graph the function \n$y = \\frac{6}{5} \\cot x$\n\noptions: a, b, c, d (each with a graph)

Explanation:

Step1: Analyze the general form of cotangent function

The general form of the cotangent function is (y = A\cot(Bx - C)+D). For the function (y=\frac{6}{5}\cot x), we have (A = \frac{6}{5}), (B = 1), (C=0), (D = 0). The period of the cotangent function (y = A\cot(Bx)) is (\frac{\pi}{|B|}), so the period of (y=\frac{6}{5}\cot x) is (\pi) (since (B = 1)). The vertical asymptotes of the basic cotangent function (y=\cot x) are at (x = n\pi), (n\in\mathbb{Z}). For (y=\frac{6}{5}\cot x), the vertical asymptotes are also at (x=n\pi), (n\in\mathbb{Z}). When (x=\frac{\pi}{2}), (y=\frac{6}{5}\cot(\frac{\pi}{2})=0). The amplitude - like factor (|A|) affects the "steepness" of the graph. Since (A=\frac{6}{5}), the graph of (y = \frac{6}{5}\cot x) is a vertical stretch of the graph of (y=\cot x) by a factor of (\frac{6}{5}).

Step2: Eliminate wrong options

• Option A: The basic cotangent function (y=\cot x) has a graph that decreases in each period. If we consider the sign, when (x\in(0,\pi)), (\cot x>0) for (x\in(0,\frac{\pi}{2})) and (\cot x < 0) for (x\in(\frac{\pi}{2},\pi)). Option A has positive values on both sides of the (y) - axis in the interval ((-\pi,\pi)) which is wrong for (y=\frac{6}{5}\cot x).
• Option B: The basic cotangent function (y = \cot x) is positive in ((0,\frac{\pi}{2})) and negative in ((\frac{\pi}{2},\pi)). Option B has the wrong sign pattern.
• Option D: When (x=\frac{\pi}{2}), (y = 0). Option D does not pass through the point ((\frac{\pi}{2},0))

Answer:

C.