back to trigonometry
# The Graphs of Sine Function and Transformation

**The Graph of sine function y = sin x**- The domain of y = sin x is the set of all real numbers, and the range is the interval [-1, 1], which has a period of 2pi, that is, sin ( x + 2pi n ) = sin x for all integer n. The key five points of y = sin x are: ( 0 , 0 ), ( pi/2 , 1 ), ( pi , 0 ), ( 3pi/2 , -1 ), ( 2pi , 0 ). Based on y = sin x is odd and period function, you can draw the rest of the curve.

**The Graph of sine function y = (3/2) sin x**- The domain of y = (3/2)sin x is the set of all real numbers, and the range is the interval [-3/2, 3/2], it has a period of 2pi. The key five points are: ( 0 , 0 ), ( pi/2 , 3/2 ), ( pi , 0 ), ( 3pi/2 , - 3/2 ), ( 2pi , 0 ). Based on y = (3/2)sin x is odd and period function, you can draw the rest of the curve.

**The Graph of sine function y = (2/3) sin ( x + pi/4 )**- The graph of y = (2/3)sin(x + pi/4) is the graph of y = (2/3)sin x move left pi/4. Thus, we need to draw y = (2/3)sin x first, then move it left pi/4 unit.

**The Graph of sine function y = (2/3) sin ( x - pi/4 )**- The graph of y = (2/3)sin(x - pi/4) is the graph of y = (2/3)sin x move right pi/4. Thus, we need to draw y = (2/3)sin x first, then move it right pi/4 unit.

**The Graph of sine function y = sin x , y = sin 2x , and y = - sin x/2**- The graph of y = - sin(x/2) has a period of 4pi. The graph of y = sin x has a period of 2pi. The graph of y = sin2x has a period of pi.

**Draw the Graph of y = A sin ( Bx + C )**- Steps:
- 1. Start from drawing the graph of y = A sin x, its range is [-A, A], period is 2pi.
- The constant factor A is the amplitude of the sine function. Its five key points are: (0, 0), (pi/2, A), (pi, 0), (3pi/2, - A), (2pi, 0). Connect these points and extend it, you get the graph of y = A sin x.
- 2. (i). If C > 0, move the graph of y = A sin x left C unit.
- (ii). If C < 0, move the graph of y = A sin x right C unit. In this step, you get the grapg of y = A sin( x + C ).
- 3. (i). If B > 1, shrinks all horizontal coordinates to 1/B times.
- (ii). If B < 1, extend all horizontal coordinates to B times.
- Keep all vertical coordinates unchanged.
- In this step, you get the graph of y = A sin( Bx + C ), its period is 2pi/B.