Graphing and

Functions of this form can be graphed by studying the effects of  k,  A,  B, and  C
on the basic graphs of y = sin x and y = cos x.


                                                   and
The constant
                              
is called the amplitude of the graph .
Both graphs are periodic with period , since
                                                                                          
                                                                                          

          See Example 1, page 478, of the textbook.



                                                   and

           and both have amplitude equal to 1.

The period of the sine function is ,
so   completes one cycle as varies from
                                                                                                                 to  
or, in other words, as varies from

                                                                                                                     to      

So the period of is
                                                            

Checking:

If , then


          


                                      

Similarly, the period of

          
                                                   is
                                                            

          See Example 2, page 479, of the textbook.


                                         and

                                                            Summarizing our results so far:

For or
                                                                                                           
                                                                                                                     

If   ,
the basic sine or cosine curve is stretched.
If   ,
the basic sine or cosine curve is compressed.

          
See Examples 3 – 4, pages 480 – 481, of the textbook.


The graphs of   and  
can be obtained by
                                                          translating
the graphs of
                                      and   

vertically,
                                                  up units
                                                                            if is positive
and
                                           down units
                                                                             if is negative.


                                                   and
Since has period ,
                                                                    
must complete one cycle as
                                                                                      
                                                                                       varies from
                                                                                     
                                                                                               to
                                                                                     


In other words, as
                                                                                         
                                                                                varies from
                                                                                        
                                                                                              to
                                                                                         


                                                   has a period of
Its graph is translated
                                                             units to the right if is positive

                                                             units to the left if is negative

The quantity is called the phase shift.

Similar results apply to



Summary of Properties of   y = A sin (Bx + C)   and   y = A cos (Bx + C)
For B > 0:
          Amplitude =            Period =            Phase Shift =

The graph completes one full cycle as varies over the interval

                                        


See Examples 6 – 8, pages 485 – 487, of the textbook.



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