Modeling Our World

1. Modeling Our World 9

2. Unit 9A Functions: The Building Blocks of Mathematical Models

3. Functions • A function describes how a dependent variable changes with respect to one or more independent variables. When there are only two variables, they are often summarized as an ordered pair with the independent variable first: (independent variable, dependent variable) • The dependent variable is a function of the independent variable. If x is the independent variable and y is the dependent variable, write the function as y = f(x).

4. Example For each situation, express the given function in words. Write the two variables as an ordered pair and write the function with the notation y = f(x). a. You are riding in a hot-air balloon. As the balloon rises, the surrounding atmospheric pressure decreases (causing your ears to pop). Pressure is the dependent variable and altitude is the independent variable, so the ordered pair of variables is (altitude, pressure). Let A stand for altitude and P stand for pressure, then we write the function as P = f(A)

5. Example (cont) b. You’re on a barge headed south down the Mississippi River. You notice that the width of the river changes as you travel southward with the current. River width is the dependent variable and distance from the source is the independent variable, so the ordered pair of variables is (distance from source, river width). Let d represent distance and w represent river width, we have the function w = f(d).

6. Representing Functions There are three basic ways to represent a function. • Use a data table. • Draw a picture or graph. • Write an equation.

7. Domain and Range • The domain of a function is the set of values that both make sense and are of interest for the independent variable. • The rangeof a function consists of the values of the dependent variable that correspond to the values in the domain.

8. Creating and Using Graphs of Functions Step 1: Identify the independent and dependent variables of the function. Step 2: Identify the domain and range of the function. Use this information to choose the scale and labels on the axes. Zoom in on the region of interest to make the graph easier to read. Step 3: Make a graph using the given data. If appropriate, fill in the gaps between data points. Step 4: Before accepting any predictions of the model, be sure to evaluate the data and assumptions from which the model was built.

9. Example Hours of Daylight as a Function The number of hours of daylight varies with the seasons. Use the following data for 30°N latitude to model the change in the number of daylight hours with time. • The number of hours of daylight is greatest on the summer solstice (around June 21), when there are about 14 hours of daylight. • The number of hours of daylight is smallest on the winter solstice (around December 21), when there are about 10 hours of daylight. • On the spring and fall equinoxes (around March 21 and September 21, respectively), there are about 12 hours of daylight of daylight.

10. Example (cont) • Since time passes regardless of other events, let time, denoted by t, be the independent variable. • Let the hours of daylight, denoted by h, be the dependent variable. • The function is h = f(t). • The domain is the 365 days of a single year. • The range is from 10 to 14 hours of daylight. • The number of hours of daylight changes smoothly with the seasons. • The same pattern repeats from one year to the next.

11. Example (cont) A graph of the function h = f(t) is shown below.