**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!!**

* *

* *