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Chapter 4: Problem 34

Graph two periods of the given cosecant or secant function. $$y=3 \sec x$$

Short Answer

Expert verified
The graph of the function \(y=3 sec x\) has two periods from \(-2\pi\) to \(2\pi\) with an amplitude of 3. The graph approaches asymptotes where \(x = k\pi/2\), where \(k\) is an odd integer, and repeats the pattern every \(2\pi\).

Step by step solution

01

Recognize the Parent Function and Amplitude

The parent function is \(sec(x)\) which is the reciprocal of \(cos(x)\). The given function \(y = 3sec(x)\) has an amplitude of 3, which scales the secant function vertically by a factor of 3.
02

Identify the Key Points.

The key points for the secant function are the maximum, minimum, and undefined points that occur at \( x = k\pi/2 \), where \(k\) is an odd integer. We then need to multiply the y-values of these key points by the amplitude of 3 in the given function.
03

Sketch the Function.

First graph the cosine function like a dashed line between \(-2\pi\) and \(2\pi\), where \(x = -3\pi/2, -\pi/2, \pi/2, 3\pi/2\) are the zeroes of the function. Then plot the key points calculated in Step 2. Connect the points with a continuous curve that approaches asymptotes where the functioning is undefined. Repeat the process for a second period of \(2\pi\) until \(x = 2\pi\). This graph represents two periods of the given secant function \(y = 3sec(x)\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the graph of \(y=3 \cos 2 x\) to obtain the graph of \(y=3 \csc 2 x\)

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