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). 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)

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

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

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

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

year population 10000(1.05) year population 10000(1.05) year population 10000(1.05) 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)

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

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

tP=f(t)P(t,f(t)) 0f(0) = 10,000 x (1.05) (0,10000) 1f(1) = x (1.05)10500(1,10500) 2f(2) = x (1.05) (2,11025) 3f(3) = x (1.05) (3,11576) 10f(10) = x (1.05) (10,16289) 15f(15) = x (1.05) (15,20789) 20f(20) = x (1.05) (20,26533) 40f(40) = x (1.05) (40,70400) EXAMPLE: In Powertown, the initial population is 10,000 and growing at a rate of 5% per year. table of data

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).

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

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

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

Coordinate Plane 9-A

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)

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

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

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

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

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

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

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

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

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 inches of mercury. 9-A

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

Making predictions from a graph 9-A

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

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)

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)

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.

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) (least) December 21 st (Winter Solstice) March 21 st (Spring Equinox) 80 12September 21 st (Fall Equinox) A

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 days.  The range is hours of daylight. 9-A

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

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

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

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

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

Watch for Deceptions: # 25 YearTobacco (billions of lb) YearTobacco (billions of lb) A

Watch for Deceptions: 9-A

Watch for Deceptions: 9-A

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