Graph y=8csc(x)

Problem

y=8*csc(x)

Solution

1. Identify the parent function and its properties. The function y=8*csc(x) is a transformation of y=csc(x) which is the reciprocal of sin(x)

2. Determine the vertical asymptotes. Since csc(x)=1/sin(x) the function is undefined where sin(x)=0 This occurs at x=n*π for any integer n

3. Analyze the amplitude and range. The coefficient 8 acts as a vertical stretch. While csc(x) has no amplitude, its local minima and maxima are shifted. The range of y=8*csc(x) is (−∞,−8]∪[8,∞)

4. Locate key points within one period (0,2*π) A local minimum occurs at x=π/2 where y=8*csc(π/2)=8*(1)=8 A local maximum occurs at x=(3*π)/2 where y=8*csc((3*π)/2)=8*(−1)=−8

5. Sketch the curves. Draw vertical asymptotes at x=0,π,2*π Plot the vertex points (π/2,8) and ((3*π)/2,−8) then draw the U-shaped curves approaching the asymptotes.

Final Answer

y=8*csc(x)

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