# How do you graph y=2cos(2x-pi)+2?

Nov 27, 2017

First factor, then pick apart the equation.

#### Explanation:

Factor out $\left(2 x - \pi\right)$ to get $2 \left(x - \frac{\pi}{2}\right)$.

Then, before analyzing the equation, draw the root equation which is $y = \cos x$

Now start picking out the important stuff.

Amplitude :
a in the equation.

$y = \underline{2} \cos \left(2 \left(x - \frac{\pi}{2}\right)\right) + 2$

The amplitude of 2 will stretch $y = \cos x$ by a factor of 2.

Period :
Find the period by looking at the second 2 in the equation.

$y = 2 \cos \left(\underline{2} \left(x - \frac{\pi}{2}\right)\right) + 2$

Now calculate period:
$p e r i o d = \frac{2 \pi}{k}$

$p e r i o d = \frac{2 \pi}{2}$

$p e r i o d = \pi$

Apply this to your graph, which will compress it from a period of $2 \pi$ to now just $\pi$. (Also can be seen as horizontal compression by factor of $\frac{1}{2}$)

Translations :
The last 2 parts of graphing is the translations.

$y = 2 \cos \left(2 \left(x - \underline{\frac{\pi}{2}}\right)\right) + \underline{2}$

The first translation is a horizontal translation right $\frac{\pi}{2}$ units. So move the graph right $\frac{\pi}{2}$.
The second translation is a vertical translation up 2 units. Finally, move graph up 2 units.

If you think you may still have it wrong, refer to this handy tool for checking: https://www.desmos.com/calculator

Voila! All done!
Please correct me if I am wrong!