Question

Graph each function over a one-period interval.y = 2 tan (¼ x)

Accepted Answer

Identify the basic function and its transformation: The given function is $$y = 2 	an\left(\frac{1}{4}xight). $$Here, the coefficient 2 is a vertical stretch, and the argument $$\frac{1}{4}x$$ affects the period of the tangent function. Recall the period of the basic tangent function $$	an(x) is$$ $$\pi. $$For $$y = 	an(bx)$$, the period is given by $$\frac{\pi}{|b|}. In $$this case, $$b = \frac{1}{4}$$, so the period is $$\frac{\pi}{\frac{1}{4}} = 4\pi. $$Determine the one-period interval for the function: Since the period is $$4\pi$$, one period can be taken as $$x in$$ $$[0, 4\pi] or $$any interval of length $$4\pi. $$For example, $$[-2\pi, 2\pi] is $$also valid. Identify the vertical asymptotes of the tangent function within one period. The tangent function has vertical asymptotes where its argument equals $$\frac{\pi}{2} + k\pi$$, for integers $$k. $$Solve $$\frac{1}{4}x = \frac{\pi}{2} + k\pi to $$find the asymptotes in terms of $$x. $$Plot key points: At $$x=0$$, $$y=2 	an(0) = 0. At$$ $$x$$ values where $$\frac{1}{4}x = \frac{\pi}{4} or$$ $$\frac{3\pi}{4}$$, calculate $$y$$ values to understand the shape. Use the vertical asymptotes to sketch the graph of $$y=2 	an\left(\frac{1}{4}xight)$$ over one period.

Graph each function over a one-period interval.
y = 2 tan (¼ x)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Period of the Tangent Function

Amplitude and Vertical Stretch

Vertical Asymptotes of Tangent