Transcript Section 3B Putting Numbers in Perspective

Section 9A
Functions: The Building Blocks of
Mathematical Models
Pages 532-539
9-A
Functions (page 533)
A function describes how a dependent
variable (output) changes with
respect to one or more independent
variables (inputs).
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)
9-A
Functions (page 533)
A function describes how a dependent
variable (output) changes with
respect to one or more
independent variables (inputs) .
input (x)
function
DOMAIN
page 536
output (y)
RANGE
page 536
9-A
Representing Functions
There are three basic ways to represent
functions:

Formula

Graph

Data Table
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
formula: P = 10000(1.05)t
The population(P) varies with respect to time(t).
P = f(t)
f(t) = 10000(1.05)t
INPUT: year
OUTPUT: population
year
t
10000(1.05)t
P
f(t)
population
year
0
10000(1.05)0
10000
population
P=f(0) = 10000(1.05)0
P=f(0) = 10000
(0,10000)
year
1
10000(1.05)1
10500
population
P=f(1) = 10000(1.05)1
P=f(1) = 10500
(1,10500)
P=f(3) = 10000(1.05)3
year
3
10000(1.05)3
11576
population
P=f(3) = 11576
(3,11576)
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
formula: P = 10000(1.05)t
t
P=f(t)
P
(t,f(t))
0
f(0) = 10,000 x (1.05)0
10000
(0,10000)
1
f(1) = 10000 x (1.05)
10500
(1,10500)
2
f(2) = 10000 x (1.05)2
11025
(2,11025)
3
f(3) = 10000 x (1.05)3
11576
(3,11576)
10
f(10) = 10000 x (1.05)10
16829
(10,16289)
15
f(15) = 10000 x (1.05)15
20789
(15,20789)
20
f(20) = 10000 x (1.05)20
26533
(20,26533)
40
f(40) = 10000 x (1.05)40
70400
(40,70400)
The population (dependent variable) varies with respect to time (independent variable).
P=f(t)
RANGE: populations of 10000 or more and DOMAIN: nonnegative years
9-A
Representing Functions
There are three basic ways to represent
functions:

Formula

Graph

Data Table
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
table of data
t
P=f(t)
P
(t,f(t))
0
f(0) = 10,000 x (1.05)0
10000
(0,10000)
1
f(1) = 10000 x (1.05)
10500
(1,10500)
2
f(2) = 10000 x (1.05)2
11025
(2,11025)
3
f(3) = 10000 x (1.05)3
11576
(3,11576)
10
f(10) = 10000 x (1.05)10
16829
(10,16289)
15
f(15) = 10000 x (1.05)15
20789
(15,20789)
20
f(20) = 10000 x (1.05)20
26533
(20,26533)
40
f(40) = 10000 x (1.05)40
70400
(40,70400)
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
table of data
t
P=f(t)
(t,f(t))
0
10000
(0,10000)
1
10500
(1,10500)
2
11025
(2,11025)
3
11576
(3,11576)
10
16829
(10,16289)
15
20789
(15,20789)
20
26533
(20,26533)
40
70400
(40,70400)
The population (dependent variable) varies with respect to time (independent variable).
P=f(t)
RANGE: populations of 10000 or more and DOMAIN: nonnegative years
9-A
Representing Functions
There are three basic ways to represent
functions:

Formula

Graph

Data Table
9-A
Graphs
(1, 2) , (-3, 1) , (2, -3) , (-1, -2) , (0, 2) , (0, -1)
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
graph
(t,f(t))
Powertown Population
(0,10000)
(1,10500)
population
(2,11025)
(3,11576)
(10,16289)
(15,20789)
year
(20,26533)
(40,70400)
The population (dependent variable) varies with respect to time (independent variable).
P=f(t)
RANGE: populations of 10000 or more and DOMAIN: nonnegative years
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
graph
(t,f(t))
Powertown Population
(0,10000)
80000
(1,10500)
70000
(2,11025)
population
60000
50000
(3,11576)
40000
30000
(10,16289)
20000
(15,20789)
10000
0
0
5
10
15
20
25
year
30
35
40
45
(20,26533)
(40,70400)
The population (dependent variable) varies with respect to time (independent variable).
P = f(t)
RANGE: populations of 10000 or more and DOMAIN: nonnegative years
EXAMPLE: In Powertown, the initial population is 10,000
and growing at a rate of 5% per year.
graph
Powertown Population
80000
70000
population
60000
50000
40000
30000
20000
10000
0
0
5
10
15
20
25
30
35
40
year
Use the graph to determine the population after 25 years.
Use the graph to determine when the population will be 60,000.
45
9-A
Representing Functions
There are three basic ways to represent
functions:

Formula

Graph

Data Table
EXAMPLE/533 Temperature Data for One Day
9-A
table of data
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
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.
9-A
EXAMPLE/533 Temperature Data for One Day
graph
temperature
Temperature for One Day
time of day
Domain: Time of Day from 6:00 am to 6:00 pm.
Range: Temperatures from 50°
to 73°F.
EXAMPLE/533 Temperature Data for One Day
9-A
graph
(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)
9-A
EXAMPLE/533 Temperature Data for One Day
graph
temperature
Temperature for One Day
hours since 6am
Domain: hours since 6 am.
Range: Temperatures from 50°
to 73°F.
EXAMPLE/533 Temperature Data for One Day
graph
Graph of Temperature vs time
(0, 50°F)
(1, 52°F)
75
(2, 55°F)
Temperature
70
(3, 58°F)
65
(4, 61°F)
60
(5, 65°F)
55
50
(6, 70°F)
45
(7, 73°F)
40
(8, 73°F)
0
2
4
6
8
Hours after 6 A.M.
10
12
(9, 70°F)
(10, 68°F)
(11, 65°F)
Domain: Hours since 6am from 0 to 12.
Range: Temperatures from 50°
to 73°F.
(12, 61°F)
9-A
9-A
EXAMPLE/533 Temperature Data for One Day
graph
Graph of Temperature vs time
75
Temperature
70
65
60
55
50
45
40
0
2
4
6
8
10
Hours after 6 A.M.
OBSERVATION from graph
The temperature rises and then falls between 6am and 6 pm.
12
9-A
(EXAMPLE/537) 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.
9-A
(EXAMPLE/537) 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)
pressure
Pressure vs Altitude
(5000, 25)
(10000, 22)
(20000, 16)
(30000, 10)
altitude
Domain: altitudes from 0 to 30,000 ft.
Range: pressure from 10 to 30 inches of mercury.
9-A
(EXAMPLE2a/538) Pressure Altitude Function
Use the data to create a graph.
(0, 30)
(5000, 25)
(10000, 22)
(20000, 16)
(30000, 10)
Pressure (inches of mercury)
Graph of Pressure vs Altitude
35
30
25
20
15
10
5
0
0
5,000
10,000
15,000
20,000
25,000
Altitude (ft)
OBSERVATION from graph
As altitude increases, atmospheric pressure decreases.
30,000
9-A
(EXAMPLE2a/538) Pressure Altitude Function
Use the graph to predict the pressure at 15,000 feet.
(0, 30)
(5000, 25)
(10000, 22)
(20000, 16)
(30000, 10)
Pressure (inches of mercury)
Graph of Pressure vs Altitude
35
30
25
20
15
10
5
0
0
5,000
10,000
15,000
20,000
25,000
Altitude (ft)
OBSERVATION from graph
As altitude increases, atmospheric pressure decreases.
30,000
9-A
(EXAMPLE2a/538) Pressure Altitude Function
Use the graph to predict when the pressure will be 12 (in. of merc.)
(0, 30)
(5000, 25)
(10000, 22)
(20000, 16)
(30000, 10)
Pressure (inches of mercury)
Graph of Pressure vs Altitude
35
30
25
20
15
10
5
0
0
5,000
10,000
15,000
20,000
25,000
Altitude (ft)
OBSERVATION from graph
As altitude increases, atmospheric pressure decreases.
30,000
9-A
More Practice (25/541)
Year
Tobacco
(billions of lb)
Year
Tobacco
(billions of lb)
(1975, 2.2)
1975
2.2
1986
1.2
(1982, 2.0)
1980
1.8
1987
1.2
(1985, 1.5)
1982
2.0
1988
1.4
1984
1.7
1989
1.4
1985
1.5
1990
1.6
(1980, 1.8)
(1984, 1.7)
(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.
9-A
More Practice (25/541)
(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)
9-A
More Practice (25/541)
(1975, 2.2)
Correct Graph
Tobacco (billions of lb)
(1980, 1.8)
2.5
(1982, 2.0)
2
(1984, 1.7)
1.5
(1985, 1.5)
1
(1986, 1.2)
0.5
(1987, 1.2)
(1988, 1.4)
0
75
85
80
90
Year
Observation from graph
The production of tobacco has slowly decreased from 1975 to
1986 and then slowly increased from 1986 to 1990.
(1989, 1.4)
(1990, 1.6)
9-A
Homework:
Pages 540-542
# 19a-b, 22, 23, 24, 26