Name____________________________
Sheilah Chason
Intro to Sinusoidal Curves
Date______________
Math 433
Aim: To explore the stretching and shrinking of sinusoidal curves and to graph these curves with different amplitudes
and periods.
1. Modeling with the Sine Function
Take a look at the following graph, which shows the approximate average daily high temperature in New York's Central Park.
#
#
Source: National Weather Service/The New York Times, January 7, 1996, p. 36.
Each year, the pattern repeats over and over, resulting in the following graph.
Here, the x-coordinate represents time in years with x = 0 representing August 1, while the y-coordinate represents the temperature in
o
F. This is an example of cyclical or periodic behavior.
Cyclical behavior is common in the business world; just as there are seasonal fluctuations in the temperature in Central Park, there are
seasonal fluctuations in the demand for surfing equipment, swimwear, snow shovels, and the list goes on. The following graph even
suggests a cyclical behavior in employment at securities firms in the United States. Ý
Next year we will derive these types of sinusoidal curves. For now we will explore changing the amplitude and frequency.
Directions: Fill out the chart. Plot all points. Do not connect the points.
Θ
Deg.
tanθ
-360
-270
Y=tanθ
-180
90
0
90
180
270
Domain:
360
30
45
60
120
135
150
210
225
240
300
315
Range:
2
y
1
x
-360 -330 -300 -270 -240 -210 -180 -150 -120 -90
-60
-30
30
-1
-2
60
90
120
150
180
210
240
270
300
330
360
330
Name____________________________
Sheilah Chason
Trig Graphs
Y=sinθ
Range:
Domain:
2
Date______________
Math 433
Critical Values:
y
1
x
-360 -330 -300 -270 -240 -210 -180 -150 -120
-90
-60
-30
30
60
90
120
150
180
210
240
270
300
330
360
-1
-2
Y=cosθ
Domain:
Range:
Critical Values:
2
y
1
x
-360 -330 -300 -270 -240 -210 -180 -150 -120
-90
-60
-30
30
60
90
120
150
180
210
240
270
300
330
360
-1
-2
Y= tanθ
Domain:
Range:
Critical Values:
2
y
1
x
-360 -330 -300 -270 -240 -210 -180 -150 -120
-90
-60
-30
30
-1
-2
60
90
120
150
180
210
240
270
300
330
360
Amplitude: On a sinusoidal curve, the difference between the maximum and minimum points divided by two.
Θ
-2π
Y=sinθ
Y=2sinθ
Y=0.5sinθ
Y=-sinθ
Y=cosθ
Y=-2cosθ
Y=1.5cosθ
-3 π/2
-π
- π/2
0°
π/2
π
3 π/2
2π

y
y





x













x









y







Summary of Amplitude:

y

x




















x















Period:
Frequency:
Θ
Y=sinθ
Y=-sin2 θ
Y=sin2θ
Y=1.5sin2θ
Y=cosθ
Y=cos2θ
Y=-0.5cos2θ
0
π/4
π /2
3 π/4
π
5 π/4
3 π/2
7 π/4
2π

y





x






y






x




















y




x
x









y





















Θ
Y=sinθ
Y=sin0.5θ
Y=2sin0.5θ
Y=-sin2θ
Y=cosθ
Y=cos0.5θ
-2π
-3 π/2
-π
- π/2
0°
π/2
π
3 π/2
2π
2.0
y
2.0
1.5
1.5
1.0
1.0
0.5
y
0.5
x
-2p
-3p
2
-p
-p
2
p
2
p
3p
2
2p
x
-2p
-3p
2
-p
-p
2
-0.5
-0.5
-1.0
-1.0
-1.5
-1.5
-2.0
-2.0
2.0
2.0
p
2
p
3p
2
2p
p
2
p
3p
2
2p
p
2
p
3p
2
2p
y
y
1.5
1.5
1.0
1.0
0.5
x
0.5
x
-2p
-3p
2
-p
-p
2
p
2
p
3p
2
-2p
-3p
2
-p
-p
2
-0.5
2p
-0.5
-1.0
-1.0
-1.5
-1.5
-2.0
-2.0
2.0
y
2.0
y
1.5
1.5
1.0
1.0
0.5
0.5
x
-2p
-3p
2
-p
-p
2
p
2
-0.5
p
3p
2
x
2p
-2p
-3p
2
-p
-p
2
-0.5
-1.0
-1.0
-1.5
-1.5
-2.0
-2.0
Y=-2cos0.5θ
Graph the following, over the given domain, indicating the amplitude, period and frequency, and critical values.
1)Y=-1.5 sinx;
0≤x≤2π
y
Amplitude:
Frequency:
Period:
x
Critical Values:
2)Y=cos2x
;
-π≤x≤π
y
Amplitude:
Frequency:
Period:
x
Critical Values:
3)Y=2sin0.5x;
[-π,π]
y
Amplitude:
Frequency:
Period:
Critical Values:
x
1)Graph Y= -sin0.5x and
Y= 2cosx over the interval [0,2π]
Amplitude:
Amplitude:
Frequency:
Frequency:
Period:
Period:
Critical Values:
Critical Values:
y
x
2)Graph Y= 1.5sin2x and
Y= -cos0.5x over the interval [-2 π,2π]
Amplitude:
Amplitude:
Frequency:
Frequency:
Period:
Period:
Critical Values:
Critical Values:
y
x
Graph the following, over the given domain, indicating the amplitude, period and frequency, and critical values.
1)Y=0.5 sinx;
0≤x≤2π
y
Amplitude:
Frequency:
Period:
x
Critical Values:
2)Y=2cos(0.5x) ;
[-2π,2π]
y
Amplitude:
Frequency:
Period:
Critical Values:
x
y
3)Y=-1.5sin(2x);
[-π,π]
Amplitude:
Frequency:
Period:
Critical Values:
x
Graph both functions on the same coordinate axis and determine the number of points of intersection.
1)Graph Y= sin2x and
Y= -0.5cosx over the interval [0,2π]
Amplitude:
Amplitude:
Frequency:
Frequency:
Period:
Period:
Critical Values:
Critical Values:
y
x
2)Graph Y= -1.5sinx and
Y= 2cos0.5x over the interval [-π,π]
Amplitude:
Amplitude:
Frequency:
Frequency:
Period:
Period:
Critical Values:
Critical Values:
y
x
Graph the following, over the given domain, indicating the amplitude, period and frequency, and critical values.
1)Y= tan2x;
0≤x≤2π
y
Frequency:
Period:
Critical Values:
x
2)Y=-tanx
;
[-2π,2π]
y
Frequency:
Period:
Critical Values:
x
y
3)Y=-tan (2x);
[-π,π]
Frequency:
Period:
Critical Values:
x
Name__________________________________
Sheilah Chason
Date__________________
Math 442
Aim: How to graph y=asin(bx)+c and y=acos(bx)+c.
To Do: If sec 40°=1.3054, find the following:
Cos 40° =
Sec 140° =
Sec -40° =
Cos 220° =
Name that Graph
y
y
y
4
4
1.5
3
3
2
1.0
2
1
x
-2p
-3p/2
-p
-p/2
p/2
p
3p/2
0.5
2p
1
-1
x
x
-2
-2p
-3p/2
-p
-p/2
p/2
p
3p/2
-2p
2p
-3p/2
-p
-p/2
p/2
p
3p/2
2p
-1
-3
-0.5
-4
-2
-1.0
-5
-3
-1.5
-4
-5
-2.0
Domain:
Range:
Amplitude:
Domain:
Range:
Amplitude:
Domain:
Range:
Amplitude:
y
y
2.8
1.6
y
2.4
1.5
2.0
1.2
1.6
1.0
1.2
0.8
0.8
0.5
0.4
0.4
x
x
-p
x
-2p
-3p/2
-p
-p/2
p/2
p
3p/2
-p/2
2p
-2p
-3p/2
-p
-p/2
-0.5
p/2
p
3p/2
p/2
-0.4
2p
p
-0.8
-1.2
-0.4
-1.6
-1.0
-2.0
-0.8
-2.4
-1.5
-2.8
-1.2
-2.0
Domain:
Range:
Amplitude:
Domain:
Range:
Amplitude:
y
Domain:
Range:
Amplitude:
y
2.8
y
2.4
2.0
1.0
1.6
0.5
1.2
0.8
0.5
0.4
x
x
-3p/2
-p
-p/2
p/2
p
3p/2
x
-3p/2
-p
-p/2
p/2
p
3p/2
-2p
-3p/2
-p
-p/2
-0.4
-0.8
-1.2
-0.5
-1.6
-2.0
-0.5
-1.0
-1.5
Domain:
Range:
-2.4
-2.8
-1.0
Domain:
Range:
Domain:
Range:
p/2
p
3p/2
2p
Amplitude:
Amplitude:
Amplitude:
Vertical Shift of main axis(c):
y
G(x)=sin(x)+1
Amplitude:
Frequency:
Period:
x
Critical Values:
Vertical Shift:
Domain: [-2π,2π]
Range:
G(π)=
y
G(x)=sin(2x)+1
Amplitude:
Frequency:
Period:
Critical Values:
x
Vertical Shift:
Domain: [-2π,2π]
Range:
G(π)=
f(x)=cos(2x)-2
y
Amplitude:
Frequency:
Period:
Critical Values:
x
Vertical Shift:
Domain: [0,2π]
Range:
F(π)=
H(x)= -sin(3x)-2
y
Amplitude:
Frequency:
Period:
Critical Values:
x
Vertical Shift:
Domain: [-2π/3,2π/3]
Range:
H(-π)
G(x)=2cos(0.5x)+3
y
Amplitude:
Frequency:
Period:
Critical Values:
Vertical Shift:
Domain: [-2π,2π]
x
Range:
G(π/2)=
F(x) = -2sin (0.5x)+2
y
Amplitude:
Frequency:
Period:
x
Critical Values:
Vertical Shift:
Domain: [-2π,2π]
Range:
F(π)=
M(x)=-2.5sin(2x)-1
y
Amplitude:
Frequency:
Period:
Critical Values:
x
Vertical Shift:
Domain: [-π,π]
Range:
F(π)=
S(x)= -1cos(0.5x)-2
y
Amplitude:
Frequency:
Period:
Critical Values:
x
Vertical Shift:
Domain: [-π,π]
Range:
s(2π)=
E(x)= 2sin3x-4
y
Amplitude:
Frequency:
Period:
Critical Values:
x
Vertical Shift:
Domain: [-2π/3,2π/3]
Range:
s(π)=
Sketch y  sin( ) and y  sin( )  1 on the axis below.
y
x
Sketch y  cos( ) and y  cos( )  2 on the axis below.
y
x
Graph the following, over the given domain, indicating the amplitude, period and frequency, and critical values.
1)Y=-1.5 sin(x)-2;
0≤x≤2π
y
Amplitude:
Frequency:
Period:
x
Critical Values:
y
2)Y=cos(2x)+1 ;
-π≤x≤π
Amplitude:
Frequency:
Period:
x
Critical Values:
y
3)Y=-2sin(0.5x)-3
[-π,π]
Amplitude:
Frequency:
x
Period:
Critical Values:
1)Graph
and
Y=
Amplitude:
Amplitude:
Frequency:
Frequency:
Period:
Period:
Critical Values:
Critical Values:
over the interval [
]
y
x
2) Graph
and
Y=
Amplitude:
Amplitude:
Frequency:
Frequency:
Period:
Period:
Critical Values:
Critical Values:
over the interval [
y
x
]