Question 1143426: The graph of cos(T) is shifted left 270°, has the period decreased to 180°, and is shifted up 2 units. What is the transformed equation?
Found 2 solutions by KMST, greenestamps:
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The graph of  (with  in degrees) has a period of  .
In every period there is a single maximum value of  , such as at  .
goes through a maximum at , and as increases,
it has the next maximum at  .
After that, the next maximum is at  , and so on.
Here is a piece of the graph of  with  in degrees:
 ,
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  to the left, we would end with a maximum at  .
A way textbooks suggest to do that shift is replacing the variable with another, such as replacing  .
hat way the point for  is at  .
We would have  .
The graph of  as a function of is shown below.
If after that we want to change the period to  ,
we can do that by changing to variable , so that it increases twice as fast as  .
If we still want the maximum at , we need to make  ,
and change the function from to  . The graph of as a function of is shown below.
Finally, to shift an x-y graph up by two units, we just add  to the expression for  .
The graph of  as a function of is shown below.
A DIFFERENT WAY:
With  , we change the function from to  ,
and that changes the period from to  .
The graph of  as a function of is shown below.
To shift the graph to the left we change  to  to get 
Finally, we add to shift the graph up by units, and get .
NOTE: There are other expressions with the same graph, such as  :
 .
Answer by greenestamps(13200)  (Show Source):
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