Question

How do you graph the function `y=cos[2x-2pi/3]+1/2`?

Answer 1

Here is a procedure one can use to graph `y=cos(2x-2pi/3)+1/2`.

1. Make a small transformation of the original function to
   `y=cos[2(x-pi/3)]+1/2`.

2. Graph of this function can be obtained by horizontally right-shifting by `pi/3` a graph of function
   `y=cos(2x)+1/2`.

3. Graph of `y=cos(2x)+1/2` can be obtained by vertically up-shifting by `1/2` a graph of function
   `y=cos(2x)`.

4. Graph of `y=cos(2x)` can be obtained by horizontally squeezing towards 0 by a factor `2` a graph of function
   `y=cos(x)`.
   "Squeezing" means that every point `(x,y)` of the graph is transformed into `(x/2,y)`.

So, the steps to graph the original function are:

(a) start from a graph of `y=cos(x)`;
(b) squeeze this graph horizontally towards 0 by a factor of `2`.
(c) shift up by `1/2`
(d) shift right by `pi/3`.