graph two periods of the given cotangent function.\n\n( y = - 2 cot \frac { pi } { 3 } x )\n\nchoose the…

Explanation:

Step1: Find the period

For the cotangent function (y = A\cot(Bx - C)+D), the period is (T=\frac{\pi}{|B|}). Here (B = \frac{\pi}{3}), so (T=\frac{\pi}{\frac{\pi}{3}}=3).

Step2: Analyze the transformation

The function (y=- 2\cot(\frac{\pi}{3}x)) has a vertical stretch by a factor of (2) and a reflection about the (x) - axis compared to the basic cotangent function (y = \cot(x)). The basic cotangent function (y=\cot(x)) has vertical asymptotes at (x = n\pi), (n\in\mathbb{Z}). For (y=-2\cot(\frac{\pi}{3}x)), the vertical asymptotes are found by solving (\frac{\pi}{3}x=n\pi), (x = 3n), (n\in\mathbb{Z}). When (x=\frac{3}{2}), (y=-2\cot(\frac{\pi}{3}\times\frac{3}{2})=-2\cot(\frac{\pi}{2}) = 0)

Answer:

C.