Sullivan Algebra and Trigonometry: Section 7.6
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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 cosx 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