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Chapter 4: Problem 41

Use a graphing utility to graph the function. (Include two full periods.) $$y=-2 \sec 4 x$$

Short Answer

Expert verified
The graph of \(y=-2 \sec 4x\) should resemble a series of 'U'-shaped curves, each with an amplitude of 2, reflected vertically, repeating every \(\pi / 4\), and with asymptotes locating points where cosine is zero.

Step by step solution

01

Understand the Function

Notice that \(y=-2 \sec 4 x\) is a transformation of the standard secant function. The magnitude of the coefficient in front of the secant function gives the amplitude (2 in this case), while the coefficient of x gives the frequency of the function (4 in this case). Moreover, the function is vertically reflected due to the negative sign.
02

Identify the Period

The period of a secant function is \(\pi / |b|\), where \(b\) is the coefficient of \(x\). In this function, \(b=4\), so the period would be \(\pi / 4\). However, because we want to graph two full periods, we will graph the function from \(x=0\) to \(x=\pi / 2\).
03

Plot the Function

Use a graphing utility to plot the function for \(x\) values from 0 to \(\pi / 2\). It's important to remember that secant has vertical asymptotes wherever its corresponding cosine function has zeros.
04

Analyze the Result

Observe the graph obtained from the graphing utility, ensuring that it contains two full periods of the secant function and follows the characteristics identified earlier for a vertically reflected secant function.

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