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Chapter 5: Problem 6

Draw the graph of the given function for \(0 \leq x \leq 2 \pi\). \(y=-\cot x\)

Short Answer

Expert verified
The graph of \ y = -\cot{x} \ has vertical asymptotes at \ x = 0, \pi, 2\pi \ and zeros at \ x = \frac{\pi}{2}, \frac{3\pi}{2} \.

Step by step solution

01

Understand the Function

The function given is \( y = -\cot{x} \). This is the negative of the cotangent function, which means \( y \) is the negative reciprocal of the tangent function.
02

Identify Key Points

Identify the key points where \( \cot{x} \) has vertical asymptotes and zeros. \( \cot{x} \) has zeros at \( x = \frac{\pi}{2}, \frac{3\pi}{2}, \) and vertical asymptotes at \( x = 0, \pi, 2\pi \). The values are reversed in the range to make it negative.
03

Plot Basic Points

Create a table of values for \( x \) and corresponding \( y \) values, considering the function鈥檚 behavior near zeros and asymptotes. For instance: \x = \frac{\pi}{4}, y = -1 \; x = \frac{3\pi}{4}, y = 1\; x = \frac{5\pi}{4}, y = -1\; x = \frac{7\pi}{4}, y = 1\.
04

Draw the Asymptotes

Draw vertical dashed lines to represent the asymptotes at \( x = 0, \pi, 2\pi \). The graph will approach these lines but never touch them.
05

Sketch the Curve

Using the key points and asymptotes, sketch the curve of \( y = -\cot{x} \) between \( x = 0 \) and \( x = 2\pi \). The curve will go from negative to positive values between each pair of asymptotes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cotangent function
The cotangent function, denoted as \(\text{cot}(x)\), is the reciprocal of the tangent function. Mathematically, \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\). Unlike the sine and cosine functions, the cotangent function has both zeros and vertical asymptotes within one period. Understanding this function is essential for graphing its negative counterpart, \(-\text{cot}(x)\).
vertical asymptotes
Vertical asymptotes are critical when graphing the cotangent function. They occur where the function is undefined, i.e., where the sine denominator is zero: \(x = 0, \pi, 2\pi\), within the specified interval \(0 \leq x \leq 2\pi\). For \(-\text{cot}(x)\), the behavior near these asymptotes is flipped vertically. Be sure to draw dashed vertical lines at these key points to denote the asymptotes in your graph.
plotting key points
To sketch accurate graphs, it is important to plot key points. For \(-\text{cot}(x)\), identify where the function has zeros and assess its behavior around vertical asymptotes. Major points include:
  • Zeros at \(x = \frac{\pi}{2}, \frac{3\pi}{2}\).
  • Intermediate values, such as \( \frac{\pi}{4} \rightarrow y = -1 \, \frac{3\pi}{4} \rightarrow y = 1 \, \frac{5\pi}{4} \rightarrow y = -1, \, \frac{7\pi}{4} \rightarrow y = 1 \).
Be sure to also plot points on either side of the asymptotes to understand the function's behavior better.
graph sketching
Graph sketching involves putting it all together. Start by plotting the vertical asymptotes and key points you calculated. Ensure that the curve approaches but never touches the vertical asymptotes. The \(-\text{cot}(x)\) curve will descend from positive to negative near each asymptote and transition through zero at key points. With these steps, you can produce a clear, accurate graph of the function within the interval \(0 \leq x \leq 2\pi\).

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