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Chapter 4: Problem 23

Sketch one full period of the graph of each function. $$y=\frac{3}{2} \cot x$$

Short Answer

Expert verified
The graph of the function \(y = \frac{3}{2} \cot x\) is a periodic function that decreases from positive infinity to negative infinity within each period, with asymptotes at \(x = n\cdot \pi)\), where n is any integer. It oscillates above and below the x-axis with a period of \(\pi\). The key points and asymptotes for one full period can be plotted to sketch a graph.

Step by step solution

01

Identify Key Points

Set \(x\) equal to values where the cotangent function has easily found y-values. For example, at \(x = \pm \frac{\pi}{4}\), \(x = \pm \frac{3\pi}{4}\). Calculate the corresponding y-values.
02

Identify the Asymptotes

The cotangent function has vertical asymptotes where the tangent function is undefined, which is when the denominator is zero. This happens at \(x = n\cdot \pi)\), where n is any integer.
03

Plot the Points and Asymptotes

Plot the points calculated from Step 1 and graph the asymptotes from Step 2 on the same set of axes.
04

Sketch the Graph

Draw the curve of \(\frac{3}{2}\cot x\) that fits the points and asymptotes. The cotangent function decreases from positive infinity to negative infinity within each period.
05

Check the Graph

Make sure that the computation of the points was done correctly and that the graph accurately represents the function \(\frac{3}{2}\cot x\). It should decrease from positive infinity to negative infinity within each period, and it should have asymptotes at \(x = n\cdot \pi)\), where n is any integer.

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Most popular questions from this chapter

Write csc \(t\) in terms of \(\cot t, \frac{\pi}{2}

Find an equation of the cosecant function with period \(\frac{3 \pi}{2}\) and phase shift \(\frac{\pi}{4}\)

Write an equation for a sine function using the given information. Amplitude \(=2 ;\) period \(=3 \pi\)

Write an equation for a cosine function using the given information. Amplitude \(=3 ;\) period \(=2.5\)

Graph at least one full period of the function defined by each equation. $$y=3 \cos \frac{3 \pi x}{2}$$
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