Question

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

a. graph
b. graph
c. graph
d. graph

Explanation:

Step1: Analyze the properties of the cotangent function

The general form of the cotangent function is \(y = A\cot x\). The period of \(y=\cot x\) is \(\pi\), and for \(y = A\cot x\), the vertical stretch factor is \(|A|\). Here \(A=\frac{6}{5}\). The function \(y = \cot x\) has vertical asymptotes at \(x = n\pi\), \(n\in\mathbb{Z}\), and passes through the point \((\frac{\pi}{2},0)\). The function \(y=\frac{6}{5}\cot x\) is a vertical stretch of \(y = \cot x\) by a factor of \(\frac{6}{5}\). Also, \(y=\cot x\) is an odd function (\(\cot(-x)=-\cot x\)), and its graph is symmetric about the origin. But when we consider the basic shape: the cotangent function \(y = \cot x\) has a graph that decreases from \(+\infty\) to \(-\infty\) in the interval \((0,\pi)\).

Step2: Eliminate wrong - option based on key - points

• For the cotangent function \(y=\frac{6}{5}\cot x\), when \(x = \frac{\pi}{2}\), \(y=\frac{6}{5}\cot(\frac{\pi}{2})=0\).
• Option A and B: The cotangent function \(y = \cot x\) (and its vertical - stretch \(y=\frac{6}{5}\cot x\)) does not pass through the origin (\(x = 0,y=\frac{6}{5}\cot(0)\) is undefined as \(\cot(0)\to\pm\infty\)). So, options B and D (which seem to pass through the origin) are incorrect.
• Option C: The function \(y=\frac{6}{5}\cot x\) has the form of a cotangent function. The cotangent function \(y = \cot x\) has vertical asymptotes at \(x = n\pi\), \(n\in\mathbb{Z}\), and in the interval \((0,\pi)\) it goes from \(+\infty\) to \(-\infty\). The vertical stretch \(y=\frac{6}{5}\cot x\) has the same general shape as \(y = \cot x\) but is stretched vertically.

Answer:

C.