Hornsby/Lial College Algebra

CHAPTER RESOURCES : AUTHOR TIPS

Chapter 1 [ TOP ]

Welcome to what we hope will be your most rewarding experience in studying algebra. In case you have never studied algebra with a graphing calculator, you should realize that while your text covers the same material that traditional texts cover, it is organized in such a way that a graphing calculator can best be utilized to support the mathematical concepts. The most important thing that you should realize throughout your study is that you must understand the mathematical concepts first so that the graphing calculator can provide support for them.

The material in Chapter 1 may be mostly a review for you, but you should not underestimate its importance. In this chapter you are introduced to the basic ideas of setting windows for graphing purposes, entering functions on your calculator, tracing and zooming in on a graph, using the logic capabilities, constructing tables, and entering statistical data. We do not provide exact keystrokes because of the many different types of graphing calculators available. You should refer to your owner's manual and/or the keystroke guide that accompanies your text. (See the preface for more information on this guide.)

As authors, we spend a great deal of time writing explanations and examples that will help you understand college algebra. When you read the material, have pencil and paper and your graphing calculator at hand. Try to duplicate the screens you see in the text as this will help you become more proficient with your calculator while you are learning the algebra at the same time.

In Chapter 1, we focus on the linear function and its associated equations, inequalities, and applications. We also introduce the idea of a comprehensive graph of a function, which will be seen throughout the book. On pages 63 and 64, you are introduced to the intersection-of-graphs method and the x-intercept method of graphical solution of equations. In this chapter, we focus strictly on linear equations, but these methods will be used throughout the book when we study other types. It is essential that you understand these two methods, as they will be repeated over and over in the following chapters.

There is no substitute for working problems, and this text has many exercises to help you understand the concepts. In addition to the exercises at the end of each section, there is a Chapter Summary (see pages 110-112), a set of Chapter Review Exercises (see pages 112-117), and a sample Chapter Test (see pages 117-118). Chapters 1-9 all have these features.

There is a solutions manual for students written by Norma James that provides complete worked-out solutions for the odd numbered exercises in this book. If it is not available in your bookstore, you may wish to order it.

Do you know the name of your instructor? His or her office hours? If not, find out, as your instructor is your first source of help. It does make a difference if you get to know him or her.

Chapter 2 [ TOP ]

In Chapter 1, we focused strictly on linear functions. The first section of Chapter 2 introduces some of the basic nonlinear algebraic functions and relations with emphasis on domain, range, and symmetry. It is true that a graphing calculator will graph these functions and relations for you quite easily, but you should also be able to draw rough sketches of them with paper and pencil. It is important that you not become too dependent on your calculator for graphing and that you know the basic shapes of the graphs and information concerning the functions and relations.

Look at the boxes on pages 124 and 125 summarizing the identity and squaring functions. Throughout the text, each time a new type of function is introduced, a box such as this will be provided. You may wish to mark these pages in some way so that you can refer back to them when necessary.

In Sections 2.2 and 2.3, you will learn how the graphs of the basic functions can be transformed by shifting, stretching, and reflecting. The "For Group Discussion" boxes on pages 137, 138, 148, 150, and 151 should be covered in class. If for some reason they are not covered or you are absent that day, be sure to go through them on your own with your calculator at hand.

You may remember studying absolute value equations and inequalities in high school or in intermediate algebra. Perhaps you never really understood why they are solved the way they are. In Section 2.4 we cover these, and using a graphing calculator to support these solutions just may allow you to understand why for the very first time! Again, we see the intersection-of-graphs and x-intercept methods used, this time, with absolute value functions rather than linear functions.

The topics covered in Sections 2.5 and 2.6 are especially important for students who plan to study calculus. Remember that piecewise-defined functions and the greatest integer function are graphed with a calculator in dot mode due to the discontinuities. While graphing calculators are incredibly powerful tools, they can lead to erroneous conclusions if used incorrectly. Graphing the greatest integer function in connected mode leads to an inaccurate graph. Look at Figure 52(b) on page 176, and you'll see why.

Have you investigated the videotape series that accompanies this text? Because the mathematics department in your school uses this text, the tapes are available to the department. Ask your instructor where you can view them.

Have you been attending class regularly? Mathematics is challenging enough - missing class certainly doesn't help, wouldn't you agree?

Chapter 3 [ TOP ]

This chapter covers polynomial functions, a very important class of functions. The linear function of Chapter 1 is the simplest of this type, and it is of degree 1. Polynomial functions of degree 2, 3, 4, and higher are discussed here.

Section 3.1 deals with complex numbers. You may have seen complex numbers in an earlier course. The arithmetic operations with complex numbers are not difficult, and the latest models of graphing calculators can perform these operations. You can check your paper-and-pencil work in this way. Complex numbers are important in the study of polynomial functions because these functions may have zeros that are not real.

Sections 3.2-3.4 deal with quadratic, or second degree, functions. Graphs of these functions are vertical parabolas, and it is important that you be able to find the x-intercepts (if any), the y-intercept, and the coordinates of the vertex both analytically and graphically. The quadratic formula in Section 3.3 is one of the most important formulas you will encounter in the study of algebra.

Throughout your study of equations in this course, it is essential to remember that the real solutions of f(x) = 0 are the x-intercepts of the graph of y = f(x). This is the x-intercept method of solution introduced in Chapter 1. Remember that in Chapter 1, you studied graphs, equations, inequalities, and applications of linear functions. In Section 2.4, you did the same thing for absolute value functions. In Sections 3.2-3.4, you're doing it again for quadratic functions. In Sections 3.5 and 3.8, here we go again - this time for higher degree polynomial functions. Do you see what is happening? We are doing the same kind of analysis over and over but applying it to a different kind of function each time. This is how this book is organized which is different from the traditional college algebra books written over the past few decades. Graphing calculators give us the freedom to approach algebra in this new and exciting way.

If you feel like you need additional help, ask your instructor where on campus you can access the tutorial program that accompanies this text or log on through this Web site.

Have you been working enough homework problems? Students who say "I can do it at home, but I blank out on a test" are often just kidding themselves. Do your homework regularly.

Chapter 4 [ TOP ]

This chapter is a bit shorter than the previous ones. We cover rational functions, root functions, and inverse functions.

Working with rational functions on a graphing calculator can be tricky due to discontinuities in the graphs. In the text, we explain about using dot mode and carefully chosen viewing windows to obtain accurate comprehensive graphs of rational functions. Pay close attention to the viewing windows that accompany the calculator-generated graphs.

You may remember from your earlier experience that rational equations and equations involving radicals may have extraneous values appear when solving them analytically. That is why an analytic check is essential when using the traditional analytic methods to solve them. By graphing the appropriate function(s), graphical methods will indicate only the actual solutions; extraneous values will not show up as solutions on the graphs. Mathematics teachers love to give equations with extraneous values, so don't get caught!

This chapter concludes with inverse functions (Section 4.5) as it is a natural lead-in to Chapter 5. A function has an inverse function if and only if it is one to one. Yet certain graphing calculators will "draw" the graph of the "inverse" of a function that is not one to one. Once again, we must understand the mathematical concepts, or we may draw incorrect conclusions from what we see on a graphing calculator screen.

Have you been using your solutions manual? Videotapes?

Chapter 5 [ TOP ]

Without a doubt, the concept of the logarithm is one of the most difficult for algebra students to grasp. The authors of your text admit that even they did not fully understand the concept until taking follow-up courses and teaching logarithms in their classes! So if you find this topic difficult, don't feel as if you're alone. If you remember the following, things will be a lot easier: A logarithm is an exponent, and it follows the usual rules of exponents.

The chapter begins with the study of exponential functions and then progresses on to logarithms. Keep in mind that an exponential function has as it inverse a logarithmic function. Be sure that you know where to find the keys on your calculator that allow you to raise 10 to a power, raise e to a power, find a common (base 10) logarithm, and find a natural (base e) logarithm.

We continue to stress that it is essential that you understand the concepts, or you may misinterpret a calculator screen. For example, look at Figure 26(b) on page 421. If you do not understand the concept of the logarithmic function, you might conclude that the graph touches the y-axis or has an endpoint. Neither of these is correct. The limited resolution of calculator screens can impair our conclusions if we are not careful. The truth is that the y-axis is a vertical asymptote, and as x approaches 0 from the right, the graph approaches the y-axis getting closer and closer as x gets smaller and smaller.

By this time you may be approaching your final exam in this course. If so, it would be a good idea to collect all of your graded tests and start going over them to prepare for the exam. Ask your instructor a few weeks before the exam what would be a good way to prepare for it. Cramming is simply not the way to get the best results, especially in mathematics. Mathematics is a cumulative subject, and concepts continually build on one another. We've always told our students that a good strategy in preparing for a test is to get a good night's sleep. Try it - you'll like it!

Chapter 6 [ TOP ]

One of the first shapes you probably recognized as a young child is the circle. Throw a ball up into the air, and it travels in the path of a parabola. Comets, like Hale-Bopp, often travel in elliptical paths. Look at the shadow that a lampshade casts against a wall and you will see one branch of a hyperbola. These four geometric figures are examples of conic sections, the topic of this chapter. Mathematicians and astronomers have studied conic sections for thousands of years.

Many of the conic sections are not examples of functions. In order to graph them in function mode using a graphing calculator, you will need to be able to do the appropriate algebraic manipulations. For example, a circle can be graphed by graphing the union of two semicircles, each of which is a function. See Example 3 parts (a) and (b) and the accompanying Figure 6 on page 469.

Using a square viewing window is important in getting the correct perspective when graphing circles. If you were to graph the circle in Example 3(b) using the standard viewing window, it would appear to be elliptical (oval-shaped). Don't always believe what you see on a calculator screen as it may be distorted if an appropriate window is not used.

In this chapter, you will also learn how to graph parametric equations. As mentioned earlier, when graphing functions, you cannot directly graph a circle because y is not a function of x. To do so, you have to graph two semicircles. Parametric equations allow you to graph circles directly. When studying parametric equations using a graphing calculator, you will have to use a different mode for your calculator so keep that owner's manual handy in case you have to refer to it.

Did you ever say to yourself "I can follow it when my teacher does it at the board, but when I try to do it myself I get stuck"? If so, you are not alone. Your teacher has probably been working at mathematics for most of his or her life, and there is no substitute for lots of practice. We certainly do not expect that everyone studying from this book will become a mathematician. But we do feel that if you put forth the required effort, both in class and at home, your experience can be rewarding and successful.

Chapter 7 [ TOP ]

This chapter covers systems of equations and matrices. It's likely that you have studied solving systems of equations in earlier courses. However, if you have not used graphing calculators in those courses, you will find that these calculators will provide great visual insight into this topic. If a solution of a system in two variables consists of an ordered pair of real numbers, that solution will appear as a point of intersection of the graphs of the equations in the system. So, once you have solved a system using the traditional elimination or substitution methods using paper and pencil, you can graph the equations and direct your calculator to find the points of intersection to support your solutions.

Matrices provide another tool for solving systems. Traditional matrix work has always involved extensive computation (some people call it number crunching) and careless arithmetic errors have always been a problem for students and teachers alike. Graphing calculators with matrix capabilities have changed all that.

Students seem to have less trouble studying the material in this chapter than in many others. If you find this to be true for you, that's great, but don't get complacent. As always, the more you practice; the better you will get. Remember this: if an application involves solving a system of equations, a calculator will help you solve that system. But that calculator will not set up the system for you. Students often say, "Once I have the equation(s) set up, solving the problem is easy. But I have trouble setting it up." This is a common problem, and practice is essential when it comes to setting up the system. A graphing calculator will do you no good if you cannot use your own brain to set up the problem.

Chapter 8 [ TOP ]

(Note: This is also Chapter 11 in A Graphical Approach to College Algebra and Trigonometry/2 and A Graphical Approach to Precalculus/2.)

This chapter contains an introduction to sequences and series, a method of proof known as mathematical induction, the binomial theorem, and also a few topics studied in more detail in finite mathematics: counting and probability. There is less emphasis on graphing in this chapter than in the other chapters of this text although there is some graphing in the first few sections. Many current models of graphing calculators have a sequence-graphing mode, which differs from the function and parametric graphing illustrated earlier in the text.

When studying sequences and series, you will need to learn formulas for specific terms and sums. You will have to be able to distinguish between arithmetic and geometric sequences and series since there are different formulas for the different kinds. Graphing calculators can help find sums, but you will have to consult your owner's manual for the proper syntax for your particular model.

Problems involving counting (permutations and combinations) and probability can be very difficult even at this elementary level. Students often have trouble determining whether to use the permutation or the combination formulas. See pages 653-654 for help with this. Your calculator will not help you set up these problems, but once they are set up, it will make the computations much easier.