Transform the graph of y = sin x into the graph of y = sin (2x + 3pi/4)

Solution:

Step1: draw the graph of y = sin x

For the standard sine function y = Asin (Bx + C), its minimum period T is: T = 2pi/B

The minimum period of y = sin x is: T = 2pi/1 = 2pi

Now, we will find the five important points in a period of y = sin x

when x = 0, y = sin x = sin 0 = 0. The first point is: (0, 0)

when x = pi/2, y = sin x = sin (pi/2) = 1. The second point is: (pi/2, 1)

when x = pi, y = sin x = sin pi = 0. The third point is: (pi, 0)

when x = 3pi/2, y = sin x = sin (3pi/2) = -1. The fourth point is: (3pi/2, -1)

when x = 2pi, y = sin x = sin 2pi = 0. The fifth point is: (2pi, 0)

Step2: Transform the graph of y = sin x into y = sin2x.

The minimum period of y = sin 2x is: T = 2pi/2 = pi.

The minimum period of y = sin2x is half of the minimum period of y = sin x. Therefore, we need to keep the y-coordinate of the graph of y = sin x and shrink the x-coordinate of the
graph of y = sin x to half to get the graph of y = sin 2x.

Step3: Transform the graph of y = sin 2x into the graph of y = sin (2x + 3pi/4)

y = sin (2x + 3pi/4) = sin2(x + 3pi/8)

Because there is a plus sign, we need to move the graph of y = sin2x left 3pi/8 units to get the graph of y = sin (2x + 3pi/4).