Unit 5 – Graphs of the other Trigonometric Functions Tangent and Cotangent MM4A3. Students will investigate and use the graphs of the six trigonometric.

Unit 5 – Graphs of the other Trigonometric Functions

Tangent and Cotangent

MM4A3. Students will investigate and use the graphs of the six trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.

Warm Up1-9 on the trig graphs

review page done

In this section, you will learn to:

• Sketch the graphs of tangent and cotangent functions (Thursday/Friday)

• Sketch the graphs of secant and cosecant functions (Monday/Tuesday)


The graph of the tangent curve:•The graph of tan :y x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


The graph of the tangent curve:

• The graph of tan :y x Period :

Domain : all real numbers

except2

x n

Range : ,

Asymptotes :2

x n


The graph of the cotangent curve:•The graph of cot :y x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


The graph of the cotangent curve:

Period :


except x n

Range : ,

Asymptotes : x n

• The graph of cot :y x


Question: Is there an amplitude for a tangent or a cotangent function? Why or why not?

No amplitude, since the two curves extend infinitely in both directions.

Graphical effects of constants , , and in

tan and cot

functions :

a b c d

y a bx c d y a bx c d


Period of Tangent and Cotangent Functions:

The period of tangent and cotangent functions

tan and cot

is .


b

1 a) Period of 2 tan 1:

2 4y x

21

2b

THIS IS DIFFERENT


Asymptotes of the tangent graph function:

Where do you think you need to set the left and right endpoints for a tangent graph below?


Asymptotes of the tangent graph function:

Asymptotes for a tangent function can be foundby solving for the two equations:

2 2bx c and bx c

Find the asymptotes of 4 tan 3 1:4

y x


Find the asymptotes of

2 2bx c and bx c

4tan 3 1:4

y x

3 34 2 4 2

x and x

33 3

4 4x and x

12 4x and x



Solve for the period and verify that the distance

between the asymptotes is equal to the period:

4tan 3 1:4

y x

3 4Distance

4 12 12 12 12 3

Period3b


Asymptotes of the cotangent graph function:

Where do you think you need to set the left and right endpoints for a cotangent graph below?


Asymptotes of the cotangent graph function:

Asymptotes for a cotangent function can be found

by solving for the two equations:

0bx c and bx c

Find the asymptotes of 4cot 3 1:4

y x


Find the asymptotes of4cot 3 1:

4y x

0bx c and bx c

3 0 34 4

x and x

53 3

4 4x and x

5

12 12x and x



Solve for the period and verify that the distance

between the asymptotes is equal to the period:

4cot 3 1:4

y x

5 4Distance

12 12 12 3

Period3b


Reflection:

• If a is positive, then there is no reflection about the x-axis.

• If a is negative, then there is a reflection about the x-axis.


Reflection : tany a bx c d

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

tany x tany x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Reflection : tany a bx c d

andan nt tay xy x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Reflection : coty a bx c d

coty x coty x

45 90 135 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90 135 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Reflection : coty a bx c d

andot tc coy xy x

45 90 135 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Effects of a on the tangent and cotangent graphs:2tany x

1tan

2y x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Effects of a on the tangent and cotangent graphs:

a) If a > 1, then the graph rises faster.

b) If 0< a < 1, then the graph rises slower.


Effects of c on the tangent and cotangent graphs:

tan4

y x

tan

4y x

45 90-45-90-135

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90 135-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Horizontal Translation or Phase Shift:

The constant c determines the phase shift of the graph.

Phase Shift = - c/b (or –c/b)

a) If c is positive, then the shift is toward the left.

b) If c is negative, then the shift is toward the right.


Effects of d on the tangent and cotangent graphs:tan 2y x tan 2y x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Vertical Translation:

The constant d determines the verticaltranslation of the graph.

a) If d is positive, then the vertical shift is upward.

b) If d is negative, then the vertical shift is downward.


Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and

sketch the graph.EXAMPLE : 2cot 2 3

4y x


a) Amplitude:

b) Period:

c) Horizontal Translation:

d) Vertical Shift:

e) Reflection:

2b

to the right8

3 units downward

about the -axisx

EXAMPLE : 2cot 2 34

y x

none


e) Endpoints:

Verify distance with the period:

Problem : 2cot 2 34

y x

2 0 24 4

x and x

5

8 8 2

52 2

4 45

8 8

x x

x x


Graph of 2cot 2 34

y x


The graph of the secant curve:• The graph of sec :y x

45 90 135 180 225 270 315 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


The graph of the secant curve:

The graph of sec :y x Period : 2


except2

x n

Range : , 1 1,and

Asymptotes :2

x n


The graph of the cosecant curve:• The graph of csc :y x

45 90 135 180 225 270 315 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


The graph of the cosecant curve:The graph of csc :y x

Period : 2


except x n

Range : , 1 1,and

Asymptotes : x n


a) Is there an amplitude for a secant or a cosecant function? Why or why not?


sec and csc

functions :

a b c d



b) Period is

c) The horizontal translation, vertical translation, and reflection all stay the same.

d) It is best to sketch the cosecant and secant graph by first graphing the reciprocal functions of sine and cosine.

2.

b


sec and csc

functions :

a b c d



Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and

sketch the graph.

EXAMPLE : 2csc 2 34

y x


a) Amplitude:

b) Period:

c) Horizontal Translation:

d) Vertical Shift:

e) Reflection:

2 2

2b

to the right8

3 units downward

about the -axisx

EXAMPLE : 2csc 2 34

y x

none


e) Endpoints:

Verify distance with the period:

Problem : 2csc 2 34

y x

2 0 2 24 4

x and x

9

8 8

92 2

4 49

8 8

x x

x x


Graph of 2csc 2 34

y x


-axis Reflection : siny y a bx c d

siny x siny x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y


-axis Reflection : siny y a bx c d

andin ns siyy x x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y


-axis Reflection : cosy y a bx c d

cosy x cosy x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y


-axis Reflection : cosy y a bx c d

andos sc coyy x x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y


-axis Reflection : tany y a bx c d

tany x tany x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


andan nt tay xy x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-axis Reflection : tany y a bx c d


coty x

90 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

coty x

-axis Reflection : coty y a bx c d


-axis Reflection : coty y a bx c d

andot tc coyy x x

90 180-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


secy x secy x

90 180 270 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-axis Reflection : secy y a bx c d


-axis Reflection : secy y a bx c d

andec cs seyy x x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


cscy x cscy x -axis Reflection : cscy y a bx c d

90 180 270 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


-axis Reflection : cscy y a bx c d

andsc cc csyy x x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Reflection:• If a is positive, then there is no reflection about the x-axis.

• If a is negative, then there is a reflection about the x-axis.

• If b is positive, then there is no reflection about the y-axis.

• If b is negative, then there is a reflection about the y-axis.


Example : 2sin 2 1y x

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


Example : 2sin 2 1y x

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


2sin 2 1y x

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

2sin 2 1y x

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y


HOMEWORK

• Graphing Trig Functions– Kuta

• Trig graphs review page 1 and 2

Unit 5 – Graphs of the other Trigonometric Functions Tangent and Cotangent MM4A3. Students will investigate and use the graphs of the six trigonometric.

Documents

phase shift

vertical shift

graphs of tangent

changing period

period of tangent

tangent graph function

cotangent mm4a3

graphs of secant