Copyright © Cengage Learning. All rights reserved. 4.5 Graphs of Sine and Cosine Functions.
4.5 Sinusoidal Graphs
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4.5 Sinusoidal Graphs Sketching and Writing Equations
description
4.5 Sinusoidal Graphs. Sketching and Writing Equations. Graphs of sine and cosine functions. Sinusoidal Graphs. Periodic Function. Functions whose graphs have a repeating pattern. Sinusoidal Graphs. Period. The horizontal length of each cycle in a periodic graph. Midline. - PowerPoint PPT Presentation
Transcript of 4.5 Sinusoidal Graphs
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
