How to draw the graph of y = 2 sin (x/2 - pi/3) + 1?

Question: Draw the graph of y = 2 sin (x/2 - pi/3) + 1

Solution:

Step 1:

Draw the graph of y = 2 sin (x/2) in one period using five points method. Compare the standard Sine function y = A sin (Bx + C) with the given sine function, we get A = 2, B = 1/2 and C = 0. The minimum period T = 2pi/B = 2pi ÷ 1/2 = 4pi. The five points are:

When x = 0, y = 2 sin (x/2) = 2 sin 0 = 0
when x = pi, y = 2 sin (x/2) = 2 sin (pi/2) = 2 × 1 = 2
when x = 2pi, y = 2 sin (x/2) = 2 sin (2pi/2) = 2 sin pi = 2 × 0 = 0
when x = 3pi, y = 2 sin (x/2) = 2 sin (3pi/2) = 2 × -1 = -2
when x = 4pi, y = sin (x/2) = 2 sin (4pi/2) = sin 2pi = 2 × 0 = 0
The five point are: (0, 0), (pi, 2), (2pi, 0), (3pi, -2), and (4pi, 0).
That is,

	point 1	point 2	point 3	point 4	point 5
x :	0	pi	2pi	3pi	4pi
y :	0	2	0	-2	0

Draw the graph of y = 2 sin x/2 in one period using the five points method

Step 2:

Draw the graph of y = 2 sin (x/2 - pi/3) = 2 sin [(1/2) (x - 2pi/3)] by shift the graph of y = 2 sin x/2 toward right 2pi/3 units.

Draw the graph of 2 sin (x/2 - pi/3) by shift the graph of y = 2 sin x/2 toward right 2pi/3 units.

Step 3:

Draw the graph of y = 2 sin (x/2 - pi/3) + 1 by move the graph of y = 2 sin (x/2 - pi/3) upward one unit.