Graphs of the Tangent
Precalculus 2
Agenda
• Do Now: Whattaya
Remember about PEMDAS
and RADicals?
• CW: Work on Portfolio!
Yes, YOU NEED To be
able to graph sine, cosine
and tangent
• HW: UNIT Exam TUESDAY
and WEDNESDAY!!
SWBAT:
• Accurately sketch a SINE,
COSINE And TANGENT
transformation, ID-ing
amplitude, period,
horizontal and vertical
shifts!!
Precalculus 1 and 2
Agenda
• Do Now: Write the
COSINE transformations!
• Class work – Geometry
MCAS Review!
• HW: Review ALL Sine
and Cosine
Packets…remember our
Unit Exam is
THURSDAY!!!!
SWBAT:
• Review Geometry for
MCAS
• Accurately sketch a SINE,
COSINE And TANGENT
transformation, ID-ing
amplitude, period,
horizontal and vertical
shifts!!
The Graph of y = tan x
x
y
1.73


1
3


(0, 0)
0
6
1.73

6

1
3

What happens as x approaches 2 ?
….it’s UNDEFINED!!

What happens as x approaches  2 ?
….it’s UNDEFINED!!!
5
3
  3 5
asymptotes  

,
,
,
,
2
2
2 2
2
2
3
 2
5

0
3

5
2
This is the graph for y = tan x.
y = - tan x
 2 
3
2



2
0

2

3
2
2
Consider the graph for y = - tan x
In this equation a, the numerical coefficient for the
tangent, is equal to -1. The fact that a is negative
causes the graph to “flip” or reflect about the xaxis.
y = a tan b (x - h)+k
b affects the period of the tangent graph.
For tangent graphs, the period can be determined by

period  .
b
Conversely, when you already know the period of a
tangent or cotangent graph, b can be determined by

b
.
period
For all tangent graphs, the period is equal to the
distance between any two consecutive vertical
asymptotes.
 2 
3
2



2
0

2

3
2
2
The distance between the asymptotes in this graph
is…

Therefore, the period of this graph is also

.
y = tan x has no phase shift.
 2 
3
2



2
0

2

3
2
2
We designated the y-intercept, located
at (0,0), as the key point.
It is important to be able to draw a tangent graph when
you are given the corresponding equation. Consider the
equation


y   tan 3 x   .
6

Begin by looking at a, b, and c.
a  1 b  3
c

6


y   tan 3 x   .
6

a  1
The negative sign here means that the tangent graph
reflects or “flips” about the x-axis. The graph will look
like this.


y   tan 3 x   .
6

b=3
Use b to calculate the period. Remember
that the period is
 
period  
the distance
b 3
between vertical asymptotes.


y   tan 3 x   .
6


c
6

This phase shift means the key point has shifted

spaces to the right. It’s x-coordinate is 6 . Also, 6
notice that the key point is an x-intercept.


y   tan 3 x   .
6

The distance between
the x-intercept and the asymptotes on

either side is 6 , because it is half the period!!!!
Caution: the distance to the asymptotes will not always be the
same as the phase shift.
0

6


y   tan 3 x   .
6

0
X-intercept

6

3
  
 
6 6 3
Vertical
asymptote


y   tan 3 x   .
6

Continue to add or subtract half of the period,  , to
6
determine the
labels for additional x-intercepts and vertical asymptotes.

2
3


2


3

Vertical
asymptote

6
0

6

3

2
  
 
3 6 2
2
3
x-intercept
http://www.analyzemath.com/Tang
ent/Tangent.html