The A-B-C’s of the Generic Sine Curve 

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:________________.

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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,  ].

.

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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_______________..

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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  .