### Sinusoidal Graphs

```4.5
Sinusoidal Graphs
Sketching and Writing Equations
Sinusoidal Graphs
Sinusoidal Graphs Graphs of sine and
cosine functions.
Periodic Function 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 x
x
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
y = sinx
3
2
pattern starts
pattern starts
 2
3

2
pattern starts

2
1



2
1 period
Period= 2

-1
-2
-3
1 period
3
2
2
The Graph of y = cos x
x
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
y = cosx
3
pattern starts
pattern starts
pattern starts
2

2
1
 2
3

2

1 period


2

3
2
-1
-2
1 period
Period = 2
-3
2
Period
3
pattern starts
2
pattern starts

2
1
 2
3

2



2
-1
1 period
Period= 
-2
-3

3
2
2
Period
3
pattern starts
pattern starts
2

2
1
 2
3

2

3
Period =
2


2

-1
-2
-3
1 period
3
2
2
Amplitude
Amplitude Half the distance between the
maximum and minimum of a
sinusoidal graph.
Amplitude
3
max.
2

2
1
Amplitude
 2
3

2



2
-1
min.
-2
Amplitude = 1
-3

3
2
2
Amplitude
max.
3
2

2
1
 2
3

2

Amplitude = 2


2
Amplitude
3
2

2
-1
-2
min.
-3
Amplitude
max.
3
2
Amplitude

2
1
 2
3

2

Amplitude = 1


2
-1
-2
-3

min.
3
2
2
Amplitude
Amplitude = 2
3
2
max.

2
1
 2
3

2


2

3

3
2
2
-1
-2 Amplitude
-3
min.
Sinusoidal Graphs
General Form of Equations
y  a sin bx
Amplitude = a
2
Period =
|b|
y  a cosbx
Sinusoidal Graphs
y = 3sin4x
y = -4cos5x
Amplitude = 3  3
Amplitude =  4  4
2 

Period =
4
2
2
Period =
5
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
2
b=
period
Write an equation:
cosine graph
amplitude = 4
Period = 
y  a cosbx
a=4
2
b=
=2

y  4 cos 2 x
Writing Equations
2
b=
period
Write an equation:
sine graph
amplitude = 3
Period = 4
y  a sin bx
a=3
2


b=
4
2

y  3 sin x
2
Writing Equations
2
b=
period
y  a sin bx
Write an equation:
a = ±2
sine graph
2
b=
3
amplitude = 2
Period = 3
Reflected over x-axis
2
y  2 sin
x
3
Writing Equations
2
b=
period
Write an equation:
cosine graph
amplitude = ½

Period =
2
Reflected over x-axis
y  a cosbx
Invert and
1
multiply!
a = 2
2  2 2
4
b=    
1 

2
4
1
y   cos 4 x
2
Writing Equations
Write an Equation
Sine graph
Amplitude = 5
2
Period =
3
Shifted up 2
y  a sin bx
a= 5
2
6
2 3

b = 2  
2
1 2
3
3
Changing the midline
causes vertical shifts
y  5 sin 3x  2
Sinusoidal Graphs
“sin”
a =1
Period = 
3
Amplitude = 1
y = sin2x
2

2
1
 2
3

2

2
2
b=



2
-1
-2
-3

3
2
2
Sinusoidal Graphs 2 
“sin”
Period = 2
b = 2  1
a =3
Amplitude = 3
3
y = 3sinx
2

2
1
 2
3

2



2
-1
-2
-3

3
2
2
“cos”
Sinusoidal Graphs

Period =
2
Amplitude = 1
a =1
2  2 2
4
b=    
3
1 
2
4
2

2
1
 2
3

2



2

3
2
-1
-2
-3

y  cos 4 x
2
Sinusoidal Graphs
“cos”
Period = 3
a = -2
3
Amplitude = 2
2

2
1
 2
3

2

Reflected
over x-axis!
2 2

b=
3 3


2
-1
-2
-3

3
2
2
2
y  2 cos x
3
Sinusoidal Graphs
“cos”
Period = 6
Amplitude = 3
2 

b=
6 3
a = -3
3
2
1
1
4
3
2
Reflected
over x-axis!
1
-1
-2
-3
2
3

y  3 cos x
3
4
Graphing Sinusoidal Functions
3
y = 3sin4x
2
amp  3
2 
period 

4
2

2
1
3

2



2
-1
-2
-3

3
2
2
Graphing Sinusoidal Functions
3
y = 2sinx
2
amp  2
2
2
Period 

1
1
4
3
2
1
-1
-2
-3
2
3
4
Graphing Sinusoidal Functions
3
y = -3sin2x
2
amp  3
2
Period 

2

2
1
 2
3

2



2
-1
-2
-3

3
2
2
Graphing Sinusoidal Functions
3

y  cos x
2
2
amp  1
2  2 2 4 
Period 
  1    4
2
1
1
4
3
2
1
-1
-2
-3
2
3
4
Graphing Sinusoidal Functions
3
amp  2
y  2 cos x
2
Period 
2

2
1
1
4
3
2
1
-1
-2
-3
2
3
4
Graphing Sinusoidal Functions
3
y  2 cos3x
2
amp  2
2 2
Period 

3 3
1
1
4
3
2
1
-1
-2
-3
2
3
4
Phase Shift
• Horizontal translations of sine and cosine
graphs.
y  a sin(bx  c)
c

y  a sin b x  
b

y  a sin(bx  c)
c

y  a sin b x  
b

Left
Right
c
Phase Shift  
b
c
Phase Shift 
b
Phase Shift
y  3 sin(2 x  )
right
Amp = 3
2
T=

2

P.S. =
2
Phase Shift
y  2 cos(4 x  3)
left
Amp = 2
2 

T=
4
2
3
P.S. = 
4
Phase Shift
“Cosine”
Amplitude = 5
Period = 

Phase shift = 
2
y  a cosbx  P.S.
a5
b
2
2



y  5 cos 2 x  
2

y  5 cos2 x  
Phase Shift
“sine”
Amplitude = 3
Period = 4
Phase shift = 3
y  a sin bx  P.S.
a3
b
2 

4
2

y  3 sin  x  3
2
3 

y  3 sin  x  
2 
2
```