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Question

# View at least two cycles of the graphs of the given functions on a calculator. $y=-2 \cot \left(2 x+\frac{\pi}{6}\right)$

Solution

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We are given:

$y=-2\cot \left(2x+\dfrac{\pi}{6}\right)$

The function $y=a\cot (bx+c)$ has a period of $\dfrac{\pi}{b}$. Two asymptotes can be located at $x=0$ and $x= \dfrac{\pi}{b}$. The displacement is $-\dfrac{c}{b}$. The range is all real numbers.

From the given, $b=2$ and $c=\frac{\pi}{6}$ so the period is $\frac{\pi}{2}$ and there is an asymptote at $x=0$. The displacement $-\frac{\pi/6}{2}=-\frac{\pi}{12}$. So, we use the following window setting:

• Xmin = $-0.3$ (The asymptote is shifted by the displacement at $x=0-\frac{\pi}{12}\approx -0.26$)
• Xmax = $3.5$ (For two periods, the curve ends at $x=0+2(\pi/2)=\pi\approx 3.1$)
• Ymin = $-10$, Ymax = $10$ (this shows enough of the curve)

$\color{white}\tag{1}$

Graph $y_1=-2(\tan (2x+\pi/6))^{-1}$. Using a graphing calculator in RADIAN mode, the graph will be:

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