Question

find the graph of the given function over a one - period interval.
which of the following is the graph of ( y = - 2\tan\frac{1}{2}x )?
a.
b.
c.

Explanation:

Step1: Analyze the function \(y = - 2\tan\frac{1}{2}x\)

The general form of the tangent function is \(y = A\tan(Bx - C)+D\). For \(y=-2\tan\frac{1}{2}x\), \(A=-2\), \(B = \frac{1}{2}\), \(C = 0\), \(D = 0\). The period of the tangent function \(y=\tan(Bx)\) is \(T=\frac{\pi}{|B|}\). Here, \(T=\frac{\pi}{\frac{1}{2}} = 2\pi\). The amplitude of the tangent function is not defined in the traditional sense (since the range of \(y = \tan x\) is \((-\infty,\infty)\)), but the coefficient \(A=-2\) affects the vertical stretch and reflection. A negative \(A\) value reflects the graph of \(y = \tan x\) about the \(x\) - axis.

Step2: Check key points

For the basic tangent function \(y=\tan x\), key points in the interval \((-\frac{\pi}{2},\frac{\pi}{2})\) are \((-\frac{\pi}{4}, - 1)\), \((0,0)\), \((\frac{\pi}{4},1)\). For \(y=-2\tan\frac{1}{2}x\), when \(x = 0\), \(y=-2\tan(0)=0\). As \(x\to\pi^{-}\), \(y=-2\tan\frac{x}{2}\to-\infty\) and as \(x\to-\pi^{+}\), \(y=-2\tan\frac{x}{2}\to+\infty\). The graph of \(y = \tan x\) is reflected about the \(x\) - axis (due to \(A=-2\)) and has a period of \(2\pi\) (due to \(B=\frac{1}{2}\)).

Answer:

A.