C 23 - Identify periodic relations and describe their characteristics

 What is a Periodic Relationship? What is a Periodic Relationship?  Activity: Identifying Periodic Behaviour Activity: Identifying Periodic Behaviour  What is a Sinusoidal Relationship? What is a Sinusoidal Relationship?  Sound waves Sound waves  Characteristics of Sinusoidal Relationships Characteristics of Sinusoidal Relationships  Activities: Real World Applications Activities: Real World Applications  References 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

A Periodic Relationship Example 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. ( Microsoft Table of Contents

(Barry et al., 2004, p. 94) Table of Contents

 A special type of periodic behaviour  Looks like waves  Any portion of the curve can be translated onto another portion of the curve. Table of Contents

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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. Table of Contents

en.wikipedia.org Table of Contents

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Sinusoidal Axis The horizontal line halfway between the local maximum and local minimum. Table of Contents Return to Vocabulary

Local Maximum/Minimum  Local Maximum – the tops of each “wave”, all equal in value.  Local Minimum – the bottoms of each “wave”, all equal in value. Table of Contents Return to Vocabulary

Amplitude The vertical distance from the sinusoidal axis to a local minimum or a local maximum. Table of Contents Return to Vocabulary

Period The change in the “x” corresponding to one cycle. Video – Laps in the Pool ( Table of Contents Return to Vocabulary

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. ( Microsoft Table of Contents Amplitude (radius of wheel) Sinusoidal Axis (centre of wheel) Period (time for 1 revolution)

More about Ferris Wheels… What is the period of this function? How high above the ground is the Ferris wheel at 10 seconds? What is the diameter of this Ferris wheel? Table of Contents

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 Table of Contents

Characteristics of a sinusoidal relationship Equation of sinusoidal axis Amplitude Period Table of Contents

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. Table of Contents