May 50.2oF 66.7oFJun 59.4oF 76.6oF82.
o
65.5 F
o
Jul FAug 64.5oF 80.1oFSept 56.8oF 72.5oFOct 46.4oF 61.8oFNov 37.9oF 51.8oF41.
o
27.8 F
o
Dec FThe graphbelowshowsthe averagelow temperatures.
Noticethat the graphincludesa full year of data,and then endswith December, the 12thmonth.It is possible
that the curvesuggestedby this graphcan be approximatedby a functionin one of the familiesof functions
we’vediscussed.Not all naturalphenomenacan be modeledwith mathematicalfunctions,but manycan.
Supposethis data was representativeof Bostonweatherin general.We couldmakea functionwhoseinput
is the time in monthsfrom the presentand whoseoutputis the averagetemperatureexpected.For example
f(1)= 22.1,f(5)=50.2. The functionwill repeatafter one year. Whatdoes13 represent?Whatis the temper-
atureexpectedto be basedon this data?
Becausethe monthscycleeachyear, 13 representsJanuaryof the next year, and in general,we can predict
the weatherin Januarygivenour knowledgeof the usualclimatein a location.For example,Januaryis the
coldestmonthof the year in the city of Boston.For the yearsshownin the table,the averagelow temperature
was about22 degrees.We can thereforepredictthat the averagelow temperaturein Januaryin Bostonwill
be about22 degrees.We coulduse sucha functionto comparecurrentweatherto past weatherand test
for climatechangesover time.
Becausethe monthsof the year and the weatherpatternsare cyclicalin nature,we needto modelthis situ-
ationwith a functionthat is also cyclicalin nature.Suchfunctionsare referredto asperiodic.A functionis
periodicif thereexistssomevaluepsuchthatf(x+p) =f(x)for allxin the domainof the function.The
trigonometricfunctionsyou will learnaboutin this chapterare one type of periodicfunction,and we can use
certaintrigonometricfunctionsto modelthe weatherdata shownabove.We will returnto this topicat the