Download 4.5 – Graphs of Sine and Cosine Functinos

Unit 5 – Graphs of the other
Trigonometric Functions
Tangent and Cotangent
Warm Up
1-9 on the trig graphs
review page done
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.
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.
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)
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.
The graph of the tangent curve:
•The graph of y  tan x :
y
5
4
3
2
1
x
-360
-270
-180
-90
90
-1
-2
-3
-4
-5
180
270
360
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.
The graph of the tangent curve:
• The graph of y  tan x :
Period : 
Domain : all real numbers
except x 

2
n
Range :  ,  
Asymptotes : x 

2
n
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.
The graph of the cotangent curve:
•The graph of y  cot x :
y
5
4
3
2
1
x
-360
-270
-180
-90
90
-1
-2
-3
-4
-5
180
270
360
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.
The graph of the cotangent curve:
• The graph of y  cot x :
Period : 
Domain : all real numbers
except x   n
Range :  ,  
Asymptotes : x   n
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.
Graphical effects of constants a, b, c and d in
y  a tan  bx  c   d and y  a cot  bx  c   d
functions :
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.
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.
Period of Tangent and Cotangent
Functions:
The period of tangent and cotangent functions
y  a tan  bx  c   d and y  a cot  bx  c   d
is

b
.
THIS IS DIFFERENT

1
a) Period of y   2 tan  x    1:
4
2

b


1
2
 2
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.
Asymptotes of the tangent graph
function:
 Where do you think you need to set the left
and right endpoints for a tangent graph
below?
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.
Asymptotes of the tangent graph
function:
Asymptotes for a tangent function can be found
by solving for the two equations:
bx  c  

2
and bx  c 

2


Find the asymptotes of y  4 tan  3x    1:
4

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.
Find the asymptotes of


y  4 tan  3 x    1:
4

bx  c  
3x 

4

3x  
x

2

2

and bx  c 
and 3x 

4

2

4
3
and 3x 
4


12
and
x
4

2
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.
Find the asymptotes of


y  4 tan  3 x    1:
4

Solve for the period and verify that the distance
between the asymptotes is equal to the period:
   
3  4 
Distance    
 


4  12  12 12 12 3
Period 

b


3
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.
Asymptotes of the cotangent graph
function:
 Where do you think you need to set the left
and right endpoints for a cotangent graph
below?
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.
Asymptotes of the cotangent graph
function:
Asymptotes for a cotangent function can be
found
by solving for the two equations:
bx  c  0 and bx  c  


Find the asymptotes of y  4cot  3x    1:
4

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.
Find the asymptotes of


y  4cot  3x    1:
4

bx  c  0 and bx  c  
3x 

4
3x 
x

4

12
 0 and 3x 


4
5
and 3x 
4
and
5
x
12
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.
Find the asymptotes of


y  4cot  3x    1:
4

Solve for the period and verify that the distance
between the asymptotes is equal to the period:
5  4 
Distance 
 

12 12 12 3
Period 

b


3
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.
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.
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.
Reflection : y  a tan  bx  c   d
y   tan x
y  tan x
y
y
5
5
4
4
3
3
2
2
1
1
x
-90
-45
45
90
x
-90
-45
45
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
90
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.
Reflection : y  a tan  bx  c   d
y  tan x and y   tan x
y
5
4
3
2
1
x
-90
-45
45
-1
-2
-3
-4
-5
90
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.
Reflection : y  a cot  bx  c   d
y   cot x
y  cot x
y
y
5
5
4
4
3
3
2
2
1
x
45
-1
-2
90
135
1
x
180
45
-1
-2
-3
-3
-4
-4
-5
-5
90
135
180
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.
Reflection : y  a cot  bx  c   d
y  cot x and y   cot x
y
5
4
3
2
1
x
45
-1
-2
-3
-4
-5
90
135
180
Effects of a on the tangent and
cotangent graphs:
1
y  tan x
2
y  2 tan x
y
y
5
5
4
4
3
3
2
2
1
1
x
x
-90
-45
45
90
-90
-45
45
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
90
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.
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.
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.
Effects of c on the tangent and
cotangent graphs:


y  tan  x  
4



y  tan  x  
4

y
y
5
5
4
4
3
3
2
2
1
1
x
x
-90
-45
45
-1
-2
-3
-4
-5
90
135
-135
-90
-45
45
-1
-2
-3
-4
-5
90
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.
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.
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.
Effects of d on the tangent and
cotangent graphs:
y  tan x  2
y  tan x  2
y
y
5
5
4
4
3
3
2
2
1
1
x
x
-90
-45
45
90
-90
-45
45
-1
-1
-2
-2
-3
-3
-4
-5
-4
-5
90
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.
Vertical Translation:
The constant d determines the vertical
translation 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.
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.
Find the amplitude, period,
reflections, horizontal shift,
vertical shift, endpoints and
sketch the graph.


EXAMPLE : y   2cot  2 x    3
4

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.


EXAMPLE : y   2cot  2 x    3
4

a) Amplitude:
b) Period:
none
 

b 2
c) Horizontal Translation:
d) Vertical Shift:
e) Reflection:

to the right
8
3 units downward
about the x-axis
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.


Problem : y   2cot  2 x    3
4

e) Endpoints:


 0 and 2 x   
4
4

5
2x 
2x 
4
4

5
x
x
8
8
5  
Verify distance with the period:
 
8 8 2
2x 
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.


Graph of y   2cot  2 x    3
4

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.
The graph of the secant curve:
• The graph of y  sec x :
y
5
4
3
2
1
x
45
-1
-2
-3
-4
-5
90
135
180
225
270
315
360
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.
The graph of the secant curve:
The graph of y  sec x :
Period : 2
Domain : all real numbers
except x 

2
n
Range :  , 1 and 1,  
Asymptotes : x 

2
n
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.
The graph of the cosecant curve:
• The graph of y  csc x :
y
5
4
3
2
1
x
45
-1
-2
-3
-4
-5
90
135
180
225
270
315
360
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.
The graph of the cosecant curve:
The graph of y  csc x :
Period : 2
Domain : all real numbers
except x   n
Range :  , 1 and 1,  
Asymptotes : x   n
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.
Graphical effects of constants a, b, c and d in
y  a sec  bx  c   d and y  a csc  bx  c   d
functions :
a) Is there an amplitude for a secant or a
cosecant function? Why or why not?
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.
Graphical effects of constants a, b, c and d in
y  a sec  bx  c   d and y  a csc  bx  c   d
functions :
b) Period is 2 .
b
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.
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.
Find the amplitude, period,
reflections, horizontal shift,
vertical shift, endpoints and
sketch the graph.


EXAMPLE : y   2csc  2 x    3
4

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.


EXAMPLE : y   2csc  2 x    3
4

a) Amplitude:
2

2

b) Period:


b
2
c) Horizontal Translation:
d) Vertical Shift:
e) Reflection:
none

to the right
8
3 units downward
about the x-axis
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.


Problem : y   2csc  2 x    3
4

e) Endpoints:
2x 

 0 and 2 x 
4

2x 
4

x
8

 2
4
9
2x 
4
9
x
8
9 
Verify distance with the period:
 
8 8
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.


Graph of y   2csc  2 x    3
4

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.
y-axis Reflection : y  a sin  bx  c   d
y  sin   x 
y  sin x
y
2
y
2
1
1
x
-360 -270 -180 -90
90
180 270 360
x
-360 -270 -180 -90
90
-1
-1
-2
-2
180 270 360
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.
y-axis Reflection : y  a sin  bx  c   d
y  sin x and y  sin   x 
y
2
1
x
-360 -270 -180 -90
90
-1
-2
180 270 360
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.
y-axis Reflection : y  a cos  bx  c   d
y  cos   x 
y  cos x
y
y
2
2
1
1
x
-360 -270 -180 -90
90
-1
-2
180 270 360
x
-360 -270 -180 -90
90
-1
-2
180 270 360
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.
y-axis Reflection : y  a cos  bx  c   d
y  cos x and y  cos   x 
y
2
1
x
-360 -270 -180 -90
90
-1
-2
180 270 360
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.
y-axis Reflection : y  a tan  bx  c   d
y  tan   x 
y  tan x
y
y
5
5
4
4
3
3
2
2
1
1
x
-90
-45
45
-1
-2
-3
-4
90
x
-90
-45
45
-1
-2
-3
-4
-5
-5
90
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.
y-axis Reflection : y  a tan  bx  c   d
y  tan x and y   tan x
y
5
4
3
2
1
x
-90
-45
45
-1
-2
-3
-4
-5
90
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.
y-axis Reflection : y  a cot  bx  c   d
y  cot x
y   cot x
y
y
5
5
4
4
3
3
2
2
1
1
x
90
180
x
-180
-90
-1
-1
-2
-2
-3
-3
-4
-5
-4
-5
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.
y-axis Reflection : y  a cot  bx  c   d
y  cot x and y  cot   x 
y
5
4
3
2
1
x
-180
-90
90
-1
-2
-3
-4
-5
180
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.
y-axis Reflection : y  a sec  bx  c   d
y   sec x
y  sec x
y
5
y
5
4
4
3
3
2
2
1
1
x
x
90
180
270
360
-360
-270
-180
-90
-1
-1
-2
-2
-3
-3
-4
-5
-4
-5
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.
y-axis Reflection : y  a sec  bx  c   d
y  sec x and y  sec   x 
y
5
4
3
2
1
x
-360
-270
-180
-90
90
-1
-2
-3
-4
-5
180
270
360
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.
y-axis Reflection : y  a csc  bx  c   d
y   csc x
y  csc x
y
y
5
5
4
4
3
3
2
2
1
1
x
90
180
270
360
x
-360
-270
-180
-90
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
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.
y-axis Reflection : y  a csc  bx  c   d
y  csc x and y  csc   x 
y
5
4
3
2
1
x
-360 -270 -180 -90
90
-1
-2
-3
-4
-5
180 270 360
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.
Reflection:
• If a is positive, then there is no reflection about the xaxis.
• If a is negative, then there is a reflection about the xaxis.
• If b is positive, then there is no reflection about the yaxis.
• If b is negative, then there is a reflection about the yaxis.
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.
Example : y  2sin   2 x   1
y
5
4
3
2
1
x
-180
-90
-1
-2
-3
-4
-5
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.
Example : y   2sin   2 x   1
y
5
4
3
2
1
x
-180
-90
-1
-2
-3
-4
-5
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.
y  2sin   2 x   1
y   2sin   2 x   1
y
y
5
5
4
4
3
3
2
2
1
1
x
-180
-90
x
-180
-90
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
HOMEWORK
• Graphing Trig Functions
– Kuta
• Trig graphs review page 1 and 2