Sketch the Graph of f(x) = tan²x
Question:
Sketch the graph of the following function :
\[ f(x)=\tan^2x \]
Solution:
We know that
\[ \tan^2x=(\tan x)^2 \]
Since square of tangent is always non-negative, the graph always lies above the x-axis.
Whenever
\[ \cos x=0 \]
the tangent function becomes undefined.
Thus vertical asymptotes occur at
\[ x=\frac{\pi}{2}+n\pi \]
Important properties:
- Period \(=\pi\)
- Range \(y\ge0\)
- Vertical asymptotes at \(x=\frac{\pi}{2}+n\pi\)
- The graph is symmetric about the y-axis
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=\tan^20=0\\[8pt] x=\frac{\pi}{4} &\Rightarrow y=\tan^2\frac{\pi}{4}=1\\[8pt] x=-\frac{\pi}{4} &\Rightarrow y=\tan^2\left(-\frac{\pi}{4}\right)=1 \end{aligned} \]
Thus the graph passes through the points
\[ (0,0),\quad \left(\frac{\pi}{4},1\right),\quad \left(-\frac{\pi}{4},1\right) \]
Plot these points and draw smooth curves approaching the vertical asymptotes.
Hence, the required graph is shown above.
Graph Features:
- Period \(=\pi\)
- Range \(y\ge0\)
- Vertical asymptotes at \(x=\frac{\pi}{2}+n\pi\)
- The graph always lies above the x-axis
- The curve approaches infinity near the asymptotes