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Section 9A Functions: The Building Blocks of Mathematical Models

Section 9A Functions: The Building Blocks of Mathematical Models. Pages 560-570. 9-A. Functions (page 561). A function describes how a dependent variable (output) changes with respect to one or more independent variables (inputs) .

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Section 9A Functions: The Building Blocks of Mathematical Models

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  1. Section 9AFunctions: The Building Blocks of Mathematical Models Pages 560-570

  2. 9-A Functions (page 561) 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. 9-A Functions (page 561) 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. 9-A Representing Functions 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 Thetemperature(dependent variable)varies with respect totime(independent variable).T = f(t) RANGE: temperatures from 50 to 73 andDOMAIN: time of day from 6am to 6pm.

  6. 9-A Graphs (1, 2) , (-3, 1) , (2, -3) , (-1, -2) , (0, 2) , (0, -1)

  7. 9-A EXAMPLE/560 Temperature Data for One Day graph Domain: Time of Day from 6:00 am to 6:00 pm. Range: Temperatures from 50° to 73°F.

  8. 9-A EXAMPLE/564 Temperature Data for One Day 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. 9-A EXAMPLE/533 Temperature Data for One Day graph Domain: hours since 6 am. Range: Temperatures from 50° to 73°F.

  10. 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.

  11. 9-A EXAMPLE/533 Temperature Data for One Day graph OBSERVATION from graphThe temperature rises and then falls between 6am and 6 pm.

  12. 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. Theatmospheric pressure (dep. variable)varies with respect toaltitude (indep. variable).P = f(A) RANGE: pressures from 10 to 30 andDOMAIN: altitudes from 0 to 30000 ft.

  13. 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.

  14. 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.

  15. 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.

  16. 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

  18. 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) Annual tobacco production (dep. variable)varies with respect toyear (indep. variable). RANGE: annual tobacco production from 1.2 to 2.2 billions of lbs DOMAIN: years from 1975 to 1990.

  19. 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.

  20. 9-A Homework: Page 568-569 # 22,24,26,28,32,34,36

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