Graph y=4csc(x)

Problem

y=4*csc(x)

Solution

1. Identify the parent function and its properties. The function y=4*csc(x) is a transformation of y=csc(x) which is the reciprocal of y=sin(x)

2. Determine the vertical stretch. The coefficient 4 indicates a vertical stretch by a factor of 4 While csc(x) has a range of (−∞,−1]∪[1,∞) the range of y=4*csc(x) is (−∞,−4]∪[4,∞)

3. Locate the vertical asymptotes. The cosecant function is undefined where sin(x)=0 These vertical asymptotes occur at x=n*π for any integer n

4. Identify key points for one period. Within the interval (0,2*π) the relative minimum occurs at x=π/2 where y=4*csc(π/2)=4*(1)=4 The relative maximum occurs at x=(3*π)/2 where y=4*csc((3*π)/2)=4*(−1)=−4

5. Sketch the curves. Draw the vertical asymptotes at x=0,π,2*π Plot the vertex points (π/2,4) and ((3*π)/2,−4) then draw the U-shaped branches approaching the asymptotes.

Final Answer

y=4*csc(x)

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