c 23 - identify periodic relations and describe their characteristics
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C 23 - Identify periodic relations and describe their characteristics
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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
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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
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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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y
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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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y
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The vertical distance
from the sinusoidal axis
to a local minimum
or a local maximum.
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y
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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
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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)
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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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