The document discusses graphs of composite trigonometric functions. It covers combining trigonometric and algebraic functions, sums and differences of sinusoids, and damped oscillation. Examples are provided of combining cosine with x^2, adding a sinusoid to a linear function, and identifying sinusoid functions. Damped oscillation is described as oscillation between two graphs where the amplitude reduces over time.

1. 4.6
Graphs of
Composite
Trigonometric
Functions
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Combining Trigonometric and Algebraic Functions
 Sums and Differences of Sinusoids
 Damped Oscillation
… and why
Function composition extends our ability to model
periodic phenomena like heartbeats and sound waves.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 2

3. Example Combining the Cosine
Function with x2
Graph y  cos x2
and state whether the function
appears to be periodic.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 3

4. Example Combining the Cosine
Function with x2
and state whether the function
The function appears
to be periodic.
Graph y  cos x2
appears to be periodic.
Quic kTime™ and a
decompressor
are needed to see this picture.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 4

5. Example Combining the Cosine
Function with x2
Graph y  cos x2   and state whether the function
appears to be periodic.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 5

6. Example Combining the Cosine
Function with x2
Graph y  cos x2   and state whether the function
appears to be periodic.
The function appears
not to be periodic.
Quic kTime™ and a
decompressor
are needed to see this picture.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 6

7. Example Adding a Sinusoid to a
Linear Function
Graph f x cos x 
x
3
and state its domain and range.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 7

8. Example Adding a Sinusoid to a
Linear Function
Graph f x cos x 
x
3
and state its domain and range.
The function f is the sum of the
functions gx cos x
and hx 
x
3
.
Here's the graph of f  g  h.
Domain: ,
Range: ,
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9. Sums That Are Sinusoids Functions
If y1  a1 sin(b(x  h1 )) and y2  a2 cos(b(x  h2 )), then
y1  y2  a1 sin(b(x  h1 ))  a2 cos(b(x  h2 ))
is a sinusoid with period 2 b .
Copyright © 2011 Pearson, Inc. Slide 4.6 - 9

10. Example Identifying a Sinusoid
Determine whether the following function is or
is not a sinusoid.
f (x)  3cos x  5sin x
Copyright © 2011 Pearson, Inc. Slide 4.6 - 10

11. Example Identifying a Sinusoid
Determine whether the following function is or
is not a sinusoid.
f (x)  3cos x  5sin x
Yes, since both functions in the sum have period 2 .
Copyright © 2011 Pearson, Inc. Slide 4.6 - 11

12. Example Identifying a Sinusoid
Determine whether the following function is or
is not a sinusoid.
f (x)  cos3x  sin5x
Copyright © 2011 Pearson, Inc. Slide 4.6 - 12

13. Example Identifying a Sinusoid
Determine whether the following function is or
is not a sinusoid.
f (x)  cos3x  sin5x
No, since cos3x has period 2 / 3 and
sin5x has period 2 / 5.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 13

14. Damped Oscillation
The graph of y  f (x)cosbx (or y  f (x)sinbx) oscillates
between the graphs of y  f (x) and y   f (x). When
this reduces the amplitude of the wave, it is called
damped oscillation. The factor f (x) is called
the damping factor.
Copyright © 2011 Pearson, Inc. Slide 4.6 - 14

15. Quick Review
State the domain and range of the function.
1. f (x)  3sin2x
2. f (x)  x  2
3. f (x)  2cos3x
4. Describe the behavior of y  e3x as x  .
5. Find f og and g o f , given f (x)  x2  3 and g(x)  x
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16. Quick Review Solutions
State the domain and range of the function.
1. f (x)  3sin2x Domain: , Range: 3,3
2. f (x)  x  2 Domain: , Range: 2,
3. f (x)  2cos 3x Domain: , Range: 2,2
4. Describe the behavior of y  e3x as x  . lim
x
e3x  0
5. Find f og and g o f , given f (x)  x2  3 and g(x)  x
f og  x  3; g o f  x2  3
Copyright © 2011 Pearson, Inc. Slide 4.6 - 16