C 23 - Identify periodic relations and describe their characteristics

What is a Periodic Relationship?

Activity: Identifying Periodic Behaviour

What is a Sinusoidal Relationship?

Sound waves

Characteristics of Sinusoidal Relationships

Activities: Real World Applications

References

A function for which the dependent variable takes on the same set of values over and over again as the independent variable changes.Table of Contents

For a pendulum, one cycle starts in the center, swings to one direction,

swings back to the center, and then

continues to the other direction, and returns to the center. The following graph shows four cycles

of a pendulum.

It takes 2.5

seconds to

complete

each cycle.

(http://motor1.physics.wayne.edu/~cinabro/cinabro/education/conceptest02_ans.html)

Microsoft

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(Barry et al., 2004, p. 94)

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A special type of periodic behaviour

Looks like waves

Any portion of the curve can be translated onto another portion of the curve.

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http://www.mediacollege.com/audio/01/sound-waves.html

• Created by vibration of objects• As the loudspeaker vibrates, the air molecules vibrate in a

pattern.• The air causes the eardrum to vibrate in the same pattern.• Some sound waves are periodic, others are sinusoidal, and some

are neither.

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The horizontal line halfway between the

local maximum and

local minimum.

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Local Maximum – the tops of each “wave”, all equal in value.

Local Minimum – the bottoms of each “wave”, all equal in value.

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The vertical distance

from the sinusoidal axis

to a local minimum

or a local maximum.

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The change in the “x”

corresponding to one cycle.

Video – Laps in the PoolVideo – Laps in the Pool(http://www.youtube.com/watch?v=S-

88Y2QxZPo)Table of Contents

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y

A Ferris wheel has a diameter of 30 m, with the centre 18 m above the ground. It makes one complete rotation every 60 seconds.

(http://www.bmlc.ca/Math40S/Pre-Calculus%20Math%2040s%20-%20Trigonometry%20Lesson%208.pdf)

Microsoft

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Amplitude

(radius of wheel)

Sinusoidal Axis (centre of wheel)

Period (time for 1 revolution)

More about Ferris Wheels…

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A trapeze artist is swinging back and forth on her trapeze. Her distance from a support beam with respect

to time is modeled by the graph below.

A different application

Using the graph, label the following:

Amplitude

Period

Sinusoidal Axis

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Characteristics of a sinusoidal relationship

Equation of sinusoidal axis

Amplitude

Period

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Barry, M., Hope, J., Pilmer, C.D., Small, M., & Avard-Spinney, A. (2004). Mathematical Modeling, Book 2. Scarborough, Ontario: Nelson Thomson Learning.

New Brunswick Department of Education. (2003). Trigonometry and 3-Space 121/122.

Wempen, F. (2010). PowerPoint® 2010 Bible. Indianapolis, IN: Wiley Publishing Inc.

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