Question

graph two periods of the given cotangent function.

( y = - 2 cot \frac { pi } { 3 } x )

choose the correct graph of two periods of ( y = - 2 cot \frac { pi } { 3 } x ) below.

a.
b.
c.

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.