Questions: Choose the correct graph of y=4 csc(2 x-pi/6).

Transcript text: Choose the correct graph of $y=4 \csc \left(2 x-\frac{\pi}{6}\right)$.

Solution

Solution Steps

Step 1: Identify the function and its properties

The given function is $ y = 4 \csc \left( 2x - \frac{\pi}{6} \right) $. The cosecant function, $ \csc(x) $, is the reciprocal of the sine function, $ \sin(x) $. The function $ y = 4 \csc \left( 2x - \frac{\pi}{6} \right) $ will have vertical asymptotes where the sine function is zero.

Step 2: Determine the period and phase shift

The period of the cosecant function $ \csc(bx - c) $ is $ \frac{2\pi}{b} $. Here, $ b = 2 $, so the period is $ \frac{2\pi}{2} = \pi $. The phase shift is determined by solving $ 2x - \frac{\pi}{6} = 0 $, which gives $ x = \frac{\pi}{12} $.

Step 3: Identify the vertical asymptotes and key points

The vertical asymptotes occur where $ \sin \left( 2x - \frac{\pi}{6} \right) = 0 $. Solving $ 2x - \frac{\pi}{6} = n\pi $ (where $ n $ is an integer), we get $ x = \frac{\pi}{12} + \frac{n\pi}{2} $. This gives asymptotes at $ x = \frac{\pi}{12}, \frac{13\pi}{12}, \frac{25\pi}{12}, \ldots $.

Final Answer

The correct graph is C. It shows the vertical asymptotes at $ x = \frac{\pi}{12} $ and $ x = \frac{25\pi}{12} $, and the correct shape of the $ 4 \csc \left( 2x - \frac{\pi}{6} \right) $ function.

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