Question 1143426
The graph of {{{y=cos(T)}}} (with {{{T}}} in degrees) has a period of {{{360^o}}} .
In every period there is a single maximum value of {{{1}}} , such as at {{{T=0^o}}} .
{{{y=cos(T)}}} goes through a maximum at {{{T=0^o}}} , and as {{{T}}} increases,
it has the next maximum at {{{T=360^o}}} .
After that, the next maximum is at {{{T=2(360^o)=720^o}}} , and so on.
Here is a piece of the graph of {{{y=cos(x)}}} with {{{x}}} in degrees:
{{{graph(900,200,-72,478,-1.1,1.1,cos(pi*x/180))}}},
 
We want to do 3 changes to get to the "transformed equation", and there often is more than one way to get to a destination, but some ways may be "smoother sailing" compared to others.
 
CHANGES IN THE ORDER LISTED:
If we want shift the graph {{{270^o}}} to the left, we would end with a maximum at {{{-270^o}}} .
A way textbooks suggest to do that shift is replacing the variable with another, such as replacing {{{T=U+270^o}}} . 
hat way the point for {{{T=U+270^o=0^o}}} is at {{{U=-270^o}}} .
We would have {{{y=cos(U+270^o)}}} .
The graph of {{{y=cos(x+270^o)}}} as a function of {{{x}}} is shown below.
{{{graph(900,200,-342,208,-1.1,1.1,cos(pi*(x+272)/180))}}}  
 
If after that we want to change the period to {{{180^o=(1/2)*(360^o)}}} ,
we can do that by changing to variable {{{x}}} , so that it increases twice as fast as {{{U}}} .
If we still want the maximum at {{{270^o}}} , we need to make {{{U+270^o=2(x+270^o)}}} , 
and change the function from {{{y=cos(U+270^o)}}} to {{{y=cos(2x+540^o)}}} . The graph of {{{y=cos(2x+540^o)}}} as a function of {{{x}}} is shown below.
{{{graph(900,200,-342,208,-1.1,1.1,cos(pi*(2x+540)/180))}}} 
 
Finally, to shift an x-y graph up by two units, we just add {{{2}}} to the expression for {{{y}}} .
The graph of {{{y=cos(2x+540^o)+2}}} as a function of {{{x}}} is shown below.
{{{graph(900,450,-342,208,-0.2,3.1,cos(pi*(2x+540)/180)+2)}}} 
 
 
A DIFFERENT WAY:
With {{{T=2W}}} , we change the function from {{{y=cos(T)}}} to {{{y=cos(2W)}}} ,
and that changes the period from {{{360^o}}} to {{{(360^o)*(1/2)=180^o}}}.
The graph of {{{y=cos(2x)}}} as a function of {{{x}}} is shown below.
{{{graph(900,200,-342,208,-1.1,1.1,cos(pi*(2x)/180))}}} 

To shift the graph {{{270^o}}} to the left we change {{{W}}} to {{{x+270^o}}} to get {{{y=cos(2(x+270^o))=cos(2x+540^o)}}}

Finally, we add {{{2}}} to shift the graph up by {{{2}}} units, and get {{{y=cos(2x+540^o)+2}}} .
 
NOTE: There are other expressions with the same graph, such as {{{y=cos(2x-180^o)+2}}} :
{{{graph(900,450,-342,208,-0.2,3.1,cos(pi*(2x-180)/180)+2)}}} .