### Sullivan Algebra and Trigonometry: Section 7.6

```Sullivan Algebra and
Trigonometry: Section 7.6
Objectives of this Section
• Graph Transformations of the Sine Function
• Graph Transformations of the Cosine Function
• Determine the Amplitude and Period of Sinusoidal
Functions
• Graph Sinusoidal Functions: y = Asin(wx)
• Find an Equation for a Sinusoidal Graph
The Graph of y = sin x
y
x
0
0
 6
12
 3
3 2
 2
1
5 6
12

0
3 2
1
2
0
 2 ,1
1.5
 6 , 12 
 ,0 
2  , 0 
(0, 0)
0
1.5
2
4
6
3 2 , 1
Characteristics of the Sine Function
1. The domain is the set of all real numbers.
2. The range consists of all real numbers from
-1 to 1, inclusive.
3. The sine function is an odd function
(symmetric with respect to the origin).
4. The sine function is periodic, with period 2 .
5. The x - intercepts are
the y - intercept is 0.
,-2 ,- ,0,  ,2 ,
;
Characteristics of the Sine Function
6. The maximum value is 1 and occurs at
x
, 3 2 ,  2 , 5 2,
; the minimum
value is -1 and occurs at x 
7 2 ,
  2 , 3 2 ,
Use the graph of y  sin x to graph


y  2 sin  x  .
4

Begin with the basic sine function:
 2 ,1
1.5
 6 , 12 
 ,0 
2  , 0 
(0, 0)
0
1.5
2
4
6
3 2 , 1
4
4
 

,1 

 2

y  2sin x
0 , 0 
0
5
0 , 0 
y  sin(x)
4
0
5


, 2 

 2

4


y  2 sin  x  
4

4


,0 

 4

0
5
 3

, 2 

 4

4
The Graph of y = cos x
y
x
0
1
 6
3 2
 3
12
 2
0
2 3
1 2

1
3 2
0
2
1
1.5
2  ,1
1 

,


2 
 3
(0, 1)
 3

,0 

 2

0
2
4
 , 1
1.5
6
y  cos x
Characteristics of the Cosine Function
1. The domain is the set of all real numbers.
2. The range consists of all real numbers from
-1 to 1, inclusive.
3. The cosine function is an even function
(symmetric with respect to the y-axis).
4. The cosine function is periodic, with
period 2 .
5. The x - intercepts are
3 2 ,
,- 3 2 ,-  2 ,  2 ,
; the y - intercept is 1.
Characteristics of the Cosine Function
6. The maximum value is 1 and occurs at
x
,2 ,0,2 ,4 ,
; the minimum
value is - 1 and occurs at x 
3 ,5
 ,  ,
1 .5
y  sin x
0


5
2


y  cos x  
2

11 .. 55
y  cos x

0

2
1 .5
5
The graphs of the sine and cosine
functions are called sinusoidal graphs.
If   0, the amplitude and period of
y  A sin x and y  A cosx are given by
Amplitude = A
Period = T 
2

Determine the amplitude and period of
y  2 sin 2 x , and graph the function.
y  2 sin 2 x
y  A sin x
A  2 ,   2
Amplitude   2  2
T
2


2
2

1 .5
y  sin x
0

5

2
1 .5
2.5


1
2
2

1.57
-1
2.5
2
2.36

3
2
y  sin 2 x
6.28
2.5
2


1.57
2

2.36
2
-2
2.5
y  2 sin 2 x
3
2
2
6.28
Find an equation for the graph.
5
4
3
-1
0
1
22
-4
5
Period = T  2
Amplitude  4
Period = T  2
T

Amplitude  A  4
2

2
T

2
2

y   A sin x
y  4 sin x
```