Write an expression for sine and cosine function?

Question

Can someone explain this to me and show me how to do this. I just need an explanation so I can do the other 16 problems

Some characteristics of a sine curve y=g(x) are given using these abbreviations: A=amplitude, p=period, f=frequency, m=minimum value, and M=maximum value. Find an expression for g(x)

1. A= 1, p = 4pi, g(0)=0
2. A=4, p=3, g(0)= -4

Help would be appreciated, thanks.

Answer 1

The sine function has a general form y = Asin(Bx - C) + D. The A drives the amplitude, the B deals with the period, the C deals with the horizontal shift of the graph, and the D deals with the vertical shift. We have a lot of information given already - let's try to make sense of it all.

For (1), the amplitude A is 1, so that's easy enough. To find B, we take our period and divide by 2pi:

B = 2pi/4pi
B = 1/2

Since g(0) = 0, this implies that the sine curve starts and ends where a "normal" sine curve would, so C and D don't come into play here. So for (1), y = sin(x/2).

Same for (2): A = 4 and B = 2pi/3. At g(0), the sine curve starts at -4, so D would be -4. So we would have y = 4sin(2pi/3) - 4.