Sketch the Graph of f(x) = tan 2x
Question:
Sketch the graph of the following function :
\[ f(x)=\tan2x \]
Solution:
We know that
\[ \tan\theta=\frac{\sin\theta}{\cos\theta} \]
Therefore
\[ f(x)=\tan2x \]
The tangent graph increases from \(-\infty\) to \(+\infty\) between consecutive asymptotes.
Whenever
\[ \cos2x=0 \]
the function becomes undefined.
Thus vertical asymptotes occur at
\[ 2x=\frac{\pi}{2}+n\pi \Rightarrow x=\frac{\pi}{4}+\frac{n\pi}{2} \]
Important properties:
- Period \(=\dfrac{\pi}{2}\)
- Vertical asymptotes at \(x=\dfrac{\pi}{4}+\dfrac{n\pi}{2}\)
- The graph increases continuously in each interval
Now calculate some important points:
\[ \begin{aligned} x=-\frac{\pi}{8} &\Rightarrow y=\tan\left(-\frac{\pi}{4}\right)=-1\\[8pt] x=0 &\Rightarrow y=\tan0=0\\[8pt] x=\frac{\pi}{8} &\Rightarrow y=\tan\frac{\pi}{4}=1 \end{aligned} \]
Thus the graph passes through the points
\[ \left(-\frac{\pi}{8},-1\right),\quad (0,0),\quad \left(\frac{\pi}{8},1\right) \]
Plot these points and draw smooth tangent curves approaching the vertical asymptotes.
Hence, the required graph is shown above.
Graph Features:
- Period \(=\dfrac{\pi}{2}\)
- Vertical asymptotes at \(x=\dfrac{\pi}{4}+\dfrac{n\pi}{2}\)
- The graph increases continuously in each interval
- Zeros occur at \(x=\dfrac{n\pi}{2}\)