Tangent and Cotangent Graphs

1
Tangent and Cotangent Graphs

• Reading and Drawing
• Tangent and Cotangent Graphs

Some slides in this presentation contain
animation. Slides will be more meaningful if you
allow each slide to finish its presentation
before moving to the next one.
2
This is the graph for y tan x.
This is the graph for y cot x.
3
One definition for tangent is .
Notice that the denominator is cos x. This
indicates a relationship between a tangent graph
and a cosine graph.
4
To see how the cosine and tangent graphs are
related, look at what happens when the graph for
y tan x is superimposed over y cos x.
5
In the diagram below, y cos x is drawn in gray
while y tan x is drawn in black. Notice that
the tangent graph has VERTICAL asymptotes
(indicated by broken lines) everywhere the cosine
graph touches the x-axis.
6
One definition for cotangent is .
Notice that the denominator is sin x. This
indicates a relationship between a cotangent
graph and a sine graph.
This is the graph for y sin x.
7
To see how the sine and cotangent graphs are
related, look at what happens when the graph for
y cot x is superimposed over y sin x.
8
In the diagram below, y sin x is drawn in gray
while y cot x is drawn in black. Notice that
the cotangent graph has VERTICAL asymptotes
(indicated by broken lines) everywhere the sine
graph touches the x-axis.
9
y tan x.
y cot x.
For tangent and cotangent graphs, the distance
between any two consecutive vertical asymptotes
represents one complete period.
10
y tan x.
One complete period is highlighted on each of
these graphs.
y cot x.
For both y tan x and y cot x, the period is
p. (From the beginning of a cycle to the end of
that cycle, the distance along the x-axis is p.)
11
For y tan x, there is no phase shift.
The y-intercept is located at the point (0,0).
We will call that point, the key point.
12
A tangent graph has a phase shift if the key
point is shifted to the left or to the right.
13
For y cot x, there is no phase shift.
Y cot x has a vertical asymptote located along
the y-axis. We will call that asymptote, the key
asymptote.
14
A cotangent graph has a phase shift if the key
asymptote is shifted to the left or to the right.
15
For a cotangent graph which has no vertical
shift, the equation for the graph can be written
as
For a tangent graph which has no vertical shift,
the equation for the graph can be written as
y a tan b (x - c).
y a cot b (x - c).
c indicates the phase shift, also known as the
horizontal shift.
a indicates whether the graph reflects about the
x-axis.
b affects the period.
16
Unlike sine or cosine graphs, tangent and
cotangent graphs have no maximum or minimum
values. Their range is (-8, 8), so amplitude is
not defined. However, it is important to
determine whether a is positive or negative.
When a is negative, the tangent or cotangent
graph will flip or reflect about the
x-axis.
y a tan b (x - c) y a cot b (x - c)
17
Notice the behavior of y tan x.
Notice what happens to each section of the graph
as it nears its asymptotes. As each section
nears the asymptote on its left, the y-values
approach - 8. As each section nears the asymptote
on its right, the y-values approach 8.
18

Notice the behavior of y cot x.
Notice what happens to each section of the graph
as it nears its asymptotes. As each section
nears the asymptote on its left, the y-values
approach 8. As each section nears the asymptote
on its right, the y-values approach - 8.
19
This is the graph for y tan x.
y - tan x
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 x-axis.
20
This is the graph for y cot x.
y - 2cot x
Consider the graph for y - 2 cot x In this
equation a, the numerical coefficient for the
cotangent, is equal to -2. The fact that a is
negative causes the graph to flip or reflect
about the x-axis.
21
y a tan b (x - c) y a cot b (x - c)
b affects the period of the tangent or cotangent
graph. For tangent and cotangent graphs, the
period can be determined by
Conversely, when you already know the period of a
tangent or cotangent graph, b can be determined by
22
A complete period (including two consecutive
vertical asymptotes) has been highlighted on the
tangent graph below.
For all tangent graphs, the period is equal to
the distance between any two consecutive vertical
asymptotes.
The distance between the asymptotes in this graph
is . Therefore, the period of this
graph is also .
23
Use , the period of this tangent graph,
to calculate b.
We will let a 1, but a could be any positive
value since the graph has not been reflected
about the x-axis.
An equation for this graph can be written as
or .
24
A complete period (including two consecutive
vertical asymptotes) has been highlighted on the
cotangent graph below.
For all cotangent graphs, the period is equal to
the distance between any two consecutive vertical
asymptotes.
The distance between the asymptotes is
. Therefore, the period of this graph is also
.
25
Use , the period of this cotangent graph,
to calculate b.
We will let a 1, but a could be any positive
value since the graph has not been reflected
about the x-axis.
An equation for this graph can be written as or
.