1

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 of sine and cosine functions

Sinusoidal curve fitting

2

3

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.

4

Take a look at the graph of y = sin x:

(one cycle)

sin 12

1sin

2.

60 5

2

sin2

.4

0 7

3

sin2

.3

0 9

sin 0 0

,x y (0, )0

0.5,6

0.7,4

1,2

0.9,3

Some points on the graph of y = sin x

5

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.

For example, are on the graph of y = sin x.

,1 , 12 2

and

, 12

, 12

More about the graph of y = sin x

6

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 point

minimum point

7

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 _______.

8

Take a look at the graph of y = cos x:

(one cycle)

cos 0 1 cos 02

1

cos2

.3

0 5

2

cos2

.4

0 7

3

cos2

.6

0 9

,x y (0, )1

0.9,

6

0.7,4

0,2

0.5,3

Some points on the graph of y = cos x

9

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

graph.

For example, are on the graph of y = cos x., 0 , 02 2

and

, 02

, 02

More about the graph of y = cos x

10

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 point2 x-interceptsminimum point

11

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.

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 ________.

Max min

2 2amplitude

12

13

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)

14

I. Reflection in x-axis

sin cosy x y x

Example:

15

II. Vertical Stretch or Compression

(Amplitude change)

sin cos

Amplitude

A A

A

y x y x

Example

cosy x

1Amplitude

3cosy x

3Amplitude

1cos

2y x

1

2Amplitude

16

II. Vertical Stretch or Compression

sin cos

Amplitude

A A

A

y x y x

*Note:

If the curve is vertically stretched

if the curve is vertically shrunk

1A

1A

17

The graph of a function in the form y = A sinx or y = A cosx is shown.Determine the equation of the specific function.

Example

18

19

sin cos

2Period

y A x y A x

T

III. Horizontal Stretch or Compression (Period change)

Example

siny x

2period T

sin(2 )y x

2

2period T

1sin

2y x

12

24period T

20

sin cos

2Period

y A x y A x

T

*Note:

If the curve is horizontally stretched

If the curve is horizontally shrunk

1

1

III. Horizontal Stretch or Compression (Period change)

21

Graphs of

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)

11. 5cos

3y x

sin( ) and cos( )y A x y A x

12. sin 2

4y x 3. cosy x

22

x

y

23

24

25

26

V. Vertical Shifts

sin cos

Vertical Shift

y A x B y A x B

nBu its

Example

cos 3 Shift 3 units downw

cos 2 Shift 2 uni

c

ts up

ar

os

d

ward

y x Parent G

y

raph

x

y x

27

sin cos

Phase shift

y A x B y A x B

units

V. Phase Shifts

Example

sin4

Phase shift units

sin

to right.4

y x

y x

28

For , determine the amplitude, period, and phase shift. Then sketch at least one cycle of the function by hand.

3sin 2y x

x

y

EXAMPLE

29

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.

31. sin 2

4y x

EXAMPLE

30

31. sin 2

4y x

x

y

EXAMPLE (CONTINUED)

31

12. sin 2

3 6y x

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.

32

12. sin 2

3 6y x

x

y

EXAMPLE (CONTINUED)

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

43

End of Sections 7.6 & 7.8