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Chapter 4: Problem 27

Sketch the graph of the function. (Include two full periods.) $$y=3 \cot 2 x$$

Short Answer

Expert verified
First identify key attributes including the period and range. Then identify where the asymptotes will be, plot key points, and sketch the graph using these. Repeat for the second period.

Step by step solution

01

Identify Key Attributes of the Function

Start by identifying the key attributes of the cotangent function, \(y = 3 \ cot(2x)\). The '3' is a vertical stretch factor implying that the range of the function will be \(-3 鈮� y 鈮� 3\). The '2x' in \(\ cot(2x)\) is a horizontal compression factor which means that the period of the function will be \(蟺/2\).
02

Identify Asymptotes

The cotangent function has vertical asymptotes wherever the angle \(x\) is a multiple of \(蟺\) (since cotangent is undefined at these points). However, since there's a '2x' inside the cotangent function, the asymptotes will be at \(x = 蟺/(2*2k)\), where \(k\) is an integer.
03

Plot Basic Points

Plot points using the period and the asymptotes. At midway between the asymptotes, the cotangent function will cross through the x-axis (i.e., \(y=0\)). And it approach negative and positive infinity as \(x\) tends to each asymptote.
04

Sketch the Graph

Draw the graph using the identified asymptotes and plotted points. Remember that the graph should approach the asymptotes (without touching them), pass through the x-axis midway between asymptotes, and have the identified range \(-3 鈮� y 鈮� 3\).
05

Repeat for Next Period

Repeat these steps for the next period since two full periods are required.

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