SOLUTION: compare the graph of y2 =4sin[3(x+3pi/4)]-3 with the graph of y1=s

Compare the graph of 

y2 = 4sin[3(x + 3p/4)] - 3 with the graph of y1 = sin x 

--------------------

To transform the graph of

y1 = sin x

into the graph of 

y2 = A·sin[B(x + C) + D]

requires this set of 5 transformations in this order

#1. A horizontal stretch by a factor of 2p/B if 2p/B > 1
    A horizontal shrink by a factor of 2p/B if 2p/B < 0

    This involves dividing the x coordinate of any point by B. 

#2. A vertical stretch by a factor of |A| if |A| > 1
    A vertical shrink by a factor of |A| if |A| < 1

    This involves multiplying the y-coordinate of any point by A.   

#3. A reflection in the x-axis if A < 0, (none if A > 0)

    This involves multiplying the y-coordinate by -1 if A < 0, or doing
    nothing if A > 0 (as in this case)

#4. A horizontal shift left |C| units if C > 0
    A horizontal shift right |C| units if C < 0

    This involves:
    subtracting |C| from the x-coordinate of any point if C positive
    or
    adding |C| to the x-coordinate of any point if C is negative.

#5. A vertical shift upward |D| units if D > 0
    A vertical shift downward |D| units if D < 0

    This involves adding |D| to the y-coordinate of any point if D is positive
    or
    subtracting |D| from the y-coordinate if D is negative. 

The 5 important points of the basic cycle of y1 = sin x are

1.    (0,0) , an x intercept, or "node"
2.  (p/2,1) , a "peak", or "maximum"
3.    (p,0) , an x-intercept, or "node"
4. (3p/2,-1), a "valley", or "minimum"
5.   (2p,0 ), an x-intercept, or "node" 

In your problem:

y2 = 4sin[3(x + 3p/4)] - 3

A = 4, B = 3, C = 3p/4, D = -3 

Applying the 5 transformation to each the 5 important points of the
basic cycle of y1 = sin x

                              transformation
          #1          #2          #3          #4               #5
   (0,0)  ->    (0,0) ->    (0,0) ->   (0,0)  ->    (-3p/4,0)  ->    (-3p/4,-3)
 (p/2,1)  ->  (p/6,1) ->  (p/6,4) -> (p/6,4)  -> (p/6-3p/4,4)  -> (p/6-3p/4,1)  
   (p,0)  ->  (p/3,0) ->  (p/3,0) -> (p/3,0)  -> (p/3-3p/4,0)  -> (p/3-3p/4,-3)
(3p/2,-1) ->  (p/2,-1)->  (p/2,-4)-> (p/2,-4) -> (p/2-3p/4,-4) -> (p/2-3p/4,-7) 
  (2p,0)  -> (2p/3,0) -> (2p/3,0) ->(2p/3,0)  ->(2p/3-3p/4,0)  -> (2p/3-3p/4,-3)

Simplifying the 5 important points on y2

1.  (-3p/4,-3) approximately (-2.4,-3)
2.  (-7p/12,1)               (-1.8,1) 
3   (-5p/12,-3)              (-1.3,-3 
4.  (-p/4, -7)               (-0.8,-7) 
5.  (-p/12, -3)              (-0.3,-3)

Plot these 5 important points of y1 and y2 and extend their graphs.

The red graph is y1 and the green graph is y2: 
 
 

Edwin
AnlytcPhil@aol.com