Sinusoids and Transformations

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Sinusoids and Transformations

• Sec. 4.4b
• HW p. 395 43-51odd, 57-69 odd

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Definition Sinusoid
A function is a sinusoid if it can be written in
the form
where a, b, c, and d are constants and neither a
nor b is 0.
In general, any transformation of a sine function
(or the graph of such a function such as
cosine) is a sinusoid.
This is the format that we are used to seeing,
thus it is OK to continue using this formatI
use this format.
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Transformations
There is a special vocabulary for describing our
traditional graphical transformations when
applied to sinusoids
Horizontal stretches and shrinks affect the
period and the frequency.
Vertical stretches and shrinks affect the
amplitude.
Horizontal translations bring about phase shifts.
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Definition Amplitude of a Sinusoid
The amplitude of the sinusoid
is
Similarly, the amplitude of
is
Graphically, the amplitude is half the height of
the wave.
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Transformations
Find the amplitude of each function and use the
language of transformations to describe how the
graphs are related.
(a)
(b)
(c)
Amplitudes (a) 1, (b) 1/2, (c) 3 3
The graph of y is a vertical shrink of the
graph of y by 1/2.
1
2
The graph of y is a vertical stretch of the
graph of y by 3, and a reflection across the
x-axis, performed in either order.
3
1
Confirm these answers graphically!!!
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Definition Period of a Sinusoid
The period of the sinusoid
is
Similarly, the period of
is
Graphically, the period is the length of one full
cycle of the wave.
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Transformations
Find the period of each function and use the
language of transformations to describe how the
graphs are related.
Periods
(a)
(b)
(c)
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Transformations
Find the period of each function and use the
language of transformations to describe how the
graphs are related.
(a)
The graph of y is a horizontal stretch of the
graph of y by 3, a vertical stretch by 2, and
a reflection across the x-axis, performed in any
order.
2
1
(b)
(c)
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Transformations
Find the period of each function and use the
language of transformations to describe how the
graphs are related.
(a)
The graph of y is a horizontal shrink of the
graph of y by 1/2, a vertical stretch by 3, and
a reflection across the y-axis, performed in any
order.
3
1
(b)
Confirm these answers graphically!!!
(c)
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Definition Frequency of a Sinusoid
The frequency of the sinusoid
is
Reciprocal of the period!!!
Similarly, the frequency of
is
Graphically, the frequency is the number of
complete cycles the wave completes in a unit
interval.
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Transformations
Find the frequency of the given function, and
interpret its meaning graphically.
Sketch the graph of the function in
by
Interpretation
Frequency
The graph completes 1 full cycle per interval of
length .
Period
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Transformations
How does the graph of
differ from the graph of ?
? A translation to the left by c units when c gt 0
New Terminology When applied to sinusoids,
we say that the wave undergoes a phase shift of
c.
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Transformations
Write the cosine function as a phase shift of the
sine function.
Write the sine function as a phase shift of the
cosine function.
Confirm these answers graphically!!!
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Reminder Graphs of Sinusoids
The graphs of these functions have the following
characteristics
Amplitude
Period
Frequency
A phase shift of
A vertical translation of
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Guided Practice
Graph one period of the given function by hand.
Period
Amplitude
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Guided Practice
Graph one period of the given function by hand.
Period
Amplitude
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Practice Problems
Construct a sinusoid with the given information.
Amplitude 2, Period , Point (0,0)
One possibility
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Practice Problems
Construct a sinusoid with the given information.
Amplitude 3.2, Period , Point (5,0)
One possibility
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Guided Practice
Identify the maximum and minimum values and the
zeros of the given function in the interval
? no calculator!
Maximum
At
Minimum
At
Zeros
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Practice Problems
Construct a sinusoid with period and
amplitude 6 that goes through (2,0).
First, solve for b
Find the amplitude
Lets just take the positive value again.
To pass through (2,0), we need a phase shift of 2
? h 2
Either will work!!!
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Practice Problems
Construct a sinusoid y f(x) that rises from a
minimum value of y 5 at x 0 to a maximum
value of y 25 at x 32.
First, sketch a graph of this sinusoid
Amplitude is half the height
The period is 64
We need a function whose minimum is at x 0. We
could shift the sine function horizontally, but
its easier to simply reflect the cosine
functionby letting a 10
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Practice Problems
Construct a sinusoid y f(x) that rises from a
minimum value of y 5 at x 0 to a maximum
value of y 25 at x 32.
But since cosine is an even function
This function ranges from 10 to 10, but we need
a function that ranges from 5 to 25vertical
translation by 15
Support this answer graphically???
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Finally, a couple of whiteboard problems
Find the amplitude of the function and use the
language of transformations to describe how the
graph of the function is related to the graph of
the sine function.
1.
Amplitude 2 Vertical stretch by 2
Amplitude 4 Vertical stretch by 4, Reflect
across x-axis
3.
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More whiteboard
Find the period of the function and use the
language of transformations to describe how the
graph of the function is related to the graph of
the cosine function.
Period Horizontal shrink by 1/3
7.
9.
Period Horizontal shrink by 1/7,
Reflect across y-axis
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Whiteboard
Graph three periods of the given function by hand.
Amplitude
Period
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Whiteboard
Graph three periods of the given function by hand.
Amplitude
Period
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Whiteboard
Graph three periods of the given function by hand.
Amplitude
Period
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Whiteboard
State the amplitude and period of the given
sinusoid, and (relative to the basic function)
the phase shift and vertical translation.
Amplitude
Period
Phase Shift
Vertical Translation 7 units up
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Whiteboard
Identify the maximum and minimum values and the
zeros of the given function in the interval
? no calculator!
Maximum
At
Minimum
At
and
Zeros
30
Whiteboard
State the amplitude and period of the given
sinusoid, and (relative to the basic function)
the phase shift and vertical translation.
Amplitude
Period
Phase Shift
Vertical Translation 1 unit down