Section 9ASection 9AFunctions: The Building Functions: The Building Blocks of Mathematical Blocks of Mathematical

ModelsModels

Pages 532-539Pages 532-539

FunctionsFunctionsA function describes how a A function describes how a dependent dependent

variable (output) variable (output) changes changes with respect with respect toto one or more one or more independent variables independent variables (inputs)(inputs)..

When there are only When there are only twotwo variables, we often variables, we often summarize them as an summarize them as an ordered pairordered pair with the with the independent variable first: independent variable first:

((independent variableindependent variable, , dependent variabledependent variable) ) ((inputinput, , outputoutput))

((xx, , yy) )

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FunctionsFunctionsA function describes how a A function describes how a dependent dependent

variable (output) variable (output) changes with respect changes with respect toto one or more one or more independent variables (inputs)independent variables (inputs) ..

((timetime, , temperaturetemperature) ) ((altitudealtitude, , pressurepressure))((growth rategrowth rate, , populationpopulation))

((interest rateinterest rate, , monthly mortgage monthly mortgage paymentpayment) ) ((relative energyrelative energy, , magnitude (of magnitude (of earthquake)earthquake)))

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FunctionsFunctions

We say that the We say that the dependent variable is a function of the independent variable. .

If If xx is the is the independent variableindependent variable and and yy is the is the dependent variabledependent variable, we write , we write the function asthe function as

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( ).y f x

( )T f t ( )P f A ( )PMT f APR

( )E f M

Representing FunctionsRepresenting Functions

There are There are three basic ways to three basic ways to represent functionsrepresent functions..

DataData TableTable or List or List

Draw a Draw a picturepicture or or graphgraph

Write an Write an equationequation

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Coordinate PlaneCoordinate Plane9-A

Coordinate PlaneCoordinate Plane Draw 2 Draw 2

perpendicular lines perpendicular lines ((xx-axis, -axis, y-y-axisaxis))

Numbers on the Numbers on the lines increase lines increase upup and and to the right.to the right.

The intersection of The intersection of these lines is the these lines is the origin (0,0)origin (0,0)

Points are described by 2 coordinates (x,y)

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Temperature Data for One Temperature Data for One Day Day

Time Temp Time Temp

6:00 am6:00 am 50°F50°F 1:00 pm1:00 pm 73°F73°F

7:00 am7:00 am 52°F52°F 2:00 pm2:00 pm 73°F73°F

8:00 am8:00 am 55°F55°F 3:00 pm3:00 pm 70°F70°F

9:00 am9:00 am 58°F58°F 4:00 pm4:00 pm 68°F68°F

10:00 am10:00 am 61°F61°F 5:00 pm5:00 pm 65°F65°F

11:00 am11:00 am 65°F65°F 6:00 pm6:00 pm 61°F61°F

12:00 pm12:00 pm 70°F70°F

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Domain and RangeDomain and RangeThe The domaindomain of a function is the set of of a function is the set of

values values

that both make sense and are of that both make sense and are of interest for the interest for the

input (independent)input (independent) variable. variable.The The rangerange of a function consists of the of a function consists of the

values values

of the of the output (dependent)output (dependent) variable that variable that correspond to the values in the domain.correspond to the values in the domain.

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Temperature Data for One Temperature Data for One Day Day

Time Temp Time Temp

6:00 am6:00 am 50°F50°F 1:00 pm1:00 pm 73°F73°F

7:00 am7:00 am 52°F52°F 2:00 pm2:00 pm 73°F73°F

8:00 am8:00 am 55°F55°F 3:00 pm3:00 pm 70°F70°F

9:00 am9:00 am 58°F58°F 4:00 pm4:00 pm 68°F68°F

10:00 am10:00 am 61°F61°F 5:00 pm5:00 pm 65°F65°F

11:00 am11:00 am 65°F65°F 6:00 pm6:00 pm 61°F61°F

12:00 pm12:00 pm 70°F70°F

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Domain and RangeDomain and Range

The The domaindomain is the hours from is the hours from 6 6 am to 6 pmam to 6 pm..

The The rangerange is temperatures from is temperatures from 50-73°F.50-73°F.

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Temperature as a Function of TimeTemperature as a Function of TimeT = f(t)T = f(t)

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Temperature as a Function of TimeTemperature as a Function of TimeT = f(t)T = f(t)

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Time

Tem

pera

ture

(degre

es

F)

6:00 PM4:00 PM2:00 PM12:00 PM10:00 AM8:00 AM6:00 AM

75

70

65

60

55

50

Plot of Temperature vs Time

Temperature as a Function of TimeTemperature as a Function of TimeT = f(t)T = f(t)

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Time

Tem

pera

ture

(degre

es

F)

6:00 PM4:00 PM2:00 PM12:00 PM10:00 AM8:00 AM6:00 AM

80

70

60

50

40

30

20

10

0

Plot of Temperature vs Time

Temperature as a Function of TimeTemperature as a Function of Time T = f(t)T = f(t)

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Graph of Temperature vs time

01020304050607080

0 2 4 6 8 10 12

Hours after 6 A.M.

Tem

per

atu

re

Temperature as a Function of TimeTemperature as a Function of TimeT = f(t)T = f(t)

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Graph of Temperature vs time

01020304050607080

0 2 4 6 8 10 12

Hours after 6 A.M.

Tem

per

atu

re

Pressure as a Function of AltitudePressure as a Function of Altitude P = f(A)P = f(A)

Altitude Pressure (inches of mercury)

0 ft0 ft 3030

5,000 ft5,000 ft 2525

10,000 ft10,000 ft 2222

20,000 ft20,000 ft 1616

30,000 ft30,000 ft 1010

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Pressure as a Function of AltitudePressure as a Function of AltitudeP = f(A)P = f(A)

The The independent variableindependent variable is is altitudealtitude..

The The dependent variabledependent variable is is atmospheric pressure.atmospheric pressure.

The The domaindomain is is 0-30,000 ft0-30,000 ft..The The rangerange is is 10-30 inches of 10-30 inches of

mercury.mercury.

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Pressure as a Function of AltitudePressure as a Function of Altitude P = f(A)P = f(A)

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Graph of Pressure vs Altitude

0

5

10

15

20

25

30

35

0 5,000 10,000 15,000 20,000 25,000 30,000

Altitude (ft)

Pre

ss

ure

(in

ch

es

of

me

rcu

ry)

Making predictions from a Making predictions from a graphgraph

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Pressure as a function of AltitudePressure as a function of Altitude P = f(A)P = f(A)

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Hours of Daylight as a Function of Hours of Daylight as a Function of Day of the YearDay of the Year (40°N latitude) (40°N latitude)

Hours of Daylight Date Day of year

1414

(greatest)(greatest)June 21June 21stst

(Summer (Summer Solstice)Solstice)

172172

1010

(least)(least)December 21December 21stst

(Winter Solstice)(Winter Solstice)355355

12 12 March 21March 21stst

(Spring Equinox)(Spring Equinox)8080

1212 September 21September 21stst

(Fall Equinox) (Fall Equinox) 264264

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Hours of daylight as a function of day of Hours of daylight as a function of day of the yearthe year ( h = f(d) ) ( h = f(d) )

The The independentindependent variablevariable is is day day of the year.of the year.

The The dependentdependent variablevariable is is hours hours of daylight.of daylight.

The The domaindomain is is 0-365 days0-365 days..The The rangerange is is 10-14 hours of 10-14 hours of

daylight.daylight.

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Hours of daylight as a functionHours of daylight as a function of day of the year of day of the year h = f(d) h = f(d)

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Latitude 40 Degrees North

0

2

4

6

8

10

12

14

16

0 100 200 300 400

Day of Year

Ho

urs

of

Da

yli

gh

t

Hours of daylight as a function of day of Hours of daylight as a function of day of the yearthe year h = f(d) h = f(d)

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Latitude 40 degrees North

0

2

4

6

8

10

12

14

16

0 100 200 300 400

Day of Year

Ho

urs

of

Da

yli

gh

t

Hours of daylight as a function of day of Hours of daylight as a function of day of the yearthe year h = f(d) h = f(d)

Latitude 40 degrees North

0

2

4

6

8

10

12

14

16

0 500 1000 1500

Day of Year(s)

Ho

urs

of

da

yli

gh

t

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Hours of daylight as a function of day of Hours of daylight as a function of day of the yearthe year h = f(d) h = f(d)

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Latitude 40 degrees North

9

10

11

12

13

14

15

-100 100 300 500 700 900 1100

Day of Year(s)

Hour

s of

day

light

Hours of daylight as a function of day Hours of daylight as a function of day of the yearof the year h = f(d) h = f(d)

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Watch for Deceptions: # 25Watch for Deceptions: # 25

Year Tobacco (billions

of lb)

Year Tobacco (billions of

lb)

19751975 2.22.2 19819866

1.21.2

19801980 1.81.8 19819877

1.21.2

19821982 2.02.0 19819888

1.41.4

19841984 1.71.7 19819899

1.41.4

19851985 1.51.5 19919900

1.61.6

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Watch for Deceptions:Watch for Deceptions:

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Watch for Deceptions:Watch for Deceptions:

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Correct Graph

0

0.5

1

1.5

2

2.5

75 77 79 81 83 85 87 89

Year

To

bac

co (

bil

lio

ns

of

lb)

Homework for Wednesday:Homework for Wednesday:

Pages 540-542Pages 540-542# 19a-b, 20a-c, 22, 24, 26# 19a-b, 20a-c, 22, 24, 26

You may use Excel to graph the You may use Excel to graph the functions in # 24, 26.functions in # 24, 26.

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