Question

graph the function
$y = \frac{6}{5} \cot x$

options: 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 \(\frac{\pi}{|B|}\). Since \(B = 1\), the period is \(\pi\). The vertical asymptotes of \(y=\cot x\) 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 \(x=\frac{\pi}{2}\), \(y=\frac{6}{5}\cot(\frac{\pi}{2})=0\).

Step2: Check the key - point and asymptotes for each option

• Option A: The graph has incorrect behavior (not passing through \((\frac{\pi}{2},0)\) as per the function \(y = \frac{6}{5}\cot x\))
• Option B: The graph has incorrect asymptotes and shape (cotangent function has a different pattern than what is shown in this option)
• Option C: The function \(y=\frac{6}{5}\cot x\) has a period of \(\pi\), vertical asymptotes at \(x = n\pi\) (\(n\in\mathbb{Z}\)) and passes through \((\frac{\pi}{2},0)\). When \(x\) approaches \(0^{+}\), \(\cot x\to+\infty\) and when \(x\) approaches \(\pi^{-}\), \(\cot x\to-\infty\). The amplitude \(|A|=\frac{6}{5}\approx1.2\) (but for cotangent function, we mainly focus on asymptotes and passing - through points). The graph in option C has the correct vertical asymptotes (\(x = 0\) and \(x=\pi\) in the basic period \((0,\pi)\)) and passes through \((\frac{\pi}{2},0)\)
• Option D: The graph has incorrect sign and behavior (cotangent function \(y=\cot x\) is positive in \((0,\frac{\pi}{2})\) and negative in \((\frac{\pi}{2},\pi)\) for the basic period, and this option does not follow that)

Answer:

C.