PowerPoint Presentation

5/7/13 Obj: SWBAT apply properties of periodic functionsBell Ringer: Construct a sinusoid with amplitude 2, period 3, 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!

1To find the phase or horizontal shift of a sinusoidGo to phase shift pdfhttp://www.analyzemath.com/trigonometry/sine.htm

To find the phase or horizontal shift of a sinusoidGo to phase shift pdfhttp://www.analyzemath.com/trigonometry/sine.htm

Horizontal Shift and Phase Shift (use Regent)

Go to phase shift pdf

4.3.10Determining 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

Therefore, the period is

..

The amplitude of the function is | a |. Therefore, the amplitude is 3.y = 3sin 4x4.3.3Graphing a Periodic Function

Period: 2pRange: y-intercept: 0x-intercepts: 0, p, 2p, ...Graph y = sin x.Amplitude: 11Domain: all real numbers-1 y 14.3.4Graphing a Periodic Function

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

Period: 2pDomain: all real numbersRange: -1 y 1Amplitude: 1

Graph y = cos x.14.3.5

Graphing a Periodic FunctionGraph y = tan x.

Asymptotes:

Domain:

Range: all real numbersPeriod: pDetermining the Period and Amplitude of y = a sin bx

Sketch the graph of y = 2sin 2x. The period is p. The amplitude is 2.

4.3.11Determining the Period and Amplitude of y = a sin bxSketch the graph of y = 3sin 3x.The period is . The amplitude is 3.

4.3.124.3.13Writing the Equation of the Periodic Function

Amplitude

= 2Period

p

b = 2Therefore, the equation as a function of sine isy = 2sin 2x.4.3.14Writing the Equation of the Periodic Function

AmplitudePeriod

= 34 p

b = 0.5Therefore, the equation as a function of cosine isy = 3cos 0.5x.Summary of Transformationsa = vertical stretch or shrink amplitudeb = horizontal stretch or shrink period/frequencyc = horizontal shift (phase shift) phaseh = horizontal shift (phase shift) phased = vertical translation/shiftk = vertical translation/shiftExit Ticket pg 439 #61-64 Horizontal Shift and Phase Shift (use Regent)

Domain:Range:Continuity:Increasing/DecreasingSymmetry:Bounded:Max./Min.Horizontal AsymptotesVertical AsymptotesEnd Behavior15

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? =

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

18

http://curvebank.calstatela.edu/unit/unit.htm 19

Read page 388 last paragraphVertical Stretch and Shrink

On your calculatorbaseline

Vertical Stretch and ShrinkAmplitude 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 ShrinkAmplitude 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

On your calculator

Horizontal Stretch/Shrink y = f(cx) stretch if c< 1 factor = 1/cshrink if c > 1 factor = 1/cb = number complete cycles in 2 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.

26

http://curvebank.calstatela.edu/unit/unit.htm 2728

Read page 388 last paragraphVertical Stretch and Shrink

On your calculatorHorizontal Stretch and Shrink

On your calculator

Horizontal Stretch/Shrinky = f(bx) stretch if |b| < 1 shrink if |b |> 1Both cases factor = 1/|b|

4.3.2Periodic FunctionsFunctions 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 xy = sin xy = 0.5sin x4.3.6

Period

Amplitude

Domain

Rangey = sin xy = 2sin xy = 0.5sin x2p2p2p120.5all real numbersall real numbersall real numbers-1 y 1-2 y 2-0.5 y 0.5Comparing the Graphs of y = a sin xThe 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.4.3.74.3.8Determining the Period for y = sin bx, b > 0

y = sin xGraph y = sin 2x

y = sin 2xy = sin x

y = sin x

Comparing the Graphs of y = sin bxPeriod

Amplitude

Domain

Rangey = sin xy = sin 2 xy = sin 0.5 x2pp4p111all real numbersall real numbersall real numbers-1 y 1-1 y 1-1 y 1The period for y = sin bx is

When b > 1, there is a horizontal shrink.When 0 < b < 1, there is a horizontal stretch.4.3.9