SOLUTION: 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?

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): You can put this solution on YOUR website!

With the amplitude unchanged, the general equation is

where c is the horizontal (phase) shift and d is the vertical shift. b is the normal period of 360 degrees, divided by the actual period.

(1) The basic graph:

(2) Shifted 270 degrees left --> c = -270:

(3) Period changed from 360 degrees to 180 degrees --> b = 360/180=2: or

(4) Shifted up 2 --> d = 2:
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