y = tan x  Recall from the unit circle:  that tan  =  tangent is undefined when x = 0.  y=tan x is undefined at x = and x =.

Domain/Range of the Tangent Function  The tangent function is undefined at + k .  Asymptotes are at every multiple of + k .  The domain is (- ,  except + k  ).  Graphs must contain the dotted asymptote lines. These lines will move if the function contains a horizontal shift, stretch or shrink.  The range of every tan graph is (- ,  ).

Period of Tangent Function  This also means that one complete cycle occurs between and.  The period is .

Critical Points  The range is unlimited; there is no maximum.  The range is unlimited; there is no minimum.

y = tan x Key Points  : asymptote. The graph approaches -  as it near this asymptote  (, -1), (0,0), (, 1)  : asymptote. The graph approaches  as it nears this asymptote

Graph of the Parent Function

Parent Function: (- ,  )

The Graph: y = a tan b (x - c)+ d  a = vertical stretch or shrink  If |a| > 1, there is a vertical stretch.  If 0<|a|<1, there is a vertical shrink.  If a is negative, the graph reflects about the x-axis.

y = 4 tan x

y = a tan b (x - c) + d  b= horizontal stretch or shrink  Period =  If |b| > 1, there is a horizontal shrink.  If 0 < |b| < 1, there is a horizontal stretch.  If b<0, the graph reflects about the y-axis.

y = tan 2x

y = a tan b (x - c ) + d  c = horizontal shift  If c is negative, the graph shifts left c units. (x - (-c)) = (x + c)  If c is positive, the graph shifts right c units. (x - (+c)) = (x - c)

y = tan (x -  /2)

y = a tan b (x-c) + d  d= vertical shift  If d is positive, graph shifts up d units.  If d is negative, graph shifts down d units.

y = tan x + 3

y = 3 tan ( 2 x-  ) - 3