Sketch the Graph of f(x) = 3 sec x
Question:
Sketch the graph of the following function :
\[ f(x)=3\sec x \]
Solution:
We know that
\[ \sec x=\frac{1}{\cos x} \]
Therefore
\[ f(x)=3\sec x=\frac{3}{\cos x} \]
The graph of secant is obtained from the graph of cosine.
Whenever
\[ \cos x=0 \]
the function becomes undefined.
Thus vertical asymptotes occur at
\[ x=\frac{\pi}{2},\ \frac{3\pi}{2},\ \frac{5\pi}{2},\dots \]
Important properties:
- Period \(=2\pi\)
- Range \(y\le -3\) or \(y\ge 3\)
- Vertical asymptotes where \(\cos x=0\)
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=3\sec0=3\\[8pt] x=\pi &\Rightarrow y=3\sec\pi=-3\\[8pt] x=2\pi &\Rightarrow y=3\sec2\pi=3 \end{aligned} \]
Thus the graph passes through the points
\[ (0,3),\quad (\pi,-3),\quad (2\pi,3) \]
Plot these points and draw the secant curves approaching the vertical asymptotes.
Hence, the required graph is shown above.