In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec x

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1. Recall that the secant function is defined as the reciprocal of the cosine function: \(y = 2 \sec x = \frac{2}{\cos x}\). This means the graph of \(y = 2 \sec x\) is related to the graph of \(y = \cos x\).
2. Identify the period of the basic secant function. Since \(\sec x\) is the reciprocal of \(\cos x\), it has the same period as \(\cos x\), which is \(2\pi\). Therefore, two periods of \(y = 2 \sec x\) will span an interval of length \(4\pi\).
3. Determine the key points of the graph by first considering the zeros of \(\cos x\), because \(\sec x\) has vertical asymptotes where \(\cos x = 0\). These occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. Mark these vertical asymptotes on the graph.
4. Find the maximum and minimum values of \(y = 2 \sec x\) by looking at the maximum and minimum values of \(\cos x\). Since \(\cos x\) ranges between \(-1\) and \(1\), \(\sec x\) will have values \(\geq 1\) or \(\leq -1\). Multiplying by 2 scales these values to \(\geq 2\) or \(\leq -2\). Plot these points accordingly.
5. Sketch the graph between the vertical asymptotes, making sure the branches of the secant curve approach the asymptotes and reflect the reciprocal behavior of the cosine function, repeating this pattern for two full periods (from \(x=0\) to \(x=4\pi\)).

Key Concepts

Secant Function and Its Properties

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It is undefined where cos(x) = 0, leading to vertical asymptotes. Understanding its periodicity and behavior near these asymptotes is essential for graphing.

Amplitude and Vertical Stretch

In the function y = 2 sec x, the coefficient 2 vertically stretches the secant graph by a factor of 2. This affects the distance of the graph's branches from the x-axis, making the minimum and maximum values twice as far from zero compared to the basic sec(x) graph.

Period of the Secant Function

The secant function has a fundamental period of 2π, meaning its pattern repeats every 2π units along the x-axis. Graphing two periods involves plotting the function from 0 to 4π (or an equivalent interval), including all asymptotes and key points within this range.