“A” controls the height or the amplitude of the curve as shown in the 3 examples below:
Practice:
y = 0.9 sin(x) will stretch/shrink the height of the sine graph by a factor of __________.
The amplitude of this graph is:________________.
y = ¼ sin(x) will stretch/shrink the height of the sine graph by a factor of __________.
The amplitude of this graph is:________________.
y = 5 sin(x) will stretch/shrink the height of the sine graph by a factor of __________.
The amplitude of this graph is:________________.
===========================================================
“B”
controls the length of one cycle, or the period of the curve. The
period of the sine curve y = sin(x) is 
. The period of the sine curve y = sin Bx
is 
. This is because the number B stretches or shrinks the curve
in the x direction. If the number B is bigger than 1, the period
will shrink. If the number B is smaller than 1, the period will stretch.
Notice that B also gives the number of cycles in the interval from [0, 
].
Practice:
The length of one period in y = sin(x) is__________________________
since there is/are _________ cycles in the interval from [0, 
].
.
The length of one
period in y = sin (3x) is ________________________ since there is/are _________
cycles in the interval from [0, 
].
.
The length of one
period in y = sin [(1/2)x] is______________________ since there is/are
_________ cycles in the interval from [0, 
].
.
==============================================================
“C”
controls the phase shift, or the amount the graph shifts to the left or
to the right. The phase shift is calculated by: 
shifts 
units to the left.
shifts 
units to the right.
It looks the same as the prior graph because it is the same!
= 
= 
What if shifts and stretches or shrinks occur
simultaneously? Example: 
= 
period shrinks by a factor of 5. This means that the new period is 
. The graph shifts 
units to the left.
Practice:
The phase shift in the curve y = sin(3x-6) is ____. It shifts the curve ____ units to the _________. Also the period will shrink/stretch by a factor of ____. The new period is__________.
The phase shift in the curve y = sin( 0.5x + 
) is__________. It shifts the curve ______________ units to
the ___________. Also the period will shrink/stretch by a factor
of___________. The new period is__________.
The phase shift in the curve y = sin( 6x - 3
) is_________. It shifts the curve __________ units to the
____________. Also the period will shrink/stretch by
a factor of _____________. The new period is_______________..
========================================================
Answers to Practice:
A. is for Amplitude.
y = 0.9 sin(x) will stretch/shrink the height of the sine graph by a factor of 0.9.
The amplitude of this graph is: 0.9
y = ¼ sin(x) will stretch/shrink the height of the sine graph by a factor of ¼
The amplitude of this graph is: ¼.
y = 5 sin(x) will stretch/shrink the height of the sine graph by a factor of 5 .
The amplitude of this graph is: 5
B. is for Bulge (stretch or shrink the size of the Period)!
The length of one period in y = sin(x) is 
since there is 1 cycle in the interval from [0, 
].
The length of one period in y = sin (3x) is 
since there are 3 cycles in the interval from [0, 
].
The length of one period in y = sin [(1/2)x] is 
since there is ½ of a cycle in the interval from [0, 
].
C. is for Change (shift) in location:
The phase shift in the curve y = sin(3x-6) is
2. It shifts the curve 2 units to the right. Also the period
will shrink/stretch by a factor of 3. The new period is 
.
The phase shift in the curve y = sin( 0.5x + 
) is 
. It shifts the curve 
units to the left (which means it doesn’t go anywhere!) . Also the
period will shrink/stretch by a factor of 2.
The new period is 
.
The phase shift in the curve y = sin( 6x - 3
) is 
. It shifts the curve 
units to the right. Also the period will shrink/stretch
by a factor of 6. The new period is 
.