Graphing Sine Functions|Please help!

Question

Sketch the graph of the given function:

y = 2 sin 1/2 (x - pi/4) + 1

I know that my amplitude is 2, my B-value is 1/2, my period is 4pi, my phase shift is pi/4 to the right, and that my vertical shift is 1, but I'm having trouble translating my information onto a graph. Would appreciate any help regarding how to sketch my graph as well as how to sketch the phase shift!

Answer 1

Since the amplitude is 2 and the vertical shift is 1 unit upward, the graph lies between the horizontal lines

y = -1 and y = 3 with midline y = 1.

Since the phase shift is π/4 units to the right and the period is 4π, one cycle lies between x = π/4 and

x = π/4 + 4π = 17π/4.

Divide the interval [π/4, 17π/4] into four congruent parts each of length 4π/4 = π.

At x = π/4, the graph lies on the midline. Point on graph: (π/4, 1)

At x = π/4 + π = 5π/4, the graph is at a highest point. Point on graph: (5π/4, 3)

At x = 5π/4 + π = 9π/4, the graph lies on the midline. Point on graph: (9π/4, 1)

At x = 9π/4 + π = 13π/4, the graph is at a lowest point. Point on graph: (13π/4, -1)

At x = 13π/4 + π = 17π/4, the graph lies on the midline. Point on graph: (17π/4, 1)

The graph is periodic with period 4π (I.e., the graph repeats every 4π.