Sums and Differences of

SinusoidsSec. 4.6b

Exploration 1: Investigating SinusoidsGraph these functions, one at a time, in the viewing window

. Which ones appear to be sinusoids? 2 ,2 by 6,6

3sin 2cosy x x 2sin 3cosy x x

2sin 3 4cos 2y x x 2sin 5 1 5cos5y x x

7 2 7cos sin

5 5

x xy

3cos2 2sin 7y x x

Only these functions appear to be sinusoids!!!

Exploration 1: Investigating SinusoidsWhat relationship between sine and cosine functions ensuresthat their sum or difference will again be a sinusoid?

The rule is simple: Sums and differencesof sinusoids with the same period are

again sinusoids!!!Definition: A sinusoid is a function that can be written as sin[ ( )]y a b x h k

Sums That Are Sinusoid Functions

1 1 1siny a b x h If 2 2 2cosy a b x h and

1 2 1 1 2 2sin cosy y a b x h a b x h then

2

b

is a sinusoid with period

Does this rule also work for the sum of two sinefunctions or the sum of two cosine functions???

Yes indeed!!!

Practice ProblemsDetermine whether each of the following functions is or is nota sinusoid.

5cos 3sinf x x x (a)

2Both functions in the sum have the same period:

A sinusoid

cos5 sin3f x x x (b)

2

5

Period of the first function:

Not a sinusoid

2

3

Period of the second function:

Practice ProblemsDetermine whether each of the following functions is or is nota sinusoid.

2cos3 3cos 2f x x x (c) Not a sinusoid

2

3

Period of the first function:

Period of the second function:

3 3 3cos cos sin

7 7 7

x x xf x a b c

(d)

14

3

All three functions have the same period:

A sinusoid

Practice Problems

2sin 5cosf x x x Let

(a) Is f a sinusoid?

2Yes, both functions have the same period

(b) Find the period of f. Period =

2 ,2 by 10,10

(c) Estimate the amplitude and phase shift graphically.

Graph the function in

5.385Calculate the maximum value of f:

This is the approximate value of the amplitude!!!

1.190Calculate the x-intercept closest to x = 0:

This is the approximate phase shift of the sine function!!!

Practice Problems

2sin 5cosf x x x Let

(d) Give a sinusoid sina b x h

2 ,2 by 6,6 To verify our answer graphically, graph f and g in thesame viewing window:

Plug in the previously-discovered values:

that approximates f x

5.385sin 1.190g x x

Practice Problems

sin 2 cos3f x x x

Show that the given function is periodic but not a sinusoid.Graph one full period of the function.

Let’s check to see if is a period of our function:

Since the two smaller functions have different periods, theirsum is not a sinusoid.

2 sin 2 2 cos 3 2f x x x 2

sin 2 4 cos 3 6x x sin 2 cos3x x

f x

Practice Problems

sin 2 cos3f x x x

Show that the given function is periodic but not a sinusoid.Graph one full period of the function.

This proves that either is a period of f or that the periodis an exact divisor of .2

2

Check the graph in the window

2 0,2 by 2,2

Evidence that the period must be !!!

What happens if we expand thex-coordinates of our window???

Damped OscillationOur last topic for

Section 4.6

Let’s experiment…

Graph 21 5 cos 6y x x

in the window 2 ,2 by 40,40

In the same window, also graph:2

2 5y x 23 5y x

What do you notice about these graphs???

The “squeezing” of the first function between theother two is called damping…

Damped OscillationThe graph of cosy f x bx (or ) siny f x bxoscillates between the graphs of and y f x

. When this reduces the amplitude of the y f xwave, it is called damped oscillation. The factor

is called the damping factor. f x

Identifying Damped OscillationFor each of the following functions, determine if the graph showsdamped oscillation. If it does, identify the damping factor and tellwhere the damping occurs.

(a) 2 sin 4xf x x Graph in , by 5,5

Does this graph show damped oscillation? Yes!!!

What is the damping factor? 2–x

Where does the damping occur? As x 8

How does this all relate to the graphs of the twosmaller functions that comprise f ???

Identifying Damped OscillationFor each of the following functions, determine if the graph showsdamped oscillation. If it does, identify the damping factor and tellwhere the damping occurs.

(b) 3cos 2f x x Graph: , by 5,5 No damping occurs

(c) 2 cos 2f x x x Graph: 2 ,2 by 12,12

Damped oscillation

Damping factor: –2x

The damping occurs as x 0

A Damped Oscillating SpringA physics class collected data for an air table glider that oscillatesbetween two springs. The class determined from the data thatthe equation 0.0650.22 cos 2.4ty e tmodeled the displacement y of the spring from its original positionas a function of time t.

(a) Identify the damping factor and tell where the damping occurs.

Damping factor: 0.0650.22 te

The damping occurs as t

More Guided PracticeTell whether the given functions exhibits damped oscillation. If so,identify the damping factor and tell whether the damping occursas x approaches zero or as x approaches infinity.

Damped oscillation, damping factor: x(a) sin 4f x x xThe damping occurs as 0xNo damping (amplitude is constant)(b) 2 cosf x x

Damped oscillation, damping factor:2

3

x

(c) 2 2sin

3 3

xx

f x

The damping occurs as x

More Guided PracticeFor each function, graph both f and plus or minus its dampingfactor in the same viewing window. Describe the behavior of thefunction f for x > 0. What is the end behavior of f ?

(a) 2 sin 4xf x x Window: 0,2 by 1,1

The function f oscillates up and down between and2 x

End Behavior:

2 x lim 0

xf x

(b) cos3xf x e x Window: 0,1.5 by 1.5,1.5

The function f oscillates up and down between andxe

End Behavior:

xe lim 0

xf x