Y tan x
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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  .

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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 (2x-) - 3


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