EXERCISE 10E 1 a Sketch the following functions for $0 \leqslant x \leqslant 3 \pi$ : - $y=\tan \left(x-\frac{\pi}{2}\right)$ if $y=-\tan x$ fil $y=\tan 3 x$

Step 1 ：First, we need to sketch the function \(y = \tan(x - \frac{\pi}{2})\) for \(0 \leq x \leq 3\pi\).

Step 2 ：Recall that the tangent function has a period of \(\pi\), so we can focus on the interval \(0 \leq x \leq \pi\) and then repeat the pattern for the rest of the interval.

Step 3 ：The graph of \(y = \tan(x)\) has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\) for any integer \(k\).

Step 4 ：Since the function \(y = \tan(x - \frac{\pi}{2})\) is a horizontal shift of \(y = \tan(x)\) to the right by \(\frac{\pi}{2}\), the vertical asymptotes will be at \(x = k\pi\) for any integer \(k\).

Step 5 ：The graph of \(y = \tan(x - \frac{\pi}{2})\) will have the same shape as \(y = \tan(x)\), but shifted to the right by \(\frac{\pi}{2}\).

Step 6 ：Now, we can sketch the graph of \(y = \tan(x - \frac{\pi}{2})\) for \(0 \leq x \leq 3\pi\) by drawing the tangent function with vertical asymptotes at \(x = k\pi\) and shifted to the right by \(\frac{\pi}{2}\).