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Introduction
Test Anxiety
Chapter 1: Introduction to Functions and Graphs
Chapter 2: Linear Functions and Equations
Chapter 3: Quadratic Functions and Equations
Chapter 4: Nonlinear Functions and Equations
Chapter 5: Exponential and Logarithmic Functions
Chapter 6: Trigonometric Functions
Chapter 7: Trigonometric Identities and Equations
Chapter 8: Further Topics in Trigonometry
Chapter 9: Systems of Equations and Inequalities
Chapter 10: Conic Sections
Chapter 11: Further Topics in Algebra
Chapter R: Reference: Basic Concepts from Algebra and Geometry

Introduction

Welcome to what we hope will be a rewarding and enjoyable experience studying algebra using the text COLLEGE ALGEBRA through Modeling and Visualization. This text uses applications and technology to make algebra more relevant and meaningful to students. Mathematical concepts are discussed along with real-life applications. Concepts are often introduced using a concrete example first. The graphing calculator is frequently used to model real data and solve problems that would be difficult to solve with only pencil and paper.

There are several steps that you can take to ensure that your experience in algebra is a successful one.

1. Attend class regularly. If you must miss a class, find out what the assignment is. Learn your instructor's name, office number, and office hours.

2. Bring your book, graphing calculator, and notebook to class every day. Read the textbook because it will greatly enhance your understanding of mathematics. Be sure to practice using your graphing calculator before the test. Take notes so that you know what topics your instructor considers important.

3. Set aside a regular time for studying. It is more productive to study mathematics one hour on four different days than to study four hours all at once. Try to complete each assignment. If you do not understand an example or exercise, ask a question the next class day. There is a Student's Solutions Manual that will help you solve the odd-numbered exercises. If this manual is not available at your bookstore, it can be ordered from the publisher on the Internet.

4. Read the chapter and section introductions. They will give you a better understanding of the coming discussion and give an overview of the relevance of mathematics. Be sure to read Putting It All Together at the end of each section. This subsection summarizes important points and can be used to review for a test.

5. If you are getting lost in the course, be sure to get help from your instructor or a tutor before you are too far behind.

Test Anxiety

It is quite common for students to experience test anxiety in mathematics. On page 125 of the text there is a Critical Thinking exercise that compares how students breathe during a lecture versus during an exam. Students exhale carbon dioxide in significantly larger amounts during an exam than during a lecture, indicating hyperventilation, which can cause difficulty concentrating. When preparing for an exam, you may find the following suggestions helpful.

1. Start studying several days before the test. Being prepared will help reduce your anxiety.

2. Rather than "pulling an all nighter", get a good night's sleep before the exam.

3. Avoid caffeine before an exam. Also, it may be better not to eat a large meal before an exam.

4. Try to breathe through your nose rather than your mouth in a slow, relaxed manner from the lower abdomen. Anxiety is frequently increased by becoming excited and breathing incorrectly.

5. Visualize yourself doing well on the test. Talk to yourself positively about the exam.

6. If you feel that your anxiety is more than normal, there may be an office on campus that can assist you in addressing this issue. Test anxiety is common and you can overcome it with a little guidance.

CHAPTER 1: Introduction to Functions and Graphs

This chapter is important because it develops concepts and skills that will be used throughout the course. An essential concept presented in this chapter is a function. Functions are used throughout this course, so it is essential that you spend time learning this concept. Functions can be represented verbally, symbolically, graphically, and numerically. A verbal representation involves words, a symbolic representation involves a formula, a graphical representation involves a graph, and a numerical representation involves a table of values.

In Chapter 1 some of the important features of a graphing calculator are introduced. You are not expected to be an expert with a graphing calculator when you are finished with this chapter. However, when you have completed this chapter you should be able to evaluate arithmetic expressions and create scatterplots, tables, and graphs on your graphing calculator. There may be other calculator features that your instructor asks you to learn.

Remember that a graphing calculator cannot give right answers if the wrong keys are pressed. A graphing calculator cannot replace mathematical understanding, because it cannot read problems, set up equations, and interpret solutions. However, a graphing calculator can be an enormous aid for tedious calculations and complicated graphs. Our goal is to use the graphing calculator to help us gain mathematical understanding and insight. To learn specific keystrokes on either the TI-83 or TI-83 Plus graphing calculators use Appendix B. There are Graphing Calculator Helps located in the margins throughout the text. These margin notes refer you to a page number in Appendix B whenever the keystrokes for a new graphing calculator skill are needed. You might also want to refer to your owner's manual.

In Section 1.1 number systems, scientific notation, and problem solving skills are introduced. Scientific notation is used in applications to represent numbers that are either very large or very small in absolute value. Two important problems solving strategies are to make a sketch and apply a formula.

In Section 1.2 the number line and the xy-plane are introduced. Number lines, scatterplots, and line graphs can be used to visualize data. Make sure you understand the difference between one-variable data and two-variable data. Graphing calculators can be used to make scatterplots and number lines. Be sure that you understand the viewing rectangle (or window) for a graphing calculator as discussed on page 19.

In Section 1.3 functions are defined. Functions are used to model a wide variety of phenomena and information. It is essential to understand the four main representations of a function: verbal, numerical, symbolic, and graphical. Study Examples 6 and 7 on pages 33-34, and Putting It All Together at the end of the section.

In Section 1.4 various data are presented, which require different types of functions to model them. Some basic types of functions (constant, linear, and nonlinear) are introduced in this section. Slope and average rate of change are used to describe how graphs of functions change. Be sure to review Putting It All Together on pages 52-53, where the characteristics of each type of function are summarized.

CHAPTER 2: Linear Functions and Equations

In this chapter we increase our mathematical skills and understanding by learning how to model data with linear functions and solve linear equations and inequalities. Linear equations and inequalities can always be solved symbolically, graphically and numerically.

In Section 2.1 we learn how to model linear data with linear functions. It is important to remember that most mathematical models are not exact; they are approximate. Two distinguishing features of the graph of a linear function are its slope and its intercepts. Linear functions can be written in the form f(x)= ax + b, where a represents the slope of the graph of f and b represents the y-intercept. The slope corresponds to the constant rate of change in the data, and the y-intercept corresponds to the initial amount of the quantity being modeled. See Examples 3 and 4.

In Section 2.2 several equations of lines are discussed. The point-slope form of a line can be used to determine a line given a point and its slope. If we are given two points, we can determine the slope first and then apply the point-slope form. For each line, the slope-intercept form, y = mx + b, is unique because a line has exactly one slope and one y-intercept. Parallel lines have the same slope, and the product of the slopes of two perpendicular lines is equal to -1. Study Example 9 on pages 82-83. This example gives a good review of many concepts in this section. See also Putting It All Together on page 85.

In Section 2.3 we learn how to solve linear equations graphically, symbolically, and numerically. See Example 6. Linear equations have one solution and can always be solved symbolically. An important graphical method is the intersection-of-graphs method. Be sure to understand this method because it will be used throughout the text. See pages 92-94. Linear equations are used to model several different types of applications. Some important steps for solving application problems are outlined on pages 98-99. These steps are used throughout the text.

In Section 2.4 linear inequalities are solved. Be sure to review the Properties of Inequalities on page 111, particularly Item 3. Both the x-intercept method and the intersection-of-graphs method are also used to solve inequalities graphically. Remember that interval notation is just a fast way of writing the solution set to an inequality rather than graphing the solution set on a number line.

In Section 2.5 piecewise-defined functions are introduced and used to model data. Piecewise-defined functions use more than one formula in their definition. The greatest integer function and the absolute value function are examples of functions that can be defined piecewise. Equations involving absolute values of linear expressions often have two solutions. See the box on page 129. To better understand how to solve inequalities involving absolute values, study Figures 2.68 and 2.69, along with the box on page 131.

In Section 2.6 linear approximation is discussed. The midpoint formula can be used as a form of linear approximation. Extrapolation and interpolation are introduced on page 140-141. Be sure that you understand the difference between extrapolation and interpolation. Interpolation is generally more accurate. Linear regression based on least-squares is a method for modeling data. Graphing calculators are capable of finding regression lines. The correlation coefficient gives a measure of how well the regression line models the data.

CHAPTER 3: Quadratic Functions and Equations

In this chapter we discuss quadratic functions, equations, and inequalities. Quadratic problems are slightly more involved than linear problems, but quadratic problems can also be solved symbolically, graphically, and numerically. Many of the techniques presented in Chapter 2 can be applied to quadratic problems.

In Section 3.1 we use quadratic functions to model quadratic data. Before modeling data, graphs of quadratic functions are discussed. The graph of a quadratic function is a parabola. One important feature of a parabola is the vertex. The vertex of a parabola can always be found by using the vertex formula. The leading coefficient a controls whether a parabola opens upward or downward and whether the parabola is narrow or wide. Certain types of nonlinear data can be modeled with quadratic functions. See Examples 4, 9, and 10. Quadratic regression can be used to find quadratic models.

In Section 3.2 we solve quadratic equations and applications. Quadratic equations are one of the simplest types of nonlinear equations to solve. If you have forgotten how to factor trinomials, there is a review of factoring in Chapter R. Review notes found in the margins of the text refer you to the appropriate pages in Chapter R. See Examples 1 and 2 on pages 174-176 for examples of factoring. Other techniques for solving quadratic equations include the square root property, completing the square, and the quadratic formula. The quadratic formula can always be used to find the solutions to a quadratic equation. The discriminant is discussed in the box on page 180. This section ends with a discussion of how to solve applied problems that involve quadratic equations. The procedure presented in Chapter 2 for solving linear applied problems is used in this section.

In Section 3.3 we solve quadratic inequalities. Quadratic inequalities can be solved quite easily by using graphical techniques. A first step in solving inequalities, both graphically and symbolically, is to find the boundary numbers. A procedure for solving quadratic inequalities symbolically is given on page 193. This procedure begins by finding the boundary numbers and then uses a table of test values to determine where the quadratic inequality is satisfied.

In Section 3.4 several important transformations of graphs are discussed. Vertical and horizontal shifts are summarized in the box on page 199. Graphs of functions can also be stretched or shrunk. These types of transformations are summarized in the boxes on pages 200-201. The graph of y = -f(x) is a reflection of the graph of y = f(x) about the x-axis, whereas the graph of y = f(-x) is a reflection of the graph of y = f(x) about the y-axis. A graphing calculator can be a valuable tool when learning how to transform graphs.

CHAPTER 4: Nonlinear Functions and Equations

In this chapter we discuss nonlinear equations and inequalities. Nonlinear problems are more difficult than linear problems. In real applications it is not uncommon to solve nonlinear equations graphically and numerically rather than symbolically. Many of the techniques that we learned in Chapter 2 and 3, such as the x-intercept method or the intersection-of-graphs method, are also used in Chapter 4. A common theme throughout the text is that one mathematical concept can be applied to a variety of situations.

In Section 4.1 graphs of nonlinear functions are discussed. Polynomial functions of degree 2 or higher are nonlinear functions. To determine where a function is either increasing or decreasing, move along the function's graph from left to right. Give the answer in terms of x-intervals. Be sure that you understand the concepts of local minimum, local maximum, absolute minimum, and absolute maximum. The term extrema refers to either maximums or minimums. Graphs of functions may display either symmetry across the y-axis or symmetry about the origin. Learn the definitions of odd and even functions and the type of symmetry the graph of each type of function displays. See page 228.

In Section 4.2 graphs of polynomials are discussed. It is essential to understand how increasing the degree of a polynomial changes the graph of a polynomial. Study the concepts of a zero, x-intercept, turning point, and end behavior of a polynomial. Review the boxes on page 241. Polynomial regression is a way to find polynomial functions that model real data. Be sure to carefully study Putting It All Together on pages 245-246.

Section 4.3 is slightly more theoretical than the previous sections. In this section our goal is to factor polynomials and solve polynomial equations of degree 3 or higher. To do this, division of polynomials is introduced. Either long division or synthetic division may be used to divide polynomials. An essential theorem in this section is the factor theorem, found on page 255. Study complete factored form of a polynomial. Polynomials may be factored graphically as well as symbolically. See Examples 6 and 7. Try to solve a polynomial equation symbolically, graphically and numerically, as illustrated in Example 12. Factoring is often used to solve polynomial equations.

Section 4.4 introduces the complex numbers and discusses the fundamental theorem of algebra. The fundamental theorem tells us that we can factor any polynomial in complete factored form if we allow zeros to be complex numbers. It is important to realize that complex numbers are used more frequently as our society becomes more technological.

Section 4.5 introduces rational functions. Important features on the graph of a rational function are its asymptotes. Vertical asymptotes often occur where there are discontinuities or breaks in the graph of a rational function. Horizontal asymptotes indicate what happens to the graph of a rational function as |x| becomes large. Graphing rational functions with a graphing calculator is more challenging than other functions encountered so far because of vertical and horizontal asymptotes. Sometimes it is advisable to use dot mode instead of connected mode. Another possibility is to use a friendly window.

Section 4.6 introduces techniques to solve polynomial and rational inequalities. Be sure to study the boxes on pages 304 and 306. When solving polynomial inequalities, start by finding the boundary numbers where equality occurs. Then use either a graph or a table of values to determine where inequality occurs. Rational inequalities can be solved in a similar manner, except boundary numbers include x-values where the rational expression is undefined.

Section 4.7 discusses power functions. Power functions are used to model a variety of data and are used extensively in calculus. Make sure that you review Properties of Exponents on page 313 before working with power functions. Root functions, such as the square root and cube root functions, are examples of power functions. When solving a radical equation by squaring both sides, be sure to check your answers in the given equation. This is because it is possible to obtain extraneous solutions when squaring both sides of an equation. Be sure to study Putting It All Together on pages 319-320. It provides a good review of the types of functions that we have studied thus far.

CHAPTER 5: Exponential and Logarithmic Functions

In this chapter we discuss inverse functions. Two important types of functions introduced are exponential and logarithmic functions. They are inverse functions and occur in applications in science, business, and economics.

In Section 5.1 we introduce how to combine functions. In arithmetic we add, subtract, multiply, and divide numbers, and in beginning algebra we add, subtract, multiply, and divide variables. Now in college algebra we learn how to add, subtract, multiply, and divide functions. Arithmetic operations on functions can be performed numerically, graphically, and symbolically. Make sure that you understand Example 3 on page 337. Composition of functions is also discussed in this section, and there are several applications. Make sure you understand the difference between multiplication of two functions and the composition of two functions, as illustrated in Figure 5.5. Putting It All Together on page 345 provides a summary of the concepts presented in this section.

In Section 5.2 inverse functions are discussed. Make sure that you can describe the inverse of a simple function verbally as discussed on page 353. For a function to have an inverse function, different inputs must result in different outputs. This type of function is called one-to-one function. A one-to-one function can be determined graphically by using the horizontal line test. Putting It All Together on page 363-364 summarizes how to find different representations of inverse functions.

In Section 5.3 exponential functions are introduced. Linear growth and exponential growth are compared. Make sure that you understand these two types of growth. Study Making Connections and Example 3 on page 375, where the growth for polynomial and exponential functions are compared. Several applications of exponential functions are given. The natural exponential function is used frequently in applications. Be sure to study Putting It All Together on page 383-384.

In Section 5.4 logarithms and logarithmic functions are introduced. Logarithms are challenging for many students and may require extra study time. Study Tables 5.19 and 5.20. Notice that the output from the common logarithmic function is an exponent of a power of 10. Spend time learning how to solve the equations in Examples 3 and 5. Although the equations are simple, it is important that you can solve these equations before going on to the next section.

Section 5.5 contains a wide variety of applications involving exponential and logarithmic equations. Remember that to solve an exponential equation, we often take the logarithm of both sides of the equation, and to solve a logarithmic equation we often exponentiate both sides of the equation.

Section 5.6 uses nonlinear regression to model three different types of data. The three types of models presented are exponential, logarithmic, and logistic. A graphing calculator can be very helpful when finding one of these types of modeling functions.

CHAPTER 6: Trigonometric Functions

In Section 6.2 the six trigonometric functions are introduced using right triangles. Be sure to memorize the box on page 466 that gives the definitions of the trigonometric functions. You will use it throughout Chapters 6-8. Become familiar with the triangles shown in Figure 6.37 and 6.40. You will use these triangles when you calculate the trigonometric functions by hand in Sections 6.3 and 6.4. There are many applications of right triangle trigonometry. Be sure to work some applications involving right triangles.

In Section 6.3 we extend the domains of the sine and cosine functions to include all angles and all real numbers. Be sure your calculator is in radian mode when evaluating a trigonometric function of a real number. It is important that you know what the graphs sine and cosine look like. Use Putting It All Together on page 489 to help you remember how the sine and cosine functions are defined.

Section 6.4 is really a continuation of Section 6.3, where the other four trigonometric functions (tangent, cotangent, secant, and cosecant) are defined for general angles and real numbers. To calculate the cotangent, secant, and cosecant functions with a calculator, use the reciprocal identities found on page 495. Whenever you know both the sine and cosine of an angle, you can use the reciprocal and quotient identities to find the other four trigonometric functions. Be sure to study Putting It All Together on page 503. Learn the domain, range, and graph of each trigonometric function. The unit circle on page 504 can be used to find the exact values of the trigonometric function of several common angles.

In Section 6.5 transformations of graphs are discussed. Many of these transformations were first discussed in Section 3.4. Be sure that you understand what the amplitude, period, phase shift, and vertical shift represent in a sinusoidal graph. Using these concepts, sinusoidal data can be modeled, as illustrated in Example 6.

In Section 6.6 the inverse sine, cosine, and tangent functions are introduced. Be sure to review inverse functions. You may want to refer to Section 5.2, where inverse functions are first introduced. Remember that arcsin x, arccos x, and arctan x represent angles. If you are given the sides of a right triangle and want to determine an angle, you can use an inverse trigonometric function to find it. Be sure to study Putting It All Together on pages 536-537.

CHAPTER 7: Trigonometric Identities and Equations

In Section 7.2 we learn how to verify identities by using the fundamental identities from Section 7.1. Remember that many of the techniques that you learned in algebra can be used to simplify trigonometric expressions. When verifying an identity, try to simplify one side of the identity into the other. Avoid working on both sides of the identity at the same time. Read through the suggestions for verifying identities found in Putting It All Together.

In Section 7.3 we first study reference angles. Reference angles are important because they are used to solve trigonometric equations. Unlike polynomial equations, trigonometric equations can have an infinite number of solutions. Trigonometric equations can be solved symbolically, graphically, and numerically. Many of the techniques that you learned in algebra can be used to solve trigonometric equations symbolically. Examples include factoring and the square root property. When finding all solutions to a trigonometric equation, one approach is to find all solutions on the interval 0  x < 2p . Then add 2p n to each solution.

Section 7.4 introduces a number of sum and difference identities. Identities are typically a new topic for most students and require extra study time to master. A summary of the identities is given in Putting It All Together on page 597.

In Section 7.5 several multiple angle identities are introduced. When simplifying identities, it is important to remember that sin 2q 2 sin q , cos 2q 2 cos q , and tan 2q 2 tan q . A summary of the identities presented in this section is given in Putting It All Together on page 615.

CHAPTER 8: Further Topics in Trigonometry

In Section 8.2 we discuss the law of cosines. The law of cosines (page 641) is a generalization of the Pythagorean theorem. The law of cosines can be used to solve triangles when you are given two sides and the included angle (SAS) or three sides (SSS). Both of these situations result in a unique solution for the given triangle. Methods for calculating areas of triangles are also covered.

In Section 8.3 vectors are discussed. Vectors have direction and magnitude, but not position. Two vectors can be added, subtracted. A vector can be multiplied by a scalar (real number). The dot product of two vectors can be used to calculate work and the angle between two vectors. If the dot product equals zero, then the two vectors are perpendicular. See Putting It All Together on pages 660-661.

In Section 8.4 parametric equations are discussed. Parametric curves can be used to create complicated curves that can be represented by a function. See Figures 8.58-8.60 on page 667. Parametric equations are also used in the study of electronics and the flight of an object through the air. Graphing calculators can be used to graph parametric equations.

Section 8.5 introduces polar equations. Note that the polar coordinates of a point are not unique, whereas the xy-coordinates of a point are unique. Be sure to learn how to convert coordinates between polar and rectangular. Graphing calculators can be used to graph polar equations. Refer to Putting It All Together on page 687 to help distinguish the various types of polar equations.

Section 8.6 discusses trigonometric form for complex numbers. Trigonometric form for a complex number is not unique. When complex numbers are written in trigonometric form, it is easy to multiply and divide them. De Moivre’s Theorem (page 695) can be used to find powers and roots of complex numbers. By introducing complex numbers, we can find all roots of a number.

CHAPTER 9: Systems of Equations and Inequalities

In previous chapters we have concentrated on functions of one input and equations involving only one variable. In this chapter we discuss functions of more than one input and equations involving more than one variable. We also discuss how matrices are used to solve systems of equations.

In Section 9.1 functions of more than one input are introduced. There are many examples of quantities that require more than one variable to compute. For example, we need to know both the width W and the length L to compute the area of a rectangle. Functions of more than one input often result in the need to solve systems of equations in more than one variable. One way to solve a system of equations in more than one variable symbolically is to use substitution, which may be used to solve either linear or nonlinear systems of equations. Systems of equations involving two variables can also be solved graphically and numerically. Be sure to review Putting It All Together on pages 459-460.

In Section 9.2 we concentrate on systems of linear equations in two variables. Inequalities are also discussed. A linear system can have 0, 1, or an infinite number of solutions. Two symbolic techniques to solve a system of linear equations are substitution and elimination. Be sure to understand both of these methods. Systems of linear inequalities occur frequently in applications. See for example, Figure 6.24 on page 469. Linear programming is another application of linear inequalities that is important in business. Review Putting It All Together on page 476-477.

In Section 9.3 we begin by solving systems of equations involving three variables. Before proceeding further, be sure that you can represent a linear system by an augmented matrix. See Examples 1 and 2 on page 483-484. Row-echelon form is an important form when solving linear systems with augmented matrices. Linear systems can be solved either by hand or by using technology. Gaussian elimination can be performed by hand to solve small systems of linear equations. It usually requires practice to become proficient at Gaussian elimination. Take careful notes the day your instructor discusses this method. Graphing calculators have the capability to solve linear systems. This is illustrated in Examples 8-10.

In Section 9.4 matrices are used in a variety of applications including digital photography. If you are interested in digital pictures then be sure to read pages 499-502. You will also learn how to add and subtract matrices, and take a scalar multiple of a matrix. Matrix multiplication is introduced and has applications in computer graphics and business. Matrix multiplication takes practice to perform by hand. Graphing calculators can perform matrix multiplication efficiently. See Example 8. Remember that matrix multiplication is not commutative, that is, AB does not equal BA. This is different than multiplication of numbers, variables, and functions.

In Section 9.5 we study matrix inverses, how they are used in applications, and how to determine them by hand and with a graphing calculator. Most matrix inverses cannot be found mentally. However, the matrix introduced on page 513, which is used in computer graphics to perform translations, has an inverse that is easy to find by hand. Remember that A^-1 will undo the operation performed by A. If A translates a point to the left 2 units left, then A^-1 translates a point to the right 2 units. Study the boxes on pages 451-452, where the identity matrices and inverses are defined. Matrix inverses can be used to solve systems of linear equations. See Examples 5 and 6.

In Section 9.6 we introduce determinants. A determinant of a matrix A is a real number. Determinants can be found efficiently with a graphing calculator and may be used to find the area of polygons. See Example 5. Cramer's Rule makes use of determinants to solve small linear systems. However, Cramer's rule is never used to solve large linear systems.

CHAPTER 10: Conic Sections

In Chapter 10 we discuss parabolas, ellipses, circles, and hyperbolas. Circles are examples of ellipses. Conic sections have many applications, particularly in astronomy.

In Section 10.1 a formal definition of a parabola is given. This definition allows for a parabola to open in any direction. In this section we discuss parabolas that open to the left or to the right, in addition to parabolas that open upward or downward. Two important features of a parabola are its focus and its directrix. Be sure to understand how each of these features influences the graph of a parabola. Parabolas have an important reflective property that is used in telescopes, headlights, and satellite dishes. Be sure to study Putting It All Together on pages 555-556.

In Section 10.2 a formal definition of an ellipse is given. Ellipses are used to model orbits of all types of celestial objects and satellites. Two important features of an ellipse are its foci and its major axis. Be sure to understand how each of these features influences the graph of an ellipse. The standard equation of a circle with center (h, k) is given on page 567. Like parabolas, ellipses have an important reflective property that is used in medicine and whispering galleries. The solution set to nonlinear equations and inequalities is also discussed in this section. Be sure to study Putting It All Together on pages 570-571.

In Section 10.3 a formal definition of a hyperbola is given. Hyperbolas are used to model trajectories of celestial objects and satellites. Two important features of a hyperbola are its foci and its asymptotes. Be sure to understand how each of these features influences the graph of a hyperbola. Like parabolas and ellipses, hyperbolas have an important reflective property that is used in telescopes. Be sure to study Putting It All Together on pages 582.

CHAPTER 11: Further Topics in Algebra

In this chapter we study several topics in algebra. The first two topics are sequences and series. Sequences are used in a wide variety of areas and series are used in science to make approximations. The last three sections discuss counting, the binomial theorem, and probability, which have numerous applications in everyday life.

In Section 11.1 we discuss sequences. A sequence is a function that computes an ordered list. A simple example is 1, 4, 9, 16, 25, 36, 49, 64, where f(n) = n² for n = 1, 2, 3, ... , 8. Sequences may have a finite or infinite number of terms. Because sequences are functions, we already know many properties of sequences. Sequences can be represented symbolically, numerically, and graphically. See Example 4. Sequences can be defined symbolically using a formula. Recursive formulas are sometimes used to define sequences. A recursive formula is a different type of formula and should be studied carefully. See Example 3. Arithmetic and geometric sequences are introduced. An arithmetic sequence is generated by a linear function given by f(n) = dn + c, where d is the common difference. A geometric sequence is given by f(n) = cr^n-1, where is the common ratio.

In Section 11.2 series are introduced. A series is the summation of the terms of a sequence. For example, a sequence is given by 1, 3, 5, 7, 9, 11, 13, 15, whereas an example of a series is

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15.

It is essential to understand the difference between a sequence and a series. Series can have a finite or infinite number of terms. Finite series always have a sum, whereas an infinite series may or may not have a sum. If we sum the terms of an arithmetic sequence, then an arithmetic series results. Similarly, if we sum the terms of a geometric sequence, then a geometric series results. An infinite arithmetic series has no sum, whereas an infinite geometric series has a sum if its common ratio r is less than one in absolute value. Summation notation is introduced on pages 613-616 and is an efficient way to write a series. A summary of these concepts is given in Putting It All Together on page 616-617.

In Section 11.3 the notion of counting is discussed. Counting in mathematics includes much more than counting from 1 to 100. It also includes things like counting the different ways that a lottery ticket can be filled out. An essential concept is the fundamental counting principle. Factorial notation is important to counting. Learn the meaning of n!. A permutation is an ordering or arrangement, which can be calculated using the box on page 625. A combination represents a subset of a set. The ordering of the elements is unimportant with combinations. Combinations can be calculated using the blue box on page 626.

In Section 11.4 the binomial theorem is introduced. Pascal's triangle can be used to calculate the binomial coefficients.

In Section 11.5 probability is introduced. Probability is used throughout our society and defined in the box on page 637. Venn diagrams and tables can sometimes help to determine the probability of certain events. See pages 639-640. Be sure to understand how to find the probability of a complement and the probability of two events. The probabilities of independent and dependent events are discussed. Refer to Putting It All Together on page 645-646 for a summary of these concepts.

Chapter R Reference: Basis Concepts from Algebra and Geometry

This chapter is intended to be a reference chapter. It reviews several important topics from earlier mathematics courses. Your instructor may give assignments from this chapter to review important skills needed in college algebra.

In Section R.1 formulas from geometry are discussed, beginning with geometric shapes in a plane. These formulas include areas and perimeters of rectangles, triangles, and circles. The Pythagorean theorem is reviewed. Then the volume and surface area of boxes, cylinders, and cones are discussed. Finally, similar triangles are reviewed.

In Section R.2 equations and graphs of circles are discussed. Finding the center and radius of a circle from its general form is also discussed.

In Section R.3 a review of properties of integer exponents is given. Many of these properties hold for rational exponents as well. Be sure you understand the properties of integer exponents before working with rational exponents in Section 4.7.

In Section R.4 polynomial expressions are discussed. Addition, subtraction, and multiplication of polynomials are reviewed. The distributive property is important to review because polynomial multiplication is based on this property. Some special products, such as the product of a sum and difference and squaring a binomial ,are also discussed.

In Section R.5 various types of factoring are reviewed. Although you have probably seen most of these factoring techniques in other mathematics courses, you may find it helpful to review them. Factoring is used to solve polynomial equations. The sum and difference of two cubes is sometimes used in calculus.

In Section R.6 rational expressions are discussed. Addition, subtraction, multiplication, and division of rational expressions are reviewed. Remember that before you can add or subtract two rational expressions, they must have a common denominator. To multiply two rational expressions, simply multiply the numerators and multiply the denominators. To divide two rational expressions remember to "invert and multiply."

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