SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.2 Imagineering

A Develop Understanding Task

Youareexcitedtogettovoteontheplansforaproposednewthrillrideatalocaltheme

park.Theengineerswantpublicinputonthedesignforthenewride.Youareoneoften

teenagerswhohavebeenselectedtoreviewtheplansbasedonyourgoodmathgrades!

Asyourexcitementmounts,theengineersbegintheirpresentation.Toyourdismay,there

arenomodelsorillustrationsoftheproposedrides—eachrideisdescribedonlywithequations.

Theequationsrepresentthepathariderwouldfollowthroughthecourseoftheride.

Unfortunately,yourcellphone—whichcontainsagraphingcalculatorapp—iscompletely

dischargedduetotoomuchtextingandsurfingtheinternet.So,youaretryinghardtokeepup

withthepresentationbytryingtoimaginewhatthegraphsofeachoftheseequationswouldlook

like.Whileeachequationconsistsoffunctionsyouarefamiliarwith,thecombinationoffunctions

ineachequationhasyouwonderingabouttheircombinedeffects.

Foreachofthefollowingproposedthrillrides,useyourimaginationandbestreasoning

abouttheindividualfunctionsinvolvedtosketchagraphofthepathoftherider.Letyrepresent

theheightoftheriderabovethegroundandxrepresentthedistancefromthestartoftheride.

Explainyourreasoningabouttheshapeofthegraph.(Note:Useradiansfortrigonometric

functions.)

Proposal#1:“TheMountainClimb”

TheEquation:! = 2$ + 5sin($)MyGraph: MyExplanation:

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9

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

Proposal#2:“ThePeriodicBump”

TheEquation:! = |10 sin($)|MyGraph: MyExplanation:

Proposal#3:“TheAmplifier”

TheEquation:! = $ ∙ sin($)MyGraph: MyExplanation:

Proposal#4:“TheGentleWave”

TheEquation:! = 10(0.9)3 ∙ sin($)MyGraph: MyExplanation:

10

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

Proposal#5:“TheSpinningHighDive”

TheEquation:! = 100 − $5 + 5 sin(4$)MyGraph: MyExplanation:

Whenyougothomeyourfriendswereallanxiouslywaitingtohearabouttheproposed

newrides.Afterexplainingthesituation,yourfriendsallpullouttheircalculatorsandtheybegan

comparingyourimaginedimageswiththeactualgraphs.

Someofyourfriends’graphsdifferedfromtheothersbecauseoftheirwindowsettings.

Somewindowsettingsrevealedthefeaturesofthegraphsyouwereexpectingtosee,whileother

windowsettingsobscuredthosefeatures.

Examinetheactualgraphsofeachofthethrillrideproposals.Selectawindowsettingthat

willrevealasmanyofthefeaturesofthegraphsaspossible.Explainanydifferencesbetweenyour

imaginedgraphsandtheactualgraphs.Whatfeaturesdidyougetright?Whatfeaturesdidyou

miss?

Proposal#1:“TheMountainClimb”

TheEquation:! = 2$ + 5sin($)ActualGraph: WhatfeaturesIgotrightandwhatImissed:

11

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

Proposal#2:“ThePeriodicBump”

TheEquation:! = |10 sin($)|ActualGraph: WhatfeaturesIgotrightandwhatImissed:

Proposal#3:“TheAmplifier”

TheEquation:! = $ ∙ sin($)ActualGraph: WhatfeaturesIgotrightandwhatImissed:

Proposal#4:“TheGentleWave”

TheEquation:! = 10(0.9)3 ∙ sin($)ActualGraph: WhatfeaturesIgotrightandwhatImissed:

12

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

Proposal#5:“TheSpinningHighDive”

TheEquation:! = 100 − $5 + 5 sin(4$)ActualGraph: WhatfeaturesIgotrightandwhatImissed:

Youandyourfriendsdecidetoproposeadifferentridetotheengineers.Nameyour

proposalandwriteitsequation.Explainwhyyouthinkthefeaturesofthisgraphwouldmakea

funride.

MyProposedRide:

Theequationformyride: Myexplanationofmyproposal:

13

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.2

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READY Topic:Exploringtheboundariesoffunctions

1.Thecurveinthegraphattherightshowsthegraphof!(#) = &'((#).(blue)a.Writetheequationofthedottedlinelabeled)(#).(red)b.Writetheequationofthedottedlinelabeledℎ(#).(green)c.Listeverythingyounoticeaboutthegraphsof!(#), )(#), ,(-ℎ(#).

2.Thecurveinthegraphattherightshowsthegraphof

!(#) = &'((#).a.Writetheequationofthelinelabeled0(#).(red)b.Sketchinthegraphof!(#) ∗ 0(#).c.Whatistheequationof!(#) ∗ 0(#)?d.Wouldtheliney=-3alsobeaboundarylineforyoursketch?Explain.

3.Thecurveinthegraphattherightshowsthegraphof!(#) = &'((#).(blue)

a.Writetheequationofthelinelabeled3(#).(green)b.Sketchinthegraphof!(#) ∗ 3(#).c.Whatistheequationof!(#) ∗ 3(#)?d.Howisthegraphof!(#) ∗ 3(#)differentfromthegraphof!(#) ∗ 0(#)?

e.Wouldtheliney=3alsobeaboundarylineforyoursketch?Explain.

READY, SET, GO! Name PeriodDate

3(#)

h(x)

g(x)

m(x)

f(x)

f(x)

14

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.2

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SET Topic:Combiningfunctions

4.!(#) = #)(#) = sin(#) ℎ(#) = !(#) + )(#)Fillinthevaluesforh(x)inthetable.Thengraphℎ(#) = # + &'((#)withasmoothcurve.

x f(x) g(x) h(x)

-2π -6.28 0

−3:2 -4.71 1

-π -3.14 0

−:2 -1.57 -1

0 0 0 :2 1.57 1

π 3.14 0 3:2

4.71 -1

2π 6.28 0

5.!(#) = #)(#) = sin(#)

Nowgraph<(#) = !(#) ∗ )(#)or<(#) = # ∗ &'((#)

x f(x) g(x) k(x)

-2π -6.28 0

−3:2 -4.71 1

-π -3.14 0

−:2 -1.57 -1

0 0 0 :2 1.57 1

π 3.14 0 3:2 4.71 -1

2π 6.28 0

15

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.2

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Matcheachequationbelowwiththeappropriategraph.Describethefeaturesofthegraphthathelpedyoumatchtheequations.

6.!(#) = |#> − 4| 7.)(#) = −# + 5sin(#) 8.ℎ(#) = 4|&'(#|keyfeatures: keyfeatures: keyfeatures:

9.-(#) = 10 − #> + 5&'((#) 10.C(#) = −# ∗ 2&'((#) 11.D(#) = (2# − 4) + |#|keyfeatures: keyfeatures: keyfeatures:

a)

b)

c)

d)

e) f)

16

SECONDARY MATH III // MODULE 8

MODELING WITH FUNCTIONS – 8.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

8.2

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GO Topic:Identifyingkeyfeaturesoffunctionfamilies

Thechartbelownamesfivefamiliesoffunctionsandtheparentfunction.Theparentistheequationinitssimplestform.Intherighthandcolumnisalistofkeyfeaturesofthefunctionsinrandomorder.Matcheachkeyfeaturewiththecorrectfunction.Akeyfeaturemayrelatetomorethanonefunction.

Family Parent(s) Keyfeatures12.Linear

E = #

a)Theendsofthegraphhavethesamebehavior.b)Thegraphshaveahorizontalasymptoteandaverticalasymptote.c)Thegraphonlyhasahorizontalasymptote.d)Thesefunctionseitherhavebothalocalmaximumandminimum

orneitheralocalmaximumandminimum.e)Thegraphisusuallydefinedintermsofitsslopeandy-intercept.f)Thegraphhaseitheramaximumoraminimumbutnotboth.g)Asxapproaches-∞,thefunctionvaluesapproachthex-axis.h)Theendsofthegraphhaveoppositebehavior.i)Therateofchangeofthisgraphisconstant.j)Therateofchangeofthisgraphisconstantlychanging.k)Thisgraphhasalinearrateofchange.l)Thesefunctionsareofdegree3.m)Thevariableisanexponent.n)Thesefunctionscontainfractionswithapolynomialinboththe

numeratoranddenominator.p)Theconstantwillalwaysbethey-intercept.

13.Quadratic

E = #>

14.Cubic

E = #F

15.Exponential

E = 2G E = 3G

etc.

16.Rational

E = 1#

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