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Tangent and Cotangent Graphs

Reading and Drawing

Tangent and Cotangent Graphs

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This is the graph for y = tan x.

This is the graph for y = cot x.

ππ

πππ

−π−π

−π− 22

3

20

22

32

ππ

πππ

−π−π

−π− 22

3

20

22

32

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One definition for tangent is . xcos

xsinxtan =

Notice that the denominator is cos x. This indicates a relationship between a tangent graph and a cosine graph.

ππ

πππ

−π−π

−π− 22

3

20

22

32

This is the graph for y = cos x.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

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.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

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 horizontal asymptotes (indicated by broken lines) everywhere the cosine graph touches the x-axis.

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One definition for cotangent is . xsin

xcosxcot =

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.

ππ

πππ

−π−π

−π− 22

3

20

22

32

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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.

ππ

πππ

−π−π

−π− 22

3

20

22

32

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ππ

πππ

−π−π

−π− 22

3

20

22

32

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 horizontal asymptotes (indicated by broken lines) everywhere the sine graph touches the x-axis.

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y = tan x.

y = cot x.

ππ

πππ

−π−π

−π− 22

3

20

22

32

ππ

πππ

−π−π

−π− 22

3

20

22

32

For tangent and cotangent graphs, the distance between any two consecutive vertical asymptotes represents one complete period.

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y = tan x.

y = cot x.

ππ

πππ

−π−π

−π− 22

3

20

22

32

ππ

πππ

−π−π

−π− 22

3

20

22

32

One complete period is highlighted on each of these graphs.

For both y = tan x and y = cot x, the period is π. (From the beginning of a cycle to the end of that cycle, the distance along the x-axis is π.)

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ππ

πππ

−π−π

−π− 22

3

20

22

32

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.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

A tangent graph has a phase shift if the key point is shifted to the left or to the right.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

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.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

A cotangent graph has a phase shift if the key asymptote is shifted to the left or to the right.

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y = a tan b (x - c).

For a tangent graph which has no vertical shift, the equation for the graph

can be written as

For a cotangent graph which has no vertical shift, the equation for the graph

can be written as

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.

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y = a tan b (x - c) y = a cot b (x - c)

Unlike sine or cosine graphs, tangent and cotangent graphs have no maximum or minimum values. Their range is (-∞, ∞), 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.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

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 - ∞.

As each section nears the asymptote on its right, the y-values approach + ∞.

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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 + ∞.

As each section nears the asymptote on its right, the y-values approach - ∞.

ππ

πππ

−π−π

−π− 22

3

20

22

32

Notice the behavior of y = cot x.

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This is the graph for y = tan x.

ππ

πππ

−π−π

−π− 22

3

20

22

32

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.

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This is the graph for y = cot x.

ππ

πππ

−π−π

−π− 22

3

20

22

32

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.

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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

.b

periodπ

=

Conversely, when you already know the period of a tangent or cotangent graph, b can be determined by

.period

bπ

=

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A complete period (including two consecutive vertical asymptotes) has been highlighted on the tangent graph below.

The distance between the asymptotes in this graph is .

Therefore, the period of this graph is also .

3x

π−=

3x

π=

3

2π

3

4

3

2

30

33

2

3

4 ππ

πππ−

π−π−

π−

For all tangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

3

2π

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.2

3

32

=ππ

=π

=period

b

We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis.

3

2πUse , the period of this tangent graph, to calculate b.

3

4

3

2

30

33

2

3

4 ππ

πππ−

π−π−

π−

2

31 == ba

An equation for this graph can be written as xy2

3tan1=

or . xy2

3tan=

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A complete period (including two consecutive vertical asymptotes) has been highlighted on the cotangent graph below.

The distance between the asymptotes is .

Therefore, the period of this graph is also .

0=x π=4x

π4

For all cotangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

π4

πππππ−π−π−π− 864202468

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.4

1

4=

ππ

=π

=period

b

We will let a = 1, but a could be any positive value since the graph has not been reflected about the x-axis.

π4Use , the period of this cotangent graph, to calculate b.

4

11 == ba

An equation for this graph can be written as

or .

xy4

1cot1=

πππππ−π−π−π− 864202468

xy4

1cot=

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ππ

πππ

−π−π

−π− 22

3

20

22

32

y = tan x has no phase shift.

We designated the y-intercept, located at (0,0), as the key point.

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ππ

πππ

−π−π

−π− 22

3

20

22

32

y = cot x has no phase shift.

We designated the vertical asymptote on the y-axis (at x = 0) as the key asymptote.

x = 0

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ππ

πππ

−π−π

−π− 22

3

20

22

32

ππ

πππ

−π−π

−π− 22

3

20

22

32

If the key point on a tangent graph shifts to the left or to the right,

or if the key asymptote on a cotangent graph shifts to the left or to the right,

that horizontal shift is called a phase shift.

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y = a tan b (x - c)

c indicates the phase shift of a tangent graph.

For a tangent graph, the x-coordinate of the key point is c.

ππ

πππ

−π−π

−π− 22

3

20

22

32

For this graph, c = because the key point shifted spaces to the right.

An equation for this graph can be written as .

2

π2

π

⎟⎠

⎞⎜⎝

⎛ π−=

2tan xy

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y = a cot b (x – c)

c indicates the phase shift of a cotangent graph.

For a cotangent graph, c is the value of x in the key vertical asymptote.

ππ

πππ

−π−π

−π− 22

3

20

22

32

For this graph, c = because the key asymptote shifted left to .

An equation for this graph can be written as or

2

π−

2

π−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ π−−=

2cot xy

.2

cot ⎟⎠

⎞⎜⎝

⎛ π+= xy

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Graphs whose equations can be written as a tangent function can also be written as a cotangent function.

Given the graph above, it is possible to write an equation for the graph. We will look at how to write both a tangent equation that describes this graph and a cotangent equation that describes the graph.

The tangent equation will be written as y = a tan b (x – c).

The cotangent equation will be written as y = a cot b (x – c).

8

3

480

848

3

28

5 ππππ−

π−

π−

π−

π−

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8

3

480

848

3

28

5 ππππ−

π−

π−

π−

π−

For the tangent function, the values for a, b, and c must be determined.

This tangent graph has reflected about the x-axis, so a must be negative. We will use a = -1.

The period of the graph is .

The key point did not shift, so the phase shift is 0. c = 0

4

44

=ππ

=π

=π

periodb

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8

3

480

848

3

28

5 ππππ−

π−

π−

π−

π−

041 ==−= cba

The tangent equation for this graph can be written

as or .)0(4tan1 −−= xy xy 4tan−=

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8

3

480

848

3

28

5 ππππ−

π−

π−

π−

π−

For the cotangent function, the values for a, b, and c must be determined. This cotangent graph has not reflected about the x-axis, so a must

be positive. We will use a = 1.

The period of the graph is .

The key asymptote has shifted spaces to the right , so the

phase shift is . Therefore, .

4

44

=ππ

=π

=π

periodb

8

π

8

π8

π=c

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8

3

480

848

3

28

5 ππππ−

π−

π−

π−

π−

841

π=== cba

The cotangent equation for this graph can be written

as .⎟⎠

⎞⎜⎝

⎛ π−=

84cot xy

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It is important to be able to draw a tangent graph when you are given the corresponding equation. Consider the equation

Begin by looking at a, b, and c.

.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

63

3

2 π==−= cba

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.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

The negative sign here means that the tangent graph reflects or “flips” about the x-axis. The graph will look like this.

3

2−=a

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.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

b = 3

3

π=

π=b

periodUse b to calculate the period. Remember that the period is the distance

between vertical asymptotes.

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.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

6

π=c

This phase shift means the key point has shifted spaces

to the right. It’s x-coordinate is . Also, notice that the key point is an x-intercept.

6

π

6

π

60

π

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The period is ; half of the period is . Therefore, the

distance between the x-intercept and the asymptotes on either side is .

.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

Since the key point, an x-intercept, is exactly halfway between two vertical asymptotes, the distance from this x-intercept to the vertical asymptote on either side is equal to half of the period.

3

π6

π

60

π

6

π

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.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

360

ππ

We can use half of the period to figure out the labels for vertical

asymptotes and x-intercepts on the graph. Since we already

determined that there is an x-intercept at , we can add half of the

period to find the vertical asymptote to the right of this x-intercept. 6

π

366

π=

π+

πx-intercept

Half of the period

Vertical asymptote

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.6

3tan3

2⎟⎠

⎞⎜⎝

⎛ π−−= xy

Continue to add or subtract half of the period, , to determine the

labels for additional x-intercepts and vertical asymptotes. 6

π

263

π=

π+

πVertical asymptoteHalf of the period

x-intercept

3

2

2360

ππππ3

2

2360

6323

2 πππππ−

π−

π−

π−

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It is important to be able to draw a cotangent graph when you are given the corresponding equation. Consider the equation

Begin by looking at a, b, and c.

.8

x4cot3y ⎟⎠

⎞⎜⎝

⎛ π+=

8c4b3a

π−===

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The positive sign here means that the cotangent graph does not reflect or “flip” about the x-axis. The graph will look like this.

3a =

.8

x4cot3y ⎟⎠

⎞⎜⎝

⎛ π+=

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b = 4

4bperiod

π=

π=

Use b to calculate the period. Remember that the period is the distance

between vertical asymptotes.

⎟⎠

⎞⎜⎝

⎛ π+=

8x4cot3y

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8c

π−=

This phase shift means the key asymptote has shifted

spaces to the left. The equation for this key asymptote is

.

8

π

8x

π−=

08

π−

⎟⎠

⎞⎜⎝

⎛ π+=

8x4cot3y

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The period is ; half of the period is . Therefore, the

distance between asymptotes and their adjacent x-intercepts is . This information can be used to label asymptotes and x-intercepts.

The distance from an asymptote to the x-intercepts on either side of it is equal to half of the period.

4

π8

π

⎟⎠

⎞⎜⎝

⎛ π+=

8x4cot3y

8

5

28

3

480

848

3 ππππππ−

π−

π−

8

π

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Sometimes a tangent or cotangent graph may be shifted up or down. This is called a vertical shift.

y = a tan b (x - c) +d.

The equation for a tangent graph with a vertical shift can be written as

The equation for a cotangent graph with a vertical shift can be written as

y = a cot b (x - c) +d.

In both of these equations, d represents the vertical shift.

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A good strategy for graphing a tangent or cotangent function that has a vertical shift:

•Graph the function without the vertical shift

• Shift the graph up or down d units.

Consider the graph for .

The equation is in the form where “d” equals

3, so the vertical shift is 3.

38

x4cot3y +⎟⎠

⎞⎜⎝

⎛ π+=

( ) dcxbcotay +−=

⎟⎠

⎞⎜⎝

⎛ π+=

8x4cot3yThe graph of was drawn in the previous example.

8

5

28

3

480

848

3 ππππππ−

π−

π−

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To draw , begin with the graph for .

Draw a new horizontal axis at y = 3.

Then shift the graph up 3 units.3

3+

The graph now represents .

38

x4cot3y +⎟⎠

⎞⎜⎝

⎛ π+= ⎟

⎠

⎞⎜⎝

⎛ π+=

8x4cot3y

38

x4cot3y +⎟⎠

⎞⎜⎝

⎛ π+=

⎟⎠

⎞⎜⎝

⎛ π+=

8x4cot3y

8

5

28

3

480

848

3 ππππππ−

π−

π−

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This concludesTangent and Cotangent

Graphs.