Section 9A Functions: The Building Blocks of Mathematical Models Pages

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) 9-A Functions (page 561)

A function describes how a dependent variable (output) changes with respect to one or more independent variables (inputs). 9-A input (x) output (y) function RANGE page 563 DOMAIN page 563

Representing Functions There are three basic ways to represent functions: Formula Graph Data Table 9-A

TimeTempTimeTemp 6:00 am50°F1:00 pm73°F 7:00 am52°F2:00 pm73°F 8:00 am55°F3:00 pm70°F 9:00 am58°F4:00 pm68°F 10:00 am61°F5:00 pm65°F 11:00 am65°F6:00 pm61°F 12:00 pm70°F 9-A 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 9-A ( 1, 2), ( -3, 1), ( 2, -3), ( -1, -2), ( 0, 2), ( 0, -1)

9-A 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

9-A ( 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

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

9-A 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.

9-A OBSERVATION from graph The temperature rises and then falls between 6am and 6 pm. EXAMPLE/533 Temperature Data for One Day graph

AltitudePressure (inches of mercury) 0 ft30 5,000 ft25 10,000 ft22 20,000 ft16 30,000 ft10 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. 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 ft.

9-A Domain: altitudes from 0 to 30,000 ft. Range: pressure from 10 to 30 inches of mercury. (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)

9-A ( 0, 30) ( 5000, 25) ( 10000, 22) ( 20000, 16) ( 30000, 10) (EXAMPLE2/565) Pressure Altitude Function Use the data to create a graph. OBSERVATION from graph As altitude increases, atmospheric pressure decreases.

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

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

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

YearTobacco (billions of lb) YearTobacco (billions of lb) 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) 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 (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 graph The production of tobacco has slowly decreased from 1975 to 1986 and then slowly increased from 1986 to 1990.

Homework: Page # 22,24,26,28,32,34,36 9-A