Sketch the Graph of f(x) = 2 sec πx
Question:
Sketch the graph of the following function :
\[ f(x)=2\sec \pi x \]
Solution:
We know that
\[ \sec\theta=\frac{1}{\cos\theta} \]
Therefore
\[ f(x)=2\sec \pi x=\frac{2}{\cos \pi x} \]
The graph of secant is obtained from the graph of cosine. Vertical asymptotes occur wherever the cosine function becomes zero. :contentReference[oaicite:0]{index=0}
Whenever
\[ \cos \pi x=0 \]
the function becomes undefined.
Thus vertical asymptotes occur at
\[ \pi x=\frac{\pi}{2}+n\pi \Rightarrow x=\frac12+n \]
Important properties:
- Period \(=\dfrac{2\pi}{\pi}=2\)
- Range \(y\le -2\) or \(y\ge 2\)
- Vertical asymptotes at \(x=\frac12,\frac32,\frac52,\dots\)
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=2\sec0=2\\[8pt] x=1 &\Rightarrow y=2\sec\pi=-2\\[8pt] x=2 &\Rightarrow y=2\sec2\pi=2 \end{aligned} \]
Thus the graph passes through the points
\[ (0,2),\quad (1,-2),\quad (2,2) \]
Plot these points and draw the secant curves approaching the vertical asymptotes.
Hence, the required graph is shown above.
Graph Features:
- Period \(=2\)
- Range \(y\le -2\) or \(y\ge 2\)
- Vertical asymptotes at \(x=\frac12+n\)
- The graph opens upward and downward in alternate intervals