Answer: (a) \(8\pi\). (b) \(a=4\), \(b=\frac18\), \(c=3\).

Answer: (a) \(8\pi\). (b) \(a=4\), \(b=\frac18\), \(c=3\).

(a) For a tangent graph, the distance between consecutive vertical asymptotes is one period.

The asymptotes are at \(x=-4\pi\) and \(x=4\pi\), so the period is

\(4\pi-(-4\pi)=8\pi\).

(b) For \(y=a\tan bx+c\), the period is \(\frac{\pi}{b}\). Hence

\(\displaystyle \frac{\pi}{b}=8\pi\),

so \(b=\frac18\).

Since the graph passes through \(P(0,3)\),

\(3=a\tan0+c\),

so \(c=3\).

Now use \(Q(2\pi,7)\):

\(7=a\tan\left(\frac18\cdot2\pi\right)+3\).

Thus

\(7=a\tan\frac{\pi}{4}+3=a+3\),

so \(a=4\).