# Draw the graph y = sin(3pi x)

- The graph of y = A sin Bx has the property
- (1). amplitude = |A|
- (2). period = 2pi/B
- For the function of y = sin(3pi x),
- since A = 1, so its amplitude is |1|
- since B = 3pi, so its period = 2pi/B = 2pi ÷ 3pi = 2/3
- Thus, its amplitude is 1 and period is 2/3

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

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

- Draw the graph of y = sin(3pi x) based on the five points

- The general form of Sine function is:
- y = A sin ( Bx - C ), C is the initial phase shift.
- When C = 0, y = A sin Bx.
- For the graph of y = sin(3pi x), its one period is from x = 0 to x = 2/3.

- Analysis of the graph:
- x = 0, y = 0.
- x = 1/6 is its quarter period, at this point, its value of y is 1 which is maximum.
- x = 1/3 is its half period, at this point, its value of y is 0.
- x = 1/2 is its three-fourth period, at this point, its value of y is -1 which is minimum.
- x = 2/3 is its end point of the first period, at this point, its value of y is 0.
- The curve of y = sin 3 pi x is continuous, its second period is from x = 2/3 to x = 4/3.