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

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.

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.