Question

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

Accepted Answer

Recall that the secant function is defined as the reciprocal of the cosine function: $$y = \sec x = \frac{1}{\cos x}. $$Therefore, $$y = 3 \sec x$$ can be written as $$y = \frac{3}{\cos x}. $$Identify the period of the basic secant function. Since $$\cos x$$ has a period of $$2\pi$$, the secant function also has a period of $$2\pi. So$$, one period of $$y = 3 \sec x$$ spans an interval of length $$2\pi. To $$graph two periods, determine the interval over which you will plot the function. For example, from $$x = 0 to$$ $$x = 4\pi$$ covers two full periods of $$y = 3 \sec x. $$Find the vertical asymptotes of the function by locating where $$\cos x = 0$$, because $$\sec x is $$undefined there. These occur at $$x = \frac{\pi}{2} + k\pi$$, where $$k is $$any integer. Mark these asymptotes within your chosen interval. Plot key points by evaluating $$y = 3 \sec x at $$values where $$\cos x is$$ $$\pm 1 or $$other convenient points, such as $$x = 0, \pi, 2\pi, 3\pi, 4\pi. $$Use these points and the asymptotes to sketch the two periods of the graph, noting that the graph will have branches opening upwards or downwards depending on the sign of $$\cos x.$$

Graph two periods of the given cosecant or secant function.

y = 3 sec x

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Concetti chiave

Secant Function and Its Properties

Amplitude and Vertical Stretch

Periodicity of Secant Function