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How To Study with DWFK Precalculus

Chapter 1: Functions and Graphs

This book might be the first of its kind to take advantage of a profound difference between today's precalculus student and students of the past. Thanks to graphing technology, you have at your fingertips an easily-accessed gallery of interesting functions that you can analyze geometrically from the very beginning of the course. Accordingly, this chapter is designed to get you "talking the talk" of function analysis immediately, based on what you can see, before your ability to see it gets lost in the algebraic details.

Remember throughout this chapter that the algebraic properties of the particular functions will be visited at length in future chapters. The emphasis in this chapter is on understanding what functions are and how they behave, and the key to that understanding is graphical.

Section 1.1 Section 1.3 Section 1.5

Section 1.2 Section 1.4 Section 1.6

Section 1.1 Modeling and Equation Solving

Objectives
You will understand how mathematics can model real-world behavior numerically, algebraically, and graphically. You will see how different kinds of models illustrate different aspects of that behavior, and you will begin to learn how to translate from one model to another. You will learn a general strategy for problem-solving based on the Polya four-step method. You will understand the problems of grapher failure and hidden behavior and know the difference between exploration and proof.

Key Ideas

Algebraic model Mathematical proof

Confirm (algebraically) Numerical model

Grapher failure Solve (numerically, graphically, algebraically)

Graphical model Support (graphically, numerically)

Hidden behavior Zero Factor property

Study Tips
This section sets the stage for the rest of the course in several important ways, and it will be new material for most students. Unlike many courses that begin with pages of elementary definitions, this one begins with hands-on analyses of mathematical models. You need to read the text (which was written with you in mind) and work through the explorations that have not already been done in class.

The exercises are rich and varied, providing ample practice for you in all the skills you will need. It is not necessary to spend a long time in this section, but the time that is spent should be spent working with models rather than talking about them.

Technology Tips
The section on grapher failure and hidden behavior will probably appeal to your teachers more than to you. Emphasis on subtleties is less effective when you are still trying to grasp the big picture. Keep in mind that we ask you to trust the geometric representations of functions that their graphers provide, and most graphs can, in fact, be trusted.
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Section 1.2 Functions and Their Properties

Objectives
You will be able to represent functions numerically, algebraically, and graphically. You will be able to determine domains and ranges and analyze function characteristics such as extreme values, boundedness, asymptotes, symmetry, continuity, and end behavior.

Key Ideas

Absolute maximum Implied domain

Absolute minimum Increasing on an interval

Bounded Independent variable

Bounded above Infinite discontinuity

Bounded below Jump discontinuity

Constant on an interval Local maximum

Continuity at a point Local minimum

Decreasing on an interval Mapping

Dependent variable Odd function

Domain Range

End behavior Relevant domain

Even function Removable disconinuity

Function Symmetry

Horizontal asymptote Vertical asymptote

Study Tips
Admittedly, there is a lot of material in this section. Much of it (function, domain, range) should have been encountered in previous algebra courses, but quite a bit of it will be new. Notice that the properties of functions are presented graphically first, then translated into the algebraic model. Once again, we are taking advantage of the fact that you come into the course more familiar with graphs of functions than students did in the past.

There is no need to get into complicated functions in this section, so we keep the presentation intuitive and mostly graphical. The main emphasis of the section is to get you acquainted with the language of functions early so that you can use it throughout the course.

Technology Tips
You may notice in this section that many of the functions with vertical asymptotes are graphed in the window [-4.7, 4.7] by [-3.1, 3.1]. This "decimal-friendly" window will eliminate the vertical lines that might otherwise appear at the integer-valued vertical asymptotes, which is why our illustrations do not show them. (See Example 9 in Section 1.1.) If you prefer seeing the apparent asymptotes, try changing the horizontal window to [-4.6, 4.6].
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Section 1.3 Twelve Basic Functions

Objectives
You will be able to use the vocabulary learned in Section 1.2 to analyze ten basic functions that appear on your calculator.

Key Ideas
Basic Functions
Piecewise-defined function

Study Tips
In the past, precalculus courses had to unveil functions a chapter at a time, which left classes with a shortage of interesting, simple functions to discuss during the beginning of the course. Today's students have a gallery of interesting, simple functions available at the push of a button, and this section takes maximal advantage of that. The purpose of this section is simply to make you look at the ten basic functions and talk about them using the language of functions developed in the previous section. You don't need to know everything about these functions now; after all, there is an entire book ahead of you for that. For example, the cosine function appears in this section so that you can learn about functions, not about the cosine function (although you will get a start on trigonometry in the process).

Technology Tips
When graphing the trigonometric functions, be sure the calculator is in "radian" mode.

The technology tip from Section P.6 can be modified to give a way to graph piecewise-defined functions on a graphing calculator. Remember that dividing a function by a statement like "(x = 0)" will restrict the graph to the domain described by the statement. For example, the following definitions of Y1 and Y2 will yield the graph of the function in Example 7:

[-4.7, 4.7] by [-3.1, 3.1]

This tip is intended primarily for classroom demonstrations. It is not a good idea to produce piecewise graphs this way while you are still learning about them, as it is more instructive to figure out for yourself how they piece together.
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Section 1.4 Building Functions from Functions

Objectives
You will be able to build functions from functions in several ways: by adding, subtracting, multiplying, or dividing functions, by composing functions, by defining functions parametrically, and by finding function inverses. You will be able to build up functions by composition and decompose functions into their basic components.

Key Ideas

Function algebra (addition, multiplication, etc.) Inverse function (or relation)

Function composition Inverse reflection principle

Horizontal line test Parameter

Implicitly defined function Parametrically defined relation

Inverse composition rule Relation

Study Tips
You may find this section more difficult than the previous sections because of the higher dependence on algebra, but the struggle is worth it, because the algebra is important. Remember that this course intends to prepare you to handle functions with equal facility algebraically, graphically, and numerically. In this section, algebra happens to rise to the fore.

Composition of functions should be easier to illustrate now that teachers have the "twelve basic functions" at their disposal. Go ahead and use them all. Students can see the effects of composition more clearly with examples like .

Parametrically defined relations are introduced early in the course so that we can exploit the pedagogical advantages of parametric mode on the graphing calculator (as we do in graphing inverses in this section). This is not a full-blown treatment of parametric curves, as that is a topic for Chapter 6.

Finding inverse relations algebraically can be very difficult. It is definitely not the point of emphasis in this section. You should understand what inverse relations are and how they relate to the original relations graphically (reflection principle) and numerically (switched coordinates in the ordered pair).

Technology Tips
You can learn a lot about parametrically-defined relations by working through Exploration 1 on your calculator.

It is highly recommended that you use the parametric mode of the calculators to explore inverse relations. Switching the coordinates in the ordered pairs is easy and natural, and the effect is readily observed.
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Section 1.5 Graphical Transformations

Objectives
You will be able to represent algebraically and graphically the basic transformations of functions: translations, reflections, stretches, and shrinks.

Key Ideas

Reflection Stretch (horizontal, vertical)

Rigid (or non-rigid) transformation Transformation

Shrink (horizontal, vertical) Translation (horizontal, vertical)

Study Tips
Working through the explorations and actually seeing the geometric effects of the algebraic transformations is the best way to understand these concepts, which derive their names from geometry rather than algebra. You may find it hard to keep all the transformations straight, but it is not vital that you master transformations now, as you will be seeing their effects on functions throughout the course. You do need to be familiar with the terminology.

Technology Tips
Graphing calculators have profoundly changed the way that you can (literally) view transformations. You can experiment with their equations to your heart's content and see immediately the effects of your tinkering. The only drawback is that you can only work on one side of the "y = " equation, so you are unable to apply the same transformations to y and x and appreciate the symmetric effects. This situation is dealt with explicitly in the text following Explorations 1 and 3.

It is recommended that you exploit the power of visualization (graphs) early and often as you proceed through this section.
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Section 1.6 Modeling with Functions

Objectives
You will be able to identify appropriate basic functions with which to model real-world problems. They will be able to produce specific functions to model data, formulas, graphs, and verbal descriptions.

Key Ideas

Coefficient of determination Correlation coefficient

Conversion factor Regression line

Study Tips
This section is intended as an introduction to the kinds of modeling that you will do throughout the course, not as a definitive look at the broad subject of mathematical modeling. You should read the section (which is not very long) and attempt a variety of exercises, but do not become discouraged if you find some of the exercises to be difficult.

The Quick Review exercises (as in all sections of the book) give you a chance to review skills that will be called upon in doing the section exercises. In this section, the first 20 section exercises are actually an extension of the Quick Review, designed to give you a running start leading up to the problem-solving exercises. The problems in this section are of the elementary algebra variety.

Modeling is an important thread throughout the course, so there will be plenty of additional chances for you to hone your modeling skills in future chapters.

Technology Tips
Although we mention a variety of regression types in the chart on page 149, we do so simply to illustrate all the possibilities. Each will be dealt with separately in its appropriate chapter later in the course. Beyond looking at the shapes and discussing their applications, there is really no need to study them any further here. In particular, it is not necessary to work through calculator examples of each.

Example 2 uses a "maximum finder," a feature of most modern calculators. You should learn how to use this feature, as there will be other exercises in the book that require you to find extrema on relevant domains.