copyright © 2011 pearson, inc. 4.6 graphs of composite trigonometric functions
TRANSCRIPT
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Copyright © 2011 Pearson, Inc.
4.6Graphs of Composite
Trigonometric Functions
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Slide 4.6 - 2 Copyright © 2011 Pearson, Inc.
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.
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Slide 4.6 - 3 Copyright © 2011 Pearson, Inc.
Example Combining the Cosine Function with x2
Graph y = cosx( )2 and state whether the function
appears to be periodic.
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Slide 4.6 - 4 Copyright © 2011 Pearson, Inc.
Example Combining the Cosine Function with x2
The function appears
to be periodic.
Graph y = cosx( )2 and state whether the function
appears to be periodic.
QuickTime™ and a decompressor
are needed to see this picture.
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Slide 4.6 - 5 Copyright © 2011 Pearson, Inc.
Example Combining the Cosine Function with x2
Graph y =cos x2( ) and state whether the function
appears to be periodic.
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Slide 4.6 - 6 Copyright © 2011 Pearson, Inc.
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.QuickTime™ and a
decompressorare needed to see this picture.
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Slide 4.6 - 7 Copyright © 2011 Pearson, Inc.
Example Adding a Sinusoid to a Linear Function
Graph f x( ) =cosx−x3 and state its domain and range.
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Slide 4.6 - 8 Copyright © 2011 Pearson, Inc.
Example Adding a Sinusoid to a Linear Function
Graph f x( ) =cosx−x3 and state its domain and range.
The function f is the sum of the
functions g x( ) =cosx
and h x( ) =−x3.
Here's the graph of f =g+h.
Domain: −∞,∞( )
Range: −∞,∞( )
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Sums That Are Sinusoids Functions
If y1 =a1sin(b(x−h1)) and y2 =a2 cos(b(x−h2 )), then
y1 + y2 =a1sin(b(x−h1)) +a2 cos(b(x−h2 ))
is a sinusoid with period 2π b .
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Slide 4.6 - 10 Copyright © 2011 Pearson, Inc.
Example Identifying a Sinusoid
Determine whether the following function is or
is not a sinusoid.
f (x) =3cosx + 5sinx
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Slide 4.6 - 11 Copyright © 2011 Pearson, Inc.
Example Identifying a Sinusoid
Yes, since both functions in the sum have period 2π.
Determine whether the following function is or
is not a sinusoid.
f (x) =3cosx + 5sinx
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Slide 4.6 - 12 Copyright © 2011 Pearson, Inc.
Example Identifying a Sinusoid
Determine whether the following function is or
is not a sinusoid.
f (x) =cos3x + sin5x
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Slide 4.6 - 13 Copyright © 2011 Pearson, Inc.
Example Identifying a Sinusoid
No, since cos 3x has period 2π / 3 and
sin5x has period 2π / 5.
Determine whether the following function is or
is not a sinusoid.
f (x) =cos3x + sin5x
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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 calleddamped oscillation. The factor f (x) is called the damping factor.
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Quick Review
State the domain and range of the function.
1. f (x) =−3sin2x
2. f (x) = x + 23. f (x) =2cos3x
4. Describe the behavior of y=e−3x as x→ ∞.
5. Find f og and gof , given f (x) =x2 + 3 and g(x) = x
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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) =2cos3x Domain: −∞,∞( ) Range: −2,2[ ]
4. Describe the behavior of y=e−3x as x→ ∞. limx→ ∞
e−3x =0
5. Find f og and gof , given f (x) =x2 + 3 and g(x) = x
f og=x+ 3; gof = x2 + 3