8-3 Translations of Sine and Cosine Curves
 
 
We already know how to translate a graph from our study of functions.  A translation is in the form   y - k = f(x - h), where it is translated k units vertically and h units horizontally.  This fits right into our study of the Sine and Cosine curves.

Sine and Cosine Curves
If the graphs of y = A Sin Bx and y = A Cos Bx are translated h units horizontally and k units vertically, then the resulting graphs have the equations:
y - k = A Sin B(x - h)  and   y - k = A Cos B(x -h)
Demonstration of Graph of y = A sin B(x - C) (Manipula Math)

Graphing Examples
 
1)  Graph the function  y - 3 = 2 sinp(x - 1)
 
                Solution: This graph has the same amplitude and period length without the translation.  A = 2 and period length = 2p/ p= 2  Graph this one.  Points on the graph are:  zeros (0, 0), (1, 0), (2, 0)
                             maximum at (.5, 2) and minimum at (1.5, -2)
 
Now translate the graph by moving the five points above, three units up and one unit to the right!!
This makes the following five points:
(1, 3),  (2, 3), (3, 3), (1.5, 5), and (2.5, 5)
Both graphs are shown on the next grid.  The green one is the final graph of the function.
 
2)  Graph the function:  y + 2 = 4 cos p/4(x + 1)
                    Solution:  This is the graph of y = cos (p/4)x translated 2 units down and one unit to the left.  The amplitude is 4 and the period length is
2p/(p/4) = 8.  Here is the graph of  the function in blue.
Maximum points at: (0, 4), (8, 4)
Minimum point at (4, -4)
Zeros at: (2, 0) and (6, 0)
Now translate the graph by moving each point 2 units down and 1 unit left!
The points are translated to:  (-1, 2), (7, 2), (3, -6), (1, -2) and (5, -2).  These points are marked on the green graph with black dots!  The green graph is the final solution!
 
 

3)  Graph y = 3 cos 4(x - p)
 
                    Solution:  This has A = 3 with period length 2p/ 4 = p/ 2.  It is shifted punits to the right.  Graph it first without the translation.

maximum points at:  (0, 3), (p/2, 3)
Minimum point at ( p/4, -3)
zeros at:  (p/8, 0) and (3p/8, 0)
 
Now shift all the points pi units to the right!
Maximum points at:  (p,3), (3p/2, 3)
minimum point at (5p/4, -3)
zeros at:  (9p/8, 0), (11p/8, 0)
 
This is actually the same graph as the one before we translated it.  Why?
 

4)  Give an equation for the following graph:
 
This graph has many different solutions including ones for both the sine and cosine functions.  In either case, the amplitude and period length are the same.  Look at the graph and find the maximum and minimum points.
They happen at 1 and -3.  Remember the formula to find the amplitude of a graph?  That's right, its maximum - minimum divided by two.
A = [1 - (-3)]/2 = 2
The period length is 6.  Look at the graph and count between maximum points or minimum points.  Now solve for B in the equation
Period Length = 2p/B
6 = 2p/B
6B = 2p
B = p/3
Our equation has the form y = 2 sin (p/3)x or y = 2 cos (p/3)x before the translation.
For the sine curve, the graph starts at a "zero".  When you translate a graph, the "zero" becomes the line midway between the max and min values.  Look at the graph:
The black dot represents a "zero" of the graph before the translation.  Now determine where the new point is.  It was moved 1 unit right and 1 unit down.  You now have h and k!!  Put them in the formula and voila!!
y + 1 = 2 sin [(p/3)(x - 1)]
You can do a similar translation for the cosine curve, but remember, the cosine starts a cycle at a maximum point!!
 
 
5)  Find an equation for the following graph:
 
Let's find a cosine function for this graph.
A = [5-(-3)]/2= 8/2 = 4
Period length is 4pcounting from the maximum point to the next maximum point.
4p= 2p/B
4pB = 2p
B = 1/2
Before the translation, the equation is y = 4 Cos [(1/2)x]
Now determine the translation.  Remember, the maximum point is the starting point for the cosine function.  Study this graph:
 
Look at the red lines.  The maximum point is moved punits to the right and one unit up from the origin.  The purple graph is before the translation/
The equation becomes:
y - 1 = 4 cos [(1/2)(x - p)]

Hopefully, this will give you a small glimpse into graphing with translations!!