Example: Determine the period, phase shift, and vertical asymptotes of the function, then sketch at least two periods.
\[ \Omega(t) = \tan\left(\frac{{t}}{{3}}-\frac{\pi}{{6}}\right) \]Solution
- Period: \(3\pi\)
- Phase Shift: \(\frac{\pi}{{2}}\)
- Vertical Asymptotes: Start at the phase shift (\(\pi/2\)) then subtract a half-period, \(3\pi/2\): \(\pi/2-3\pi/2 = -\pi\). Next, add or subtract the period as many times as necessary to indicate the asymptotes.\(x=-4\pi,\;x=-\pi,\;x=2\pi,\dots\)
This is a positive tangent, so going up to the right between asymptotes. The \(x\) intercepts will be at the phase shift, \(\pi/2\), plus and minus the period as many times as necessary, so \((-5\pi/2, 0),\;(\pi/2,0),\;(7\pi/2),\dots\) Graph: The \(y\) values are not required to be precise, just the values noted above, so: