We will use a sine function as an example, but the method is general:
Suppose you are asked to graph `y=-2sin(1/3*pi(x+2))-2 `
This is of the form `y=asin(b(x-h))+k `
a: a is the amplitude. This is the maximum distance the graph takes from the midline. Note if a<0 the graph is reflected over the midline.
b: b yields the period; the period p is found by `p=(2pi)/b ` . (If you were graphing tangent or cotangent the numerator is pi.) This gives the horizontal distance required for the graph to begin repeating. (If b<0 the graph is reflected over a vertical line. It is often easier to rewrite the original function since sin(-x)=-sin(x) and cos(-x)=cos(x).)
h: h is the horizontal translation or phase shift. If h>0 shift to the right, if h<0 shift to the left h units.
k: y=k is the midline.
So for our example a=-2, b=1/3pi, h=-2 and k=-2
The graph is a transformed sine wave: the amplitude is 2, the graph is reflected over the horizontal, the period is 6, the graph is shifted left 2 units, and the midline is y=-2.
Here is a graph of each of the transformations with the final answer in green:
Red is the original sine, orange has amplitude 2 and has been reflected, blue has the change in period, purple has the phase shift, and green shifts vertically (moved the midline.)