Example: Determine the period, phase shift, and the vertical asymptotes of the function. Then sketch at least two full periods.
\[ \psi(x) = 4\tan\left(x-\frac{\pi}{{2}}\right) \]Solution
- Period: \(\pi\)
- Phase Shift: \(\frac{\pi}{{2}}\)
- Vertical Asymptotes: Start at the phase shift (\(\pi/2\)) then subtract a half-period (also \(\pi/2\). Then add or subtract the period as many times as necessary to indicate the asymptotes.\(x=0,\;x=\pi,\;x=2\pi,\;x=3\pi,\dots\)
The 4 indicates that 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 \((\pi/2, 0),\;(3\pi/2,0),\dots\)
Graph: The \(y\) values are not required to be precise, just the values noted above, so: