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Chapter 4: Problem 22

Graph two periods of the given cotangent function. $$y=-2 \cot \frac{\pi}{4} x$$

Short Answer

Expert verified
The graph of the function \(y=-2 \cot \frac{\pi}{4} x\) is a cotangent curve that has been reflected about the x-axis (due to the negative sign in front of the function) and has a period of \(4\pi\) (due to the scaling factor \(\frac{\pi}{4}\)). Vertical asymptotes occur at every multiple of the period and minimum points occur halfway between each asymptote.

Step by step solution

01

Identify the properties of cotangent function

The function is of the form \(y=A \cot Bx\), where A represents the amplitude and B determines the period. Here, A=-2, that means the function will be upside down, and B=\(\frac{\pi}{4}\), which is the reciprocal of period. Therefore, the period is \(4\pi\). The function will have vertical asymptotes at \(\frac{4n\pi}{\pi}=n(4\pi)\) for an integer n.
02

Draw the axes and label them

Draw the x and y axes. Mark the x-axis with period intervals. Here, the period of the function is \(4\pi\), so the markings should be at multiples of \(4\pi\).
03

Sketch intermediate points and asymptotes

Sketch the middle points (half period points) of the function, which will be at the maximum (if A>0) or minimum (if A<0) points of the function. As A is negative here, these points will be at the minimum of the function. Mark vertical asymptotes at every period of the function.
04

Sketch the function

Join these intermediate points keeping in mind that the function approaches asymptotes but never crosses or touches them. The graph towards an asymptote is infinity in the cotangent function or negative infinity if the function is upside down.
05

Graph two periods of the function

After the graph has been drawn, make sure it covers at least two periods of the function. In this case, it should cover the periods from 0 to \(8\pi\).

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Most popular questions from this chapter

Find the slant asymptote of $$ f(x)=\frac{2 x^{2}-7 x-1}{x-2} $$ (Section \(2.6, \text { Example } 8)\)

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