Question

Graph two periods of the given cosecant or secant function.y = 2 csc x

Accepted Answer

Recall that the cosecant function is the reciprocal of the sine function, so $$\csc x = \frac{1}{\sin x}. $$This means the graph of $$y = 2 \csc x is $$related to the graph of $$y = \sin x$$, but with vertical stretches and undefined points where $$\sin x = 0. $$Identify the key points of the sine function over two periods. Since the period of $$\sin x is$$ $$2\pi$$, two periods span from $$0 to$$ $$4\pi. $$Mark points where $$\sin x is 0$$, 1, or -1 within this interval, because these will influence the shape and asymptotes of $$y = 2 \csc x. $$Determine the vertical asymptotes of $$y = 2 \csc x. $$These occur where $$\sin x = 0$$, i.e., at $$x = 0, \pi, 2\pi, 3\pi, 4\pi. $$Draw vertical dashed lines at these points to indicate the function is undefined there. Plot the points of $$y = 2 \csc x by $$taking the reciprocal of the sine values and multiplying by 2. For example, where $$\sin x = 1$$, $$y = 2 	imes \frac{1}{1} = 2$$, and where $$\sin x = -1$$, $$y = 2 	imes \frac{1}{-1} = -2. $$Between the asymptotes, the graph will form branches that go to infinity near the asymptotes. Sketch the two periods of $$y = 2 \csc x by $$connecting the plotted points with smooth curves that approach the vertical asymptotes, ensuring the graph reflects the reciprocal nature and vertical stretch of the function.

Graph two periods of the given cosecant or secant function.

y = 2 csc x

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Concetti chiave

Understanding the Cosecant Function

Amplitude and Vertical Stretch

Periodicity and Graphing Multiple Periods