Question

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

Accepted Answer

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 $$y = \cos x. $$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 $$4\pi. $$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. 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. 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 $$x=0 to$$ $$x=4\pi).$$

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

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Concetti chiave

Secant Function and Its Properties

Amplitude and Vertical Stretch

Period of the Secant Function