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# Graphs

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Below is the graph of y = a tan bx for some positive constants a and b. Find ab.

Jul 11, 2022

#1
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ab = 4.

Jul 11, 2022
#2
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The graph has period $$\frac{2 \pi}{3}.$$ The period of y = a tan bx is $$\frac{\pi}{b},$$ so $$b = \frac{3}{2}.$$ The graph is then of the form $$y = a \tan \left( \frac{3x}{2} \right).$$

Since the graph passes through $$\left( \frac{\pi}{6}, 2 \right)$$$$2 = a \tan \frac{\pi}{4} = a.$$Therefore, $$ab = 2 \cdot \frac{3}{2} = \boxed{3}.$$

Jul 11, 2022