Investigations
Precalculus: A Study of
Functions and Their Applications

Investigations are included in each exercise set in Precalculus: A Study of Functions and Their Applications. These are extended problems in which students explore either a mathematical concept or an application. Some investigations extend ideas presented in the section. Some investigations foreshadow topics that students encounter in later sections. Investigations can be used as part of a regular homework assignment, group homework, a small group inclass activity, or as a class discussion.
Four investigations from various parts of the book are given below.
Flat
Tax (This investigation is taken from Section 2.1:
Linear Functions)
The federal income tax laws are quite complicated. There have been various proposals to simplify the system. Many of these proposals are often referred to as a flat tax. To most people, a flat tax implies a single tax rate. This is usually not the case in most of the proposals. For example, a tax proposal endorsed by 1996 presidential candidate Steve Forbes proposed that a person would be taxed at a rate of 17% only on the income they earned above $13,300. The first $13,300 would not be taxed at all.

Transformations
of Linear and Exponential Functions
(This investigation is taken from Section 3.2: Function Transformations: Changes in Input) You may have noticed that some transformations can either be a horizontal or a vertical shift. For example, the following graph shows f(x) = 1.5x  6 and g(x) = 1.5x  3. The graph of g can be considered a horizontal shift of f by moving it two units to the left or a vertical shift of f by moving it three units up. When can a transformation be obtained by either a vertical or horizontal shift? We will look at this question as it pertains to linear and exponential functions.

The
Gateway Arch (This
investigation is taken from Section 3.3: Combining Functions)
The historical note in this section mentioned that the Gateway Arch (part of the Jefferson National Expansion Memorial in St. Louis, Missouri) is not a parabola but is a shape known as a catenary. In this investigation, we will look at the shape of the Gateway Arch and compare it to a parabola. The catenary is the name given to the shape formed by the graph of the hyperbolic cosine (abbreviated cosh) and is the shape of a uniform flexible cable, or chain whose ends are supported from the same height. The word catenary comes from the Latin word for chain. The hyperbolic cosine is defined as The functions e^{x} and e^{x} are exponential
functions whose base is e, a number approximately equal to 2.718.
The number e is an important constant that occurs in a variety of
mathematical contexts and so is denoted by its own symbol (similar to the
number p ). The exponential function y =
e^{x} is often the "nicest" exponential function to use, partly
because it crosses the yaxis with a slope of one.
where y is the height above the ground and x is the distance from the line of symmetry through the middle of the arch.

Projectile
Motion (This
investigation was taken from Section 5.4: Trigonometric Identities.)
Suppose a projectile was fired into the air at an angle of q . The vertical distance of the projectile (in feet) is given by v(t) = 16t^{2} + v_{0}tsin q where t is time (in seconds) and v_{0} is the initial velocity (in feet per second). The horizontal distance of the projectile (in feet) is given by h(t) = v_{0}tcos q . We are interested in determining the total horizontal distance (known as range) traveled by the projectile.
