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Chapter 4: Problem 20

Sketch the graph of the function. (Include two full periods.) $$y=\frac{1}{4} \sec x$$

Short Answer

Expert verified
The graph of the function \(y = \frac{1}{4} \sec x\) consists of U-shaped curves peaking or bottoming out at \(y = 1/4\) or \(y = -1/4\) respectively, repeating every \(2\pi\) units along the x-axis.

Step by step solution

01

Identify the period of the function

The standard secant function, \(\sec x\), has a period of \(2\pi\). Since there is no coefficient on x in this function, the period remains \(2\pi\).
02

Identify the amplitude of the function

The amplitude of the function is the absolute value of the coefficient on the secant function. In this case, the amplitude is \(1/4\).
03

Identify the asymptotes of the function

The secant function has vertical asymptotes where the cosine function is zero. For two full periods (-2蟺 to 2蟺), these are at \(-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}\).
04

Sketch the graph

Draw a horizontal line at y=1/4 and y=-1/4 to represent the maxima and minima of the function. Now, plot vertical dashed lines at the asymptote values. These lines divide the x-axis into four sections for one full period. Sketch a U shaped curve in each section, peaking at \(y = 1/4\) in the sections between \(-\frac{3\pi}{2}\) and \(-\frac{\pi}{2}\), and \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), while bottoming out at \(y = -1/4\) in the sections between \(-2\pi\) and \(-\frac{3\pi}{2}\), and \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), and \(\frac{3\pi}{2}\) and \(2\pi\). Repeat for the second period.

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