COURSE SUCCESS

STUDENT SUPPLEMENTS AUTHOR TIPS STUDENT APPENDIX RELATED LINKS SELECTED SOLUTIONS HINTS FOR SELECTED SOLUTIONS

AUTHOR TIPS Introduction Chapter 1: Introduction to Functions and Graphs Chapter 2: Linear Functions and Equations Chapter 3: Nonlinear Functions and Equations Chapter 4: Inverse Functions Chapter 5: Trigonometric Functions Chapter 6: Trigonometric Identities and Equations Chapter 7: Further Topics in Trigonometry Chapter 8: Systems of Equations and Inequalities Chapter 9: Further Topics in Algebra Introduction Welcome to what we hope will be a rewarding and enjoyable experience studying algebra and trigonometry using the text PRECALCULUS through Modeling and Visualization. This text uses applications and technology to make algebra and trigonometry more relevant and meaningful to students. Mathematical concepts are discussed along with real-life applications. 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 and trigonometry is a successful one. 1. Attend class regularly. If you must miss a class, find out what the next 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 enhance your understanding of mathematics greatly. Practice using your graphing calculator before the test. Be sure to take notes so that you know what 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 on the same day. Try to complete each assignment. If you do not understand an example or exercise, ask questions the next day in class. There is a Student's Solution Manual that will help you solve the odd-numbered exercises. If it is not available at your bookstore you can order it 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 summarizes important points from the section. 5. If you are getting lost, be sure to get help before you are too far behind. 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 perform arithmetic and make scatterplots, tables, and graphs. 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. Specific keystrokes are not provided in the text because there is a wide variety of graphing calculators. You should refer to your owner's manual or the keystroke guide that accompanies your text. See the preface of the text for details. In Section 1.1 different number systems are used to describe data. Algorithms are also introduced as a means to perform computations involving numbers. Society has created large amounts of data and needs to find ways to visualize and analyze this information. In Section 1.2 visualization and graphing calculators are used to summarize data and recognize trends. Number lines, scatterplots, and line graphs are introduced as a means to visualize data. Make sure you understand the difference between one-variable data and two-variable data. Be sure to understand the viewing rectangle (or window) of a graphing calculator as presented on page 21. 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 5 and 6 on page 33, along with Putting It All Together. In Section 1.4 different types of data are presented, which requires us to create different types of functions to model this data. 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, where characteristics of each type of function are outlined. To determine where a function is increasing or decreasing, it is important to remember to move along the function's graph from left to right and give the answer in terms of x-values. In Section 1.5 functions are used to model linear and nonlinear data. Both linear and quadratic models are introduced. It is important to understand the graph of a quadratic function. This includes finding the vertex and determining whether a parabola opens upward or downward. In Section 1.6 we invent our own functions to model data. This is done with transformations of graphs to fit both linear and quadratic data. Review the blue box on page 81 and Putting It All Together on page 89. CHAPTER 2: Linear Functions and Equations In this chapter we increase our mathematical skills and understanding by learning how to solve linear equations and inequalities. Linear equations and inequalities can be solved symbolically, graphically and numerically. In Section 2.1 we learn how to solve linear equations symbolically, graphically, and numerically. See Examples 6 and 8. Linear equations have one solution. Two graphical methods for solving equations are the x-intercept method and the intersection-of-graphs method. Be sure to understand each of these methods because they will be used throughout the course. See pages 104­107. Linear equations are used to model several different types of applications. Be sure to study Figure 2.18 on page 112, where the different ways to solve an equation are outlined. In Section 2.2 linear inequalities are solved. Make sure you review the Properties of Inequalities on page 117, particularly Item 3. The x-intercept method and intersection-of-graphs method are also used to solve inequalities graphically. Remember that interval notation is just a fast way of writing the solutions to an inequality rather than graphing them on a number line. In Section 2.3 various 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 slope-intercept form. For each line, the slope-intercept form is unique because a line has exactly one slope and one x-intercept. Parallel lines have the same slope, while the product of the slopes of two perpendicular lines is equal to ­1. Study Example 7 on pages 138­139. This is a good review of many concepts in this section. See also Putting It All Together on page 142. In Section 2.4 piecewise-defined functions are used to model data. 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 blue box on page 154. To better understand how to solve inequalities involving absolute values, study Figures 2.63 and 2.64, along with the blue box on page 155. In Sections 2.5 linear approximation is discussed. The midpoint formula can be used as a form of linear approximation. The distance formula is introduced on page 166. Extrapolation and interpolation are introduced on page 169. Be sure to understand the difference between extrapolation and interpolation. Interpolation is generally more accurate. Linear regression is a method used to model data. Calculators are capable of finding regression lines. CHAPTER 3: 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, such as the x-intercept method or the intersection-of-graphs method, are also used in Chapter 3. A common theme throughout this text is that we can apply a mathematical technique to a variety of situations. In Section 3.1 we solve quadratic equations and inequalities symbolically, graphically, and numerically. Quadratic equations are one of the simplest types of nonlinear equations. Review factoring quadratic expressions with Examples 1 and 2 on pages 184­185. Other techniques for solving quadratic equations include the square root property, completing the square, and the quadratic formula. The quadratic formula is especially important because it can always be used to find the solutions to a quadratic equation. The discriminant is discussed in the blue box on page 190. If you learned how to solve absolute value inequalities in Section 2.4, you will find the techniques to solve quadratic inequalities similar. In Section 3.2 graphs of nonlinear functions are discussed. Be sure to understand local minimums, local maximums, absolute minimums, and absolute maximums. The term extrema is sometimes used to denote maximums and minimums. 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 pages 204 and 206. 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 208 before working with power functions. In Section 3.3 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 blue boxes on page 225. Reflections of graphs across the x- and y-axis are summarized on page 227. Carefully study Putting It All Together on pages 232­233. Section 3.4 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 to understanding this section is the factor theorem on page 242. Study complete factored form of a polynomial. Polynomials may be factored graphically as well as symbolically. See Examples 4 and 5. Practice solving polynomial equations symbolically, graphically and numerically as illustrated in Examples 9 and 10. Factoring is often used to solve a polynomial equation. Section 3.5 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 3.6 introduces rational functions. Important features on the graph of a rational function are its asymptotes. Vertical asymptotes 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 asymptotes. Sometimes it is advisable to use dot mode instead of connected mode. Another possibility is to use a friendly window. Be sure to study the Chapter 3 Summary on pages 289­290. CHAPTER 4: Inverse Functions In this chapter inverse functions are discussed. Two important types of functions introduced in the chapter are exponential and logarithmic functions. They are inverse functions and occur in applications in scientific areas or in other areas such as business and economics. In Section 4.1 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 of functions can be performed numerically, graphically, and symbolically. Make sure that you understand Example 3 on page 300. Composition of functions is also discussed in this section and there are several applications. Make sure you understand the difference between addition of two functions as shown in Figure 4.1 and composition of functions as shown in Figure 4.6. Putting It All Together on page 308 summarizes many of the results in the section. In Sections 4.2 inverse functions are discussed. Make sure that you can describe the inverse of a simple function verbally as discussed on page 315. For a function to have an inverse function, different inputs must result in different outputs. This is called one-to-one and can be determined graphically using the horizontal line test. Putting It All Together on page 325 summarizes how to find different representations of inverse functions. In Section 4.3 exponential functions are introduced. Make sure that you understand Making Connections and Example 1 on page 333, where polynomial and exponential functions are compared. Spend time reviewing Properties of Exponents on page 334. The natural exponential function is used frequently in applications. See Examples 6 and 7. Be sure to study Putting It All Together on page 342. In Section 4.4 logarithms and logarithmic functions are introduced. Logarithms are challenging for many students and may require extra study time. Study Table 4.16 carefully. 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 4. Although the equations are simple, they sometimes take time to master. It is important that you understand the solution to these equations before going on to the next section. In Section 4.5 a wide variety of applications involving exponential and logarithmic equations are given. Nonlinear regression is also covered. CHAPTER 5: Trigonometric Functions In this chapter the six trigonometric functions and numerous applications are discussed. Although you may have seen trigonometry in the past, most students have not studied trigonometry as much as they have algebra. As a result, students often find that learning trigonometry requires more effort than algebra. Trigonometric functions, like other functions, have graphical, numerical, verbal, and symbolic representations. In Section 5.1 angles are presented. Angles may be measured in either degrees or radians. Radian measure is used frequently in technical fields. It is important to become familiar with radian measure, because it will be used extensively throughout Chapters 5­7. Memorize Table 5.1. Be sure to understand Figure 5.3, where standard position of an angle is illustrated. Arc length and area of a sector formulas are presented in this section. Remember, in both formulas q must be in radians, not degrees. In Section 5.2 the six trigonometric functions are defined using right triangles. It is important to remember that only the measure of the angle q affects the values of trigonometric functions and not the size of a triangle. Try to memorize the blue box on page 404. The trigonometric values of 30°, 45°, and 60° can be computed by hand. However, calculators are usually needed to determine trigonometric values. A common mistake is to have a calculator in the incorrect angle mode when evaluating trigonometric functions. Before starting a problem decide whether your calculator should be in degree or radian mode. In Section 5.3 the sine and cosine functions are defined for any angle q. Several applications including robotics and highway design are discussed. The sine and cosine functions are also defined on the unit circle for any real number. It is important to remember that the input to a trigonometric function can be an angle or a real number. Angles can be measured either in degrees or radians. When finding the trigonometric function of a real number t, be sure to set your calculator in radian mode. Study the graphs of the sine and cosine functions carefully. See Figures 5.71 and 5.77. In Section 5.4 we extend our discussion in Section 5.3 to include the tangent, cosecant, secant, and cotangent functions. Memorize the blue box on page 434. Practice finding the six trigonometric functions using a calculator. To find the cosecant, secant, and cotangent functions you may need to use the reciprocal identities found on page 435. Spend time studying Putting It All Together on page 444. It reviews the domain, range, period and graphs of each trigonometric function. The figure on page 445 can be used to find exact trigonometric values of several special angles. You may want to refer to it in the future. In Section 5.5 we model different types of periodic phenomena found in nature such as temperature, daylight hours, tides, and simple harmonic motion. Either the sine or cosine function may be used to model this type of phenomena. It is important to know how the amplitude, period, phase shift, and vertical shift affect the graphs of the sine or cosine functions. See the blue box on page 452 and Figure 5.130. In Section 5.6 we discuss the inverse trigonometric functions. Be sure to carefully review inverse functions before studying the inverse trigonometric functions. You may wish to review Section 4.2 first. Like other functions, inverse functions have verbal, symbolic, numerical, and graphical representations. It may be helpful to remember that an inverse trigonometric function outputs an angle. For example, arcsin 1 represents the angle whose sine is 1 and lies in the interval [­p/2, p/2]. See Example 2. The inverse cosine and inverse tangent functions are also discussed. Inverse trigonometric functions are used in applications to find angles. See Example 5, where the optimal angle to throw the shot in track and field is calculated. Inverse trigonometric functions are also used to solve triangles and trigonometric equations. Solving a triangle means finding the measures of each side and each angle in a triangle. Study Putting It All Together on page 479 for a summary of the inverse trigonometric angles. CHAPTER 6: Trigonometric Identities and Equations In this chapter we spend time verifying identities symbolically and give graphical and numerical support. Verifying identities requires practice and concentration. Applications are also presented. Identities are often used to make formulas simpler or to solve trigonometric equations. In Section 6.1 we discuss the fundamental identities, which are the reciprocal, quotient, Pythagorean, and negative-angle identities. Try to memorize these identities because it will make verifying other identities easier. Each of the six trigonometric functions is either an odd or even function. The cosine and secant functions are both even and their graphs are symmetric with respect to the y­axis. The other trigonometric functions are odd functions and their graphs are symmetric with respect to the origin. See Figures 6.16­6.21 and Putting It All Together. In Section 6.2 we verify identities symbolically, and give graphical and numerical support. When verifying identities, start with one side of the equation and simplify it to the other side. See Example 4. Follow the suggestion in Putting It All Together. Don't give up­identities takes practice. If you get stuck, ask for help. In Section 6.3 we solve trigonometric equations symbolically, graphically, and numerically. The intersection-of-graphs method and x-intercept method can be used to solve trigonometric equations graphically. See Examples 3-5. Be sure to learn about reference angles. They are important for solving trigonometric equations symbolically, even if you have a calculator. Some types of trigonometric equations cannot be solved symbolically. These types of equations must be solved either graphically or numerically. See Example 9. Several applications involving trigonometric equations are presented. In Sections 6.4 and 6.5 a variety of identities are presented for the sine, cosine, and tangent functions. Symbolic verification with graphical and numerical support is given. These identities have applications in music and touch-tone phones. Practice writing each identity down. There are several identities and it will take time to become familiar with them all. CHAPTER 7: Further Topics in Trigonometry In this chapter several different topics involving trigonometry are introduced. In the first two sections we learn how to solve triangles that do not contain a right angle. In the next section vectors are discussed. Vectors are an important topic because they have applications in many different areas of study. In Sections 7.4 and 7.5 we learn new ways to generate complicated graphs. Finally complex numbers are discussed in the last section. In Section 7.1 the law of sines is presented. The law of sines may be used to solve triangles when two angles and a side are given (ASA or AAS) or two sides and an angle opposite one of the sides are given (SSA). The SSA case is called the ambiguous case because it can result in 0, 1, or 2 different triangles that satisfy the conditions. See Examples 5­7 and Putting It All Together to better understand the ambiguous case. Be sure to learn the standard labeling for triangles illustrated in Figure 7.7. Memorize the law of sines shown in the blue box on page 570. In Section 7.2 the law of cosines is presented. The law of cosines may be used to solve triangles when all three sides (SSS) or two sides and the included angle (SAS) are given. Neither of these cases is ambiguous. Memorize the law of cosines shown in the blue box on page 575. Several applications are presented along with formulas for finding the area of a triangle. Heron's formula is a convenient way to find the area of a triangle when the lengths of all three sides are known. In Section 7.3 vectors are introduced and several applications are presented. Vectors are very important in mathematics because they simplify complicated concepts and have many applications. Remember that vectors have length and direction, but not position. Parallel vectors with identical lengths are equivalent. Vectors can be added, asubtracted, and multiplied by a scalar (real number). See Putting It All Together for a summary of operations on vectors. The dot product is important because it can be used to find the angle between two vectors, calculate work, and make computer graphics more efficient. In Section 7.4 parametric equations are introduced as a way to create a variety of graphs that cannot be represented by functions. See Figures 7.62­7.64. Parametric equations can be represented numerically, graphically, and verbally as shown in Example 1. Parametric equations are used frequently in applications involving computer graphics, motion, and product design. In Section 7.5 we introduce the polar coordinate system, which is different from the xy­coordinate system since it uses r and q to locate a point. See Figure 7.79 and Example 1. There are many interesting and unique curves that can be created using polar coordinates. See Figures 7.87, 7.89, and 7.92. In Section 7.6 we do an in-depth study of complex numbers. You may wish to review concepts of complex numbers presented in Section 3.5 before starting this section. Trigonometric form for complex numbers makes use of polar coordinates to represent a complex number. Products, quotients, and roots of complex numbers can be found more easily in trigonometric form than in rectangular form. Complex numbers have many applications in our society, particularly in electricity and engineering. Study Putting It All Together for a summary of the concepts and formulas presented in this section. CHAPTER 8: 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 8.1 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 have verbal, numerical, symbolic, and graphical representations. Study pages 636­638. In this text we do not emphasize graphical representations of functions of more than one variable because they are often difficult to draw by hand, and many graphing calculators do not have this capability. 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 page 645. In Section 8.2 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 8.27 on page 658. Linear programming is another application of linear inequalities that is important in business. Review Putting It All Together on page 663. In Section 8.3 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 422. 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 7 and 8. In Section 8.4 matrices are used in a variety of applications including digital photography and computer graphics. If you are interested in digital pictures then be sure to read pages 683­684. You will also learn how to add and subtract matrices, and take a scalar multiple of a matrix. Matrix multiplication is introduced and has many applications in computer graphics and business. Matrix multiplication takes practice to perform by hand. Graphing calculators can perform matrix multiplication efficiently. See Example 5. 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 8.5 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 698, which is used in computer graphics to perform translations, has an inverse that is easy to find by hand. Remember that A­1 will undo the operation performed by A. If A translates a point 2 units left, A­1 translates a point 2 units right. Study the blue boxes on pages 699­700, where the identity matrix and inverses are defined. Matrix inverses can be used to solve systems of linear systems. See Examples 4 and 5. In Section 8.6 we introduce determinants. A determinant of a matrix A is a real number. Determinants can be found efficiently using 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 9: 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. Conic sections are discussed in Section 9.3 and have applications in astronomy, construction, optics, and physics. The last two sections involve counting and probability, which have numerous applications in everyday life. In Section 9.1 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. This 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) = crn­1, where r is the common ratio. In Section 9.2 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 + 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, an arithmetic series results. Similarly, if we sum the terms of a geometric sequence, 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 752­753. It is an efficient way to write a series. A summary of these concepts is given in Putting It All Together on page 754. In Section 9.3 conic sections are discussed. There are three types of conic sections: parabolas, ellipses, and hyperbolas. See Figures 9.10­9.12. Parabolas may open to the right or the left as well as upward or downward. The important features on a parabola are its vertex, focus, and directrix. There are several applications of parabolas. Ellipses are a generalization of circles. Important features on an ellipse include the major and minor axes, two foci, and two vertices. Hyperbolas are the third type of conic section. Their graphs have two branches. Important features of hyperbolas are two foci, two asymptotes, two vertices, and a transverse axis connecting the vertices. Planets, satellites, and comets travel in orbits that may be modeled by conic sections. See Figures 9.37­9.39. The standard equation of a circle with center (h, k) is given on page 773. In Section 9.4 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 what 4! means. A permutation is an ordering or arrangement, which can be calculated using the blue box on page 784. A combination represents a subset of a set. The ordering of the elements is unimportant with combinations. Counting combinations can be done using the blue box on page 786. This section concludes with the binomial theorem, which is particularly important in probability and statistics. In Section 9.5 probability is introduced. Probability is used throughout our society and defined in the blue box on page 792. Venn diagrams and tables can sometimes help to determine the probability of certain events. See page 794. Be sure to understand how to find the probability of a complement and the probability of two events. Probability of independent and dependent events are also discussed. Refer to Putting It All Together on page 800 for a summary of these concepts. HINTS FOR SELECTED EXTENDED AND DISCOVERY EXERCISES Gary Rockswold has selected several of the more challenging Extended and Discovery Exercises from each chapter. Below are hints and suggestions for how to solve these more difficult problems. Chapter 1: Introduction to Functions and Graphs Chapter 2: Linear Functions and Equations Chapter 3: Nonlinear Functions and Equations Chapter 4: Inverse Functions Chapter 5: Trigonometric Functions Chapter 6: Trigonometric Identities and Equations Chapter 7: Further Topics in Trigonometry Chapter 8: Systems of Equations and Inequalities Chapter 9: Further Topics in Algebra CHAPTER 1: Introduction to Functions and Graphs Exercise 4, p. 100 In this exercise, think of a partial graph of a circle defined by the function f(x) = , where 0 < x < 750. Using the viewing angle [0, 1200, 100] by [-800, 0, 100], graph the points (535, -400) and (730, -570) together with the function f(x) = . The points represent the location of the cities. The function represents the path of the cold front. Does the function intersect the points? A movement of 110 miles south and 160 miles east translates to the function g(x) = f(x - 160) -110. Use the same viewing angle that you used in part (a). Graph the points (535, -400) and (730, -570) together with the functions f(x) and g(x). Do the points lie on or between the two functions? CHAPTER 2: Linear Functions and Equations Exercise 2, p. 180 The height of a human (in inches) can be estimated by measuring the length of the humerus (in inches). The approximate height of males and females was measured and compared to the length of the humerus. The corresponding data is shown in the table. Find the corresponding y (females) value that corresponds to x = 12. Plot the ordered pairs (x, y (females)) in one color. Plot the ordered pairs (x, y (males)) in a different color. Use the viewing window [7, 15, 1] by [48, 74, 2]. Compare the data sets. Consider the slope m1 = for any two points in the female data set. Consider the slope m2 = for any two points in the male data set. The data is linear. Let f(x) = m1x + b1 model the data for females. Let g(x) = m2x + b2 model the data for males. In this case, use m1 and m2 found in part (c). Determine b1 and b2 by substituting values for (x, f(x)) and (x, g(x)) from the data set. Find f(9.7) and f(10.1). These values represent the range for the height of a female. Find g(9.7) and g(10.1). These values represent the range for the height of a male. Exercise 3, p. 180 Now, using at least 7 females and 8 males, measure the length of the humerus (in inches) and the height of each individual (in inches). (a) Make a data table similar to that in Exercise #2. (b) Using the same process as in Exercise #2, parts (c) and (d), determine linear functions f(x) and g(x) that model the data. Or, use the regression feature of your graphing calculator. (c) How do the results of (b) compare to the results in Exercise #2? Consider that data in Exercise #2 was based on a large sample. Because the data in this exercise is limited to a much smaller set, it may not follow a definite linear trend. Exercise 4, page 180 Consider the age of the earth to be one year. (1) Now consider the ratio of the age of dinosaurs to the age of the earth. The earliest dinosaurs lived years or 14.6 days ago. (Hint: 0.04 years 365days/year = 14.6 days.) (2) Now consider the ratio of the age of Homo sapiens to the age of the earth. Earliest man lived years or 0.58 hours ago. (Hint: years 365days/year 24hours/day = 0.58 hours.) CHAPTER 3: Nonlinear Functions and Equations Exercises 15-20, p. 295 To determine the boundary numbers of f(x) > 0 (or f(x) < 0, or f(x) > 0, or f(x) < 0), consider the values of x for which f(x) = 0 or for which f(x) is undefined. (Factor the numerator and the denominator of f(x), and find the zeros of the numerator and denominator.) These are the boundary numbers. Consider intervals defined by the boundary numbers. Choose a test value in each interval. Determine whether the function is positive or negative in each interval. The solution to the rational inequality is the union of the intervals in which f(x) > 0 (or f(x) < 0, or f(x) > 0, or f(x) < 0). Exercises 9-14, p. 294 To determine the boundary numbers of f(x) > 0 (or f(x) < 0, or f(x) > 0, or f(x) < 0), consider f(x) = 0. Factor the polynomial, and find the zeros of the polynomial. These are the boundary numbers. Consider intervals defined by the boundary numbers. Choose a test value in each interval. Determine whether the function is positive or negative in each interval. The solution to the polynomial inequality is the union of the intervals in which f(x) > 0 (or f(x) < 0, or f(x) > 0, or f(x) < 0). CHAPTER 4: Inverse Functions Exercise 1, p. 384 Consider the data below. x (Distance) 128 181 222 422 671 1070 1883 y (Period) 0.29 0.50 0.67 1.77 3.55 7.16 16.69 Utilizing the regression capabilities of your graphing calculator, determine a function that models this data. Consider linear, quadratic, power and exponential functions. Both quadratic and power function model the data quite closely. (viz. The quadratic function f(x) = and the power function g(x) = .) Consider the fact that because the coefficients of each function are quite small, a noticeable deviance will become apparent only for large distance values (ie. x > 3000). Exercise 2, p. 384 The table presents the increase in radiative forcing over 1750 levels in 50-year increments. The function R(x) = is an approximate model for the data. To solve for C, consider that x = 0 corresponds to the year 1800. Therefore, R(0) = 0.2 = and C = 0.2. (Consider that x = 1 corresponds to the year 1850, x = 2 corresponds to the year 1900, etc.) To estimate the constant k, substitute any ordered pair (x, R(x)) into the function R(x) = and solve for k. The point (4, 2.4) will give the best approximation. To determine when radiative forcing could reach 3 w/m2 solve the equation R(x) = 3 and identify the ordered pair (x, 3). Support this result by graphing Y1 = R(x) and Y2 = 3 in the window [-1, 5, 1] by [0, 3.2, 0.2] and find the intersection of the graphs. CHAPTER 5: Trigonometric Functions Exercise 1, p. 489-490 In this exercise, you will use trigonometry to determine the coordinates of a point Q, given the coordinates of a point P, the distance between P and Q, and the bearing from P to Q. Considering the point P as a starting point, ; . a and b may be determined using trigonometry. Since the bearing between the points P and Q is , we will use its complement, . a = , b = Therefore, In this part, substitute = 152, = 186, = 23.2, and d = 208 into the formulas for and determined in part (a). Exercise 3, p. 490 The maximum average monthly temperature in Buenos Aires is 74º F and the minimum average monthly temperature is 49º F. Consider the behavior of the cosine function: f(x) = acos(b(x - c)) + d. a = amplitude of the graph = 1/2 the difference between the maximum and minimum points = 1/2(74 - 49). = the period of the graph = 12 (the cycle repeats itself every 12 months.) Solve for b. c = the phase shift of the graph. Normally, the maximum point of the cosine graph occurs at = 0. In this case, the maximum point occurs when x = 1. Therefore, the phase shift is 1. d = the vertical phase shift of the graph. A cosine graph with amplitude=a would normally have a maximum value of a and a minimum value of -a. d represents the difference between a and the actual maximum value of 74 , or the difference between -a and the actual minimum value of 49 . Use the value of a found in part 1. Using the values for a, b, c and d found in part (a), construct the function f(x) = acos(b(x - c)) + d. Consider data from a two year period. Month 1 2 3 4 5 6 7 8 9 10 11 12 Temperature (º F) 74 73 69 61 55 50 49 51 55 60 66 71 Month 13 14 15 16 17 18 19 20 21 22 23 24 Temperature (º F) 74 73 69 61 55 50 49 51 55 60 66 71 Graph f(x) along with this data in the window [0, 25, 1] by [48, 75, 1]. Does the function model the data well? c. Consider the fact that seasons in the northern and southern hemispheres are reversed. CHAPTER 6: Trigonometric Identities and Equations Exercise 1, p. 560-561 When a piano string is played, the human ear hears as one tone the sum of the upper harmonics; the fundamental frequency and the higher frequencies, , , etc. The pressure for each frequency in pounds per square foot is modeled by P = 0.002/nsin, where = 440 hertz and = 2, = 3, = 4, etc. Graph the following in the window [0, 0.01, 0.002] by [-0.005, 0.005, 0.001]. P = è = 0.002sin(2(440)x) P = + è = 0.002sin(2(440)x) + = 0.002(sin(2(440)x) + sin(2(880)x)) P = + + è = 0.002sin(2(440)x) + + sin(2(1320)x) = 0.002(sin(2(440)x) + sin(2(880)x) + sin(2(1320)x)) P = + + + è = 0.002sin(2(440)x) ++ sin(2(1320)x) + sin(2(1760)x) = 0.002(sin(2(440)x) + sin(2(880)x) + sin(2(1320)x) + sin(2(1760)x)) P = + + ++ è = 0.002sin(2(440)x) + + sin(2(1320)x) + sin(2(1760)x) + sin(2(2200)x) = 0.002(sin(2(440)x) + sin(2(880)x) + sin(2(1320)x) + sin(2(1760)x) + sin(2(2200)x)) P = + + ++ models what the human ear hears. Consider the relationship of this graph to the others. The graphs in part (a) approach a maximum value of 0.003. Compare the graph of a pure tone, , to the graph of the sound heard by the human ear, . CHAPTER 7: Further Topics in Trigonometry Exercise 4, p. 633 Certain standards to determine the position of important points have been established by the U.S. Coast and Geodetic Survey and the U.S. Geological Survey. The coordinates of two basic control monuments are A(2,101,345.1, 998,764.3) and B(2,131,667.8, 923,541.7). Find the coordinates of an unknown point P if Angle PAB = 37º 41'37" and Angle PBA = 57º 52'04". (1) Find and label a point C on the indicated vertical axis such that AC is perpendicular to BC. Find the length of AC and BC by subtracting the x and y coordinates of points A and B. AC = - and BC = - . Then find the length of AB using the Pythagorean Theorem. AB = Since the sum of the three angles of a triangle is 180º, then Angle APB = 180º - Angle PBA - Angle PAB. And, using the Law of Sines and the measure of AB and Angle APB found above, AP/sin(Angle PBA) = AB/sin(Angle APB). Solve for AP. (2) Now, find and label a point D on the indicated horizontal axis such that AD is perpendicular to PD. Angle DAB can be determined using the definition of tangent. Angle DAB = Angle ABC = tan-1(AC/BC). Angle DAP = Angle DAB - Angle PAB. By the definition of cosine, cos(Angle DAP) = AD/AP. Solve for AD. By the definition of sine, sin(Angle DAP) = DP/AP. Solve for DP. Finally, we can determine the coordinates of the point P(,). = + AD = - DP Exercise 6, p. 633 Given a = , b =, , and a - b =, , apply the Law of Cosines to the triangle and derive the equation a•b = cos .   2 = 2 + 2 - 2cos è 2 -2 - 2 = -2cos è - - = -2 cos è ( -)2 + ( -)2 - (2 + 2) - (2 + 2) = -2cos è 2 - 2+ 2 + 2 - 2 + 2 -2 - 2 - 2 - 2 = -2 cos è -2 - 2 = -2cos è + = cos è a•b = cos CHAPTER 8: Systems of Equations and Inequalities Exercise 2, p. 726 Consider the equations: f(0) = a(0) + b = 3617 f(5) = a(5) + b = 6121 f(10) = a(10) + b = 9340 f(15) = a(15) + b = 12,432 These four equations can be written as: AX = == B or AX = B. The least-squares solution is found by solving the normal equations: ATAX = ATB or X = (ATA)-1 ATB. Enter: A = and B = Use the capabilities of your graphing calculator to solve for X. This is equivalent to finding a and b such that f(x) = ax + b. Exercise 7, p. 726 Using the process outline, code the following words. HELP would initially be coded as 8 5 12 16. Enter these numbers into a matrix B. B = Using A = , find AB. AB = = and scale the elements of AB between 1 and 26 by adding or subtracting multiples of 26. 21 + 0(26) = 21 40 - 1(26) = 14 -50 + 2(26) = 2 -92 + 4(26) = 12 Therefore, the word HELP is coded as 21 2 14 12, or UBNL. LETTER would initially be coded as 12 5 20 20 5 18, so B = and AB = = Now, add or subtract multiples of 26 to scale all elements of AB between 1 and 26. 29 - 1(26) = 3 60 - 2(26) = 8 28 - 1(26) = 2 -70 + 3(26) = 8 -140 + 6(26) = 16 -61 + 3(26) = 17 3 8 8 16 2 17 finally translates to CHHPBQ. CHAPTER 9: Further Topics in Algebra Exercise 1, p. 807 Because of the frequent use of antibiotics, some haploid bacteria, which contain a genetic material called plasmids, have become immune to antibiotics. Geneticists wish to predict this resistance after many generations. Suppose that a certain bacteria contains two plasmids, R1 and R2. Plasmid R1 is resistant to ampicillin and plasmid R2 is resistant to tetracycline. When the bacteria reproduce, the daughter cells could contain either: R1R1--two plasmids of type R1, R1R2--one plasmid of each type, or R2R2--two plasmids of type R2, etc. The probability that a mother cell with k plamids of type R1 produces a daughter cell with j plasmids of type R1 is = Compute for 0 < k < 2 and 0 < j < 2 and record the results in a matrix. P = Assume that =1 and = 0 whenever k < j. Recall that = Note that the matrix is symmetric and the greatest elements lie along the diagonal, the elements , ,. What do these elements represent? Exercise 2, p. 807 Because of the frequent use of antibiotics, some haploid bacteria, which contain a genetic material called plasmids, have become immune to antibiotics. Geneticists wish to predict this resistance after many generations. Suppose that a certain bacteria contains two plasmids, and . Plasmid is resistant to ampicillin and plasmid is resistant to tetracycline. Let be the probability that the daughter cell contains--two plasmids of type. Let be the probability that the daughter cell contains--one plasmid of each type. Let be the probability that the daughter cell contains--two plasmids of type. Define a 13 matrix A = . If an entire generation of bacteria has one plasmid of each type, then A1 = [0 1 0]. The result is that the bacteria is resistant to both antibiotics. An represents the probability in the nth generation for plasmids and, and is defined as An = , where n > 1 and P = It is reasonable to infer that since all bacteria of a generation are resistant to both antibiotics, future generations will be as well. Another possibility is that there is equal probability that the bacteria will be resistant to ampicillin, tetracycline, or both. Determine the actual probabilities. A1 = [0 1 0] A2 = A1P = A3 = A2P = A4 = A3P = . . . An = As n becomes larger, An approaches [1/2 0 1/2] . What does An represent? Does this agree with your conjecture?

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