Question

Graph each function over a two-period interval.y = 1 - 2 cot [2(x + π/2)]

Accepted Answer

Identify the given function: $$y = 1 - 2 \cot \left[ 2 \left( x + \frac{\pi}{2} ight) ight]. $$Notice that the function involves a cotangent with a horizontal transformation and a vertical shift and stretch. Determine the period of the cotangent function inside the argument. The standard period of $$\cot x is$$ $$\pi. $$Since the argument is multiplied by 2, the period becomes $$\frac{\pi}{2}$$ because period $$= \frac{\pi}{|b|}$$ where $$b=2. $$Since the problem asks for a two-period interval, calculate the length of the interval to graph: $$2 	imes \frac{\pi}{2} = \pi. So$$, you will graph the function over an interval of length $$\pi in $$terms of $$x. $$Account for the horizontal shift inside the argument: $$x + \frac{\pi}{2}. $$This means the graph is shifted to the left by $$\frac{\pi}{2}. So$$, choose your $$x-$$interval accordingly, for example from $$-\frac{\pi}{2} to$$ $$\frac{\pi}{2} or $$any interval of length $$\pi$$ that reflects this shift. Plot key points by evaluating the cotangent function at values where the argument $$2(x + \frac{\pi}{2})$$ equals multiples of $$\frac{\pi}{2} ($$where cotangent has zeros and asymptotes). Then apply the vertical transformations: multiply by $$-2$$ and add 1 to get the final $$y$$ values. Sketch the graph using these points and asymptotes over the chosen two-period interval.

Graph each function over a two-period interval.
y = 1 - 2 cot [2(x + π/2)]

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Cotangent Function and Its Properties

Function Transformations (Shifts and Scaling)

Period of a Trigonometric Function