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Precalculus

Version bπc = 3

by

Carl Stitz, Ph.D. Jeff Zeager, Ph.D.

Lakeland Community College Lorain County Community College

July 15, 2011

ii

Acknowledgements

While the cover of this textbook lists only two names, the book as it stands today would simply not exist if not for the tireless work and dedication of several people. First and foremost, we wish to thank our families for their patience and support during the creative process. We would also like to thank our students - the sole inspiration for the work. Among our colleagues, we wish to thank Rich Basich, Bill Previts, and Irina Lomonosov, who not only were early adopters of the textbook, but also contributed materials to the project. Special thanks go to Katie Cimperman, Terry Dykstra, Frank LeMay, and Rich Hagen who provided valuable feedback from the classroom. Thanks also to David Stumpf, Ivana Gorgievska, Jorge Gerszonowicz, Kathryn Arocho, Heather Bubnick, and Florin Muscutariu for their unwaivering support (and sometimes defense!) of the project. From outside the classroom, we wish to thank Don Anthan and Ken White, who designed the electric circuit applications used in the text, as well as Drs. Wendy Marley and Marcia Ballinger for the Lorain CCC enrollment data used in the text. The authors are also indebted to the good folks at our schools’ bookstores, Gwen Sevtis (Lakeland CC) and Chris Callahan (Lorain CCC), for working with us to get printed copies to the students as inexpensively as possible. We would also like to thank Lakeland folks Jeri Dickinson, Mary Ann Blakeley, Jessica Novak, and Corrie Bergeron for their enthusiasm and promotion of the project. The administration at both schools have also been very supportive of the project, so from Lakeland, we wish to thank Dr. Morris W. Beverage, Jr., President, Dr. Fred Law, Provost, Deans Don Anthan and Dr. Steve Oluic, and the Board of Trustees. From Lorain County Community College, we which to thank Dr. Roy A. Church, Dr. Karen Wells, and the Board of Trustees. From the Ohio Board of Regents, we wish to thank former Chancellor Eric Fingerhut, Darlene McCoy, Associate Vice Chancellor of Affordability and Efficiency, and Kelly Bernard. From OhioLINK, we wish to thank Steve Acker, John Magill, and Stacy Brannan. We also wish to thank the good folks at WebAssign, most notably Chris Hall, COO, and Joel Hollenbeck (former VP of Sales.) Last, but certainly not least, we wish to thank all the folks who have contacted us over the interwebs, most notably Dimitri Moonen and Joel Wordsworth, who gave us great feedback, and Antonio Olivares who helped debug the source code.

• 1 Relations and Functions Preface ix
  • 1.1 Sets of Real Numbers and The Cartesian Coordinate Plane
    • 1.1.1 Sets of Numbers
    • 1.1.2 The Cartesian Coordinate Plane
    • 1.1.3 Distance in the Plane
    • 1.1.4 Exercises
    • 1.1.5 Answers
  • 1.2 Relations
    • 1.2.1 Graphs of Equations
    • 1.2.2 Exercises
    • 1.2.3 Answers
  • 1.3 Introduction to Functions
    • 1.3.1 Exercises
    • 1.3.2 Answers
  • 1.4 Function Notation
    • 1.4.1 Modeling with Functions
    • 1.4.2 Exercises
    • 1.4.3 Answers
  • 1.5 Function Arithmetic
    • 1.5.1 Exercises
    • 1.5.2 Answers
  • 1.6 Graphs of Functions
    • 1.6.1 General Function Behavior
    • 1.6.2 Exercises
    • 1.6.3 Answers
  • 1.7 Transformations
    • 1.7.1 Exercises
    • 1.7.2 Answers
• 2 Linear and Quadratic Functions iv Table of Contents
  • 2.1 Linear Functions
    • 2.1.1 Exercises
    • 2.1.2 Answers
  • 2.2 Absolute Value Functions
    • 2.2.1 Exercises
    • 2.2.2 Answers
  • 2.3 Quadratic Functions
    • 2.3.1 Exercises
    • 2.3.2 Answers
  • 2.4 Inequalities with Absolute Value and Quadratic Functions
    • 2.4.1 Exercises
    • 2.4.2 Answers
  • 2.5 Regression
    • 2.5.1 Exercises
    • 2.5.2 Answers
• 3 Polynomial Functions
  • 3.1 Graphs of Polynomials
    • 3.1.1 Exercises
    • 3.1.2 Answers
  • 3.2 The Factor Theorem and The Remainder Theorem
    • 3.2.1 Exercises
    • 3.2.2 Answers
  • 3.3 Real Zeros of Polynomials
    • 3.3.1 For Those Wishing to use a Graphing Calculator
    • 3.3.2 For Those Wishing NOT to use a Graphing Calculator
    • 3.3.3 Exercises
    • 3.3.4 Answers
  • 3.4 Complex Zeros and the Fundamental Theorem of Algebra
    • 3.4.1 Exercises
    • 3.4.2 Answers
• 4 Rational Functions
  • 4.1 Introduction to Rational Functions
    • 4.1.1 Exercises
    • 4.1.2 Answers
  • 4.2 Graphs of Rational Functions
    • 4.2.1 Exercises
    • 4.2.2 Answers
  • 4.3 Rational Inequalities and Applications
    • 4.3.1 Variation
    • 4.3.2 Exercises
    • 4.3.3 Answers Table of Contents v
• 5 Further Topics in Functions
  • 5.1 Function Composition
    • 5.1.1 Exercises
    • 5.1.2 Answers
  • 5.2 Inverse Functions
    • 5.2.1 Exercises
    • 5.2.2 Answers
  • 5.3 Other Algebraic Functions
    • 5.3.1 Exercises
    • 5.3.2 Answers
• 6 Exponential and Logarithmic Functions
  • 6.1 Introduction to Exponential and Logarithmic Functions
    • 6.1.1 Exercises
    • 6.1.2 Answers
  • 6.2 Properties of Logarithms
    • 6.2.1 Exercises
    • 6.2.2 Answers
  • 6.3 Exponential Equations and Inequalities
    • 6.3.1 Exercises
    • 6.3.2 Answers
  • 6.4 Logarithmic Equations and Inequalities
    • 6.4.1 Exercises
    • 6.4.2 Answers
  • 6.5 Applications of Exponential and Logarithmic Functions
    • 6.5.1 Applications of Exponential Functions
    • 6.5.2 Applications of Logarithms
    • 6.5.3 Exercises
    • 6.5.4 Answers
• 7 Hooked on Conics
  • 7.1 Introduction to Conics
  • 7.2 Circles
    • 7.2.1 Exercises
    • 7.2.2 Answers
  • 7.3 Parabolas
    • 7.3.1 Exercises
    • 7.3.2 Answers
  • 7.4 Ellipses
    • 7.4.1 Exercises
    • 7.4.2 Answers
  • 7.5 Hyperbolas vi Table of Contents
    • 7.5.1 Exercises
    • 7.5.2 Answers
• 8 Systems of Equations and Matrices
  • 8.1 Systems of Linear Equations: Gaussian Elimination
    • 8.1.1 Exercises
    • 8.1.2 Answers
  • 8.2 Systems of Linear Equations: Augmented Matrices
    • 8.2.1 Exercises
    • 8.2.2 Answers
  • 8.3 Matrix Arithmetic
    • 8.3.1 Exercises
    • 8.3.2 Answers
  • 8.4 Systems of Linear Equations: Matrix Inverses
    • 8.4.1 Exercises
    • 8.4.2 Answers
  • 8.5 Determinants and Cramer’s Rule
    • 8.5.1 Definition and Properties of the Determinant
    • 8.5.2 Cramer’s Rule and Matrix Adjoints
    • 8.5.3 Exercises
    • 8.5.4 Answers
  • 8.6 Partial Fraction Decomposition
    • 8.6.1 Exercises
    • 8.6.2 Answers
  • 8.7 Systems of Non-Linear Equations and Inequalities
    • 8.7.1 Exercises
    • 8.7.2 Answers
• 9 Sequences and the Binomial Theorem
  • 9.1 Sequences
    • 9.1.1 Exercises
    • 9.1.2 Answers
  • 9.2 Summation Notation
    • 9.2.1 Exercises
    • 9.2.2 Answers
  • 9.3 Mathematical Induction
    • 9.3.1 Exercises
    • 9.3.2 Selected Answers
  • 9.4 The Binomial Theorem
    • 9.4.1 Exercises
    • 9.4.2 Answers
• 10 Foundations of Trigonometry Table of Contents vii
  • 10.1 Angles and their Measure
    • 10.1.1 Applications of Radian Measure: Circular Motion
    • 10.1.2 Exercises
    • 10.1.3 Answers
  • 10.2 The Unit Circle: Cosine and Sine
    • 10.2.1 Beyond the Unit Circle
    • 10.2.2 Exercises
    • 10.2.3 Answers
  • 10.3 The Six Circular Functions and Fundamental Identities
    • 10.3.1 Beyond the Unit Circle
    • 10.3.2 Exercises
    • 10.3.3 Answers
  • 10.4 Trigonometric Identities
    • 10.4.1 Exercises
    • 10.4.2 Answers
  • 10.5 Graphs of the Trigonometric Functions
    • 10.5.1 Graphs of the Cosine and Sine Functions
    • 10.5.2 Graphs of the Secant and Cosecant Functions
    • 10.5.3 Graphs of the Tangent and Cotangent Functions
    • 10.5.4 Exercises
    • 10.5.5 Answers
  • 10.6 The Inverse Trigonometric Functions
    • 10.6.1 Inverses of Secant and Cosecant: Trigonometry Friendly Approach
    • 10.6.2 Inverses of Secant and Cosecant: Calculus Friendly Approach
    • 10.6.3 Calculators and the Inverse Circular Functions.
    • 10.6.4 Solving Equations Using the Inverse Trigonometric Functions.
    • 10.6.5 Exercises
    • 10.6.6 Answers
  • 10.7 Trigonometric Equations and Inequalities
    • 10.7.1 Exercises
    • 10.7.2 Answers
• 11 Applications of Trigonometry
  • 11.1 Applications of Sinusoids
    • 11.1.1 Harmonic Motion
    • 11.1.2 Exercises
    • 11.1.3 Answers
  • 11.2 The Law of Sines
    • 11.2.1 Exercises
    • 11.2.2 Answers
  • 11.3 The Law of Cosines
    • 11.3.1 Exercises viii Table of Contents
    • 11.3.2 Answers
  • 11.4 Polar Coordinates
    • 11.4.1 Exercises
    • 11.4.2 Answers
  • 11.5 Graphs of Polar Equations
    • 11.5.1 Exercises
    • 11.5.2 Answers
  • 11.6 Hooked on Conics Again
    • 11.6.1 Rotation of Axes
    • 11.6.2 The Polar Form of Conics
    • 11.6.3 Exercises
    • 11.6.4 Answers
  • 11.7 Polar Form of Complex Numbers
    • 11.7.1 Exercises
    • 11.7.2 Answers
  • 11.8 Vectors
    • 11.8.1 Exercises
    • 11.8.2 Answers
  • 11.9 The Dot Product and Projection
    • 11.9.1 Exercises
    • 11.9.2 Answers
  • 11.10 Parametric Equations
    • 11.10.1 Exercises
    • 11.10.2 Answers
• Index

x Preface

semester of 2009 and actually began writing the textbook on December 16, 2008. Using an open- source text editor called TexNicCenter and an open-source distribution of LaTeX called MikTex 2.7, Carl and I wrote and edited all of the text, exercises and answers and created all of the graphs (using Metapost within LaTeX) for Version 0.9 in about eight months. (We choose to create a text in only black and white to keep printing costs to a minimum for those students who prefer a printed edition. This somewhat Spartan page layout stands in sharp relief to the explosion of colors found in most other College Algebra texts, but neither Carl nor I believe the four-color print adds anything of value.) I used the book in three sections of College Algebra at Lorain County Community College in the Fall of 2009 and Carl’s colleague, Dr. Bill Previts, taught a section of College Algebra at Lakeland with the book that semester as well. Students had the option of downloading the book as a .pdf file from our website www.stitz-zeager.com or buying a low-cost printed version from our colleges’ respective bookstores. (By giving this book away for free electronically, we end the cycle of new editions appearing every 18 months to curtail the used book market.) During Thanksgiving break in November 2009, many additional exercises written by Dr. Previts were added and the typographical errors found by our students and others were corrected. On December 10, 2009, Version

2 was released. The book remains free for download at our website and by using Lulu.com as an on-demand printing service, our bookstores are now able to provide a printed edition for just under $19. Neither Carl nor I have, or will ever, receive any royalties from the printed editions. As a contribution back to the open-source community, all of the LaTeX files used to compile the book are available for free under a Creative Commons License on our website as well. That way, anyone who would like to rearrange or edit the content for their classes can do so as long as it remains free.

The only disadvantage to not working for a publisher is that we don’t have a paid editorial staff. What we have instead, beyond ourselves, is friends, colleagues and unknown people in the open- source community who alert us to errors they find as they read the textbook. What we gain in not having to report to a publisher so dramatically outweighs the lack of the paid staff that we have turned down every offer to publish our book. (As of the writing of this Preface, we’ve had three offers.) By maintaining this book by ourselves, Carl and I retain all creative control and keep the book our own. We control the organization, depth and rigor of the content which means we can resist the pressure to diminish the rigor and homogenize the content so as to appeal to a mass market. A casual glance through the Table of Contents of most of the major publishers’ College Algebra books reveals nearly isomorphic content in both order and depth. Our Table of Contents shows a different approach, one that might be labeled “Functions First.” To truly use The Rule of Four, that is, in order to discuss each new concept algebraically, graphically, numerically and verbally, it seems completely obvious to us that one would need to introduce functions first. (Take a moment and compare our ordering to the classic “equations first, then the Cartesian Plane and THEN functions” approach seen in most of the major players.) We then introduce a class of functions and discuss the equations, inequalities (with a heavy emphasis on sign diagrams) and applications which involve functions in that class. The material is presented at a level that definitely prepares a student for Calculus while giving them relevant Mathematics which can be used in other classes as well. Graphing calculators are used sparingly and only as a tool to enhance the Mathematics, not to replace it. The answers to nearly all of the computational homework exercises are given in the

xi

text and we have gone to great lengths to write some very thought provoking discussion questions whose answers are not given. One will notice that our exercise sets are much shorter than the traditional sets of nearly 100 “drill and kill” questions which build skill devoid of understanding. Our experience has been that students can do about 15-20 homework exercises a night so we very carefully chose smaller sets of questions which cover all of the necessary skills and get the students thinking more deeply about the Mathematics involved.

Critics of the Open Educational Resource movement might quip that “open-source is where bad content goes to die,” to which I say this: take a serious look at what we offer our students. Look through a few sections to see if what we’ve written is bad content in your opinion. I see this open- source book not as something which is “free and worth every penny”, but rather, as a high quality alternative to the business as usual of the textbook industry and I hope that you agree. If you have any comments, questions or concerns please feel free to contact me at [email protected] or Carl at [email protected].

Jeff Zeager Lorain County Community College January 25, 2010

Chapter 1

Relations and Functions

1.1 Sets of Real Numbers and The Cartesian Coordinate Plane

1.1.1 Sets of Numbers

While the authors would like nothing more than to delve quickly and deeply into the sheer excite- ment that is Precalculus, experience^1 has taught us that a brief refresher on some basic notions is welcome, if not completely necessary, at this stage. To that end, we present a brief summary of ‘set theory’ and some of the associated vocabulary and notations we use in the text. Like all good Math books, we begin with a definition.

Definition 1.1. A set is a well-defined collection of objects which are called the ‘elements’ of the set. Here, ‘well-defined’ means that it is possible to determine if something belongs to the collection or not, without prejudice.

For example, the collection of letters that make up the word “smolko” is well-defined and is a set, but the collection of the worst math teachers in the world is not well-defined, and so is not a set.^2 In general, there are three ways to describe sets. They are

Ways to Describe Sets

1. The Verbal Method: Use a sentence to define a set.
2. The Roster Method: Begin with a left brace ‘{’, list each element of the set only once and then end with a right brace ‘}’.
3. The Set-Builder Method: A combination of the verbal and roster methods using a “dummy variable” such as x.

For example, let S be the set described verbally as the set of letters that make up the word “smolko”. A roster description of S would be {s, m, o, l, k}. Note that we listed ‘o’ only once, even though it

(^1)... to be read as ‘good, solid feedback from colleagues’... (^2) For a more thought-provoking example, consider the collection of all things that do not contain themselves - this leads to the famous Russell’s Paradox.

2 Relations and Functions

appears twice in “smolko.” Also, the order of the elements doesn’t matter, so {k, l, m, o, s} is also a roster description of S. A set-builder description of S is:

{x | x is a letter in the word “smolko”.}

The way to read this is: ‘The set of elements x such that x is a letter in the word “smolko.”’ In each of the above cases, we may use the familiar equals sign ‘=’ and write S = {s, m, o, l, k} or S = {x | x is a letter in the word “smolko”.}. Clearly m is in S and q is not in S. We express these sentiments mathematically by writing m ∈ S and q /∈ S. Throughout your mathematical upbringing, you have encountered several famous sets of numbers. They are listed below.

Sets of Numbers

1. The Empty Set: ∅ = {} = {x | x 6 = x}. This is the set with no elements. Like the number ‘0,’ it plays a vital role in mathematics.a
2. The Natural Numbers: N = { 1 , 2 , 3 ,.. .} The periods of ellipsis here indicate that the natural numbers contain 1, 2, 3, ‘and so forth’.
3. The Whole Numbers: W = { 0 , 1 , 2 ,.. .}
4. The Integers: Z = {... , − 1 , − 2 , − 1 , 0 , 1 , 2 , 3 ,.. .}
5. The Rational Numbers: Q =

{ (^) a b |^ a^ ∈^ Z^ and^ b^ ∈^ Z

. Rational numbers are the ratios of integers (provided the denominator is not zero!) It turns out that another way to describe the rational numbersb^ is:

Q = {x | x possesses a repeating or terminating decimal representation.}

6. The Real Numbers: R = {x | x possesses a decimal representation.}
7. The Irrational Numbers: P = {x | x is a non-rational real number.} Said another way, an irrational number is a decimal which neither repeats nor terminates.c
8. The Complex Numbers: C = {a + bi | a,b ∈ R and i =

− 1 } Despite their importance, the complex numbers play only a minor role in the text.d a... which, sadly, we will not explore in this text. bSee Section 9.2. cThe classic example is the number π (See Section 10.1), but numbers like √2 and 0. 101001000100001... are other fine representatives. dThey first appear in Section 3.4 and return in Section 11.7.

It is important to note that every natural number is a whole number, which, in turn, is an integer. Each integer is a rational number (take b = 1 in the above definition for Q) and the rational numbers are all real numbers, since they possess decimal representations.^3 If we take b = 0 in the

(^3) Long division, anyone?

4 Relations and Functions

For an example, consider the sets of real numbers described below.

Set of Real Numbers Interval Notation Region on the Real Number Line

{x | 1 ≤ x < 3 } [1, 3) (^1 )

{x | − 1 ≤ x ≤ 4 } [− 1 , 4] (^) − 1 4

{x | x ≤ 5 } (∞, 5] (^5)

{x | x > − 2 } (− 2 , ∞) (^) − 2

We will often have occasion to combine sets. There are two basic ways to combine sets: intersec- tion and union. We define both of these concepts below.

Definition 1.2. Suppose A and B are two sets.

• The intersection of A and B: A ∩ B = {x | x ∈ A and x ∈ B}
• The union of A and B: A ∪ B = {x | x ∈ A or x ∈ B (or both)}

Said differently, the intersection of two sets is the overlap of the two sets – the elements which the sets have in common. The union of two sets consists of the totality of the elements in each of the sets, collected together.^4 For example, if A = { 1 , 2 , 3 } and B = { 2 , 4 , 6 }, then A ∩ B = { 2 } and A ∪ B = { 1 , 2 , 3 , 4 , 6 }. If A = [− 5 , 3) and B = (1, ∞), then we can find A ∩ B and A ∪ B graphically. To find A ∩ B, we shade the overlap of the two and obtain A ∩ B = (1, 3). To find A ∪ B, we shade each of A and B and describe the resulting shaded region to find A ∪ B = [− 5 , ∞).

A = [− 5 , 3), B = (1, ∞)

A ∩ B = (1, 3)

A ∪ B = [− 5 , ∞)

While both intersection and union are important, we have more occasion to use union in this text than intersection, simply because most of the sets of real numbers we will be working with are either intervals or are unions of intervals, as the following example illustrates.

(^4) The reader is encouraged to research Venn Diagrams for a nice geometric interpretation of these concepts.

1.1 Sets of Real Numbers and The Cartesian Coordinate Plane 5

Example 1.1.1. Express the following sets of numbers using interval notation.

1. {x | x ≤ −2 or x ≥ 2 } 2. {x | x 6 = 3}
2. {x | x 6 = ± 3 } 4. {x | − 1 < x ≤ 3 or x = 5}

Solution.

1. The best way to proceed here is to graph the set of numbers on the number line and glean the answer from it. The inequality x ≤ −2 corresponds to the interval (−∞, −2] and the inequality x ≥ 2 corresponds to the interval [2, ∞). Since we are looking to describe the real numbers x in one of these or the other, we have {x | x ≤ −2 or x ≥ 2 } = (−∞, −2] ∪ [2, ∞).

(−∞, −2] ∪ [2, ∞)

2. For the set {x | x 6 = 3}, we shade the entire real number line except x = 3, where we leave an open circle. This divides the real number line into two intervals, (−∞, 3) and (3, ∞). Since the values of x could be in either one of these intervals or the other, we have that {x | x 6 = 3} = (−∞, 3) ∪ (3, ∞)

3. For the set {x | x 6 = ± 3 }, we proceed as before and exclude both x = 3 and x = −3 from our set. This breaks the number line into three intervals, (−∞, −3), (− 3 , 3) and (3, ∞). Since the set describes real numbers which come from the first, second or third interval, we have {x | x 6 = ± 3 } = (−∞, −3) ∪ (− 3 , 3) ∪ (3, ∞).

4. Graphing the set {x | − 1 < x ≤ 3 or x = 5}, we get one interval, (− 1 , 3] along with a single number, or point, { 5 }. While we could express the latter as [5, 5] (Can you see why?), we choose to write our answer as {x | − 1 < x ≤ 3 or x = 5} = (− 1 , 3] ∪ { 5 }.

(− 1 , 3] ∪ { 5 }

1.1 Sets of Real Numbers and The Cartesian Coordinate Plane 7

reach P from the origin (0, 0) by moving 2 units to the right and 4 units downwards. Notice that the order in the ordered pair is important − if we wish to plot the point (− 4 , 2), we would move to the left 4 units from the origin and then move upwards 2 units, as below on the right.

x

y

P

− 4 − 3 − 2 − 1 1 2 3 4

− 4

− 3

− 2

− 1

1

2

3

4

x

y

P (2, −4)

(− 4 , 2)

− 4 − 3 − 2 − 1 1 2 3 4

− 4

− 3

− 2

− 1

1

2

3

4

When we speak of the Cartesian Coordinate Plane, we mean the set of all possible ordered pairs (x, y) as x and y take values from the real numbers. Below is a summary of important facts about Cartesian coordinates.

Important Facts about the Cartesian Coordinate Plane

• (a, b) and (c, d) represent the same point in the plane if and only if a = c and b = d.
• (x, y) lies on the x-axis if and only if y = 0.
• (x, y) lies on the y-axis if and only if x = 0.
• The origin is the point (0, 0). It is the only point common to both axes.

Example 1.1.2. Plot the following points: A(5, 8), B

, C(− 5. 8 , −3), D(4. 5 , −1), E(5, 0),

F (0, 5), G(− 7 , 0), H(0, −9), O(0, 0).^10

Solution. To plot these points, we start at the origin and move to the right if the x-coordinate is positive; to the left if it is negative. Next, we move up if the y-coordinate is positive or down if it is negative. If the x-coordinate is 0, we start at the origin and move along the y-axis only. If the y-coordinate is 0 we move along the x-axis only.

(^10) The letter O is almost always reserved for the origin.

8 Relations and Functions

x

y

A(5, 8)

B

C(− 5. 8 , −3)

D(4. 5 , −1)

E(5, 0)

F (0, 5)

G(− 7 , 0)

H(0, −9)

O(0, 0)

− 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 7 8 9

− 9

− 8

− 7

− 6

− 5

− 4

− 3

− 2

− 1

1

2

3

4

5

6

7

8

9

The axes divide the plane into four regions called quadrants. They are labeled with Roman numerals and proceed counterclockwise around the plane:

x

y

Quadrant I x > 0, y > 0

Quadrant II x < 0, y > 0

Quadrant III x < 0, y < 0

Quadrant IV x > 0, y < 0

− 4 − 3 − 2 − 1 1 2 3 4

− 4

− 3

− 2

− 1

1

2

3

4