Section 9AFunctions: The Building Blocks of Mathematical Models

Pages 560-570

A function describes how a dependent variable (output) changes with respect to one or more independent variables (inputs).

If x is the independent variable and y is the dependent variable, we write

y = f(x)

We summarize the input/output pair as an ordered pair with the independent variable always listed first: (independent variable, dependent variable)

(input, output) (x, y)

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Functions (page 561)

Functions (page 561)

A function describes how a dependent variable (output) changes with respect to one or more independent variables (inputs) .

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input (x)

output (y)

function

RANGEpage 563

DOMAINpage 563

Representing Functions

There are three basic ways to represent functions:

Formula

Graph

Data Table

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Time Temp Time Temp

6:00 am 50°F 1:00 pm 73°F

7:00 am 52°F 2:00 pm 73°F

8:00 am 55°F 3:00 pm 70°F

9:00 am 58°F 4:00 pm 68°F

10:00 am 61°F 5:00 pm 65°F

11:00 am 65°F 6:00 pm 61°F

12:00 pm 70°F

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The temperature(dependent variable) varies with respect to time(independent variable).T = f(t)

EXAMPLE/560 Temperature Data for One Day

table of data

RANGE: temperatures from 50 to 73 and DOMAIN: time of day from 6am to 6pm.

Graphs

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(1, 2) , (-3, 1) , (2, -3) , (-1, -2) , (0, 2) , (0, -1)

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Domain: Time of Day from 6:00 am to 6:00 pm.

Range: Temperatures from 50° to 73°F.

EXAMPLE/560 Temperature Data for One Day

graph

Temperature for One Day

time of day

tem

per

atu

re

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(6:00 am,

50°F)(7:00 am,

52°F)(8:00 am,

55°F)(9:00 am,

58°F)(10:00 am,

61°F)(11:00 am,

65°F)(12:00 pm,

70°F)(1:00 pm,

73°F)(2:00 pm,

73°F)(3:00 pm,

70°F) (4:00 pm, 68°F) (5:00 pm,

65°F) (6:00 pm, 61°F)

EXAMPLE/564 Temperature Data for One Day

graph

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Domain: hours since 6 am.

Range: Temperatures from 50° to 73°F.

EXAMPLE/533 Temperature Data for One Day

graph

Temperature for One Day

hours since 6am

tem

per

atu

re

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Graph of Temperature vs time

40

45

50

55

60

65

70

75

0 2 4 6 8 10 12

Hours after 6 A.M.

Tem

per

atu

re

EXAMPLE/533 Temperature Data for One Day

graph

(0, 50°F)(1, 52°F)(2, 55°F)(3, 58°F)(4, 61°F)(5, 65°F)(6, 70°F)(7, 73°F)(8, 73°F)(9, 70°F) (10, 68°F)(11, 65°F) (12, 61°F)

Domain: Hours since 6am from 0 to 12.

Range: Temperatures from 50° to 73°F.

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Graph of Temperature vs time

40

45

50

55

60

65

70

75

0 2 4 6 8 10 12

Hours after 6 A.M.

Tem

per

atu

re

OBSERVATION from graphThe temperature rises and then falls between 6am and 6 pm.

EXAMPLE/533 Temperature Data for One Day

graph

Altitude Pressure (inches of mercury)

0 ft 30

5,000 ft 25

10,000 ft 22

20,000 ft 16

30,000 ft 10

9-A(EXAMPLE/565) Pressure Altitude Function - (EXAMPLE/565) Pressure Altitude Function - Suppose you measure the atmospheric pressure Suppose you measure the atmospheric pressure as you rise upward in a hot air balloon. as you rise upward in a hot air balloon. Consider the data given below.Consider the data given below.

The atmospheric pressure (dep. variable) varies with respect to altitude (indep. variable).P = f(A)

RANGE: pressures from 10 to 30 and DOMAIN: altitudes from 0 to 30000 ft.

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Domain: altitudes from 0 to 30,000 ft.

Range: pressure from 10 to 30 inches of mercury.

Pressure vs Altitude

altitude

pre

ssu

re

(EXAMPLE2/565) Pressure Altitude Function - (EXAMPLE2/565) Pressure Altitude Function - Suppose you measure the atmospheric pressure Suppose you measure the atmospheric pressure as you rise upward in a hot air balloon. Use the as you rise upward in a hot air balloon. Use the data to create a graph.data to create a graph.

(0, 30)(5000,

25)(10000,

22)(20000,

16)(30000,

10)

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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)

(0, 30)(5000,

25)(10000,

22)(20000,

16)(30000,

10)

(EXAMPLE2/565) Pressure Altitude Function(EXAMPLE2/565) Pressure Altitude FunctionUse the data to create a graph.Use the data to create a graph.

OBSERVATION from graphAs altitude increases, atmospheric pressure decreases.

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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)

(0, 30)(5000,

25)(10000,

22)(20000,

16)(30000,

10) OBSERVATION from graphAs altitude increases, atmospheric pressure decreases.

(EXAMPLE2/566) Pressure Altitude Function(EXAMPLE2/566) Pressure Altitude FunctionUse the graph to predict the pressure at 15,000 Use the graph to predict the pressure at 15,000

feet.feet.

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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)

(0, 30)(5000,

25)(10000,

22)(20000,

16)(30000,

10) OBSERVATION from graphAs altitude increases, atmospheric pressure decreases.

(EXAMPLE2/566) Pressure Altitude Function(EXAMPLE2/566) Pressure Altitude FunctionUse the graph to predict when the pressure will be Use the graph to predict when the pressure will be

12 (in. of merc.)12 (in. of merc.)

More Practice More Practice

21/568 (volume of gas tank, cost to fill 21/568 (volume of gas tank, cost to fill the tank) the tank)

The cost to fill a gas tank varies with the volume of gas that the tank holds.

C = f(v)The cost increases as the volume of the

tank increases.

23/568, 25*/568 23/568, 25*/568

Year Tobacco (billions of lb)

Year Tobacco (billions of lb)

1975 2.2 1986 1.2

1980 1.8 1987 1.2

1982 2.0 1988 1.4

1984 1.7 1989 1.4

1985 1.5 1990 1.6

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More Practice (35/570)More Practice (35/570)

(1975,

2.2)(1980,

1.8)(1982,

2.0)(1984,

1.7)(1985,

1.5)(1986,

1.2)(1987,

1.2)(1988,

1.4)(1989,

1.4)(1990,

1.6)

Annual tobacco production (dep. variable) varies with respect to year (indep. variable).

RANGE: annual tobacco production from 1.2 to 2.2 billions of lbs

DOMAIN: years from 1975 to 1990.

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Correct Graph

0

0.5

1

1.5

2

2.5

75 80 85 90

Year

To

bac

co (

bil

lio

ns

of

lb)

More Practice (35/570)More Practice (35/570)

(1975,

2.2)(1980,

1.8)(1982,

2.0)(1984,

1.7)(1985,

1.5)(1986,

1.2)(1987,

1.2)(1988,

1.4)(1989,

1.4)(1990,

1.6)

Observation from graphThe production of tobacco has slowly decreased from 1975 to

1986 and then slowly increased from 1986 to 1990.

Homework:

Page 568-569# 22,24,26,28,32,34,36

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