Example: Determine the period, phase shift, and vertical asymptotes of the function, then sketch at least two periods.
\[ \kappa(x) = \tan(4x-\pi) \]Solution
- Period: \(\frac{\pi}{{4}}\)
- Phase Shift: \(\frac{\pi}{{4}}\)
- Vertical Asymptotes: Start at the phase shift (\(\pi/4\)) then subtract a half-period, \(\pi/8\): \(\pi/4-\pi/8 = \pi/8\). Next, add or subtract the period as many times as necessary to indicate the asymptotes.\(x=-\pi/8,\;x=\pi/8,\;x=3\pi/8,\dots\)
This is a positive tangent, so going up to the right between asymptotes. The \(x\) intercepts will be at the phase shift, \(\pi/4\), plus and minus the period as many times as necessary, so \((-\pi/4, 0),\;(0,0),\;(\pi/4),\dots\) Graph: The \(y\) values are not required to be precise, just the values noted above, so: