4.5 sinusoidal graphs sketching and writing equations
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Sinusoidal Graphs
Sinusoidal 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.
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
2
2
2
2
2
3
2
3
y = sinx
1 period
pattern startspattern starts
Period= 2
pattern starts
1 period
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
2
2
2
2
2
3
2
3
1 period
y = cosx
pattern startspattern starts
Period = 2
pattern starts
1 period
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 = period
2
Write an equation:
cosine graph
amplitude = 4
Period =
bxay cos
a = 4
b = 2
= 2
xy 2cos4
Writing Equations
b = period
2
Write an equation:
sine graph
amplitude = 3
Period = 4
bxay sin
a = 3
b = 4
22
xy2
sin3
Writing Equations
b = period
2 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 = period
2 bxay cos
a =
b =
Write an equation:
cosine graph
amplitude = ½
Period =
Reflected over x-axis2
2
1
2
2
2
1
2
4
4
xy 4cos2
1
Invert and multiply!
Writing Equations
Write an Equation
Sine graph
Amplitude = 5
Period =
Shifted up 23
2
bxay sin
a =
b =
5
322
2
3
1
2
2
6
3
xy 3sin5 2
Changing the midline causes vertical shifts
Sinusoidal Graphs
3
2
1
-1
-2
-3
2
2
2
2
2
3
2
3
Period =
Amplitude =
b = 2
2
y = sin2x
“sin”
1
a =1
Sinusoidal Graphs
3
2
1
-1
-2
-3
2
2
2
2
2
3
2
3
Period =Amplitude
b =
2
21
y = 3sinx
“sin”2= 3
a =3
Sinusoidal Graphs
3
2
1
-1
-2
-3
2
2
2
2
2
3
Period =
Amplitude =
“cos”
2
2
3
b = 2
2
2
1
2
4
4
xy 4cos
1
a =1
Sinusoidal Graphs
3
2
1
-1
-2
-3
2
2
2
2
2
3
Period =
Amplitude =
“cos”
2
3
b =
3
2
23
23
Reflected over x-axis!
xy3
2cos2
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
Phase Shift
• Horizontal translations of sine and cosine graphs.
)sin( cbxay )sin( cbxay
b
c
Left Right
b
cxbay sin
b
cxbay sin
Phase Shift b
cPhase Shift
Phase Shift
“Cosine”
Amplitude = 5
Period = Phase shift =
2
..cos SPxbay 5a
b2
2
2
2cos5 xy
xy 2cos5
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