y = tan x

1
y tan x

• Recall from the unit circle
• that tan ?
• tangent is undefined when x 0.
• ytan x is undefined at x and x .

2
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 (-?,? ).

3
Period of Tangent Function

• This also means that one complete cycle occurs
  between and .
• The period is ?.

4
Critical Points

• The range is unlimited there is no maximum.
• The range is unlimited there is no minimum.

5
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

6
Graph of the Parent Function
7
Parent Function (-?,?)
8
The Graph y a tan b (x - c) d

• a vertical stretch or shrink
• If a gt 1, there is a vertical stretch.
• If 0ltalt1, there is a vertical shrink.
• If a is negative, the graph reflects about the
• x-axis.

9
y 4 tan x
10
y a tan b (x - c) d

• b horizontal stretch or shrink
• Period
• If b gt 1, there is a horizontal shrink.
• If 0 lt b lt 1, there is a horizontal stretch.
• If blt0, the graph reflects about the y-axis.

11
y tan 2x
12
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)

13
y tan (x - ?/2)
14
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.

15
y tan x 3
16
y 3 tan (2x-?) - 3