Question

which function is not graphed correctly?
a. ( y = sin x )
b. ( y = cos x )
c. ( y = \tan x )
d. ( y = cot x )

Explanation:

Step1: Analyze the properties of \(y = \sin x\)

The function \(y=\sin x\) has a period of \(2\pi\), and it passes through the origin \((0,0)\). The graph of \(y = \sin x\) has a maximum value of \(1\) and a minimum value of \(- 1\). The given graph for \(y=\sin x\) in option A does not satisfy the property of passing through the origin. The standard \(y = \sin x\) graph should cross the origin \((0,0)\), but the shown graph for \(y=\sin x\) in option A does not.

Step2: Analyze the properties of \(y=\cos x\)

The function \(y = \cos x\) has a period of \(2\pi\), and \(y(0)=\cos(0) = 1\). The graph of \(y=\cos x\) has a maximum value of \(1\) and a minimum value of \(-1\). The graph in option B is consistent with the properties of \(y=\cos x\).

Step3: Analyze the properties of \(y=\tan x\)

The function \(y=\tan x\) has a period of \(\pi\), and vertical asymptotes at \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\). The graph of \(y=\tan x\) passes through the origin \((0,0)\). The graph in option C is consistent with the properties of \(y = \tan x\).

Step4: Analyze the properties of \(y=\cot x\)

The function \(y=\cot x=\frac{\cos x}{\sin x}\) has a period of \(\pi\) and vertical asymptotes at \(x = n\pi,n\in\mathbb{Z}\). The graph of \(y=\cot x\) has no \(x -\)intercepts in the intervals \((n\pi,(n + 1)\pi)\). The graph in option D is consistent with the properties of \(y=\cot x\).

Answer:

A. \(y = \sin x\)