#### Transcript 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
 3
 2
5 6

3 2
2
12
32
1
12
0
1
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    2 , 3 2 ,
7 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
(0, 0)
1.5
0
2
4
2 ,0
6
3 2 ,1
4
y  2sin x
4


,1

 2

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
 6
 3
 2
2 3

3 2
2
1
32
12
0
1 2
1
0
1
1.5
2 ,1
 1 
, 

 3 2
(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
 
2
5


y  cos x  
2

11 .. 55
y  cos x

0
1 .5

2
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
2
2
T



2
1 .5
y  sin x
0
 
2
5
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
2.5
 
2.36
2
y  2 sin 2 x
3 2
6.28
2
Find an equation for the graph.
5
4
3
-1
0
-4
5
Period = T  2
1
22
Amplitude  4
Period = T  2
2
T
Amplitude  A  4

2
2



T
2
y   A sin x
y  4 sin x
```