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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 =.

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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 (- , ).

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

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

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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

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Graph of the Parent Function

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Parent Function: (- , )

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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.

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y = 4 tan x

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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.

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y = tan 2x

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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)

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y = tan (x - /2)

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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.

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y = tan x + 3

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y = 3 tan ( 2 x- ) - 3

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