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Chapter 4: Problem 44

Graph two periods of the given cosecant or secant function. $$y=2 \sec \left(x+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The graph of the function \(y=2 \sec \left(x+\frac{\pi}{2}\right)\) includes vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\) and max/min points at \(x= n\pi\), where \(n\) is an integer. The curve connected these points, creating two complete periods of the function.

Step by step solution

01

Understand the Function

The given function is \(y=2 \sec \left(x + \frac{\pi}{2}\right)\). The \(2\) in front of the secant function indicates a vertical stretch by a factor of 2. The \(\frac{\pi}{2}\) added to \(x\) indicates a horizontal shift to the left by \(\frac{\pi}{2}\) units.
02

Identify the Key Features of the Graph

Because we鈥檙e working with a secant function, the key features to note are the vertical asymptotes and the maximum and minimum points. The standard secant function has vertical asymptotes at \(x= n\pi\), where \(n\) is an odd integer, and has maximum and minimum points midway between the vertical asymptotes. Noting the transformation in our function, the vertical asymptotes are shifted to \(\frac{\pi}{2} + n\pi\) and the max/min points are at \(x= n\pi\), where \(n\) is an integer.
03

Plot the Key Features on the Graph

Place vertical asymptote lines at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer. Mark the maximum and minimum points at \(x= n\pi\), where \(n\) is an integer. The maximum and minimum values will vary depending on whether \(n\) is odd or even due to the wave nature of the secant function.
04

Draw the Graph

Connect the maximum and minimum points with a smooth curve moving towards the vertical asymptotes but never crossing them. Continue this pattern until two complete periods of the function are graphed.

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