattern repeats after every interval
\[ 2 \]
Plot these points and draw smooth cotangent curves approaching the vertical asymptotes.
Hence, the required graph is shown above.
Graph Features:
- Period \(=2\)
- Vertical asymptotes at \(x=2n\)
- The graph decreases continuously in each interval
- Zeros occur at \(x=1,3,5,\dots\)