Sinusoidal Graphs
Download
Report
Transcript 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 = 2sinx
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 cos3x
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 cosbx P.S.
a5
b
2
2
y 5 cos 2 x
2
y 5 cos2 x
Phase Shift
“sine”
Amplitude = 3
Period = 4
Phase shift = 3
y a sin bx P.S.
a3
b
2
4
2
y 3 sin x 3
2
3
y 3 sin x
2
2