Question

In Exercises 45–52, graph two periods of each function. y = csc|x|

Accepted Answer

Understand the function given: $$y = \csc|x|. $$This means the cosecant function is applied to the absolute value of $$x. $$Recall that $$\csc 	heta = \frac{1}{\sin 	heta}$$, so the function is undefined where $$\sin|x| = 0. $$Identify the domain restrictions: Since $$\sin|x| = 0 at$$ $$|x| = n\pi$$ for integers $$n$$, the function $$y = \csc|x|$$ will have vertical asymptotes at $$x = 0, \pm \pi, \pm 2\pi, \pm 3\pi, \ldots. $$Determine the period of the function: The basic $$\csc x$$ function has period $$2\pi$$, but because of the absolute value inside, the function is symmetric about the y-axis and the period effectively becomes $$\pi. So$$, two periods correspond to an interval of length $$2\pi on $$the positive side, and by symmetry, also on the negative side. Sketch the graph over two periods: For $$x \geq 0$$, plot $$y = \csc x$$ from $$0 to$$ $$2\pi ($$excluding points where $$\sin x = 0). $$For $$x \u003c 0$$, use the symmetry $$y = \csc|x| = \csc(-x) = \csc x to $$reflect the graph about the y-axis. Mark vertical asymptotes at $$x = 0, \pm \pi, \pm 2\pi$$ and plot the characteristic 'cosecant' curves between these asymptotes, noting that the function values go to $$\pm \infty$$ near the asymptotes and have minimum or maximum values at points where $$\sin|x| is$$ $$\pm 1.$$

In Exercises 45–52, graph two periods of each function. y = csc|x|

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Understanding the Cosecant Function (csc x)

Effect of the Absolute Value on the Input (|x|)

Periodicity and Graphing Multiple Periods