Question

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

Answer 1

First factor, then pick apart the equation.

Explanation:

Factor out `(2x-pi)` to get `2(x-pi/2)`.

Then, before analyzing the equation, draw the root equation which is `y=cosx`

Now start picking out the important stuff.

Amplitude :
a in the equation.

`y=ul2cos(2(x-pi/2))+2`

The amplitude of 2 will stretch `y=cosx` by a factor of 2.

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

`y=2cos(ul2(x-pi/2))+2`

Now calculate period:
`period=(2pi)/k`

`period=(2pi)/2`

`period= pi`

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

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

`y=2cos(2(x-ul(pi/2)))+ul2`

The first translation is a horizontal translation right `pi/2` units. So move the graph right `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!