Notes 9A Functions: The building blocks of mathematical models
Example 1: Writing Functions
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).
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.
Representing Functions
There are three basic ways to represent a function. We will focus on #2.
1. We can represent a function with a data table
2. We can draw a picture, or graph, of a function
3. We can write an equation
Before we draw a graph of any function, we must determine the values that we should
show on each axis. Mathematically, each axis on a graph extends to infinity in both
directions. However, most functions are meaningful only over a small region of the
coordinate plane.
Example 2: Temperature Data for One day

Identify the independent and dependent variable and write as an ordered
pair

Write the independent and dependent variable using function notation

Identify the domain and range of the temperature function
Example 3 Pressure-altitude Function
Imagine measuring the atmospheric pressure as you rise upward in a hot-air balloon. The
following table shows typical values you might find for the pressure at different altitudes,
with the pressure given in units of inches of mercury.

Use these data to graph a function showing how atmospheric pressure depends on
altitude.

Use the graph to predict the atmospheric pressure at an altitude of 15,000 feet

Discuss the validity of your prediction.
Example 4 Hours of Daylight
The number of hours of daylight varies with the seasons. Use the following data for
40N latitude (the latitude of San Francisco, Denver, Philadelphia, and Washington,
D.C.) to model the change in the number of daylight hours with time.

The number of hours of daylight is greatest on the summer solstice (about June
21), when it is about 14 hours for latitude 40N

The number of hours of daylight is smallest on the winter solstice (about
December 21), when it is about 10 hours for latitude 40N

On the spring and fall equinoxes (about March 21 and September 21,
respectively), there are about 12 hours of daylight.
1. at what times of year does the number of daylight hours change most
gradually?
2. Most quickly?
3. Discuss the validity of the model.