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

Feb 10, 2015

Here is a procedure one can use to graph $y = \cos \left(2 x - 2 \frac{\pi}{3}\right) + \frac{1}{2}$.

1. Make a small transformation of the original function to
$y = \cos \left[2 \left(x - \frac{\pi}{3}\right)\right] + \frac{1}{2}$.

2. Graph of this function can be obtained by horizontally right-shifting by $\frac{\pi}{3}$ a graph of function
$y = \cos \left(2 x\right) + \frac{1}{2}$.

3. Graph of $y = \cos \left(2 x\right) + \frac{1}{2}$ can be obtained by vertically up-shifting by $\frac{1}{2}$ a graph of function
$y = \cos \left(2 x\right)$.

4. Graph of $y = \cos \left(2 x\right)$ can be obtained by horizontally squeezing towards 0 by a factor $2$ a graph of function
$y = \cos \left(x\right)$.
"Squeezing" means that every point $\left(x , y\right)$ of the graph is transformed into $\left(\frac{x}{2} , y\right)$.

So, the steps to graph the original function are:

(a) start from a graph of $y = \cos \left(x\right)$;
(b) squeeze this graph horizontally towards 0 by a factor of $2$.
(c) shift up by $\frac{1}{2}$
(d) shift right by $\frac{\pi}{3}$.