Section 9A Functions: The Building Blocks of Mathematical Models

1
Section 9AFunctions The Building Blocks of
Mathematical Models

• Pages 532-539

2
Functions
9-A

• A function describes how a dependent variable
  (output) changes with respect to one or more
  independent variables (inputs).
• When there are only two variables, we often
  summarize them as an ordered pair with the
  independent variable first
• (independent variable, dependent variable)
• (input, output)
• (x, y)

3
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))

4
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

5
Representing Functions
9-A

• There are three basic ways to represent
  functions.
• Data Table or List
• Draw a picture or graph
• Write an equation

6
Coordinate Plane
9-A
7
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)

8
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
9
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.

10
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
11
Domain and Range
9-A

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

12
Temperature as a Function of TimeT f(t)
9-A

13
Temperature as a Function of TimeT f(t)
9-A

14
Temperature as a Function of TimeT f(t)
9-A

15
Temperature as a Function of Time T f(t)
9-A

16
Temperature as a Function of TimeT f(t)
9-A

17
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
18
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.

19
Pressure as a Function of Altitude P f(A)
9-A

20
Making predictions from a graph
9-A

21
Pressure as a function of Altitude P f(A)
9-A

22
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
23
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.

24
Hours of daylight as a function of day of the
year h f(d)
9-A

25
Hours of daylight as a function of day of the
year h f(d)
9-A
26
Hours of daylight as a function of day of the
year h f(d)
9-A

27
Hours of daylight as a function of day of the
year h f(d)
9-A

28
Hours of daylight as a function of day of the
year h f(d)
9-A
29
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
30
Watch for Deceptions
9-A

31
Watch for Deceptions
9-A

32
9-A

• Homework for Wednesday
• Pages 540-542
• 19a-b, 20a-c, 22, 24, 26
• You may use Excel to graph the functions in 24,
  26.