Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions
                                        Using the same methods as in the last section, we summarize the results.


                                                                    and


                                                                                               Graph of 
                              
Period:
Domain: All real numbers except , an integer
Range: All real numbers
Symmetric with respect to the origin
Increasing function between consecutive asymptotes
Discontinuous at , an integer



                                                     Graph of   
                 
Period:
Domain: All real numbers except , an integer
Range: All real numbers
Symmetric with respect to the origin
Decreasing function between consecutive asymptotes
Discontinuous at , an integer


The amplitude is not defined for the tangent or cotangent functions.
The graphs of both functions are unbounded both in the positive and negative directions.
They rise upwards forever and shoot downwards forever.
The constants B and C effect a change in the period and the phase shift of both functions.

      and both have period ,
                    so
      and
                    complete one cycle as
                                                                      
                                                                      varies from
                                                                              to  
In other words,
                                                                    as  x  varies from
                                                                            to   

                       is the phase shift

                           is the period

of these two functions.

                   See Examples 1 – 2, pages 495 – 496, of the textbook.





                                                                              and

                                     Graph of  
            
Period:
Domain: All real numbers except
, an integer
Range: All real numbers such that
 or 
Symmetric with respect to the y axis
Discontinuous at Discontinuous at
, an integer




                       Graph of  
           
Period:
Domain: All real numbers except
, an integer
Range: All real numbers such that
 or 
Symmetric with respect to the origin
Discontinuous at Discontinuous at
, an integer



The amplitude is not defined for the secant or the cosecant functions.
The graphs of both functions are unbounded both in the positive and negative directions.
They rise upwards forever and shoot downwards forever.
The constants B and C effect a change in the period and the phase shift of both functions.

Both functions have period , so (as we did for the sine and the cosine functions) we can
find the period and the phase shift for each function by solving

                                                                                                                           
                                                                                                                                   and
                                                                                                                           

                   See Example 3, pages 498 – 499, of the textbook.


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