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Chapter 4: Problem 28

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

Short Answer

Expert verified
The graph of the function \(y = 3 \cot(\frac{\pi x}{2})\) will have a period of 4, vertical asymptotes at \(x = 0, 2, 4, 6, 8\) for the first two periods, and oscillates between \(-3\) and \(3\) across each period.

Step by step solution

01

Identify the period of the function

The period of the cotangent function is \(\pi\). Here, the coefficient of \(x) in the argument is \(\frac{\pi}{2}\), so the period of \(y=3 \cot(\frac{\pi x}{2})\) will be \(\frac{2\pi}{\frac{\pi}{2}} = 4\). This means the function completes one full cycle between \(x=0\) and \(x=4\).
02

Identify vertical asymptotes

The cotangent function has vertical asymptotes wherever its argument is some integer multiple of \(\pi\), in this case \(\frac{\pi}{2}x = n\pi \Rightarrow x = 2n\), where \(n\) is an integer. For the first two periods from \(x=0\) to \(x=8\), these will occur at \(x = 0, 2, 4, 6, 8\).
03

Sketch the graph

Between each pair of vertical asymptotes, the cotangent function starts from positive infinity, passes through \(y=0\), and then heads to negative infinity. Due to the scaling factor of 3, this reach will be between \(-3\) and \(3\). Repeat this behavior for each period. Lastly, update the graph to include these points and sketch the transformed cotangent function. Always remember, the cotangent function has the property that it is undefined at its asymptotes; sketch it accordingly.

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