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Section 2.4 – General Sinusoidal Graphs
Objective : Given any one of the following sets of information about a sinusoid, find the other two:
- the equation
- the graph
- the amplitude, period or frequency, phase displacement, and vertical displacement
In this section, you will put together the ideas from Sections 2.2 and 2.3 to graph functions whose equations have the
form y VDAsin B(PD)or y VDAcos B(PD).
The four constants A, B, VD, and PD have the following effects:
- A is the amplitude. The absolute value is needed since the constant A may be a negative number.
So,Amplitude A.
- B is the number of cycles the sinusoid makes in 360 , so the period. B
P
- VD is the vertical displacement.
- PD is the phase displacement.
The period is the number of degrees per cycle. It is sometimes convenient to speak of the number of cycles per degree. This quantity is called the frequency.
The frequency of a periodic function is the reciprocal of the period. So, period
(^) frequency and. frequency
period
An efficient stepwise procedure for drawing a sinusoid is:
Draw the sinusoidal axis.
Draw upper and lower bounds by going A units above and below the sinusoidal axis.
Find the starting point of a cycle at PD,the phase displacement.
Cosine functions start a cycle at a high point. Sine functions start a cycle on the sinusoidal axis, heading up.
The cycle will end one period later atPD Period.
Halfway between two high points will be a low point. Halfway between each high and low point, the graph will cross the sinusoidal axis.
After graphing the five critical points, sketch the graph through these five critical points.
Example 1: Find the period, amplitude, frequency, phase displacement, and vertical displacement. Then use this information to find critical points and sketch the graph.
y 5 3 cos^2 ( 20 )
y VDAcos B(PD)
A 3 ,B 2 ,VD 5 , and PD 20
Amplitude = A Period B
P
P
Frequency 4
P
step
180
cycle per degree 4
180 45
Phase displacement:
PD 20 shift right
20
Vertical displacement: VD 5 shift up 5 units, so the sinusoidal axis is aty 5
If B is negative, use the absolute value of B in the period calculation. Your text’s problems do not have negative B values, but you may run into negative B values in the future.
Case 1 :
Given an equation
2
4
6
8
Section 2.4 – General Sinusoidal Graphs (continued)
Example 1 continued:
You want to find values of that make the argument equal to 0 , 90 , 180 , 270 ,and 360.
Set 2 ( 20 ) 0 2 ( 20 ) 90 2 ( 20 ) 180 2 ( 20 ) 270 2 ( 20 ) 360
20 0
20 45
20 90
20 135
20 180 20
65
110
155
200
Critical Points
2 ( 20 )
y 5 3 cos 2 ( 20 )
20
0 8 65 90 5 110 180 2 155
270 5 200 360 8
Example 2: Find the period, amplitude, frequency, phase displacement, and vertical displacement. Then use this information to find critical points and sketch the graph.
y 3 2 sin
y VDAsin B(PD)
A 2 , ,
B VD 3 , and PD 30
Amplitude = A Period B
P
P
Frequency 4
P
step
720
cycle per degree 4
720
Phase displacement: (^) PD 30 shift left 30
Vertical displacement: VD 3 shift down 3 units, so the sinusoidal axis is aty 3
y 5 3 cos 2 ( 20 )
20 65 110 155 200
Sinusoidal Axisy 5
Vertical Displacement (VD) up 5 units
Phase Displacement
(PD) 20 right
Amplitude A 3
Case 1: Given an equation
Remember,
step 45 ,
so
the θ-values are
45 apart.
180
7
14
21
28
35
42
49
Section 2.4 – General Sinusoidal Graphs (continued)
Example 3: For the sinusoid sketched, determine the period, frequency, amplitude, phase displacement, and vertical
displacement. Then write an equation for the sinusoid.
Assume the graph is of a cosine function, since a cosine graph starts a cycle at a high point.
y VDAcos^ B(PD)
One complete cycle begins at 5 and ends at 65. So, the period is 65 5 60.
The frequency is Period
Frequency
cycle per degree
The sinusoidal axis is halfway between the upper bound (UB), 42, and the lower bound (LB), 28.
So, the vertical displacement is the average of 42 and 28.
UB LB
VD
7 units The sinusoidal axis isy 7.
The amplitude is the distance between the sinusoidal axis and the upper bound and it is a non-reflected
cosine graph (thus, A is positive), A 42 7
35 units
Assuming the graph to be of a cosine function, the phase displacement is^5 ,
soPD 5.
Since the period is 60 , B 360 Period
So, an equation of the sinusoid graphed isy VDAcos B(PD)
y 7 35 cos 6 ( 5 )
y
5 65
Case 2:
Given a graph
1
2
3
4
5
6
Section 2.4 – General Sinusoidal Graphs (continued)
Example 4: Draw a graph and find an equation of the sinusoid described.
Period 90 ,
amplitude 2 units, phase displacement (for a sine function) equals 30 ,
vertical displacement 3 units.
Period
B
Assuming a non-reflected sine function, soA 2.
PD 30 VD 3
B 4
So, the equation is
y VDAsin B(PD)
y 3 2 sin 4 ( 30 )
You want to find values of that make the argument equal to 0 , 90 , 180 , 270 ,and 360.
Set
4 ( 30 ) 0
4 ( 30 ) 90
4 ( 30 ) 180
4 ( 30 ) 270
4 ( 30 ) 360
30 0
30 22. 5
30 45
30 67. 5
30 90 30
52. 5
75
97. 5
120
P
step
22. 5
Critical Points
4 ( 30 )
y 3 2 sin 4 ( 30 )
30 (^03)
- 5 (^905) 75 180 3
- 5 270 1 120 (^3603)
All material has been taken from Trigonometry, by P. Foerster, 3rd^ Edition
Case 3:
Given the Amplitude, Period, PD, and VD
y 3 2 sin 4 ( 30 )
Sinusoidal Axisy 3
30
- 5 75 120
- 5
