Question

Graph each function over a one-period interval.y = (1/2) csc (2x + π/2)

Accepted Answer

Identify the function: The given function is $$ y = \frac{1}{2} \csc(2x + \frac{\pi}{2}) . $$The cosecant function, $$ \csc(	heta) $$, is the reciprocal of the sine function, $$ \sin(	heta) . $$Determine the period of the function: The standard period of $$ \csc(x)  is$$ $$ 2\pi . $$For $$ \csc(bx) $$, the period is $$ \frac{2\pi}{b} . $$Here, $$ b = 2 $$, so the period is $$ \frac{2\pi}{2} = \pi . $$Find the phase shift: The phase shift is determined by the expression inside the cosecant function, $$ 2x + \frac{\pi}{2} . $$Set $$ 2x + \frac{\pi}{2} = 0  to $$find the phase shift. Solving for $$ x $$ gives $$ x = -\frac{\pi}{4} . $$Determine the vertical shift and amplitude: The function $$ y = \frac{1}{2} \csc(2x + \frac{\pi}{2}) $$ has a vertical stretch by a factor of $$ \frac{1}{2} . $$There is no vertical shift since there is no constant added or subtracted from the function. Graph the function: Start by graphing $$ y = \sin(2x + \frac{\pi}{2}) $$ over one period $$[0, \pi]. $$Identify the asymptotes where the sine function is zero, as these are the undefined points for the cosecant function. Then, sketch the $$ \csc $$ function by drawing curves that approach the asymptotes, reflecting the reciprocal nature of the sine function.

Graph each function over a one-period interval.
y = (1/2) csc (2x + π/2)

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Key Concepts

Cosecant Function

Period of Trigonometric Functions

Phase Shift