Transcript Document

5/7/13
Education is Power!
Obj: SWBAT apply properties of periodic functions
Bell Ringer:
Construct a sinusoid with amplitude 2, period 3π, point 0,0
HW Requests:
Pg 395 #72-75, 79, 80
WS Amplitude, Period, Phase Shift Dignity without compromise!
In class: 61-68
Homework:
Study for Quiz,
Bring your Unit Circle
Read Section 5.1
Project Due Wed. 5/8
Each group staple all projects together
To find the phase or horizontal shift of a sinusoid
1. 𝑓 𝑥 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
𝑓 𝑥 = 𝑎 cos 𝑏𝑥 + 𝑐 + 𝑑
where a, b, c, and d are constants and neither a nor b is 0
Let c = -2 the shift is to the right or left
Let c = +2 the shift is to the right or left
Engineers and physicist change the nomenclature +c becomes -h
What does this change mean?
2. 𝑓 𝑥 = 𝑎 sin(𝑏 𝑥 − ℎ ) + 𝑘
𝑓 𝑥 = 𝑎 cos(𝑏 𝑥 − ℎ ) + 𝑘
where a, b, c, and d are constants and neither a nor b is 0
Let h = -2 the shift is to the right or left
Let h = +2 the shift is to the right or left
Find the relationship between h and c
Solve for h: (bx+c) = b(x-h)
Go to phase shift pdf
http://www.analyzemath.com/trigonometry/sine.htm
To find the phase or horizontal shift of a sinusoid
1. 𝑓 𝑥 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
𝑓 𝑥 = 𝑎 cos 𝑏𝑥 + 𝑐 + 𝑑
where a, b, c, and d are constants and neither a nor b is 0
2. 𝑓 𝑥 = 𝑎 sin(𝑏 𝑥 − ℎ ) + 𝑘
𝑓 𝑥 = 𝑎 cos(𝑏 𝑥 − ℎ ) + 𝑘
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 shift
For #1, the phase shift is -c/b
Note: the phase shift can be positive or negative
Go to phase shift pdf
http://www.analyzemath.com/trigonometry/sine.htm
Horizontal Shift and Phase Shift (use Regent)
Go to phase shift pdf
Determining the Period and Amplitude of y = a sin bx
Given the function y = 3sin 4x, determine the period
and the amplitude.
2
.
The period of the function is
b
2 
Therefore, the period is
 .
4
2
The amplitude of the function is | a |.
Therefore, the amplitude is 3.
y = 3sin 4x
4.3.10
Graphing a Periodic Function
Graph y = sin x.
1
Period: 2
Amplitude: 1
Domain:
all real numbers
Range:
y-intercept: 0
x-intercepts: 0, ±, ±2, ... -1 ≤ y ≤ 1
4.3.3
Graphing a Periodic Function
Graph y = cos x.
1
Period: 2
Amplitude: 1
Domain: all real numbers
Range: -1 ≤ y ≤ 1
y-intercept: 1
 3
x-intercepts:  , , ...
2
2
4.3.4
Graphing a Periodic Function
Graph y = tan x.
Asymptotes:
 3 5 
,
,
,...,

  n, n I
Period: 
2 2
2
2

Domain: {x | x    n, n I , x  R}
2
Range: all real numbers
4.3.5
Determining the Period and Amplitude of y = a sin bx
Sketch the graph of y = 2sin 2x.
The period is .
The amplitude is 2.
4.3.11
Determining the Period and Amplitude of y = a sin bx
Sketch the graph of y = 3sin 3x.
The period is 2  . The amplitude is 3.
3
2
3
4
3
5
3
4.3.12
Writing the Equation of the Periodic Function
2
Period 
b
2
 
b
b=2
Therefore, the equation as a function of sine is
y = 2sin 2x.
| maxim um
 minim um
|
Amplitude 
| 2  (2) | 2

2
=2
4.3.13
Writing the Equation of the Periodic Function
| maxim um
 minim um
|
Amplitude 
2
| 3  (3) |

2
=3
2
Period 
b
2
4 
b
b = 0.5
Therefore, the equation as a function of cosine is
y = 3cos 0.5x.
4.3.14
Summary of Transformations
𝑓 𝑥 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
𝑓 𝑥 = 𝑎 sin 𝑏(𝑥 − ℎ ) + 𝑘
•
•
•
•
•
•
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
Horizontal Shift and Phase Shift (use Regent)
Audacity
Sinusoid- Periodic Functions
A function is a sinusoid if it can be written in the form
𝑓 𝑥 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
where a, b, c, and d are constants and neither a nor b is 0
Domain:
Range:
Continuity:
Increasing/Decreasing
Symmetry:
Bounded:
Max./Min.
Horizontal Asymptotes
Vertical Asymptotes
End Behavior
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
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 =
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
Sinusoid
A function is a sinusoid if it can be written in the form
𝑓 𝑥 = 𝑎 cos 𝑏𝑥 + 𝑐 + 𝑑
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
Read page 388 – last paragraph
Vertical Stretch and Shrink
baseline
On your calculator
1. sin 𝑥
2. ½ cos (x)
3. -4 sin(x)
What are the amplitudes?
What is the amplitude of the
graph? Peak to peak value
2
Vertical Stretch and Shrink
baseline
Amplitude of a graph
Abs(max value – min value)
2
For graphing a sinusoid:
To find the baseline or middle
line on a graph
y = max value – min value
2
Use amplitude to graph.
Vertical Stretch and Shrink
baseline
Amplitude of a graph
Abs(max value – min value)
2
For graphing a sinusoid:
To find the baseline or middle
line on a graph
y = max value – amplitude
Horizontal Stretch and Shrink
b = number complete cycles in 2π rad.
On your calculator
1. sin 𝑥 T =
2. sin(2x) T =
𝑥
3. sin( ) T =
2
4. sin(5x) T =
What are the periods (T)?
Horizontal Stretch/Shrink y = f(cx)
stretch if c< 1 factor = 1/c
shrink if c > 1 factor = 1/c
See if you can write the equation for the Ferris Wheel
We can use these values to modify the basic cosine or
sine function in order to model our Ferris wheel
situation.
Audacity
Sinusoid- Periodic Functions
A function is a sinusoid if it can be written in the form
𝑓 𝑥 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑
where a, b, c, and d are constants and neither a nor b is 0
Sinusoid
A function is a sinusoid if it can be written in the form
𝑓 𝑥 = 𝑎 cos 𝑏𝑥 + 𝑐 + 𝑑
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
Read page 388 – last paragraph
Vertical Stretch and Shrink
On your calculator
1. sin 𝑥
2. ½ cos (x)
3. -4 sin(x)
28
Horizontal Stretch and Shrink
On your calculator
1. sin 𝑥
2. sin2(x)
𝑥
3. sin( )
2
4. sin3(x)
Horizontal Stretch/Shrink
y = f(bx) stretch if |b| < 1
shrink if |b |> 1
Both cases factor = 1/|b|
The frequency is the reciprocal of the period.
f=
.
𝑏
2𝜋
where T =
2𝜋
𝑏
Periodic Functions
Functions that repeat themselves over a particular interval
of their domain are periodic functions. The interval is called
the 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 values
of the angles of 300, 450, and 600. In particular, keep in mind
the quadrantal angles of the unit circle.
http://curvebank.calstatela.edu/unit/unit.htm
http://www.analyzemath.com/trigonometry/sine.htm
(0, 1)
(-1, 0)
(1, 0)
(0, -1)
The points on the unit
circle are in the form
(cosine, sine).
4.3.2
Determining the Amplitude of y = a sin x
Graph
y = 2siny x= 2sin x and y = 0.5sin x.
sinxx
yy==sin
y = 0.5sin x
4.3.6
Comparing the Graphs of y = a sin x
y = sin x
Period
Amplitude
Domain
Range
y = 2sin x
y = 0.5sin x
2
2
2
1
2
0.5
all real numbers
all real numbers
all real numbers
-1 ≤ y ≤ 1
-2 ≤ y ≤ 2
-0.5 ≤ y ≤ 0.5
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
Determining the Period for y = sin bx, b > 0
Graph y = sin 2x
x
x
y and
s in y  sin . y = sin x
y = sin x y =2 sin 2x
2
y = sin x
4.3.8
Comparing the Graphs of y = sin bx
y = sin x
Period
Amplitude
Domain
Range
y = sin 2 x
y = sin 0.5 x
2

4
1
1
1
all real numbers
all real numbers
all real numbers
-1 ≤ y ≤ 1
-1 ≤ y ≤ 1
-1 ≤ y ≤ 1
2
The period for y = sin bx is
, b  0.
b
When b > 1, there is a horizontal shrink.
When 0 < b < 1, there is a horizontal stretch.
4.3.9