5/7/13 obj: swbat apply properties of periodic functions bell ringer: construct a sinusoid with...
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5/7/13 Obj: SWBAT apply properties of periodic functionsBell Ringer: Construct a sinusoid with amplitude 2, period 3π, point 0,0HW Requests: Pg 395 #72-75, 79, 80WS Amplitude, Period, Phase ShiftIn class: 61-68 Homework: Study for Quiz,Bring your Unit CircleRead Section 5.1 Project Due Wed. 5/8Each group staple all projects together
Education is Power!
Dignity without compromise!
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To find the phase or horizontal shift of a sinusoid
where a, b, c, and d are constants and neither a nor b is 0Let c = -2 the shift is to the right or leftLet c = +2 the shift is to the right or left
Engineers and physicist change the nomenclature +c becomes -h What does this change mean?
Go to phase shift pdfhttp://www.analyzemath.com/trigonometry/sine.htm
where a, b, c, and d are constants and neither a nor b is 0Let h = -2 the shift is to the right or leftLet h = +2 the shift is to the right or left
Find the relationship between h and cSolve for h: (bx+c) = b(x-h)
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To find the phase or horizontal shift of a sinusoid
where a, b, c, and d are constants and neither a nor b is 0
Go to phase shift pdfhttp://www.analyzemath.com/trigonometry/sine.htm
where a, b, c, and d are constants and neither a nor b is 0
For #2, factor b out of the argument, the resulting h is the phase shiftFor #1, the phase shift is -c/bNote: the phase shift can be positive or negative
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Horizontal Shift and Phase Shift (use Regent)
Go to phase shift pdf
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4.3.10
Determining the Period and Amplitude of y = a sin bx
Given the function y = 3sin 4x, determine the period and the amplitude.
The period of the function is2b
Therefore, the period is24
2
.
.
The amplitude of the function is | a |. Therefore, the amplitude is 3.
y = 3sin 4x
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4.3.3
Graphing a Periodic Function
Period: 2p
Range: y-intercept: 0x-intercepts: 0, ±p, ±2p, ...
Graph y = sin x.
Amplitude: 1
1
Domain:
all real numbers
-1 ≤ y ≤ 1
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4.3.4
Graphing a Periodic Function
y-intercept: 1x-intercepts: , ...
Period: 2p Domain: all real numbersRange: -1 ≤ y ≤ 1Amplitude: 1
2
,32
Graph y = cos x.
1
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4.3.5
Graphing a Periodic FunctionGraph y = tan x.
Asymptotes:
2
,32
,52
,...,2 n, n I
Domain: {x | x
2 n, n I , x R}
Range: all real numbers
Period: p
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Determining the Period and Amplitude of y = a sin bx
Sketch the graph of y = 2sin 2x.
The period is p.
The amplitude is 2.
4.3.11
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Determining the Period and Amplitude of y = a sin bxSketch the graph of y = 3sin 3x.
The period is . The amplitude is 3.23
23
53
43
4.3.12
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4.3.13
Writing the Equation of the Periodic Function
| maximum minimum|2
Amplitude
| 2 ( 2) |
2= 2
Period 2b
p 2b
b = 2
Therefore, the equation as a function of sine isy = 2sin 2x.
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4.3.14
Writing the Equation of the Periodic Function
| maximum minimum|2
Amplitude Period 2b
| 3 ( 3) |2
= 3
4 p 2b
b = 0.5
Therefore, the equation as a function of cosine isy = 3cos 0.5x.
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Summary of Transformations
• a = vertical stretch or shrink amplitude• b = horizontal stretch or shrink
period/frequency• c = horizontal shift (phase shift) phase• h = horizontal shift (phase shift) phase• d = vertical translation/shift• k = vertical translation/shift
Exit Ticket pg 439 #61-64
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Horizontal Shift and Phase Shift (use Regent)
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AudacitySinusoid- Periodic Functions
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
Domain:Range:Continuity:Increasing/DecreasingSymmetry:Bounded:Max./Min.Horizontal AsymptotesVertical AsymptotesEnd Behavior
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Sinusoid – a function that can be written in the form below.
Sine and Cosine are sinusoids.
The applet linked below can help demonstrate how changes in these parameters affect the sinusoidal graph:
http://www.analyzemath.com/trigonometry/sine.htm
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For each sinusoid answer the following questions.What is the midline? X = What is the amplitude? A =What is the period? T = (radians and degrees)What is the phase? ϴ =
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Definition: A function y = f(t) is periodic if there is a positive number c such that f(t+c) = f(t) for all values of t in the domain of f. The smallest number c is called the period of the function.
- a function whose value is repeated at constant intervals
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SinusoidA 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
Why is the cosine function a sinusoid?
http://curvebank.calstatela.edu/unit/unit.htm
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Read page 388 – last paragraphVertical Stretch and Shrink
1. ½ cos (x)2. -4 sin(x) What are the amplitudes?
What is the amplitude of thegraph? 2
On your calculatorbaseline
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Vertical Stretch and Shrink
Amplitude of a graph
Abs(max value – min value) 2For graphing a sinusoid:To find the baseline or middleline on a graphy = max value – min value 2Use amplitude to graph.
baseline
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Vertical Stretch and Shrink
Amplitude of a graph
Abs(max value – min value) 2For graphing a sinusoid:To find the baseline or middleline on a graphy = max value – amplitude
baseline
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Horizontal Stretch and Shrink
1. T = 2. sin(2x) T = 3. sin) T = 4. sin(5x) T = What are the periods (T)?
On your calculator
Horizontal Stretch/Shrink y = f(cx) stretch if c< 1 factor = 1/cshrink if c > 1 factor = 1/c
b = number complete cycles in 2π rad.
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See if you can write the equation for the Ferris Wheel
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We can use these values to modify the basic cosine or sine function in order to model our Ferris wheel situation.
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AudacitySinusoid- Periodic Functions
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
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SinusoidA 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
Why is the cosine function a sinusoid?
http://curvebank.calstatela.edu/unit/unit.htm
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28
Read page 388 – last paragraphVertical Stretch and Shrink
1. ½ cos (x)2. -4 sin(x)
On your calculator
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Horizontal Stretch and Shrink
1. sin2(x)2. sin)3. sin3(x)
On your calculator
Horizontal Stretch/Shrinky = f(bx) stretch if |b| < 1 shrink if |b |> 1Both cases factor = 1/|b|
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The frequency is the reciprocal of the period.
f =
.
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4.3.2
Periodic Functions
Functions that repeat themselves over a particular intervalof their domain are periodic functions. The interval is calledthe period of the function. In the interval there is one complete cycle of the function.
To graph a periodic function such as sin x, use the exact valuesof the angles of 300, 450, and 600. In particular, keep in mindthe quadrantal angles of the unit circle.
(1, 0)(-1, 0)
(0, 1)
(0, -1)
The points on the unitcircle are in the form(cosine, sine).
http://curvebank.calstatela.edu/unit/unit.htmhttp://www.analyzemath.com/trigonometry/sine.htm
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Determining the Amplitude of y = a sin x
Graph y = 2sin x and y = 0.5sin x.
y = sin x
y = 2sin x
y = sin x
y = 0.5sin x
4.3.6
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Period
Amplitude
Domain
Range
y = sin x y = 2sin x y = 0.5sin x
2p 2p 2p
1 2 0.5
all real numbers all real numbers all real numbers
-1 ≤ y ≤ 1 -2 ≤ y ≤ 2 -0.5 ≤ y ≤ 0.5
Comparing the Graphs of y = a sin x
The amplitude of the graph of y = a sin x is | a |.When a > 1, there is a vertical stretch by a factor of a.When 0 < a < 1, there is a vertical shrink by a factor of a.
4.3.7
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4.3.8
Determining the Period for y = sin bx, b > 0
y = sin x
Graph y = sin 2x
and y sin
x
2 .y = sin 2xy = sin x y = sin xy sin
x
2
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Comparing the Graphs of y = sin bx
Period
Amplitude
Domain
Range
y = sin x y = sin 2 x y = sin 0.5 x
2p p 4p
1 1 1
all real numbers all real numbers all real numbers
-1 ≤ y ≤ 1 -1 ≤ y ≤ 1 -1 ≤ y ≤ 1
The period for y = sin bx is
2b
, b 0.
When b > 1, there is a horizontal shrink.When 0 < b < 1, there is a horizontal stretch.
4.3.9