GRAPHS OF TRIGONOMETRIC FUNCTIONS
a sin(bx  c)  d
If y  
, then
a cos(bx  c)  d
Domain is (, ) and Range is [ | a |  d , | a |  d ]
2
is called the period of function
|b|
|a| is called Amplitude
c
 is called Phase Shift
b
d is called vertical translation
If y  a sec(bx  c)  d , then

c
Domain is all real numbers except x  (2n  1) 
2b b
Range is (,  | a |  d ]  [| a |  d , )
2
is called the period of function
|b|
No Amplitude for the functions sec, csc, tan and cot
c
 is called Phase Shift
b
d is called vertical translation
 c
x  (2n  1)  are the asymptotes of the function
2b b
If y  a csc(bx  c)  d , then
n c
Domain is all real numbers except x 

b b
Range is (,  | a |  d ]  [| a |  d , )
n c
x
 are the asymptotes of the function
b b
2
is called the period of function
|b|
c
 is called Phase Shift
b
d is called vertical translation
If y  a tan(bx  c)  d , then
Domain is all real numbers except x  (2n  1)
Range is (, ), Period is

2b

c
b

|b|

c
x  (2n  1)  are the asymptotes of the function
2b b
If y  a cot(bx  c)  d , then
Domain is all real numbers except x 
Range is (, ), Period is
n c

2b b

|b|
n c
x
 are the asymptotes of the function
2b b
EXAM QUESTION
2x
The period of f ( x)  3cos
is
3
3
4
A)2 B)
C )3 D)4 E )
4
3
EXAM QUESTION
Let f ( x)  a sin(bx), where b  0.
If the periond of f is 12 and f (3)  4, then f (25)  ?
A)2
B)6
C )4
D)0
E )8
EXAM QUESTION
Let f ( x)  a tan(bx), where a  0, b  0.
3
If the periond of f is 3, then f ( )  is
4
a
a
A)equal to 
B)undefined C ) equal to 
b
b
D)equal to  a
E )equal to b
EXAM QUESTION
Let f ( x)  a cos(bx).
If the periond of f is 8 and f (4)  3, then f (12)  ?
A)3
B)4
C )12
D)8
E )0
EXAM QUESTION
The range of f ( x)  1  4sec x is
A)(, 3]  [5, )
B)(, 3)  (5, )
C )(, 1)  (1, )
D)(, 1]  [1, )
E )(1,4)  (5, )
EXAM QUESTION
Let n be any integer, then the equation of
the vertical asymptote of the function
x 
f ( x)  2csc 
 is
 2 
A) x  2n
B ) x  2n  1
C ) x  4n
D) x  (2n  1)
E ) x  2n
EXAM QUESTION
If f ( x)  2cot 2 x, then the number of
the vertical asymptotes over the interval
  3
 ,
 4 4
A)2

 is equal to

B)1
C )3
D)4
E )0
EXAM QUESTION
2x
If f ( x)  3cot , then the number of
3
the vertical asymptotes over the interval
 3 15 
  4 , 4  is equal to
A)3
B)2
C )4
D)5
E )6
EXAM QUESTION
x
If f ( x)  3cot , then the number of
2
the vertical asymptotes over the interval
  
  ,   is equal to
 2 
A)1
B)2
C )0
D)3
E )4
EXAM QUESTION
3x
If f ( x)  2cot , then the number of
2
the vertical asymptotes over the interval
 

  ,3  is equal to
 6

A)5
B)9
C )3
D)2
E )4
EXAM QUESTION
If f ( x)  3  2cot
x
, then the number of
3
the vertical asymptotes over the interval
 4,4 is equal to
A)6
B)3
C )2
D)1
E )4
EXAM QUESTION
The number of the x  intercepts of the graph
f ( x)  2cot 2 x on the interval (  ,  ) is
A)4
B)3
C )2
D)1
E )5
y  a cos bx a  0
y
a
x- intercepts


4
2b

3
2b
2
2b


0
2b

2b
a
2
2b
3
2b
4
2b
x
y  a cos bx
a0
y
a
4

2b
4
2b
3

2b
2

2b


0

2b
2b
x- intercepts
a
2
2b
3
2b
x
y  3 cos 2 x
a3 b2
y
3


3

4


2
2


0
4

4
3
3
4

x
y  2 cos 3x
a  2 b  3
y
2
2

3


2


3


0
6
2


6
3

2
2
3
x
y  a sin bx
a0
y
a
0
a
3
2b

2b
2
2b
4
2b
5
2b
x
y  a sin bx a  0
y  a sin bx a  0
y
a
5
2b

2b
0
a
2
2b
3
2b
4
2b
x
Draw one full period of y=2sin(3x–π/2)
Period = 2π/|b| = 2π/3
Amplitude = |a| = 2
2
Phase shift = -c/b = π/6
π/2
π/6
2π/3
5π/6
π/3
-2
This is the graph of 2sin(3x). Now click to see the phase shift and to get 2sin(3x–π/2)
Graph one full period of sin(x–π /2) –1/2
a =1, b =1,c = – π/2 and d = –1/2
Section 5.7 Question 43
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b = π/2
Vertical translation
½ units down
Vertical translation: 1/2 units down
1
1/2
y = sin(x–π /2)

2
y = sin(x)
–1
–3/2
Phase shift
π/2 units right
y = sin(x–π /2) –1/2
Graph one full period of 2sin(3x–π /2) +1
y
a =2, b =3,c = – π/2 and
d=1
Amplitude = |a| =2
3
2
Period = 2π/b = 2π/3
Phase shift = – c/b =
π/6
1
π/2
π/6
Vertical translation:
1 unit up
2π/3
π/3
–1
–2
This is the graph of 2sin(3x). Now click to see the phase shift , vertical translation
and to get 2sin(3x–π/2)+1
x
Graph one full period of sin(x+π /6)
y
a =1, b =1,c = π/6
Section 5.7 Question 18
Amplitude = |a| =1
Period = 2π/b = 2π
1
Phase shift = – c/b
= –π/6
3π/2
–π/6
π/2
2π
π
–1
This is the graph of sin(x). Now click to see the phase shift and to get sin(x+π/6)
x
Graph one full period of cos(2x–π/3)
y
Section 5.7 Question 20
a =1, b = 2,c = – π/3
Amplitude = |a| =1
Period = 2π/b = π
1
Phase shift = – c/b
= π/6
π/2
π/6
π/4
3π/4
π 7π/6
–1
This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-π/3)
x
Graph one full period of y=(1/2)sin(πx/3)
y
a =1/2, b = π/3
Amplitude = |a| =1/2
Period = 2π/b = 6
1/2
9/2
3/2
–1/2
3
6
x
3
y  3 sin x
2
3
a  3 b 
2
2b  3
y
3
0
3
5
3

3
2
3

4
3
x
Graph one full period of y=2sinx and y= sinx
In [0 , π] ,
0≤ sinx ≤ 2sinx
2
In [π , 2π] ,
1
2sinx ≤ sinx ≤ 0

2
–1
–2

3
2
2
y  a sec(bx)

5
2b
4

2b
3

2b
a
2
2b


0

2b
2b
a
2
2b
3
2b
4
2b
5
2b
y  a csc(bx)
a
5
2b
4

2b
3

2b
2

2b


0

2b
2b
a
2
2b
3
2b
4
2b
5
2b
y

2


4b
4b
a
a

4b
2
4b
y  a tan bx a  0
x
y

2
4b
a

4b


4b
2
4b
a
y  a tan bx a  0
x
y
y  a cot bx a  0
a
a

4b
2
4b
3
4b
4
4b
x
y
y  a cot bx a  0
a

4b
a
2
4b
3
4b
4
4b
x
Draw one full period of y = 2tan(x/2)
y
a = 2 and b = 1/2 , 4b = 2
Asymptotes:
x = ±2π/4b = ± 2π/2 = ± π
Lets draw
asymptotes

Mark 2 and –2 on the y-axis
and ±π/4b = ±π/2 on the xaxis
Now we can draw the graph
Section 5.6 Question 29

2
2


–2
2

x
Graph one full period of (3/2)csc(3x)
a =3/2, b = 3
Period = 2π/b = 2π/3
3/2
π/2
π/6
–3/2
Section 5.6 Question 34
π/3
2π/3
Graph one full period of (1/3)tanx
a =1/3, b =1→4b = 4
Period = π/|b| = π
–π/4
–π/2
Section 5.6 Question 22
1/3
–1/3
π/4
π/2
Graph one full period of 2cscx
y
a =2, b = 1
Period = 2π/b = 2π
2
π/2
–2
Section 5.6 Question 28
π
3π/2
2π
x
Graph one full period of -3sec(2x/3)
y
a = –3 , b = 2/3
Period = 2π/b = 3π
3
3π/4
–3
Section 5.6 Question 36
3π/2
9π/4
3π
x
Draw one full period of y = –3tan(3x)
y
a = –3 and b = 3 , 4b = 12
Asymptotes:
x = ±2π/4b = ± 2π/12 = ± π/6
Period = π/b = π/2
Lets draw asymptotes
3
Mark 3 and –3 on the y-axis
and ±π/4b = ±π/12 on the x-axis
Now we can draw the graph


6

12

12

–3

6
x
y
Draw one full period of y = (1/2)cot(2x)
a = 1/2 and b = 2 , 4b = 8
Asymptotes:
x = π/b = π/2 and x = 0(y-axis)
Period = π/b = π/2
Lets draw asymptotes
Mark 1/2 and –1/2 on the y-axis
and π/8, 2π/8, 3π/8 and 4π/8 on the
x-axis
Now we can draw the graph
1/2
0
–1/2
3
8

8
2
8
4
8
x
Graph one full period of y=3/2sin(x /4+3π /4)
y
a =3/2, b = 1/4,c = 3π/4
Amplitude = |a| = 3/2
3/2
Period = 2π/b = 8π
Phase shift = – c/b = –3π
6π
–3π
2π
8π
4π
–3/2
This is the graph of y=3/2sin(x/4). Now click to see the phase shift and to get
y=3/2sin(x/4+3π/4)
x
Graph one full period of y= sec(x − π/2 )+1
a = 1 , b = 1, c = −π/2, d = 1
y
sec(x)
Period = 2π/|b| = 2π
Phase shift = −c/b = π/2
2
Vertical translation :
1
cos(x)
(d =) 1 unit up
π
|
π/2
Click to shift π/2 unit to right
Click to shift 1 unit up
–1
|
|
3π/2
|
2π
x
Graph one full period of y= csc(x/3-π/12)+4
y
a = 1, b = 1/3, c = π/3, d = 4
Period = 2π/|b| = 6π
5
Phase shift = -c/b = π/4
Vertical translation: 4 unit up
sin(x/3)
3
1
π/4
|
|
3π/2
3π
9π/2
|
–1
csc(x/3)
|
6π
x
Sketch the graph of y = |(1/2)sin(3x)|
1/2sin(3x) ≥ 0
1/2sin(3x) ≥ 0
1/2
-2π/3
π/3
0
-π/3
1/2sin(3x) ≤ 0
-1/2
Section 5.5 Question 48
2π/3
1/2sin(3x) ≤ 0
Sketch the graph of y = cos2(x)
1
1/2
-3π/2
-π
-π/2
-π/4
π/4
Section 5.5 Question 65
π/2
π
3π/2
Sketch the graph of y = sin|x|
sin(x) if x ≥ 0
y = sin|x| =
−sin(x)
1
-2π
-π
π
0
-1
Section 5.5 Question 68
if x ≤ 0
2π