Introduction:
We have asked you to read carefully the following sample sections
of the text:
First Edition: 1) Sections 1.1 through 1.5 of Chapter 1; 2)
Sections 2.2 through 2.6 of Chapter 2; 3) Sections 4.4 and 4.6 of
Chapter 4.
Second Edition: 1) Sections 1.1 through 1.5 of Chapter 1;
2) Sections 2.2 through 2.8 of Chapter 2; 3) Sections 4.5 through
4.8 of Chapter 4.
Chapter 1 Reading Notes
The principal topic of the sample sections for Chapter 1 is
functions of one variable as defined by formulas, graphs and tables.
It includes the usual "rule" definition of function and the recitation
of the standard function terminology: domain, range, dependent and
independent variables, etc.
These are standard topics in virtually any advanced algebra course.
The only indication that this book might have a different emphasis
than other advanced algebra books is in the problems for Section 1.5
which repeatedly ask the student to draw a reasonable graph of a
function that is described by some "real world" situation. Drill
problems covering the terminology and definitions are included, but
emphasized less than in traditional algebra books.
Chapter 2 Reading Notes
The principal topics of the sample sections, linear, exponential,
logarithmic and power functions and their properties, are certainly
familiar topics in virtually any advanced algebra course. However, as
you read these sample sections, you will find that the treatment of
these topics is much different than that found in traditional advanced
algebra courses. In standard courses, linear, exponential and power
functions are treated as separate, self-contained packages, each with
its own application and with its applications usually placed at the
end of each package. In this book, these three types of functions are
introduced as means to describe different types of growth (increase)
and decay (decrease) of a quantity or function.
Linear functions are those with a constant rate of increase (or
decrease).
Exponential functions are those with a constant percentage rate
of increase (or decrease).
Power functions have no similar description in terms of rates of
increase or decrease. However, the main points that made are:
1) The values of any exponential function with a positive
growth factor will always eventually become and stay larger than the
values of any power function (including any linear function)
with a positive power, regardless of the relative sizes of their
(positive) initial values.
2) The values of an exponential function with a large positive
growth factor will always eventually become and stay larger than the
values of an exponential function with a smaller positive growth
factor regardless of the relative sizes of their (positive) initial
values.
3) The values of a power function with a large positive power
will always eventually become and stay larger than the values of
a power function with a smaller positive power, regardless of the
relative sizes of their (positive) initial values.
The practical implications of these comparisons are explored in a
variety of contexts including population growth and comparative
economic growth.
The section on logarithmic functions makes similar comparisons about
the relative growth of power and logarithmic functions.
Chapter 4 Reading Notes
Chapter 4 deals with polynomial functions, a standard topic in
advanced algebra, and with transformations of functions (vertical and
horizontal shifts, vertical and horizontal dilations, etc.), an
important topic that is frequently neglected in advanced algebra
courses.
You are not required to read Section 4.1, 4.2, 4.3 of the First
Edition or Sections 4.1 through 4.4 of the Second Edition for any of
the work specified for this module, but you may find it interesting to
scan the content because you will find that it focuses much more on
the graphical characteristics of polynomial functions than typical
advanced algebra books.
The focus of the sample section, Finding Polynomial Patterns,
is on recognizing polynomial functions by successive differences, an
important topic for applications that is normally omitted in standard
advanced algebra courses.
The sample section on transformations of functions, Building New
Functions From Old, is a bit misplaced in this chapter because
the content is not restricted to polynomial functions. Our experience
with teaching this material is that it is quite accessible to just
average students, and that it helps all students to begin to recognize
and use relationships between an algebraic formula for a function and
its graph.
