1. Trigonometryand Right Angles
BasicFunctions
Learningobjectives
A studentwill be able to:
- Determineif a relationis a function.
- Statethe domainand rangeof a function.
- Categorizea functionaccordingto a functionfamily.
- Identifykey characteristicsof functions,includingthe conceptof a periodicfunction.
Introduction
This chapterwill introduceyou to a particularfamilyof functions,the trigonometricfunctions,whichare the
basisfor this book.In this first lesson,we will reviewthe basiccharacteristicsof functionsin general:what
a functionis, whatthe graphof a functionlookslike, and the characteristicsof severalfamiliesof functions.
Whilethis lessonwill not definetrigonometricfunctions,we will considerone of their basiccharacteristics,
and someimportantapplicationsof thesefunctions.
The basicsof functions
Considertwo situationsshownin the boxesbelow:
Situation1: Your car can travel30 mileson one gallonof gasolineat 55mph.For every
mile per hour fasteryou drive,the car travelshalf a mile less per gallonof gasoline.Situation2: You collectdata fromseveralstudentsin your classon their agesand
their heightsin inches:Age 18 17 18 18 17
Height 65" 64" 67" 68" 66"In the first situation,let the variablexrepresentthe speedof your car, and letyrepresentthe numberof
milesit can travelusingone gallonof gasoline.If you travelatxmilesper hour, you will goy= 30 - .5(x-
55) mileson one gallonof gasoline.For example,if you travelat 60 mph,you will travel30 - .5(60- 55) =
27.5 mileson one gallonof gasoline.Noticethat you can use your speedto “predict”how far you can travel
with one gallonof gasoline.
Now considerthe secondsituation.Can you use the data to “predict”height,giventhe age of a student?
This is not the casein the secondsituation.For example,if a studentis 18 yearsold, thereare several
heightsthat the studentcouldbe.
Bothsituationsare relations.Arelationis simplya relationshipbetweentwo sets of numbersor data.For
example,in the secondsituation,we createda relationshipbetweenstudents’agesand heights,just by
writingeachstudent’s informationas an orderedpair. In the first situation,thereis a relationshipbetween