f  x   sin x
0
Starts at ZERO
Ends at ZERO
Half-way thru at ZERO
PERIOD = 2
2
Quarter-way thru at MAX =1
ONE above the x-axis
Three quarters-way thru at MIN = – 1
ONE below the x-axis
f  x   a sin b  x  c   d
a  amplitude
y  10 sin x
a  amplitude  10  10
y  10 sin x
Starts at ZERO
Ends at ZERO
Half-way thru at ZERO
Period = 2
y  10 sin x
Quarter-way thru at MAX = 1 x 10 = Ten above the x-axis
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
Three quarters-way thru at MIN = – 1 x 10 = Ten below the x-axis
y  10 sin x
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
y  10 sin x
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
y  10 sin x
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
1
y   sin x
3
a  amplitude   13  13
1
y   sin x
3
Starts at ZERO
Ends at ZERO
Half-way thru at ZERO
Period = 2
1
y   sin x
3
 13 
Quarter-way thru at MAX = 1  1
2
1
1
3
below the x-axis
3
3
1
2
3
3
1– 1   13 
Three quarters-way thru at MIN =
1
3
above the x-axis
1
y   sin x
3
1
2
1
3
3
1
2
3
3
1
1
y   sin x
3
1
3
1
3
1
y   sin x
3
1
3
1
3
f  x   a sin b  x  c   d
a  amplitude
d  vertical shift
d0
d 0
y  sin  x   1
d  vertical shift
d 1
d0
y  sin  x   1
y  sin  x   1
Period = 2
Starts at ZERO on the Translated x-axis
Ends at ZERO on the Translated x-axis
Half-way thru at ZERO on the Translated x-axis
y  sin  x   1
Quarter-way thru at MAX = ONE above the Translated x-axis
Three quarters-way thru at MIN = ONE below the Translated x-axis
y  sin  x   1
y  sin  x   1
y  sin  x   1
y  sin  x   2
d  2
d 0
y  sin  x   2
y  sin  x   2
Starts at ZERO on the Translated x-axis
Ends at ZERO on the Translated x-axis
Half-way thru at ZERO on the Translated x-axis
Period = 2
y  sin  x   2
Quarter-way thru at MAX = ONE above the Translated x-axis
Three quarters-way thru at MIN = ONE below the Translated x-axis
y  sin  x   2
y  sin  x   2
y  sin  x   2
f  x   a sin b  x  c   d
a  amplitude
d  vertical shift
d0
d 0
c  phase shift
example : sin  x  c 
c units 
example : sin  x  c 
c units 
y  sin  t   
c  phase shift
example : sin  x     units 
y  sin  t   
Vertical Axis
Shifts  units
to the Left
y  sin  t   
Starts at the Translated ZERO
Ends at ZERO
Half-way thru at ZERO
Period = 2
y  sin  t   
Quarter-way thru at MAX = ONE above the x-axis
Three quarters-way thru at MIN = ONE below the x-axis
y  sin  t   
y  sin  t   
3  2
2
3

5
2
Period = 2
y  sin  t   
3

5
2
y  sin  t   
3

5
2
2


y  sin x 
c  phase shift


example : sin x 
2

 units 
2

y  sin x  
2

Vertical Axis
Shifts  units
to the Left

y  sin x  
2

Starts at the Translated ZERO
Ends at ZERO
Half-way thru at ZERO
5
2
Period = 2

y  sin x  
2

Quarter-way thru at MAX = ONE above the x-axis
5
2
Three quarters-way thru at MIN = ONE below the x-axis

y  sin x  
2

Period = 2
5
2

y  sin x  
2

5
2

y  sin x  
2

5
2



y  3 sin x 
4
4
a  amplitude  3  3
d  vertical shift
d4
d0
c  phase shift

example : sin x  
4

 units 
4


y  3 sin x    4
4
7
6
5
4
3
Vertical Axis
2

Shifts units
1
4
to the Right
-1
-2
-3
-4
-5
-6
-7

4


y  3 sin x    4
4
7
6
Starts at the Translated ZERO
5
4
Ends at ZERO
3
Period = 2
2
Half-way thru at ZERO
1
-1
-2
-3
-4
-5
-6
-7

4
3
4
5
4
7
4
9
4


y  3 sin x    4
4
7
6
5
4
3
2
1
-1
-2
Quarter-way thru at MAX = 1 x 3

4
3
4
5
4
-3
= 3 units above the translated -4horizontal x-axis
Three quarters-way thru at MIN
-5 = – 1 x 3
-6
= 3 units below the translated horizontal x-axis
-7
7
4
9
4


y  3 sin x    4
4
7
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
-7

4
3
4
5
4
7
4
9
4


y  3 sin x    4
4
7
6
5
4
3
Period = 22
1
7

4
5

4
3

4


-1
4 -2
-3
-4
-5
-6
-7

4
3
4
5
4
7
4
9
4


y  3 sin x    4
4
7
6
5
4
3
2
1
7

4
5

4
3

4


-1
4 -2
-3
-4
-5
-6
-7

4
3
4
5
4
7
4
9
4


y  3 sin x    4
4
7
6
5
4
3
2
1
7

4
5

4
3

4


-1
4 -2
-3
-4
-5
-6
-7

4
3
4
5
4
7
4
9
4
f  x   a sin b  x  c   d
a  amplitude
d  vertical shift
d0
d 0
c  phase shift
example : sin  x  c 
c units 
example : sin  x  c 
c units 
f  x   a sin b  x  c   d
d  vertical shift
a  amplitude
d0
What About b
d 0
c  phase shift
example : sin  x  c 
c units 
example : sin  x  c 
c units 
f  x   a sin b  x  c   d
b  determines the period
2
Period =
b
b must be positive
Given sin  2 x  must rewrite  sin 2x

Given sin  2 x    must rewrite sin 2 x  
2

y  sin 3x
Amplitude : 1
Vertical Shift : 
Phase Shift : 
2 2
Period :

b
3
y  sin 3x
2 2
Period :

b
3
3
2
1
Use 8 Spaces for One Period

6

3
-1
-2
-3
Period 

2
2
3
2
3
y  sin 3x
2 2
Period :

b
3
Starts at ZERO
3
Ends at ZERO
2
Half-way thru at ZERO
1

6

3
-1
-2
-3
Period 

2
2
3
2
3
y  sin 3x
2 2
Period :

b
3
Quarter-way thru at MAX = ONE above the x-axis
3
2
1

6

3

2
2
3
-1
-2
Three quarters-way thru at MIN =-3ONE below the x-axis
y  sin 3x
2 2
Period :

b
3
3
2
1

6
-1
-2
-3

3

2
2
3
y  sin 3x
2 2
Period :

b
3
3
2
1
Use 8 Spaces for One Period

2
3


2


3



6
6
-1
Period 
2
3
-2
-3

3

2
2
3
y  sin 3x
2 2
Period :

b
3
3
2
1

2
3


2


3



6
6
-1
-2
-3

3

2
2
3
y  sin 3x
2 2
Period :

b
3
3
2
1

2
3


2


3



6
6
-1
-2
-3

3

2
2
3
y  sin  12 x 
Amplitude : 1
Vertical Shift : 
Phase Shift : 
f  x   a sin b  x  c   d
b must be positive
Given sin  12x  must rewrite  sin12x
(sin x is an ODD Function)
2 
Period :

12 6
y   sin 12 x 
Period :
2 2 


b
12 6
3
2
1
Use 8 Spaces for One Period


6


8


12


24 -1
-2
-3

24


8
12
Period 

6

6
y   sin 12 x 
Period :
2 2 


b
12 6
Starts at ZERO
3
Ends at ZERO
2
Half-way thru at ZERO


6


8


12
1


24

24


8
12
-1
-2
-3
Period 

6

6
y   sin 12 x 
Period :
2 2 


b
12 6
Quarter-way thru at MAX = 1 x (–1) = One below the x-axis
3
2
1


6


8


12


24

24

12

8

6
-1
-2
-3
Three quarters-way thru at MIN = – 1 x (–1) = One above the x-axis
y   sin 12 x 
Period :
2 2 


b
12 6
3
2
1
Use 8 Spaces for One Period


6


8


12


24

24
-1
Period 

6
-2
-3

12

8

6
y   sin 12 x 
Period :
2 2 


b
12 6
3
2
1


6


8


12


24

24
-1
-2
-3

12

8

6
y   sin 12 x 
Period :
2 2 


b
12 6
3
2
1


6


8


12


24

24
-1
-2
-3

12

8

6
1
y  sin  x   
4
Amplitude : 1
Vertical Shift : 
Phase Shift :  
2
2
Period :

 8
1
b
4
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
3
2
1
Use 8 Spaces for One Period
8
6
4
2
-1
-2
-3
2
4
6
Period  8
8
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
Vertical Shift : 
3
2
1
8
6
7
5
4
3
Period  8
2
2

-1
-2
-3
Phase Shift :  

4
3
8
6
5
7
9
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
Starts at the Translated ZERO
3
Ends at ZERO
2
Half-way thru at ZERO
8
7
6
5
4
3
1
Period  8
2
2

-1
-2
-3

4
3
8
6
5
7
9
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
Quarter-way thru at MAX = ONE above the x-axis
3
2
1
8
7
6
5
4
3
2
2

-1

4
3
8
6
5
7
9
-2
-3
Three quarters-way thru at MIN
= ONE below the x-axis
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
3
2
1
Use 8 Spaces for One Period
8
7
6
5
4
3
2
2

-1
Period  8
-2
-3

4
3
8
6
5
7
9
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
3
2
1
8
7
6
5
4
3
2
2

-1
-2
-3

4
3
8
6
5
7
9
1
y  sin  x   
4
Period :
2
2

 8
1
b
4
3
2
1
8
7
6
5
4
3
2
2

-1
-2
-3

4
3
8
6
5
7
9


y  2 sin  2 x    1
2

Vertical Shift : 1
Amplitude : 2
f  x   a sin b  x  c   d

Given sin 2 x  
2
Phase Shift :

4



must rewrite sin 2 x  
4
2
Period :

2



y  2 sin 2 x    1
4
3
2
1
-1
-2
-3


y  2 sin 2 x    1
4
Period :
3
2
1
Use 8 Spaces for One Period


3
4


2


4
-1
-2
-3

4

2
3
4
Period  

2

2


y  2 sin 2  x    1
4

Period :
2

2
3

Phase Shift : 2 
4
1


3
4


2



4
4

2
3
4

-1
Vertical Shift : 1 
-2
-3
Period  
5
4


y  2 sin 2  x    1
4

Period :
2

2
Starts at the Translated ZERO
3
Ends at ZERO
2
1
Half-way thru at ZERO


3
4


2



4
4

2
3
4

-1
-2
-3
Period  
5
4


y  2 sin 2  x    1
4

Period :
2

2
Quarter-way thru at MAX = 1 x 2
3
= 2 units above the translated horizontal
x-axis
2
1


3
4


2



4
4

2
-1
-2
Three quarters-way thru at MIN = – 1 x 2
-3
= 2 units below the translated horizontal x-axis
3
4

5
4


y  2 sin 2  x    1
4

Period :
2

2
3
2
1
Use 8 Spaces for One Period


3
4


2



4
4
-1
Period  
-2
-3

2
3
4

5
4


y  2 sin 2  x    1
4

Period :
2

2
3
2
1


3
4


2



4
4
-1
-2
-3

2
3
4

5
4


y  2 sin 2  x    1
4

Period :
2

2
3
2
1


3
4


2



4
4
-1
-2
-3

2
3
4

5
4