Graph y=9csc(x)

Problem

y=9*csc(x)

Solution

1. Identify the parent function and its properties. The function y=9*csc(x) is a transformation of y=csc(x) which is the reciprocal of y=sin(x)

2. Determine the vertical stretch. The coefficient 9 indicates a vertical stretch by a factor of 9 While y=sin(x) has a range of [−1,1] the function y=9*sin(x) has a range of [−9,9] Consequently, the range of y=9*csc(x) is (−∞,−9]∪[9,∞)

3. Locate the vertical asymptotes. Vertical asymptotes occur where the sine function is zero. For y=csc(x) these occur at x=n*π for any integer n

4. Find key points for one period. Within the interval (0,2*π) the relative minimum occurs at x=π/2 where y=9*csc(π/2)=9*(1)=9 The relative maximum occurs at x=(3*π)/2 where y=9*csc((3*π)/2)=9*(−1)=−9

5. Sketch the curves. Draw the vertical asymptotes at x=0,π,2*π Between 0 and π draw a U-shaped curve opening upward with its vertex at (π/2,9) Between π and 2*π draw a U-shaped curve opening downward with its vertex at ((3*π)/2,−9)

Final Answer

y=9*csc(x)

Want more problems? Check here!