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Periodic!Functions!
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Activity!1!
The!“London!Eye”!is!the!world’s!largest!Ferris!wheel!which!measures!450!feet!in!diameter,!and!caries!up!to!800!
passengers!in!32!capsules.!!It!turns!continuously,!completing!a!single!rotation!once!every!30!minutes.!!This!is!slow!
enough!for!people!to!hop!on!and!off!while!it!turns.!
Suppose!you!hop!on!this!Ferris!wheel!at!! = 0!and!ride!it!for!two!full!turns.!!The!wheel!is!turning!in!a!counterJclockwise!
direction!and!let!!(!)!be!your!height!above!the!ground,!measured!in!feet!as!a!function!of!t,!the!number!of!minutes!you!
have!been!riding.!
Find!some!values!of!f(t)!and!complete!the!table.!!
t#minutes!
f(t)!feet!
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Graph!the!function!below.!
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The!above!function!is!a!periodic!function.!!In!your!groups!develop!a!definition!of!a!periodic!function.!!List!characteristics!
you!think!the!function!has.!
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Periodic!Function:!
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Activity!2!
Are!the!following!functions!periodic?!!If!the!function!is!periodic!identify!the!amplitude,!period,!and!midline.!!If!the!
function!is!not!periodic,!explain!why.!
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Activity!3!
40!ft.!
5!ft.!
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1. Use!the!diagram!above.!!Suppose!you!and!a!friend!are!the!last!two!people!seated!on!a!Ferris!wheel.!!Once!the!
ride!begins,!the!wheel!moves!at!a!constant!speed.!!It!takes!36!seconds!to!complete!one!revolution.!!You!enter!
the!Ferris!wheel!at!the!3!o’clock!position!and!the!Ferris!wheel!is!heading!upward!when!you!first!get!on.!
a. At!0!seconds,!when!the!ride!starts,!how!high!above!the!ground!are!you?!
b. At!what!height!are!you!at!9!seconds?!!18!seconds?!!27!seconds?!
c. At!what!height!are!you!at!126!seconds?!!How!many!revolutions!have!you!made?!
d. Predict!where!you!will!be!at!3!minutes.!
2. Sketch!a!graph!showing!the!relationship!between!your!height!above!the!ground!and!the!time!since!the!ride!
began.!!Use!0 ≤ ! ≤ 144!for!the!domain,!where!! = 0!is!the!time!at!which!the!ride!starts.!
3. How!far!(in!feet)!have!you!traveled!after!one!revolution!of!the!wheel?!!How!far!have!you!traveled!after!144!
seconds?!
Activity!4!
For!the!graph!below,!describe!the!height,!ℎ = !(!),!above!the!ground!of!the!Ferris!wheel,!where!h!is!in!meters!and!t!is!
in!minutes.!!You!board!the!wheel!at!! = 0.!!Determine!the!diameter!of!the!wheel,!at!what!height!above!the!ground!!you!
board!the!wheel,!how!long!it!takes!to!complete!one!revolution,!and!the!length!of!time!the!graph!shows!you!riding!the!
wheel.!!Use!complete!sentences!and!make!it!a!story.!
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Activity!5!
Many!natural!phenomena!display!periodic!behavior!similar!to!the!upJandJdown!motion!of!a!Ferris!wheel!seat.!
Sketch!a!graph!of!what!the!following!situations!might!look!like.!!Label!what!determines!things!like!amplitude,!midline,!
and!period.!
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BackJandJforth!motion!of!a!pendulum!or!swing!
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UpJandJdown!motion!of!a!weight!bouncing!on!a!spring!
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Hours!of!daylight!each!day!over!the!course!of!a!year!
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Depth!of!the!tide!at!a!particular!location!during!the!course!of!a!day!
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Flickering!of!a!fluorescent!light!
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Vibration!of!a!guitar!or!piano!string!