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Section 9A Functions: The Building Blocks of Mathematical Models Pages 532-539

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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)

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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 536 DOMAIN page 536

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Functions 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)) 9-A

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Functions 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 9-A

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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 population 10,000(1.05) t f(t) P t

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year population 10000(1.05) 1 10500 1 year population 10000(1.05) 3 11576 3 year population 10000(1.05) 0 10000 0 P=f(0) = 10,000(1.05) 0 P=f(0) = 10,000 (0,10000) P=f(1) = 10,000(1.05) 1 P=f(1) = 10,500 (1,10500) P=f(3) = 10,000(1.05) 3 P=f(3) = 11,576 (3,11576) year population 10000(1.05) 3 11576 3 0

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tP=f(t)P(t,f(t)) 0f(0) = 10,000 x (1.05) 0 10000(0,10000) 1f(1) = 10,000 x (1.05)10500(1,10500) 2f(2) = 10,000 x (1.05) 2 11025(2,11025) 3f(3) = 10,000 x (1.05) 3 11576(3,11576) 10f(10) = 10,000 x (1.05) 10 16829(10,16289) 15f(15) = 10,000 x (1.05) 15 20789(15,20789) 20f(20) = 10,000 x (1.05) 20 26533(20,26533) 40f(40) = 10,000 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. formula: P = 10,000(1.05) t The population (dependent variable) varies with respect to time (independent variable).

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

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tP=f(t)P(t,f(t)) 0f(0) = 10,000 x (1.05) 0 10000(0,10000) 1f(1) = 10000 x (1.05)10500(1,10500) 2f(2) = 10000 x (1.05) 2 11025(2,11025) 3f(3) = 10000 x (1.05) 3 11576(3,11576) 10f(10) = 10000 x (1.05) 10 16829(10,16289) 15f(15) = 10000 x (1.05) 15 20789(15,20789) 20f(20) = 10000 x (1.05) 20 26533(20,26533) 40f(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

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tP=f(t)(t,f(t)) 010,000(0,10000) 110,500(1,10500) 211,025(2,11025) 311,576(3,11576) 1016,829(10,16289) 1520,789(15,20789) 2026,533(20,26533) 4070,400(40,70400) EXAMPLE: In Powertown, the initial population is 10,000 and growing at a rate of 5% per year. table of data The population (dependent variable) varies with respect to time (independent variable).

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Domain and Range 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. 9-A

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tP=f(t)(t,f(t)) 010,000(0,10000) 110,500(1,10500) 211,025(2,11025) 311,576(3,11576) 1016,829(10,16289) 1520,789(15,20789) 2026,533(20,26533) 4070,400(40,70400) EXAMPLE: In Powertown, the initial population is 10,000 and growing at a rate of 5% per year. table of data The population (dependent variable) varies with respect to time (independent variable). RANGE: populations of 10,000 or more and DOMAIN: nonnegative years

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

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Coordinate Plane 9-A

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Coordinate Plane 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) 9-A ( 1, 2), ( -3, 1), ( 2, -3), ( -1, -2), ( 0, 2), ( 0, -1)

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Temperature Data for One Day 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

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Domain and Range The domain is the hours from 6 am to 6 pm. The range is temperatures from 50- 73°F. 9-A

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

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

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

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

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

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Pressure as a Function of Altitude P = f(A) AltitudePressure (inches of mercury) 0 ft30 5,000 ft25 10,000 ft22 20,000 ft16 30,000 ft10 9-A

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Pressure as a Function of Altitude P = f(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. 9-A

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

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Making predictions from a graph 9-A

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

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EXAMPLE: In Powertown, the initial population is 10,000 and growing at a rate of 5% per year. graph RANGE: populations of 10,000 or more and DOMAIN: nonnegative years (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)

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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) RANGE: populations of 10,000 or more and DOMAIN: nonnegative years The population (dependent variable) varies with respect to time (independent variable). P = f(t)

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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.

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Hours of Daylight as a Function of Day of the Year (40°N latitude) Hours of Daylight DateDay of year 14 (greatest) June 21 st (Summer Solstice) 172 10 (least) December 21 st (Winter Solstice) 355 12March 21 st (Spring Equinox) 80 12September 21 st (Fall Equinox) 264 9-A

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

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

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

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

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

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

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Watch for Deceptions: # 25 YearTobacco (billions of lb) YearTobacco (billions of lb) 19752.219861.2 19801.819871.2 19822.019881.4 19841.719891.4 19851.519901.6 9-A

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Watch for Deceptions: 9-A

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Watch for Deceptions: 9-A

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Homework: Pages 540-541 # 19a-b, 20a-c, 22, 24*, 26* *use graph paper! 9-A

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Discrete and Continuous Functions

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