Reviewing Sine WavesHonors Physics

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

1sin2

.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 CurveOnce 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

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–18

Section 4.5, Figure 4.43, Graph of Cosine Curve, pg. 287

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

(one cycle)

,x ySome points on the graph:

cos 02

1cos2

.3

0 5 2cos

2.

40 7

3cos2

.6

0 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 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 min2

amplitude

9

Transformations 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)

11

I. Reflections over x-axis

sin cosy x y x

Example:

Note: Calculator Mode must be in Radians

12

II. Vertical Stretching or Compression (Amplitude change)

sin cos

Amplitude

a a

a

y x y x

Example

cosy x

1Amplitude

3cosy x

3Amplitude

1 cos2

y x

12

Amplitude

13

ExampleThe graph of a function y = asinx or y = acosx is shown. Determine the equation of the specific function.

14

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

22

period

1sin2

y x

12

2 4period

15

x

y

17

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

11. 5cos3

y x

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

18

12. sin 24

3. cos

y x

y x

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

19

20

21

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

22

IV. Phase Shifts (continued)

Example

1sin2

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 sc nitb

siny x1sin2 2

y x

2 3

23

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

1) 3sin 2y x

x

y

24

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 24

y 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. sin3 6

y x

x

y

26

Modeling Using a Sinusoidal Function

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.