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 sine function \(y=\sin x\) has a period of \(2\pi\), and its values range from \(- 1\) to \(1\). At \(x = 0\), \(y=\sin(0)=0\); at \(x=\frac{\pi}{2}\), \(y = \sin(\frac{\pi}{2})=1\); at \(x=\pi\), \(y=\sin(\pi)=0\); at \(x=\frac{3\pi}{2}\), \(y=\sin(\frac{3\pi}{2})=-1\); at \(x = 2\pi\), \(y=\sin(2\pi)=0\). The given graph of \(y = \sin x\) does not match these key - point values (for example, the shape and key - point values are incorrect as per the standard sine function).

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

The cosine function \(y = \cos x\) has a period of \(2\pi\), and its values range from \(-1\) to \(1\). At \(x = 0\), \(y=\cos(0)=1\); at \(x=\frac{\pi}{2}\), \(y=\cos(\frac{\pi}{2})=0\); at \(x=\pi\), \(y=\cos(\pi)=-1\); at \(x=\frac{3\pi}{2}\), \(y=\cos(\frac{3\pi}{2})=0\); at \(x = 2\pi\), \(y=\cos(2\pi)=1\). The graph of \(y=\cos x\) shown is consistent with these properties.

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

The tangent function \(y=\tan x=\frac{\sin x}{\cos x}\) has a period of \(\pi\) and vertical asymptotes at \(x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}\). The graph of \(y = \tan x\) shown is consistent with its periodicity and asymptote properties.

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

The cotangent 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 \(y=\cot x\) shown is consistent with its periodicity and asymptote properties.

Answer:

A. \(y = \sin x\)