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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)
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 tan :y x
90 180 270 360-90-180-270-360
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
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 tan :y x Period :
Domain : all real numbers
except2
x n
Range : ,
Asymptotes :2
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.
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
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:
Period :
Domain : all real numbers
except x n
Range : ,
Asymptotes : x n
• The graph of cot :y x
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.
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
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
tan and cot
is .
y a bx c d y a bx c d
b
1 a) Period of 2 tan 1:
2 4y x
21
2b
THIS IS DIFFERENT
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 foundby solving for the two equations:
2 2bx c and bx c
Find the asymptotes of 4 tan 3 1:4
y x
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
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
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
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
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:
0bx c and bx c
Find the asymptotes of 4cot 3 1:4
y x
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 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
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
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
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 : 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
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 : 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
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 : 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
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 : 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
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: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.
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:
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
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: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
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 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.
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 : 2cot 2 3
4y x
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.
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
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.
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
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 2cot 2 34
y x
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 sec :y x
45 90 135 180 225 270 315 360
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
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 sec :y x Period : 2
Domain : all real numbers
except2
x n
Range : , 1 1,and
Asymptotes :2
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.
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
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 csc :y x
Period : 2
Domain : all real numbers
except x n
Range : , 1 1,and
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.
a) Is there an amplitude for a secant or a cosecant function? Why or why not?
Graphical effects of constants , , and in
sec and csc
functions :
a b c d
y a bx c d y a bx c d
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.
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
Graphical effects of constants , , and in
sec and csc
functions :
a b c d
y a bx c d y a bx c d
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 : 2csc 2 34
y x
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.
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
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.
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
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 2csc 2 34
y x
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.
-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
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.
-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
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.
-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
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.
-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
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.
-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
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.
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
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.
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
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.
-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
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.
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
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.
-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
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.
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
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.
-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
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.
• 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.
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 : 2sin 2 1y x
-90-180
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
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 : 2sin 2 1y x
-90-180
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
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.
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
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.