4.5Sinusoidal Graphs

Sketching and Writing Equations

Sinusoidal GraphsSinusoidal Graphs

Periodic Function

Graphs of sine and cosine functions.

Functions whose graphs have a repeating pattern.

Period The horizontal length of each cycle in a periodic graph.

Midline The horizontal line halfway between a sinusoid’s max/min values.

The Graph of y = sin xx y

Characteristics of the Sine Function

• The domain is the set of all real numbers.• The range consists of all real numbers

from -1 to 1, inclusive.• The sine function is an odd function

(symmetric with respect to the origin).• The sine function has a period of 2π• Midline is the x-axis

Period

3 2 1 -1 -2 -3

22

2

2

23

23

y = sinx

1 period

pattern startspattern starts

Period= 2

pattern starts

1 period

The Graph of y = cos xx y

Characteristics of the Cosine Function

• The domain is the set of all real numbers.• The range consists of all real numbers

from -1 to 1, inclusive.• The cosine function is an even function

(symmetric with respect to the y-axis).• The cosine function has a period of 2π• Midline is the x-axis

Period

3 2 1 -1 -2 -3

22

2

2

23

23

1 period

y = cosx

pattern startspattern starts

Period = 2

pattern starts

1 period

Period

3 2 1 -1 -2 -3

22

2

2

23

23

1 period

pattern starts

pattern starts

Period=

Period

3 2 1 -1 -2 -3

22

2

2

23

23

1 period

pattern startspattern starts

Period =32

Amplitude

Amplitude Half the distance between the maximum and minimum of a sinusoidal graph.

Amplitude

3 2 1 -1 -2 -3

22

2

2

23

23

Amplitude = 1

Amplitude

max.

min.

Amplitude

3 2 1 -1 -2 -3

22

2

2

23

23

max.

min.

Amplitude

Amplitude = 2

Amplitude

3 2 1 -1 -2 -3

22

2

2

23

23

max.

min.

Amplitude

Amplitude = 1

Amplitude

3 2 1 -1 -2 -3

22

2

2

23

23

max.

min.

Amplitude

Amplitude = 2

3

Sinusoidal Graphs

Amplitude =

Period = ||2b

a

General Form of Equations

bxay cosbxay sin

Sinusoidal Graphs

y = 3sin4x

Amplitude = 3 3

Period = 42

2

y = -4cos5x

Amplitude = 4 4

Period = 52

Writing Equations

Answer these questions first:1. Is it a “sin” or “cos” graph?2. What is the amplitude? 3. What is the period? 4. Is the graph reflected? 5. Is it shifted up/down?6. Is it shifted left/right?

Writing Equations

b = period2

Write an equation:

cosine graph

amplitude = 4

Period =

bxay cos

a = 4

b = 2

= 2

xy 2cos4

Writing Equations

b = period2

Write an equation:

sine graph

amplitude = 3

Period = 4

bxay sin

a = 3

b = 4

22

xy2

sin3

Writing Equations

b = period2 bxay sin

Write an equation:

sine graph

amplitude = 2

Period = 3

Reflected over x-axis

a = ±2

b = 3

2

xy3

2sin2

Writing Equations

b = period2 bxay cos

a =

b =

Write an equation:

cosine graph

amplitude = ½

Period =

Reflected over x-axis2

21

2

2

21

2

4

4

xy 4cos21

Invert and multiply!

Writing Equations

Write an EquationSine graphAmplitude = 5Period = Shifted up 2

32

bxay sin

a =

b =

5

322

23

12

26

3

xy 3sin5 2

Changing the midline causes vertical shifts

Sinusoidal Graphs

3 2 1 -1 -2 -3

22

2

2

23

23

Period =Amplitude =

b = 2

2

y = sin2x

“sin”

1

a =1

Sinusoidal Graphs

3 2 1 -1 -2 -3

22

2

2

23

23

Period =Amplitude

b =

22

1

y = 3sinx

“sin”2= 3

a =3

Sinusoidal Graphs

3 2 1 -1 -2 -3

22

2

2

23

Period = Amplitude =

“cos”

2

23

b = 2

2

21

2

4

4

xy 4cos

1

a =1

Sinusoidal Graphs

3 2 1 -1 -2 -3

22

2

2

23

Period =

Amplitude =

“cos”

23

b =

32

232

3

Reflected over x-axis!

xy32cos2

a = -2

Sinusoidal Graphs

3 2 1 -1 -2 -3

442

2 1

1

3

Period = Amplitude =“cos”

3

b = 6

2

36

Reflected over x-axis!

a = -3 3

xy3

cos3

Graphing Sinusoidal Functions

y = 3sin4x 3 2 1 -1 -2 -3

2

2

2

23

23

amp 3period

42

2

Graphing Sinusoidal Functions

y = 2sinx 3 2 1 -1 -2 -3

422 1

1 3

34

amp 2Period

2

2

Graphing Sinusoidal Functions

y = -3sin2x 3 2 1 -1 -2 -3

2

2

2

23

23

2

amp 3Period

22

Graphing Sinusoidal Functions

3 2 1 -1 -2 -3

422 1

1 3

3

xy2

cos

4

amp 1Period

2

2

21

2

4

4

Graphing Sinusoidal Functions

3 2 1 -1 -2 -3

422 1

1 3

3

xy cos2

4

amp 2 Period 2

2

Graphing Sinusoidal Functions

3 2 1 -1 -2 -3

422 1

1 3

3

xy 3cos2

4

amp 2Period

32

32

Phase Shift

• Horizontal translations of sine and cosine graphs.

)sin( cbxay )sin( cbxay

bc

Left Right

bcxbay sin

bcxbay sin

Phase Shift bc

Phase Shift

Phase Shift)2sin(3 xy

Amp = 3

T =2

2

P.S. = 2

right

Phase Shift)34cos(2 xy

Amp = 2

T =

P.S. =

42

43

2

left

Phase Shift

“Cosine”Amplitude = 5Period = Phase shift =

2

..cos SPxbay 5a

b2 2

2

2cos5 xy

xy 2cos5

Phase Shift

“sine”Amplitude = 3Period = 4Phase shift = 3

..sin SPxbay 3a

b4

2

32

sin3

xy

2

2

32

sin3 xy