Question

How do I graph a function based off the unit circle?

Answer 1

So if you remember what the normal y = sin(x) function looks like (a wave), y = 2 sin(4x) is just changed a little.

The standard format for sine/cosine function
y = a sin(bx− c) + d

a = amplitude, distance from center of the wave to the highest point. This function a = 2 so the height of the sine wave reaches 2 instead of 1.
"c" and "d" shift the graph left/right and up/down respectively. These equal zero so the sine wave is not shifted.

The range (y-values) is then just the amplitude -2 ≤ y ≤ 2

The domain (x-value) is all real numbers because the wave just keeps going on to infinity in both directions.

2π / |b| = period, distance per wave
this equation b = 4
period is then π/2
this is the distance before a wave repeats.

Graph
     x  |  y
-π/8    -2
    0      0
  π/8    2
3π/8   -2
5π/8    2

see the pattern? I'm using the amplitude or peaks and bottoms of the wave y = 2 and -2  then using the x-distance between like points is the period so you add π/2

(π/8 , 2) 
+ π/2
(5π/8 , 2)

Same for the minumums of the wave (y = -2)
(-π/8 , -2)
+ π/2
(3π/8 , -2)

Hope this helps, otherwise there are youtube videos you can watch or try an online graphing calculator like Desmos.com