1

Graphs of sine and cosine curves

Sections 10.1 – 10.3

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.

• We say that the range of the sine curve is

Each cycle (wave) corresponds to one revolution (360 or

2 radians) of the unit circle.

• We say that the period of the sine curve is 2.

1 1y

3

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

(one cycle)

sin 12

1

sin2

.6

0 5

2sin

2.

40 7

3sin

2.

30 9

sin 0 0

,x ySome points on the graph:

0,0 56

,0.

74

,0.

12

,

93

,0.

4

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 the intercepts, the maximum point, and the minimum point.

5

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 also 2.

1 1y

6

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

(one cycle)

,x ySome points on the graph:

cos 02

1cos

2.

30 5

2

cos2

.4

0 7

3cos

2.

60 9

cos 0 1

0,1 96

,0.

74

,0.

12

,

53

,0.

7

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.

8

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

2amplitude

9

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 shifts/displacement)

10

I. Reflections over x-axis

sin cosy x y x

Example:

11

II. Vertical Stretching 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

12

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

13

sin cos

2Period

y a x y ab xb

b

III. Horizontal Stretching or Shrinking/Compression (Period change)

Example

siny x

2period

sin(2 )y x

2

2period

1sin

2y x

12

24period

14

x

y

15

Graphs of

Examples

State the amplitude and period for each function. Then graph one cycle of each function by hand. Verify using your graphing calculator.

11. 5cos

3y x

sin( ) and cos( )y a bx y a bx

16

Graphs of

12. sin 2

4

3. cos

y x

y x

sin( ) and cos( )y a bx y a bx

17

18

19

sin cos

Phase shift

y a bx y a bx

unitsb

c c

c

IV. Phase Shifts

Example

4Phase shift = =1 4

Thus, graph shifts units to right.4

sin4

y x

siny x

sin4

y x

20

IV. Phase Shifts (continued)

Example

1sin

2

22Phase shift = =

1 2 12

Thus, graph shifts units to the le

1s

t

n

f

i2 2

.

y x

y x

Phase shift u scnit

b

siny x1sin

2 2y x

2 3

21

Example:

Determine the amplitude, period, and phase shift of the function. Then sketch the graph of the function by hand.

1) 3sin 2y x

22

Example:

1) 3sin 2y x

x

y

23

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. sin 2

4y x

24

32. sin 2

4y x

x

y

25

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.

13. sin

3 6y x

26

13. sin

3 6y x

x

y

27

Modeling using a sinusoidal function

P. 299 #56

On a Florida beach, the tides have water levels about 4 m between low and high tides. The period is about 12.5 h. Find a cosine function that describes these tides if high tide is at midnight of a given day.

28

29

Modeling using a sinusoidal functionA region that is 30° north of the equator averages a minimum

of 10 hours of daylight in December. Average hours of daylight are at a maximum of 14 hours in June.

Let x represent the month of the year with 1 for January, 2 for February, 3 for March, …through 12 for December.

If y represents the average number of hours of daylight in month x, use a sine function of the form y = a sin(bx + c) +d to model the daylight hours.

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End of Section