# Draw the graph of y = csc4(x 3pi/2)

- The graph of y = A sin Bx has the property
- (1). amplitude = |A|
- (2). period = 2pi/B

- Step 1: Draw the graph of y
_{1} = sin(6pi x)
- since A = 1, so its amplitude is |1|
- since B = 6pi, so its period = 2pi/B = 2pi/6pi = 1/3
- thus, its amplitude is 1 and period is 1/3

- Find the five points in one period
- one period is 1/3, half period is 1/6, quarter period is 1/12
- divide the five points equally in a period [0, 1/3]
- The five points in x-axis are:
- x
_{1} = 0
- x
_{2} = 1/12
- x
_{3} = 1/6
- x
_{4} = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4
- x
_{5} = 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3
- so the five points in xy-plane are: (0, ?), (1/12, ?), (1/6, ?), (1/4, ?), (1/3, ?)

- Now find the values of the function y = sin(6pi x) in the five points
- when x = 0, y = sin(6pi x) = sin(6pi × 0) = sin(0) = 0, so the point is (0, 0)
- when x = 1/12, y = sin(6pi x) = sin(6pi × 1/12) = sin(pi/2) = 1, so the point is (1/12, 1)
- when x = 1/6, y = sin(6pi x) = sin(6pi × 1/6) = sin(pi) = 0, so the point is (1/6, 0)
- when x = 1/4, y = sin(6pi x) = sin(6pi × 1/4) = sin(3pi/2) = -1, so the point is (1/4, -1)
- when x = 1/3, y = sin(6pi x) = sin(6pi × 1/3) = sin(2pi) = 0, so the point is (1/3, 0)
- The five points are (0, 0), (1/12, 1), (1/6, 0), (1/4, -1), (1/3, 0)

- Draw the graph of y = sin(6pi x) based on the five points

- Step 2: Draw the graph of y
_{2}= 1

- Step 3: Draw the graph of y = 1/sin(6pi x)
- x = 0, y = 1/0 = infinity
- x = 1/12 = 0.0833, y = 1/1 = 1
- x = 1/6 = 0.1667, y = y = 1/0 = infinity
- x = 1/4 = 0.25, y = 1/(-1) = -1
- x = 1/3 = 0.3333, y = 1/0 = y = 1/0 = infinity
- x = 5/12 = 0.4167, y = 1/1 = 1
- x = 1/2 = 0.5, y = 1/0 = infinity