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How To Study with DWFK Precalculus

Chapter P: Prerequisites

We have gathered into Chapter P the topics that we assume have been covered (although not necessarily mastered) in your previous algebra courses. In many cases, we put them here in order to trim some redundancy out of the later chapters. Most instructors can (and should) move quickly through this material, perhaps using selected exercises to familiarize your class with the basic notation the book will use or to "dust away the cobwebs" that have accumulated over the summer. Some teachers might choose to skip it entirely. Whether or not the chapter is considered part of the course, you should read it to get a feeling for the knowledge base upon which the rest of the course is built.

Section P.1	Section P.3	Section P.5
Section P.2	Section P.4	Section P.6

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Section P.1 Real Numbers

Key Ideas

Additive inverse (opposite)	Distributive property	Natural number	Repeating decimal
Algebraic expression	Exponent	Negative number	Scientific notation
Associative Properties	Identity properties	nth power of a number	Set notation
Base	Inequality	Ordering of the real numbers	Terminating decimal
Bounded and unbounded intervals	Integer	Origin (of the real number line)	Trichotomy property
Closed and open intervals	Interval notation	Positive number	Variable
Commutative properties	Inverse properties	Rational number	Whole number
Constant	Irrational number	Real number
Coordinate of a point	Multiplicative inverse (reciprocal)	Real number line

Study Tips
The importance of understanding the real numbers as a complete, ordered number system satisfying the field axioms of algebra has been gradually de-emphasized since its heyday in the "New Math" era of the 60's. It is true that the real numbers are characterized by their properties; however, you do not need to appreciate that fact in order to use them. We list the properties here to give a complete introduction to the real numbers, not to imply that they should be memorized.

The properties of exponents in this section refer to integer exponents. They will be revisited later (as properties of real exponents) when you study exponential functions, so it is not necessary to spend much time on them here.

Be sure that you understand interval notation. It is the preferred way to describe domains and ranges of functions throughout this book.

Technology Tips
Your calculator does not really know what (or any irrational number) really is. That is because your calculator treats numbers as either finite decimal expansions or as fractions, and irrational numbers are neither. There are other limitations of technology that we will mention later in the course.

Note that calculators have their own version of "scientific notation" that you must be able to convert to the usual form (Figure P.5).
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Section P.2 Cartesian Coordinate System

Key Ideas

Absolute value as distance from zero	Ordered pair (x, y)
Absolute value of a real number	Origin (of the Cartesian plane)
Cartesian plane	Pythagorean theorem
Center of a circle	Quadrants I, II, III, IV
Coordinates of a point	Radius of a circle
Distance formulas (line and plane)	Rectangular coordinate system
Equation of a circle	x-axis and y-axis
Midpoint formulas (line and plane)	x-coordinate and y-coordinate

Study Tips
One of the most important "pre-calculus" features of this course is its emphasis on moving comfortably among algebraic, numerical, and graphical representations to model function behavior. The Cartesian coordinate system is the fundamental concept that connects them all. This is, therefore, a very important (and very "prerequisite") section.

Happily, you are likely to have seen this material already, probably in at least two previous courses. It is still a good idea to look over the exercises so that any misconceptions can be exposed before you carry them any further into the course.

One thing we want to work on throughout the course is the notion of a mathematical proof. Take a good look at the derivation of the distance formula from the Pythagorean theorem and the equation of a circle from the distance formula. You should be able to follow (and perhaps even produce) the steps easily, and it will lay the groundwork for later proofs. (There are also some very accessible analytic geometry proofs in the exercises.)

Although it may be unfamiliar to you, note carefully the distance interpretations of absolute value, i.e., as the distance of a from zero and as the distance between a and b. This is the most important use of absolute value in future courses.

Technology Tips
If you have experience writing programs for your calculator, it is a nice exercise to write a program that will find the midpoint and length of a segment determined by a given pair of points. On the other hand, there is very little value in borrowing such a program from someone else, as it actually inhibits your understanding of the formulas if you use a calculator to do them.
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Section P.3 Linear Equations and Inequalities

Key Ideas

Double inequality	Linear inequality in x
Equation	Number line graph of a solution set
Equivalent equations	Solution of an equation
Equivalent inequalities	Solution of an inequality
Linear equation in x	Solution set

Study Tips
As was the case with the properties of real numbers in Section P.1, the properties of equality and inequality are given here for their pedagogical value in teaching the rules of equation-solving. It is not intended that they be memorized.

You should actually be well-acquainted with the concepts of this section after two previous courses in algebra, but be sure to try a variety of exercises in order to determine what you might need to review.

Technology Tips
Many calculators have equation-solving capabilities, but you should not be using them in this particular section, the emphasis of which is on algebraic manipulation. This book will welcome the use of technology for solving equations in other contexts later in the course.

Your teacher should establish technology "ground rules" for some assignments, and perhaps for some tests and quizzes. Exercises in this book will occasionally specify that they are to be done "with calculators" or "without calculators," depending on what skill or concept is being assessed.
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Section P.4 Lines in the Plane

Key Ideas

Graph of an equation	Slope of a line
Linear equation in x and y	Slope-intercept form
Parallel lines	Square viewing window
Perpendicular lines	Vertical and horizontal lines
Point-slope form	y-intercept of a line

Study Tips
Linear equations have been moved from the polynomial chapter to the prerequisites chapter in order to lay the general groundwork for graphing and equation-solving in a familiar context. You should have seen the concepts of this section in at least two previous courses, so it should not be necessary to spend much time here.

Linear equations, linear graphs, and the concept of slope are the main foundations upon which all of differential calculus is built. That is why it is worth seeing this material for the third time. (Indeed, you will continue to encounter linear equations and graphs throughout this book, right up to the last chapter.)

Technology Tips
Since vertical lines are not graphs of functions, you can not produce vertical line graphs on your calculator in the usual way. We will soon see how important it is for functions to be single-valued. You can "fool" the grapher into graphing the linear equation x = a by entering a line through s with very large slope, e.g.,

Even if you have used graphing calculators before, you might never have noticed the effect that the viewing window has on the apparent slope of a line. Exercises 31, 32, and 37 through 40 illustrate the phenomenon nicely. Only a "square" viewing window can be trusted to show you the true shape of a graph.

The convention shown in this section for labeling viewing windows will be used throughout the book.
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Section P.5 Solving Equations Graphically, Numerically, and Algebraically

Key Ideas

Completing the square	Quadratic formula
Extraneous solution	Root of an equation
Point of intersection (of two graphs)	Zero of an equation*

*Technically, equations have roots and functions have zeros, but "root" and "zero" are used interchangeably in this section to postpone the mention of functions until Chapter 1. A more careful treatment of these terms is given in Section 1.1 under the subheading "The Importance of Zeros."

Study Tips
You will probably recall that it took many weeks to cover these equation-solving techniques in your earlier courses, and you still might fear that you can't use them very well. Do not worry if your instructor appears insensitive to your plight and seems to race through this material; no class can afford to get bogged down in this section. You will need these equation-solving techniques on a regular basis as the course progresses, and it is probably best to review them, quickly and matter-of-factly, as the need arises.

Although specific algebraic manipulations have been placed in this section as prerequisite material, the general concept of solving equations algebraically, numerically, and graphically is so essential that it is covered rather extensively in Chapter 1.

Technology Tips
Once you have learned how to solve equations with your calculator, you might find yourself reluctant to use algebraic techniques. Nonetheless, it might be important to do so. Bear in mind that most of the exercises in this book are designed to teach something, so simply getting the correct answer is usually not the goal. For example, Exercises 1 to 24 specify paper-and-pencil solutions precisely so that you can practice the algebraic techniques reviewed in this section.

It is also true that calculators can not be trusted to give "exact" answers, but we do not offer that as a good argument to use on behalf of algebraic methods, especially since we will frequently use technology to solve equations in this book. We defend algebraic methods on their own merits, as significant mathematics that all students should understand and appreciate. Also, such techniques as completing the square and factoring will show up in different contexts later on, both in this course and in calculus.
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Section P.6 Solving Inequalities Algebraically and Graphically

Key Ideas

Absolute value inequalities	Inequalities involving fractions
Cubic inequalities	Quadratic inequalities

Study Tips
The same warning sounded in the previous section applies here. Do not assume that your instructor should crawl slowly through this material so that you and your classmates can master all of these algebraic skills. Even if that were a reachable goal, getting bogged down in this section amounts to repeating a previous course, and we have work to do. On the other hand, feel free to do as much review as you want on your own. That's why Chapter P is here.

Technology Tips
A crude version of a "number line" graph of an inequality can be produced on most calculators. In the "Y=" screen, enter the inequality in parentheses and take the reciprocal. The number line graph will appear along the line y= 1. For example, the inequality

can be entered as shown on the left below, with the result as shown on the right:

This works because the expression inside the parentheses is interpreted by the calculator as a function, taking on the value 1 when the statement is true and 0 when the statement is false. The reciprocal, then, takes on the value 1 when the statement is true and becomes undefined when the statement is false. The grapher turns on a pixel along the line y= 1 above every x value for which the statement is true, resulting in a number line graph.

Whether or not you find this technology nugget interesting, you should not use it to do the exercises that specifically call for an algebraic solution. Those exercises are intended as practice on the fields of algebra.