Transcript Document

7.6 Graphs of the Sine and Cosine Functions
7.8 Phase shift; Sinusoidal Curve Fitting
In these sections, we will study the following topics:
The
graphs of basic sine and cosine functions
The
amplitude and period of sine and cosine functions
Transformations
Sinusoidal
of sine and cosine functions
curve fitting
1
2
The graph of y = sin x
The graph of y = sin x is a cyclical curve that takes on values
between –1 and 1.
The range of y = sin x is _____________.
Each cycle (wave) corresponds to one revolution of the unit circle.
The period of y = sin x is _______ or _______.
Graphing the sine wave on the x-y axes is like “unwrapping” the
values of sine on the unit circle.
3
Take a look at the graph of y
= sin x:
(one cycle)
Some points on the graph of y = sin x
sin 0  0

 x, y 
(0, 0 )
sin


6
1
 0 .5
2



,
0.5


 6

sin


4
2
 0 .7
2



 , 0 .7 
 4

sin


3
3
2



 , 0 .9 
 3

 0 .9
sin

1
2

 
 ,1 
 2 
4
More about the graph of y
= sin x
Notice that the sine curve is symmetric about the origin.
Therefore, we know that the sine function is an ODD function; that is,
for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

 

,1  and   ,  1 
 2 
 2

For example, 

are on the graph of y = sin x.


 , 1
2


 

  ,  1
2


5
Using Key Points to Graph the Sine Curve
Once you know the basic shape of the sine curve, you can use the key
points to graph the sine curve by hand.
The five key points in each cycle (one period) of the graph are:
3
x-intercepts
maximum
minimum
point
point
6
The graph of y = cos x
The graph of y = cos x is also a cyclical curve that takes on
values between –1 and 1.
The range of the cosine curve is ________________.
The period of the cosine curve is _______ or _______.
7
S e c tio n 4 .5 , F ig u re 4 .4 3 , G ra p h o f
C o s in e C u rve , p g . 2 8 7
Take a look at the graph of y = cos x:
(one cycle)
Some points on the graph of y = cos x
co s 0  1

cos


6
3
 0 . 9 cos
2

C o p y rig ht © H o ug hto n M ifflin C o m p a n y. A ll rig hts re se rve d .
 x, y 
(0, 1)


,
0.9


 6



4
2
 0 .7
2



,
0
.7


 4

cos


3
1
 0 .5
2

cos

0
2

D ig ita l F ig u re s , 4 – 18


,
0
.5


 3



,
0


 2

8
More about the graph of y
= cos x
Notice that the cosine curve is symmetric about the y-axis.
Therefore, we know that the cosine function is an EVEN function; that
is, for every point (x, y) on the graph, the point (-x, y) is also on the
S e c tio n 4 .5 , F ig u re 4 .4 3 , G ra p h o f
graph.

 C o sin e
 C u rve, p g . 287
,
0
and

,
0
For example,  2 

 are on the graph of y = cos x.
2




 

 , 0
 2



 , 0
 2

9
Using Key Points to Graph the Cosine Curve
Once you know the basic shape of the cosine curve, you can use the
key points to graph the cosine curve by hand.
The five key points in each cycle (one period) of the graph are:
maximum point
2 x-intercepts
minimum point
10
Characteristics of the Graphs of y = sin x and y = cos x
Domain:
____________
Range:
____________
Amplitude:
The amplitude of the sine and cosine functions is half the
distance between the maximum and minimum values of the function.
am plitude 
M ax  m in
2


2
The amplitude of both y= sin x and y = cos x is ______.
Period:
The length of the interval needed to complete one cycle.
The period of both y= sin x and y = cos x is ________.
11
12
Transformations of the graphs of y = sin x and y = cos x

Reflections over x-axis

Vertical Stretches or Shrinks

Horizontal Stretches or Shrinks/Compression

Vertical Shifts

Phase shifts (Horizontal)
13
I. Reflection in x-axis
y   sin x
y   cos x
Example:
14
II. Vertical Stretch or Compression
(Amplitude change)
y  A sin x
y  A cos x
A m plitude  A
Example
y  cos x
Am plitude  1
y  3 cos x
A m plitude  3
y
1
cos x
2
A m plitude 
1
2
15
II. Vertical Stretch or Compression
y  A sin x
y  A cos x
Am plitude  A
*Note:

If A  1 the curve is vertically stretched

if A  1 the curve is vertically shrunk
16
Example
The graph of a function in the form y = A sinx or y = A cosx
is shown.
Determine the equation of the specific function.
17
18
III. Horizontal Stretch or Compression (Period change)
y  A sin  x
y  A cos  x
P eriod  T 
2

Example
y  sin x
period  T  2 
y  sin(2 x )
period  T 
2
2

1 
y  sin  x 
2 
period  T 
2
1
2
 4
19
III. Horizontal Stretch or Compression (Period change)
y  A sin  x
P erio d  T 
y  A co s  x
2

*Note:

If   1 the curve is horizontally stretched

If   1 the curve is horizontally shrunk
20
Graphs of y  A sin( x ) and y  A cos( x )
Examples
State the amplitude and period for each function. Then graph
each of function using your calculator to verify your answers.
(Use radian mode and ZOOM 7:ZTrig)
1.
1 
y  5 cos  x 
3 
2.
y
1
4
sin  2 x 
3.
y  co s   x 
21
22
y
x
23
24
25
V. Vertical Shifts
y  A sin   x   B
y  A cos   x   B
V ertical S hift  B u n its
Example
y  cos x  2
S hift 2 uni ts up w ard
y  cos x
P arent G raph
y  cos x  3
S hift 3 units dow nw ar d
26
V. Phase Shifts
y  A sin   x     B
y  A cos   x     B
P hase shift 


units
Example
y  sin x
 

y  sin  x  
4

P hase shift

units to right.
4
27
EXAMPLE
For y  3 sin  2 x    , determine the amplitude, period, and phase
shift. Then sketch at least one cycle of the function by hand.
y
x
28
EXAMPLE
List all of the transformations that the graph of y = sin x has undergone
to obtain the graph of the new function. Graph the function by hand.
1. y  
3
4
sin  2 x 
29
EXAMPLE (CONTINUED)
1. y  
3
4
sin  2 x 
y
x
30
EXAMPLE
List all of the transformations that the graph of y = sin x has undergone
to obtain the graph of the new function. Graph the function by hand.
 
1
2. y  sin  x    2
6
3
31
EXAMPLE (CONTINUED)
 
1
2. y  sin  x    2
6
3
y
x
32
33
34
35
36
37
38
*
*NOTE: In 2005, summer solstice was on June 21 (172nd day of the year).
39
40
Use a graphing calculator to graph the scatterplot of the data in the table
below. Then find the sine function of best fit for the data. Graph this
function with the scatterplot.
41
42
End of Sections 7.6 & 7.8
43