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

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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)

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

Horizontal Shift and Phase Shift (use Regent)

Go to phase shift pdf

4.3.10

Determining the Period and Amplitude of y = a sin bxGiven the function y = 3sin 4x, determine the period and the amplitude.The period of the function is

2b

Therefore, the period is 24

2

.

.

The amplitude of the function is | a |. Therefore, the amplitude is 3.

y = 3sin 4x

4.3.3

Graphing a Periodic Function

Period: 2

Range: y-intercept: 0x-intercepts: 0, ±, ±2, ...

Graph y = sin x.

Amplitude: 1

1

Domain: all real numbers

-1 ≤ y ≤ 1

4.3.4

Graphing a Periodic Function

y-intercept: 1x-intercepts: , ...

Period: 2 Domain: all real numbersRange: -1 ≤ y ≤ 1Amplitude: 1

2

,32

Graph y = cos x.

1

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:

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 bxSketch the graph of y = 3sin 3x.

The period is . The amplitude is 3.23

23

53

43

4.3.12

4.3.13

Writing the Equation of the Periodic Function

| maximum minimum|2

Amplitude

| 2 ( 2) |

2= 2

Period 2b

2b

b = 2Therefore, the equation as a function of sine isy = 2sin 2x.

4.3.14

Writing the Equation of the Periodic Function

| maximum minimum|2

Amplitude Period 2b

| 3 ( 3) |2

= 3

4 2b

b = 0.5

Therefore, the equation as a function of cosine isy = 3cos 0.5x.

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

Horizontal Shift and Phase Shift (use Regent)

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

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

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

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

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

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

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.

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.

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

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

28

Read page 388 – last paragraphVertical Stretch and Shrink

1. ½ cos (x)2. -4 sin(x)

On your calculator

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|

The frequency is the reciprocal of the period.

f =

.

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

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

Period

Amplitude

Domain

Range

y = sin x 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

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

4.3.8

Determining the Period for y = sin bx, b > 0

y = sin x

Graph y = sin 2x

and y sin

x2

.y = sin 2xy = sin x y = sin xy sinx2

Comparing the Graphs of y = sin bx

Period

Amplitude

Domain

Range

y = sin x 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

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