Section 9A Functions: The Building Blocks of Mathematical Models

1
Section 9AFunctions The Building Blocks of
Mathematical Models

• Pages 560-570

2
Functions (page 561)
9-A

• 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)

3
Functions (page 561)
9-A

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

input (x)
function
output (y)
DOMAINpage 563
RANGEpage 563
4
Representing Functions
9-A

• There are three basic ways to represent
  functions
• Formula
• Graph
• Data Table

5
9-A
EXAMPLE/560 Temperature Data for One Day table
of data
Time Temp Time Temp
600 am 50F 100 pm 73F
700 am 52F 200 pm 73F
800 am 55F 300 pm 70F
900 am 58F 400 pm 68F
1000 am 61F 500 pm 65F
1100 am 65F 600 pm 61F
1200 pm 70F
The temperature(dependent variable) varies with
respect to time(independent variable).T f(t)
RANGE temperatures from 50 to 73 and DOMAIN
time of day from 6am to 6pm.
6
Graphs
9-A
(1, 2) , (-3, 1) , (2, -3) , (-1, -2) , (0, 2) ,
(0, -1)
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9-A
EXAMPLE/560 Temperature Data for One Day graph

Domain Time of Day from 600 am to 600
pm. Range Temperatures from 50 to 73F.
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9-A
EXAMPLE/564 Temperature Data for One Day graph
(600 am, 50F)(700 am, 52F)(800 am,
55F)(900 am, 58F)(1000 am, 61F)(1100 am,
65F)(1200 pm, 70F)(100 pm, 73F)(200 pm,
73F)(300 pm, 70F) (400 pm, 68F) (500 pm,
65F) (600 pm, 61F)

9
9-A
EXAMPLE/533 Temperature Data for One Day graph

Domain hours since 6 am. Range Temperatures
from 50 to 73F.
10
9-A
EXAMPLE/533 Temperature Data for One Day graph
(0, 50F)(1, 52F)(2, 55F)(3, 58F)(4,
61F)(5, 65F)(6, 70F)(7, 73F)(8, 73F)(9,
70F) (10, 68F)(11, 65F) (12, 61F)

Domain Hours since 6am from 0 to 12. Range
Temperatures from 50 to 73F.
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9-A
EXAMPLE/533 Temperature Data for One Day graph

OBSERVATION from graphThe temperature rises and
then falls between 6am and 6 pm.
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9-A
(EXAMPLE/565) Pressure Altitude Function -
Suppose you measure the atmospheric pressure as
you rise upward in a hot air balloon. Consider
the data given below.
Altitude Pressure (inches of mercury)
0 ft 30
5,000 ft 25
10,000 ft 22
20,000 ft 16
30,000 ft 10
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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9-A
(EXAMPLE2/565) Pressure Altitude Function -
Suppose you measure the atmospheric pressure as
you rise upward in a hot air balloon. Use the
data to create a graph.

(0, 30)(5000, 25)(10000, 22)(20000,
16)(30000, 10)
Domain altitudes from 0 to 30,000 ft. Range
pressure from 10 to 30 inches of mercury.
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9-A
(EXAMPLE2/565) Pressure Altitude FunctionUse the
data to create a graph.

(0, 30)(5000, 25)(10000, 22)(20000,
16)(30000, 10)
OBSERVATION from graphAs altitude increases,
atmospheric pressure decreases.
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9-A
(EXAMPLE2/566) Pressure Altitude FunctionUse the
graph to predict the pressure at 15,000 feet.

(0, 30)(5000, 25)(10000, 22)(20000,
16)(30000, 10)
OBSERVATION from graphAs altitude increases,
atmospheric pressure decreases.
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9-A
(EXAMPLE2/566) Pressure Altitude FunctionUse the
graph to predict when the pressure will be 12
(in. of merc.)

(0, 30)(5000, 25)(10000, 22)(20000,
16)(30000, 10)
OBSERVATION from graphAs altitude increases,
atmospheric pressure decreases.
17
More Practice
21/568 (volume of gas tank, cost to fill 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
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9-A
More Practice (35/570)
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
(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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9-A
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.
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9-A

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