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

Functions 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) (independent variable, dependent variable) (input, output) (x, y) (x, y) 9-A

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) (interest rate, monthly mortgage payment) (relative energy, magnitude (of earthquake)) 9-A

Functions We say that the. 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

Representing Functions There are three basic ways to represent functions. There are three basic ways to represent functions. Data Table or List Data Table or List Draw a picture or graph Draw a picture or graph Write an equation Write an equation 9-A

Coordinate Plane 9-A

Coordinate Plane Draw 2 perpendicular lines (x-axis, y-axis) Draw 2 perpendicular lines (x-axis, y-axis) Numbers on the lines increase up and to the right. Numbers on the lines increase up and to the right. The intersection of these lines is the origin (0,0) The intersection of these lines is the origin (0,0) Points are described by 2 coordinates (x,y) 9-A

Temperature Data for One Day TimeTempTimeTemp 6:00 am 50°F 1:00 pm 73°F 7:00 am 52°F 2:00 pm 73°F 8:00 am 55°F 3:00 pm 70°F 9:00 am 58°F 4:00 pm 68°F 10:00 am 61°F 5:00 pm 65°F 11:00 am 65°F 6:00 pm 61°F 12:00 pm 70°F 9-A

Domain and Range The domain of a function is the set of values 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 The range of a function consists of the values of the output (dependent) variable that correspond to the values in the domain. 9-A

Temperature Data for One Day TimeTempTimeTemp 6:00 am 50°F 1:00 pm 73°F 7:00 am 52°F 2:00 pm 73°F 8:00 am 55°F 3:00 pm 70°F 9:00 am 58°F 4:00 pm 68°F 10:00 am 61°F 5:00 pm 65°F 11:00 am 65°F 6:00 pm 61°F 12:00 pm 70°F 9-A

Domain and Range The domain is the hours from 6 am to 6 pm. The domain is the hours from 6 am to 6 pm. The range is temperatures from °F. 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 ft 30 5,000 ft 25 10,000 ft 22 20,000 ft 16 30,000 ft 10 9-A

Pressure as a Function of Altitude P = f(A) The independent variable is altitude. The independent variable is altitude. The dependent variable is atmospheric pressure. The dependent variable is atmospheric pressure. The domain is 0-30,000 ft. The range is inches of mercury. 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

Hours of Daylight as a Function of Day of the Year (40°N latitude) Hours of DaylightDateDay of year 14(greatest) June 21 st (Summer Solstice) (least) December 21 st (Winter Solstice) March 21 st (Spring Equinox) September 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 independent variable is day of the year. The dependent variable is hours of daylight. The dependent variable is hours of daylight. The domain is days. The domain is days. The range is hours of daylight. 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 for Wednesday: Pages # 19a-b, 20a-c, 22, 24, 26 You may use Excel to graph the functions in # 24, A