Section 9A Functions: The Building Blocks of Mathematical Models

1
Section 9AFunctions The Building Blocks of
Mathematical Models

• Pages 532-539

2
Functions (page 533)
9-A

• 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)

3
Functions (page 533)
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 536
RANGEpage 536
4
Functions
9-A

• A function describes how a dependent variable
  (output) changes with respect to one or more
  independent variables (inputs) .
• (time, temperature)
• (altitude, pressure)
• (growth rate, population)
• (interest rate, monthly mortgage payment)
• (relative energy, magnitude (of earthquake))

5
Functions
9-A

• We say that the dependent variable is a function
  of the independent variable. If x is the
  independent variable and y is the dependent
  variable, we write the function as

6
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. formula P 10,000(1.05)t
The population(P) varies with respect to
time(t). P f(t) f(t) 10,000(1.05)t INPUT
year (time) OUTPUT population
year
P
10,000(1.05)tf(t)
t
population
7
Pf(0) 10,000(1.05)0 Pf(0) 10,000 (0,10000)
0
Pf(1) 10,000(1.05)1 Pf(1) 10,500 (1,10500)
1
Pf(3) 10,000(1.05)3 Pf(3) 11,576 (3,11576)
year
year
11576
11576
10000(1.05)3
3
10000(1.05)3
3
population
population
8
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. formula P 10,000(1.05)t
t Pf(t) P (t,f(t))
0 f(0) 10,000 x (1.05)0 10000 (0,10000)
1 f(1) 10,000 x (1.05) 10500 (1,10500)
2 f(2) 10,000 x (1.05)2 11025 (2,11025)
3 f(3) 10,000 x (1.05)3 11576 (3,11576)
10 f(10) 10,000 x (1.05)10 16829 (10,16289)
15 f(15) 10,000 x (1.05)15 20789 (15,20789)
20 f(20) 10,000 x (1.05)20 26533 (20,26533)
40 f(40) 10,000 x (1.05)40 70400 (40,70400)
The population (dependent variable) varies with
respect to time (independent variable).
9
Representing Functions
9-A

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

10
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. table of data
t Pf(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)
11
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. table of data
t Pf(t) (t,f(t))
0 10,000 (0,10000)
1 10,500 (1,10500)
2 11,025 (2,11025)
3 11,576 (3,11576)
10 16,829 (10,16289)
15 20,789 (15,20789)
20 26,533 (20,26533)
40 70,400 (40,70400)
The population (dependent variable) varies with
respect to time (independent variable).
12
Domain and Range
9-A

• The domain of a function is the set of values
  that both make sense and are of interest for the
  input (independent) variable.
• The range of a function consists of the values of
  the output (dependent) variable that correspond
  to the values in the domain.

13
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. table of data
t Pf(t) (t,f(t))
0 10,000 (0,10000)
1 10,500 (1,10500)
2 11,025 (2,11025)
3 11,576 (3,11576)
10 16,829 (10,16289)
15 20,789 (15,20789)
20 26,533 (20,26533)
40 70,400 (40,70400)
The population (dependent variable) varies with
respect to time (independent variable).
RANGE populations of 10,000 or more and
DOMAIN nonnegative years
14
Representing Functions
9-A

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

15
Coordinate Plane
9-A
16
Coordinate Plane
9-A

• Draw 2 perpendicular lines (x-axis, y-axis)
• Numbers on the lines increase up and to the
  right.
• The intersection of these lines is the origin
  (0,0)
• Points are described by 2 coordinates (x,y)

(1, 2) , (-3, 1) , (2, -3) , (-1, -2) , (0, 2) ,
(0, -1)
17
Temperature Data for One Day
9-A
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
18
Domain and Range
9-A

• The domain is the hours from 6 am to 6 pm.
• The range is temperatures from 50-73F.

19
Temperature as a Function of TimeT f(t)
9-A

20
Temperature as a Function of TimeT f(t)
9-A

21
Temperature as a Function of TimeT f(t)
9-A

22
Temperature as a Function of Time T f(t)
9-A

23
Temperature as a Function of TimeT f(t)
9-A

24
Pressure as a Function of Altitude P f(A)
9-A
Altitude Pressure (inches of mercury)
0 ft 30
5,000 ft 25
10,000 ft 22
20,000 ft 16
30,000 ft 10
25
Pressure as a Function of AltitudeP f(A)
9-A

• The independent variable is altitude.
• The dependent variable is atmospheric pressure.
• The domain is 0-30,000 ft.
• The range is 10-30 inches of mercury.

26
Pressure as a Function of Altitude P f(A)
9-A

27
Making predictions from a graph
9-A

28
Pressure as a function of Altitude P f(A)
9-A

29
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. graph
(t,f(t))
(0,10000)
(1,10500)
(2,11025)
(3,11576)
(10,16289)
(15,20789)
(20,26533)
(40,70400)
The population (dependent variable) varies with
respect to time (independent variable).Pf(t)
RANGE populations of 10,000 or more and
DOMAIN nonnegative years
30
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. graph
(t,f(t))
(0,10000)
(1,10500)
(2,11025)
(3,11576)
(10,16289)
(15,20789)
(20,26533)
(40,70400)
The population (dependent variable) varies with
respect to time (independent variable).P f(t)
RANGE populations of 10,000 or more and
DOMAIN nonnegative years
31
EXAMPLE In Powertown, the initial population is
10,000 and growing at a rate of 5 per
year. graph
Use the graph to determine the population after
25 years.
Use the graph to determine when the population
will be 60,000.
32
Hours of Daylight as a Function of Day of the
Year (40N latitude)
9-A
Hours of Daylight Date Day of year
14 (greatest) June 21st (Summer Solstice) 172
10 (least) December 21st (Winter Solstice) 355
12 March 21st (Spring Equinox) 80
12 September 21st (Fall Equinox) 264
33
Hours of daylight as a function of day of the
year ( h f(d) )
9-A

• The independent variable is day of the year.
• The dependent variable is hours of daylight.
• The domain is 0-365 days.
• The range is 10-14 hours of daylight.

34
Hours of daylight as a function of day of the
year h f(d)
9-A

35
Hours of daylight as a function of day of the
year h f(d)
9-A
36
Hours of daylight as a function of day of the
year h f(d)
9-A

37
Hours of daylight as a function of day of the
yearh f(d)
9-A

38
Hours of daylight as a function of day of the
year h f(d)
9-A
39
Watch for Deceptions 25
9-A
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
40
Watch for Deceptions
9-A

41
Watch for Deceptions
9-A

42
9-A

• Homework
• Pages 540-541
• 19a-b, 20a-c, 22, 24, 26
• use graph paper!