A Preview of College Algebra CHAPTER

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1 hal9_ch_-9.qd //9 : PM Page A Preview of College Algebra CHAPTER Chapter Outline. Solving Sstems of Linear Equations b Using Augmented Matrices. Sstems of Linear Equations in Three Variables. Horizontal and Vertical Translations of the Graphs of Functions. Reflecting, Stretching, and Shrinking Graphs of Functions. Algebra of Functions. Sequences, Series, and Summation Notation.7 Conic Sections Train Wheel Design The ale of a train is solid so that the wheels must turn together. On the arc of a circular curve in the tracks, the wheels must travel different distances while turning the same number of times. This is accomplished b using a slightl slanted wheel. This allows the wheels to slide on the rails so that the point of contact with the rail varies the turning radius of each wheel. In the figure, which wheel is the inside wheel on a curve and which is the outside wheel? This question is eamined in Eercise of Eercises.. r r

2 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Section. Solving Sstems of Linear Equations b Using Augmented Matrices The laws of mathematics can be arrived at b the principle of looking for the simplest concepts and the link between them. PHILOSOPHER, 9 ) RENÉ DESCARTES (FRENCH MATHEMATICIAN AND A Mathematical Note James Joseph Slvester ( 97) wrote a paper formulating man of the properties of matrices in and introduced the term matri. Objective:. Use augmented matrices to solve sstems of two linear equations with two variables. In Chapter, we covered two algebraic methods for solving sstems of linear equations: the substitution method and the addition method. Wh Do We Need Another Algebraic Method for Solving Sstems of Linear Equations? The answer to this question is that the substitution method and the addition method work fine for relativel simple sstems, but these methods become increasingl complicated to eecute for mess coefficients or for larger sstems of equations. Graphs and tables are also limited to linear sstems with onl two variables. Thus to handle larger sstems of linear equations, we will develop over the net two sections the augmented matri method. This method is ver sstematic and simplifies the solution of larger sstems of linear equations. This method is also well suited to implementation b computer or calculator. After we have developed the augmented matri method in this section, we illustrate in Section. how to use the TI- Plus calculator to implement this method.. Use Augmented Matrices to Solve Sstems of Two Linear Equations with Two Variables We now develop the augmented matri method, a method that has man similarities to the addition method. A matri is a rectangular arra of numbers arranged into rows and columns. Each row consists of entries arranged horizontall. Each column consists of entries arranged verticall. The dimension of a matri is given b stating first the number of rows and then the number of columns. The dimension of the following matri is, read two b three, because it has two rows and three columns. Matri c d Dimension rows b columns The entries in an augmented matri for a sstem of linear equations consist of the coefficients and constants in the equations. To form the augmented matri for a sstem, first we align the similar variables on the left side of each equation and the constants on the right side. Then we form each row in the matri from the coefficients and the constant of the corresponding equation. A should be written in an position that corresponds to a missing variable in an equation. Sstem of Linear Equations e f Coefficients of Coefficients of Augmented Matri Brackets enclose the matri. Constants Optional vertical bar to separate coefficients from the constant terms

3 hal9_ch_-9.qd //9 : PM Page. Solving Sstems of Linear Equations b Using Augmented Matrices (-) Eample Writing Augmented Matrices for Sstems of Linear Equations Write an augmented matri for each sstem of linear equations. (a) (b) (c) 9 c d 9 c 9 d c d The -coefficients and are in the first column. The -coefficients and are in the second column. The constants and are in the third column. First write each equation with the constant on the right side of the equation. Then form the augmented matri. Write each equation in the form A B C, using zero coefficients as needed. Then form the augmented matri. Self-Check Write the augmented matri for the sstem e f. To solve sstems of linear equations b using augmented matrices, we need to be able to represent these sstems b using matrices and also be able to write the sstem of linear equations represented b a given augmented matri. Eample Writing Sstems of Linear Equations Represented b Augmented Matrices Using the variables and, write a sstem of linear equations that is represented b each augmented matri. (a) c (b) c (c) c 7 d d 7 d 7 7 Each row of the matri represents an equation with an -coefficient, a -coefficient, and a constant on the right side of the equation. When the coefficient of a variable is, ou do not have to write this variable in the equation. For eample, can be written as. The last pair of equations can be written as 7,. Thus the solution of this sstem is the ordered pair (7, ). Self-Check Using the variables and, write the sstem of linear equations represented b the augmented matri c 9 d.

4 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra In order to simplif or use matrices with a calculator, we must first learn how to enter a matri into a calculator. Technolog Perspective.. illustrates how to do this with a TI- Plus calculator. Note that the MATRIX feature is the secondar function of the - ke. Technolog Perspective.. Entering a Matri into a Calculator Enter the matri from Eample (a) into a calculator. TI- Plus Kestrokes. Access the matri EDIT menu b pressing nd MATRIX and TI- Plus Screens ENTER then.. Enter the dimensions of this matri b pressing ENTER and then.. Enter each element of the matri. ENTER ENTER Press after each value. Then press nd QUIT.. To displa a matri that has been entered as Matri, ENTER nd MATRIX press. Technolog Self-Check Enter the matri from Eample (b) into a calculator. The augmented matri c 7 in Eample (c) is an ecellent eample of the d tpe of matrices that we want to produce. It follows immediatel from this matri that the solution of the corresponding sstem of linear equations has 7 and. In ordered-pair notation, the answer is (7, ). What Matri Represents the of a Sstem of Linear Equations? A matri of the form c k d represents a sstem of linear equations whose solution k is the ordered pair (k, k ). The material that follows shows how to produce this form. Equivalent sstems of equations have the same solution set. The strateg for solving a sstem of linear equations is to transform the sstem into an equivalent sstem composed of simpler equations. In Chapter, we used the properties of equalit to justif the transformations we used to solve sstems of linear equations. We review these transformations and give the corresponding elementar row operations on a matri.

5 hal9_ch_-9.qd //9 : PM Page. Solving Sstems of Linear Equations b Using Augmented Matrices (-) Transformations Resulting Elementar Row Operations in Equivalent Sstems on Augmented Matrices. An two equations in a sstem ma. An two rows in the matri ma be be interchanged. interchanged.. Both sides of an equation in a. An row in the matri ma be sstem ma be multiplied b a multiplied b a nonzero constant. nonzero constant.. An equation in a sstem ma be. An row in the matri ma be replaced b the sum of itself and a replaced b the sum of itself and a constant multiple of another equation constant multiple of another row. in the sstem. We use the elementar row operations on an augmented matri just as if the rows were the equations the represent. This is illustrated b the parallel development in Eample. Eample Solve the sstem Solving a Sstem b Using the Addition Method and the Augmented Matri Method e f. Addition Method e f Augmented Matri Method r r c d r r r c d e f e f e f e f Answer: (, ) c c c r r r r r d d d This notation means that the first row is interchanged with the second row. Replace the second row with itself minus times the first row. 9 Multipl the second row b. Replace the first row with itself plus times the second row. This form gives the answer. Does this answer check? Self-Check What is the solution of the sstem of linear equations represented b c? d

6 hal9_ch_-9.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra Wh Do We Use Arrows, Not Equal Smbols, to Denote the Flow from One Matri to the Net? The matrices are not equal because entries in them have been changed. However, if we use the elementar row operations, the do represent equivalent sstems of equations and will help us to determine the solution of the sstem of linear equations. The recommended first step in transforming an augmented matri to the reduced form c k d is to get a to occur in the row, column position. k Eample illustrates three was to accomplish this. We suggest that ou master the notation that is used to describe each step. This will help ou better understand this topic and also prepare ou to use calculator and computer commands to perform these elementar row operations. Eample Using the Notation for Elementar Row Operations Use the elementar row operations to transform c 9 d into the form c d. (a) B interchanging rows and (b) B multipling row b c 9 d c 9 d r r r r c 9 d 9 The notation r r denotes that rows and have been interchanged. r denotes the new row obtained b multipling r b. (c) B replacing row with the sum of row and times row. c 9 d r r r c 9 d r denotes the new row obtained b adding row and times row. Self-Check Perform the indicated elementar row operations to produce new matrices. a. c r c r r d d b. c 9 r c 9 r r d d r c. c 7 r c 7 d d The method used in Eample (a) ma be the easiest to appl, but it works onl when there is a coefficient of in another row to shift to this first row. The method used in part (c) is often used to avoid the fractions that can result from the method in part (b). Eample uses the elementar row operations to solve a sstem of linear equations.

7 hal9_ch_-9.qd //9 : PM Page 7. Solving Sstems of Linear Equations b Using Augmented Matrices (-7) 7 Eample Solving a Sstem of Linear Equations b Using an Augmented Matri Use an augmented matri and elementar row operations to solve e. f e f Answer: (, ) r r r r r r r r r r c c c c d c d d d d First form the augmented matri. To work toward the reduced form c k d, interchange rows k and to place a in row, column. Net add r to r to produce a in row, column. Then divide row b to produce a in row, column. To complete the transformation to the form c a in row, column b adding r to r. d, produce k Write the equivalent sstem of equations for the reduced form. Does this answer check? k Self-Check 9 7 Use an augmented matri and elementar row operations to solve e. f Note that in Eample, we first worked on column and then on column. This column-b-column strateg is recommended for transforming all augmented matrices to reduced form. This strateg also is used in Eample. Eample Solving a Sstem of Linear Equations b Using an Augmented Matri Use an augmented matri and elementar row operations to solve e 9 f. e 9 f r r r r r r r c 9 9 d First form the augmented matri. Then work on putting column in the form c r d. means to multipl row b. r r means to subtract twice row from row. Net work on column to put the matri in the reduced form c means to multipl row b r. k k d.

8 hal9_ch_-9.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra r r r r means to subtract from row. r r Write the equivalent sstem of equations for the reduced form. Answer: a, b Does this answer check? Self-Check 9 Use an augmented matri and elementar row operations to solve e. f The reduced form c alwas ields a unique solution for a consistent sstem k d k of independent equations. For an inconsistent sstem with no solution or a consistent sstem of dependent equations with an infinite number of solutions, the reduced form is different. We now eamine these two possibilities. Solving an inconsistent sstem b the addition method produces an equation that is a contradiction. Note the matri equivalent of this in Eample 7. Eample 7 Solving an Inconsistent Sstem b Using an Augmented Matri Use an augmented matri and elementar row operations to solve e f. e f Answer: r r r There is no solution. c d c d First form the augmented matri. Then work on putting column in the form c d. There is no need to proceed further because the last row in the matri corresponds to an equation that is a contradiction. Thus the original sstem of equations is inconsistent and has no solution. Self-Check 7 Write the solution of the sstem of linear equations represented b c 7. d We encourage ou to compare this Eample 7 to the Eample 7 in Section. where the same problem was worked b the addition method. Eample eamines a consistent sstem of dependent equations.

9 hal9_ch_-9.qd //9 : PM Page 9. Solving Sstems of Linear Equations b Using Augmented Matrices (-9) 9 Eample Solving a Consistent Sstem of Dependent Equations b Using an Augmented Matri Use an augmented matri and elementar row operations to solve e f. e f r r r r r c d First form the augmented matri. Then work on putting column in the form c The last row corresponds to an equation that is an identit. It cannot be used to simplif column further. These two equations form a dependent sstem of equations with an infinite number of solutions. Because the coefficient of in the first equation is, solve this equation for in terms of. You can also solve this equation for d. Answer: There are infinitel man solutions, all having the form a., b in terms of and epress the answer in the form a, b. You ma wish to confirm that both of these represent the same set of points. All the solutions are points on the same line, points that can be written in this form. Self-Check Write the solution of the sstem of linear equations represented b c. d The general solution of a sstem of dependent linear equations describes all solutions of the sstem and is given b indicating the relationship between the coordinates of the solutions. The particular solutions obtained from the general solution contain onl constant coordinates. The general solution in Eample is a B arbitraril selecting -values,, b. we can produce as man particular solutions as we wish. For and, we obtain two particular solutions a and (, )., b Self-Check Answers. c d. a. c. e 9 f b. c 9 d. (, ) c. c 7 d d.. (, ) (., ) 7. There is no solution; it is an inconsistent sstem.. A dependent sstem with a general solution (, ) ; three particular solutions are (, ), (, ), and (, ).

10 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Technolog Self-Check Answer.. Using the Language and Smbolism of Mathematics. A matri is a arra of numbers.. The entries in an augmented matri for a sstem of linear equations consist of the and in the equations.. sstems of equations have the same solution set.. The notation r r denotes that rows and for a matri are.. The notation r denotes that row is r being replaced b multipling the current row b.. The notation to represent that row is being replaced b the sum of the current row plus twice row is. 7. The solution of a sstem of dependent linear equations describes all solutions of the sstem and is given b indicating the relationship between the coordinates of the solutions.. The solutions obtained from a general solution contain onl constant coordinates.. Quick Review. Determine whether (, ) is a solution of the sstem 7 e f.. If the -coordinate of the solution of e f is, determine the -coordinate.. If both sides of the equation are multiplied b, the resulting equivalent equation is.. Solve e b the substitution method. f. Solve e b the addition method. f. Eercises Objective Use Augmented Matrices to Solve Sstems of Two Linear Equations with Two Variables In Eercises, write an augmented matri for each sstem of linear equations In Eercises 7, write a sstem of linear equations in and that is represented b each augmented matri. 7. c. c d d 9. c. c d d. c 7. d

11 hal9_ch_-9.qd /7/9 : AM Page. Solving Sstems of Linear Equations b Using Augmented Matrices (-) In Eercises, use the given elementar row operations to complete each matri In Eercises, write the solution for the sstem of linear equations represented b each augmented matri. If the matri represents a consistent sstem of dependent equations, write the general solution and three particular solutions.. c. c 9 d d. c. c 7 d d. c. c d d In Eercises 7, label the elementar row operation used to transform the first matri to the second. Use the notation developed in this section c d c d c d c d c c c c c c c d c 7 d c 9 d c d d d 7 d 9 d d d r r r r r r r r r r r r r?????? r r r r r r r r c c c c. c? c d 7 d 9 d 9 d d 9 c c c c c c c c d d d d d d d d d. In Eercises, use an augmented matri and elementar row operations to solve each sstem of linear equations Connecting Concepts to Applications In Eercises, write a sstem of linear equations using the variables and, and use this sstem to solve the problem.. Numeric Word Problem Find two numbers whose sum is and whose difference is.. Numeric Word Problem Find two numbers whose sum is if one number is times the other number.. Complementar Angles The two angles shown are complementar, and one angle is larger than the other. Determine the number of degrees in each angle.. Supplementar Angles The two angles shown are supplementar, and one angle is 7 larger than the other. Determine the number of degrees in each angle.?

12 hal9_ch_-9.qd /7/9 :9 AM Page (-) Chapter A Preview of College Algebra. Fied and Variable Costs A seamstress makes custom costumes for operas. One month the total of fied and variable costs for making costumes was $,. The net month the total of the fied and variable costs for making costumes was $,. Determine the fied cost and the variable cost per costume.. Rates of Two Bicclists Two bicclists depart at the same time from a common location, traveling in opposite directions. One averages km/h more than the other. After hours, the are km apart. Determine the speed of each bicclist. 7. Rate of a River Current A small boat can go km downstream in hour, but onl km upstream in hour. Determine the rate of the boat and the rate of the current.. Miture of Two Disinfectants A hospital needs L of a % solution of disinfectant. How man liters of a % solution and a % solution should be mied to obtain this % solution? 9. Miture of a Fruit Drink A fruit concentrate is % water. How man liters of pure water and how man liters of concentrate should be mied to produce L of miture that is % water?. Basketball Scores During one game for the Phoeni Suns, Steve Nash scored 9 points on 7 field goals. How man of these field goals were -pointers and how man were -pointers?. Radius of a Train Wheel On a curve, the radius r of the inside train wheel is less than the radius r of the outside wheel. This allows the outside wheel to travel a greater distance as the train goes around the curve. On one curve, both radii are measured in inches, and the result is r r.. The inside wheel covers. inches per revolution. Determine the radius of the inside wheel and the radius of the outside wheel. r r Group discussion questions. Challenge Question Solve e a b c f for (, ) a b c in terms of a, a, b, b, c,and c. Assume that a b a b.. Discover Question a. Etend the augmented matri notation given for sstems of two linear equations with two variables to write an augmented matri for this sstem. z z z b. Write a sstem of linear equations that is represented b this augmented matri. Use the variables,,and z. c. Write a sstem of linear equations that is represented b this augmented matri. Use the variables w,,, and z. 7 7 Cumulative. Cumulative Review. Write the first five terms of the arithmetic sequence defined b a n n.. Write the th term of the arithmetic sequence defined b a n n.. Write the first five terms of the geometric sequence defined b a n n.. Simplif (a b) (a) (b) a b assuming all bases are nonzero.. Simplif ( ) () /.

13 hal9_ch_-9.qd //9 : PM Page. Sstems of Linear Equations in Three Variables (-) Section. Sstems of Linear Equations in Three Variables Objective:. Solve a sstem of three linear equations in three variables. A first-degree equation in two variables of the form A B C is called a linear equation because its graph is a straight line if A and B are not both. Similarl, a first-degree equation in three variables of the form A B Cz D also is called a linear equation. However, this name is misleading because if A, B, and C are not all, the graph of A B Cz D is not a line but a plane in three-dimensional space. The graph of a three-dimensional space on two-dimensional paper is limited in its portraal of the third dimension. Nonetheless, we can give the viewer a feeling for planes in a three-dimensional space b orienting the -, -, and z-aes as shown in the figure. This graph illustrates the plane z, whose -intercept is (,, ), whose -intercept is (,, ), and whose z-intercept is (,, ). Drawing lines to connect these intercepts gives the view of the plane in the region where all coordinates are positive. The plane defined b z z. Solve a Sstem of Three Linear Equations in Three Variables A sstem of three linear equations in three variables is referred to as a (three-bthree) sstem. A solution of a sstem of equations with the three variables,, and z is an ordered triple (,, z) that is a solution of each equation in the sstem. Eample Determining Whether an Ordered Triple Is a of a Sstem of Linear Equations z Determine whether (,, ) is a solution of z. z First Equation Second Equation Third Equation z ( )?? is true. Answer: (,, ) is not a solution of this sstem. z z () ( )? () ( ) ()???? is true.? is false. To be a solution of this sstem, the point must satisf all three equations. Self-Check Determine whether (,.,.7) is a solution of the sstem in Eample. The graph of each linear equation A B Cz D is a plane in three-dimensional space unless A, B, and C are all. A sstem of three linear equations in three variables can be viewed geometricall as the intersection of a set of three planes. These planes ma intersect in one point, no points, or an infinite number of points. The illustrations in the following bo show some of the was we can obtain these solutions. Can ou sketch other was of obtaining these solution sets?

14 hal9_ch_-9.qd //9 :7 PM Page (-) Chapter A Preview of College Algebra Tpes of Sets for Linear Sstems with Three Equations The linear sstem A B C z D A B C z D A B C z D can have An Infinite Number One No of s II I III III II II I III I The planes intersect at a single point; the sstem is consistent and the equations are independent. The planes have no point in common; the sstem is inconsistent. The planes intersect along a line and thus have an infinite number of common points; the sstem is consistent and the equations are dependent. Can I Use Graphs to Solve Sstems of Three Linear Equations with Three Variables? No, although the figures in the bo can give us an intuitive understanding of the possible solutions to these sstems it is not practical to actuall solve these sstems graphicall. Thus we rel entirel on algebraic methods. In this section, we illustrate two methods for solving sstems of three linear equations in three variables ( sstems). Eample etends the substitution and addition methods from Sections. and. to solve a sstem. Later in the section, we use augmented matrices, which were introduced in Section.. Equivalent sstems of equations have the same solution set. The general goal of each step of a solution process is to produce an equivalent sstem that is simpler than the previous step. B eliminating some of the variables, we can reduce a sstem to a sstem with onl two variables, and then we can eliminate another variable to produce an equation with onl one variable. We can then back-substitute to obtain the values of the other two variables. This strateg is outlined in the following bo. Strateg for Solving a Sstem of Linear Equations * Step. Write each equation in the general form A B Cz D. Step. Select one pair of equations and use the substitution method or the addition method to eliminate one of the variables. Step. Repeat step with another pair of equations. Be sure to eliminate the same variable as in step. Step. Eliminate another variable from the pair of equations produced in steps and, and solve this sstem of equations. Step. Back-substitute the values from step into one of the original equations to solve for the third variable. Step. Does this solution check in all three of the original equations? * If a contradiction is obtained in an of these steps, the sstem is inconsistent and has no solution. If an identit is obtained in an step, the sstem is either dependent with infinitel man solutions or inconsistent with no solution.

15 hal9_ch_-9.qd //9 :7 PM Page. Sstems of Linear Equations in Three Variables (-) Eample Solve the sstem Solving a Sstem of Linear Equations z z. z Produce a Sstem of Equations () () () z z z The sstem of equations is Produce an Equation with Onl One Variable z e z f Back-Substitute () () z z z z z z z z e z f. () () z z z First decide which variable to eliminate. Here eliminate b working first with equations () and () and then with equations () and (). Add equations () and (). Then add equations () and (). Note that has been eliminated from both of these equations. Subtract the second equation from the first equation to eliminate. Solve this equation for z. z () z First substitute for z in the equation () () () z and solve for. Then substitute for and for z in the equation z and solve for. The answer as an ordered triple is (,, ). Check: Check (,, ). () z () z () z? ( ) ()??? is true.? is true.? is true. Answer: The solution (,, ) checks in all three equations. Self-Check z Solve the sstem z. z We can also use augmented matrices to organize and epedite our work in solving sstems of linear equations. To prepare for this, we eamine what matrices for sstems of linear equations look like. In Eample, the entries in the first column of the matri are the coefficients of, the entries in the second column are the coefficients of, the entries in the third column are the coefficients of z, and the entries in the fourth column are the constants.

16 hal9_ch_-9.qd //9 :7 PM Page (-) Chapter A Preview of College Algebra Eample Writing Augmented Matrices for Sstems of Linear Equations Write an augmented matri for each sstem of linear equations. z (a) z z (b) 7 z z 7 The augmented matri contains the coefficients and the constants from each equation. Use zero coefficients as needed for an missing terms. Self-Check Write the augmented matri for this sstem of linear equations. z z The reduced form for the augmented matri associated with a consistent sstem of k three independent linear equations with three variables is k. The solution k of this sstem is (k, k, k ). Similar forms can be obtained for dependent or inconsistent sstems. The reduced form defined here is also called reduced row echelon form. Properties of the Reduced Row Echelon Form of a Matri. The first nonzero entr in a row is a. All other entries in the column containing the leading are s.. All nonzero rows are above an rows containing onl s.. The first nonzero entr in a row is to the left of the first nonzero entr in the following row. Eample Using Elementar Row Operations Use the elementar row operations to transform the matri form. into the

17 hal9_ch_-9.qd /7/9 : AM Page 7. Sstems of Linear Equations in Three Variables (-7) 7 r r r r r r r r Place a in the upper left position b interchanging the first and third rows. Introduce s into column, rows and. To produce the new row, subtract twice row from row. To produce the new row, subtract times row from row. Self-Check Produce a in row, column of the matri the second and third rows. b interchanging The last two steps in Eample can be combined, as illustrated in Eample. Eample Solving a Sstem of Linear Equations b Using an Augmented Matri a b c Solve a b c. a b c 7 The first step is to form the augmented matri. 7 r r r r r r r r r r r r r r r r M m Transform the first column into the form. Transform the second column into the form.

18 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra r r M 7 m Transform the third column into the form. Answer: (,, ) r r r r r r The answer is displaed in the last column of the reduced form. Does this answer check? Self-Check Write an ordered triple for the solution of the sstem of linear equations represented b. The overall strateg used in Eample can be summarized as work from left to right and produce the leading s before ou produce the s. This strateg is formalized in the following bo. Transforming an Augmented Matri into Reduced Echelon Form Step. Step. M o M p o p m p o o o p m Transform the first column into this form b using the elementar row operations to a. Produce a in the top position. b. Use the in row to produce s in the other positions of column. Transform the net column, if possible, into this form b using the elementar operations to a. Produce a in the net row. b. Use the in this row to produce s in the other positions of this column. If it is not possible to produce a in the net row, proceed to the net column. Step. Repeat step column b column, alwas producing the in the net row, until ou arrive at the reduced form. For emphasis, remember that our goal is to produce the leading s before ou produce the s. You ma use shortcuts in this process whenever the are appropriate. This procedure also works for dependent and inconsistent sstems.

19 hal9_ch_-9.qd //9 : PM Page 9. Sstems of Linear Equations in Three Variables (-9) 9 Eample Solving an Inconsistent Sstem b Using an Augmented Matri r s t Solve s t. s t r r r Although this matri is not in reduced form, the last row indicates that the sstem is inconsistent with no solution. The last row represents the equation r s t, which is a contradiction. Answer: There is no solution. Self-Check Write the solution for the sstem of linear equations represented b. Eample 7 produces the general solution for a consistent sstem of dependent linear equations. Eample 7 Solving a Consistent Sstem of Dependent Equations b Using an Augmented Matri Solve. Thus r r r r r r Answer: The general solution is (,, ); three particular solutions are (,, ), (,, ), and ( 7,, ). The new matri is in reduced form. A row of s in an n n consistent sstem indicates a dependent sstem with infinitel man solutions. This is the sstem represented b the reduced matri. The sstem is dependent, since the last equation is an identit satisfied b all values of,, and. The general solution is acquired b solving the first two equations for and in terms of. These particular solutions were found b letting be,, and, respectivel.

20 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Self-Check 7 Write the general solution for the sstem of linear equations represented b. Forms That Indicate a Dependent or Inconsistent n n Sstem of Equations Let A be the augmented matri of an n n sstem of equations.. If the reduced form of A has a row of the form [...k], where k, then the sstem is inconsistent and has no solution.. If the sstem is consistent and the reduced form of A has a row of the form [...] (all zeros), then the equations in the sstem are dependent and the sstem has infinitel man solutions. The matri method is also a convenient method for solving sstems of linear equations that do not have the same number of variables as equations. Eample Solving a Sstem of Equations with More Variables Than Equations Find a general solution and three particular solutions for the 7z sstem e. z f c 7 d r r r r r r r Use the elementar row operations to transform the matri to reduced row echelon form. r r r z e z f z z Answer: The general solution is ( z, z, z) ; particular solutions are (,, ), (, 7, ), and (,, ). c d This is the sstem represented b the reduced matri. The general solution is obtained b solving these equations for and in terms of z. The particular solutions were found b letting z be,, and, respectivel.

21 hal9_ch_-9.qd //9 : PM Page. Sstems of Linear Equations in Three Variables (-) Self-Check Write the general solution and one particular solution for the sstem z e. z f Some graphing calculators have the capabilit to produce the reduced form of an augmented matri. On a TI- Plus calculator, the rref feature produces the row echelon form of a matri. This is option B in the MATH menu of MATRIX menu. Before ou work through Technolog Perspective.., first store the matri from Eample in our calculator as Matri. Note that the screen size ma not displa all entries on one view. You ma need to use the arrow kes to scroll though all the entries. Technolog Perspective.. Using a Matri to Solve a Sstem of Linear Equations Solve the sstem of equations from Eample. TI- Plus Kestrokes. First store the matri for Eample as Matri.. Access the matri MATH menu b nd MATRIX pressing and then use the down arrow ke until ou reach option B. Choose this rref feature b TI- Plus Screens pressing.. Net select Matri as the matri to nd MATRIX reduce b pressing and close with a right parenthesis.. To perform the row reduction, ENTER press. ENTER Answer: (,, ) Technolog Self-Check Use the rref feature to solve Eample. The graphical method and tables of values can be used to solve some sstems of linear equations with two variables. Eample 9 involves a sstem with four equations and four variables. Graphs and tables are not appropriate tools for these larger sstems because it is difficult to graphicall depict more than three dimensions. For these larger sstems, augmented matrices are often used with calculators or computers.

22 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Eample 9 Solving a Sstem of Four Linear Equations b Using an Augmented Matri w z w z Use an augmented matri and a calculator to solve μ. w z w z First form the augmented matri. Note that it is a matri. Enter this as matri [A] on a calculator. Then use the rref feature to transform this matri to reduced row echelon form. Obtain the answer from this reduced form. Answer: (,,, ) Does this answer check? Self-Check 9 w z w z Solve μ. w z w z Self-Check Answers. (,.,.7) is a solution of the sstem.. (,, )... (,, ). There is no solution. 7. ( z, z, z). ( z, z, z) ; (,, ) 9. (,,, ) Technolog Self-Check Answer. (,, )

23 hal9_ch_-9.qd /7/9 : AM Page. Sstems of Linear Equations in Three Variables (-). Using the Language and Smbolism of Mathematics. The graph of A B Cz D, if A, B, and C are not all, is a in three-dimensional space.. A of a sstem of linear equations with three variables is an ordered triple that satisfies each equation in the sstem.. A sstem of linear equations contains linear equations with variables.. The reduced form for an augmented matri is also called reduced row form.. In reduced row echelon form, a row of the form [...k], where k, represents an sstem of linear equations.. Quick Review. Determine whether (, ) is a solution of the sstem Choose the letter that best describes each sstem of e. 9 f equations.. e f A. A consistent sstem of independent equations. Solve b the substitution method.. e f B. A consistent sstem of dependent equations. e f C. An inconsistent sstem. Eercises Objective Solve a Sstem of Three Linear Equations in Three Variables In Eercises, solve each sstem of three linear equations in three variables.. z. z 7 z z z z. z z z. z z z. z. z z z z 7 z 7 7. z z 7 z. z z z z z z z. z 7. z z z. z. z z z z z In Eercises, write an augmented matri for each sstem of linear equations. z. z z z. z 7 z z 9 7. z. z z In Eercises 9, write a sstem of linear equations in,, and z that is represented b each augmented matri

24 hal9_ch_-9.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra 7.. In Eercises, use the given elementar row operations to complete each matri r r r r r r r r 9 r r r 7 9 r r r 7 7 r r r r r r r r r 7 r r r 9 r r r r 9 r r r r r r r r r r r r In Eercises 7, write the solution for the sstem of linear equations represented b each augmented matri. If the matri represents a dependent sstem, write the general solution and three particular solutions In Eercises, label the elementar row operation used to transform the first matri to the second. Use the notation introduced in Section In Eercises 9, use an augmented matri and elementar row operations to solve each sstem of linear equations z z z z. z. z z z z z r? r?

25 hal9_ch_-9.qd //9 : PM Page. Sstems of Linear Equations in Three Variables (-).. a b c a b c a b c a b c a b c a b c 7. r s t. r s t r s t r s t r 9s t r s t In Eercises 9 and, find a general solution and two particular solutions for each sstem. 9. a b c. a b c a b c a b c 7. Miture of Foods A zookeeper mies three foods, the contents of which are described in the following table. How man grams of each food are needed to produce a miture with g of fat, 9 g of protein, and,7 g of carbohdrates? Connecting Concepts to Applications. Numeric Word Problem The sum of three numbers is. The largest number is less than the sum of the other two numbers. The sum of the largest and the smallest is twice the other number. Find the three numbers.. Numeric Word Problem The largest of three numbers is 7 times the second number. The second number is 7 times the smallest number. The sum of the numbers is. Find the three numbers.. Dimensions of a Triangle The perimeter of this triangle is cm. The length of the longest side is twice that of the shortest side. The sum of the lengths of the shortest side and the longest side is cm more than the length of side b. Find the length of each side. B a c. Dimensions of a Triangle Triangle ABC has sides a, b, and c with side a the longest side and side b the shortest side. The length of the longest side of the triangle is cm less than the sum of the lengths of the other two sides. The length of the shortest side is cm more than one-half the length of side c. Find the length of each side if the perimeter is cm.. Angles of a Triangle Triangle ABC has angles A, B, and C with angle A the largest angle and angle B the smallest angle. Angle C is twice as large as the smallest angle. The largest angle is 9 larger than the sum of the other two angles. Find the number of degrees in each angle. (Hint: The sum of the angles of a triangle is.). Angles of a Triangle Triangle ABC has angles A, B, and C with angle A the largest angle and angle B the smallest angle. The smallest angle of this triangle is 7 less than the largest angle. Angle C is times as large as the smallest angle. How man degrees are in each angle? C b A A B C Fat (%) Protein (%) Carbohdrates (%) 7. Use of Farmland A farmer must decide how man acres of each of three crops to plant during this growing season. The farmer must pa a certain amount for seed and devote a certain amount of labor and water to each acre of crop planted, as shown in the following table. A B C Seed cost ($) Hours of labor Gallons of water 9 7 The amount of mone available to pa for seed is $,. The farmer s famil can devote, hours to tending the crops, and the farmer has access to, gal of water for irrigation. How man acres of each crop would use all these resources? Connecting Algebra to Geometr In Eercises 9 and 7, match each graph with the linear equation that defines this plane. 9. z 7. z A. z B. z

26 hal9_ch_-9.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra In Eercises 7 and 7, graph the plane defined b each linear equation. 7. z 7. z Group discussion questions 7. Discover Question a b a. Solve e a b f. b. Use the solution for part a to solve this nonlinear sstem: μ. 7. Discover Question a b c a. Solve a b c. a b c 7 { } b. Use the solution for part a to solve this nonlinear z sstem:. z z 7 7. Challenge Question Find the constants a, b, and c such that (,, ) is a solution of the linear sstem a b cz a b cz a b cz 7. Challenge Question Solve the following sstem for (,, z) in terms of the nonzero constants a, b, and c. a b cz a b cz a b cz. Cumulative Review. Use the distance formula to find the distance between the points (, ) and (, ).. The two legs of a right triangle measure 7 cm and cm. What is the length of the hpotenuse?. Multipl ( )( ).. List all integer factors of from to.. List all integer factors of 7 from to 7. Section. Horizontal and Vertical Translations of the Graphs of Functions Objective:. Analze and use horizontal and vertical translations.. Analze and Use Horizontal and Vertical Translations Each tpe of algebraic equation f () generates a specific famil of graphs. Some of the families that we have eamined in earlier chapters are linear functions, absolute value functions, quadratic functions, rational functions, square root functions, eponential functions, and logarithmic functions. Each member of a famil of functions shares a characteristic shape with all other members of this famil. Now we will use one basic graph from a famil to generate other graphs in this famil.

27 hal9_ch_-9.qd //9 : PM Page 7. Horizontal and Vertical Translations of the Graphs of Functions (-7) 7 What Is Meant b a Vertical Translation of a Graph? This is a shift up or down of the graph. To analze these shifts we start b observing the patterns illustrated b the following functions. Observing patterns is an important part of mathematics. B recognizing patterns, we can gain insights that deepen our understanding and make us more productive. The functions compared net are and. Numericall Graphicall 7 Verball Each -value is more than The graph of can be obtained b the corresponding -value. shifting the graph of up units. The graph of can be obtained b verticall shifting or translating the graph of up units. Vertical translations or vertical shifts are described in the following bo. Vertical Shifts of f() For a function f () and a positive real number c: Algebraicall Graphicall Numericall f () c To obtain the graph of f () c, shift the graph of f () up c units. f () c To obtain the graph of f () c, shift the graph of f () down c units. For the same -value, c. For the same -value, c. The graph of each quadratic equation function f () a b c is a parabola (see Section.). In Eample, note that the verte of is (, ) and the verte of is (, ).

28 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Eample Eamining a Vertical Shift of a Quadratic Function Compare the functions and numericall, graphicall, and verball. Numericall Graphicall Verball For each value of, is The parabola defined b can be obtained b units less than. shifting the parabola defined b down units. The range of is [, ). The range of is [, ). Self-Check Compare the graphs of and. Eamples and illustrate how to recognize functions that are vertical shifts of each other. Eample eamines two graphs, and Eample eamines two tables of values. Eample Identifing a Vertical Shift Use the graphs of f () and f () to write an equation for in terms of f (). ƒ () ƒ () The graph of f () can be obtained b shifting the graph of f () up units. Thus f (). Ke points shifted are as follows: (, ) is shifted up units to (, ). (, ) is shifted up units to (, ). (, ) is shifted up units to (, ). (, ) is shifted up units to (, ). Answer: f ()

29 hal9_ch_-9.qd //9 : PM Page 9. Horizontal and Vertical Translations of the Graphs of Functions (-9) 9 Self-Check Use the graph of f () to graph f (). ƒ() Eample Identifing a Vertical Shift Use the table of values for f () and f () to write an equation for in terms of f (). f () f () 7 9 For each value of, is units less than. Thus f (). Answer: f () Self-Check Use the table for f () to complete the table for f (). Since Vertical Shifts Affect the -values of a Function, Do Horizontal Shifts Affect the -values? Yes, we will start our inspection of horizontal shifts b eamining the functions and. Numericall Graphicall

30 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Verball For the same -value in and, the -value is units less for than for. The graph of can be obtained b shifting the graph of to the left units. Horizontal Shifts of f() For a function f () and a positive real number c: Algebraicall Graphicall Numericall f ( c) To obtain the graph of f ( c), For the same -value in f ( c) shift the graph of f () to the left c units. To obtain the graph of f ( c), shift the graph of f () to the right c units. and, the -value is c units less for than for. For the same -value in and, the -value is c units more for than for. Caution: Adding a positive c shifts the graph to the left. Reconsider. Note that output values of occur units sooner for than the do for the function. Two units sooner means units to the left on the number line. Eample Eamining a Horizontal Shift of a Quadratic Function Compare the functions f and f () ( ) () numericall, graphicall, and verball. Numericall 9 9 f () 9 9 f () ( ) Graphicall ƒ () 9 7 ƒ () ( ) Verball For the same -value in f and f, the The graph of f () ( ) can be -value is units more for f than for f. obtained b shifting the parabola defined b f () to the right units. The verte of f () is (, ). The verte of f () ( ) is (, ). Self-Check Compare the graphs of f () and f () ( ). Eample eamines two graphs that are horizontal translations or shifts of each other.

31 hal9_ch_-9.qd //9 : PM Page. Horizontal and Vertical Translations of the Graphs of Functions (-) Eample Identifing a Horizontal Shift Use the graphs of f () and f () to write an equation for in terms of f (). ƒ () ƒ () Note that the graph of f () can be obtained b shifting the graph of f () to the right units. Thus f ( ). Ke points shifted are as follows: (, ) is shifted right units to (, ). (, ) is shifted right units to (, ). (, ) is shifted right units to (, ). Answer: f ( ) Self-Check Use the graph of f () to graph f ( ). ƒ() Eample eamines horizontal shifts when two functions are defined b a table of values. Eample Using Translations to Report Sales Data When working with data for which the input values are ears, analsts often make use of a horizontal translation to work with smaller input values but with equivalent results. The following tables represent the sales figures for a compan founded in. Use the information in the tables to write an equation for f () as a translation of f (). Sales Year ($),, 7,, 9 7,, Sales Year ($),,,, 7,, Note that corresponds to ear, to ear, and so on. We can obtain a table of values for based on the table for b shifting the input -values to the left units. Thus f ( ). Answer: f ( ) Self-Check The following two tables show the number of students who completed a graphics design program that started in 99 at a Midwestern communit college. Year Students 99 Year Students Use the information in the tables to write an equation for f () as a translation of f ().

32 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Eample 7 gives some additional practice recognizing horizontal and vertical shifts from the equation defining the function. Eample 7 Identifing Horizontal and Vertical Shifts Describe how to shift the graph of f () to obtain the graph of each function. (a) f () Shift the graph of f () up units. (b) f () Shift the graph of f () down units. (c) f ( ) Shift the graph of f () left units. (d) f ( ) Shift the graph of f () right units. Self-Check 7 Describe the shift in the graph of f () to obtain the graph of each function. a. f ( ) b. f () c. f () d. f ( ) Can I Combine Horizontal and Vertical Shifts in the Same Problem? Yes, Eample illustrates how to do this. Eample Combining Horizontal and Vertical Shifts Use the graphs of f () and f () to write an equation for in terms of f (). Give the domain and range of each function. ƒ () ƒ () The graph of f () can be obtained b shifting the graph of f () to the right units and up unit. Note how this shift affects the domain and range. Ke points shifted are as follows: (, ) is shifted right units and up unit to (, ). (, ) is shifted right units and up unit to (, ). Answer: f ( ) Domain of : Range of : Domain of : D [, ] R [, ] D [, ] Range of : R [, ] Self-Check Use the graph of f () and f () to write an equation for in terms of f (). 7 (, ) ƒ () ƒ ()

33 hal9_ch_-9.qd //9 : PM Page. Horizontal and Vertical Translations of the Graphs of Functions (-) The parabola f () has a verte of (, ). This parabola can be shifted horizontall and verticall to produce man other parabolas. The verte is a ke point on each of these parabolas. Eample 9 gives practice in identifing the verte of a parabola from the defining equation. Eample 9 Determining the Verte of a Parabola Determine the verte of each parabola b using the fact that the verte of f () is (, ). (a) f () 7 Verte of (, 7) (b) f () ( 7) Verte of ( 7, ) (c) f () ( 7) 9 Verte of (7, 9) (d) f () ( ) Verte of (, ) This is a vertical shift 7 units up of ever point on f (). This is a horizontal shift 7 units left of ever point on f (). This is a horizontal shift 7 units right and a vertical shift 9 units up of ever point on f (). This is a horizontal shift units left and a vertical shift units down of ever point on f (). ƒ() Verte (, ) Self-Check 9 Determine b inspection the verte of the parabola defined b each function. a. f () ( ) b. f () c. f () ( ) d. f () ( ) In Eample 9(c), ou can think of the shifts following the same order as the order of operations in the epression. First we subtract inside the parentheses, producing a shift 7 units to the right. Later we add 9, producing a shift 9 units up. Self-Check Answers. For each value of, is units more than. The parabola defined b can be obtained b shifting the parabola defined b up units The parabola f () ( ) can be obtained b shifting the parabola defined b f () to the left unit. 9.. f (,99) 7. a. Shift the graph of f () to the right units. b. Shift the graph of f () down units. c. Shift the graph of f () up units. d. Shift the graph of f () to the left units.. f ( ) 9. a. (, ) b. (, ) c. (, ) d. (, )

34 hal9_ch_-9.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra. Using the Language and Smbolism of Mathematics. Each member of a famil of functions has a graph with the same characteristic.. A shift of a graph up or down is called a shift or a translation.. A shift of a graph left or right is called a shift or a translation.. For a function f () and a positive real number c: a. f () c will produce a translation c units. b. f () c will produce a translation c units. c. f ( c) will produce a translation c units. d. f ( c) will produce a translation c units.. For a function f () and positive real numbers h and k: a. f ( h) k will produce a translation h units and a translation k units. b. f ( h) k will produce a translation h units and a translation k units.. The shape of the graph of f () is called a, and the lowest point on this graph is called its.. Quick Review Make a quick sketch of the graph of each function.. f (). f (). f (). f (). f (). Eercises Objective Analze and Use Horizontal and Vertical Translations Eercises give functions that are translations of f (). Match each graph to the correct function.. f (). f (). f (). f () A. B. C. D. Eercises give functions that are translations of f (). Match each graph to the correct function.. f (). f () ( ) 7. f () ( ). f () A. B. C. D.

35 hal9_ch_-9.qd /7/9 : AM Page. Horizontal and Vertical Translations of the Graphs of Functions (-) Eercises 9 give functions that are translations of f (). Match each graph to the correct function. 9. f (). f (). f (). f () A. B. C. D. Eercises give functions that are translations of f (). Match each graph to the correct function.. f (). f (). f (). f () A. B. C. D. In Eercises 7, use the given graph of f () and horizontal and vertical shifts to graph each function. (Hint: First translate the three points labeled on the graph.) 7. f (). f () 9. f ( ). f ( ). f ( ). f ( ) ƒ() In Eercises and, each graph is a translation of the graph of Write the. equation of each graph. Identif the -intercept of each graph... In Eercises and, use the given table of values for f () to complete each table... In Eercises 7 and, use the given tables to write an equation for and. 7 f () f () 7. Write an equation for in terms of f().. Write an equation for in terms of f(). 9. Use the given table for f () to complete the table for. f() 7 f () f ( ) 7

36 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra. Use the given table for f () to complete the table for. In Eercises and, use the given tables to write an equation for and. f () f () f ( ). Write an equation for in terms of f().. Write an equation for in terms of f(). In Eercises, determine the verte of each parabola b using the fact that the verte of f () is at (, ).. f () ( ). f () ( ). f (). f () 7. f () ( ). f () ( ) In Eercises 9, determine the verte of the graph of each absolute value function b using the fact that the verte of f () is at (, ). 9. f (). f (). f (). f (). f () 7. f () 9 Eercises describe a translation of the graph f (). Match each description to the correct function.. A translation eleven units left A. f ( ). A translation eleven units right B. f ( ) 7. A translation eleven units down C. f ( ). A translation eleven units up D. f () 9. A translation eleven units right E. f ( ) and eleven units up F. f (). A translation eleven units left and eleven units down In Eercises, determine the domain and range of each function f, given that the domain of a function f is D [, ) and the range is R [, ).. The graph of f is obtained from the graph of f b shifting it units right.. The graph of f is obtained from the graph of f b shifting it units left.. The graph of f is obtained from the graph of f b shifting it 7 units down.. The graph of f is obtained from the graph of f b shifting it units up.. The graph of f is obtained from the graph of f b shifting it units left and units up.. The graph of f is obtained from the graph of f b shifting it units right and units down. In Eercises 7, match each function with its graph. Use the shape of the graph of each function and our knowledge of translations to make our choices. 7. f (). f () 9. f (). f (). f (). f () ( ). f (). f () ( ) A. B. C. D. E. F. G. H.. Write the first five terms of each sequence. (Hint: See Eample in Section.7.) a. a n n b. a n n c. a n (n ). Write the first five terms of each sequence. a. a b. a n n n n 9

37 hal9_ch_-9.qd //9 : PM Page 7. Horizontal and Vertical Translations of the Graphs of Functions (-7) 7 Connecting Concepts to Applications 7. Shifting the Temperature Scale Man properties of nature are related to the temperature of the object being studied. The Celsius temperature scale ( C) is the same incrementall as the Kelvin temperature scale (K) ecept that it is 7 lower (water freezes at C and 7 K). The equation V(t) t gives the volume in milliliters of a gas as a function of the temperature in degrees Celsius. Write an equation for the volume as a function of the temperature in kelvins and complete the following table. tºc V(t) t tºk V? 7. Shifting a Time Reference A compan has a function that approimates its revenue in terms of the ear those revenues were generated. The compan was founded in Januar 99. Thus 99 is taken as ear for the compan, 99 as ear, and so on. A formula for its revenue per ear in millions of dollars is given b R(t) t, where t is the number of ears the compan has been in business. Convert this formula to give the revenue in terms of the calendar ear. 9. Shifting a Pricing Function The price in dollars of five different tpes of networking cables is given in the table. Because of increased overhead cost, the price of each item is increased b $. Form a table that gives the new price of each item. Year t 9 Revenue R(t) t Price P() Item No. ($) Shifting the Distance for Golf Clubs Thanks to a new technolog in making golf clubs, man golfers have been able to add d to the distance the can hit with each club. Form a table that gives the new distance for each new club. Group discussion questions Iron Distance for Old Clubs (d) 9 7. Calculator Discover Use a graphing calculator to graph each pair of functions on the same graphing screen. Then describe the relationship between each pair of functions. a. Y b. Y Y c. Y d. Y Y Y 7. Calculator Discover Use a graphing calculator to graph the functions: f (), f (), f (), and f (). Can ou describe the relationship between the graph of f () and f () c for an positive constant c? 7. Calculator Discover Use a graphing calculator to graph the functions: f (),, f (), and f (). Can ou describe the relationship between the graph of f and the graph of f () c for an positive constant c? 7. Error Analsis A student graphing f () obtained the following displa. Eplain how ou can tell b inspection that an error has been made. [,, ] b [,, ]. Cumulative Review Simplif each epression..... ( ). ( ) ( )

38 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Section. Reflecting, Stretching, and Shrinking Graphs of Functions Objectives:. Recognize and use the reflection of a graph.. Use stretching and shrinking factors to graph functions. A ke concept of algebra is that there are some basic families of graphs. In Section., we learned how to shift a graph to create an eact cop of this graph at another location in the plane. We now learn how to modif a given graph to create a stretching or a shrinking of this shape. We start b eamining the reflection of a graph.. Recognize and Use the Reflection of a Graph What Is Meant b the Reflection of a Point? The reflection of a point (, ) across the -ais is a mirror image of the point on the opposite side of the -ais. The reflection of the point (, ) about the -ais is the point (, ). We now eamine the reflection of an entire graph b eamining the functions and. (, ) (, ) Point Reflection Numericall Verball Each -value is the opposite of the corresponding -value. Graphicall The graph of can be obtained b reflecting across the -ais each point of the graph of. A reflection of a graph is simpl a reflection of each point of the graph. The following bo describes a reflection of a graph across the -ais. Reflection of f() Across the -Ais Graphicall Numericall Algebraicall To obtain the graph of f (), reflect the graph of f () across the -ais. For each value of, is the additive inverse of. That is,. Original function: Reflection: f () f ()

39 hal9_ch_-9.qd //9 : PM Page 9. Reflecting, Stretching, and Shrinking Graphs of Functions (-9) 9 Eample Verball Eamining a Reflection of a Quadratic Function Compare the functions and numericall, graphicall, and verball. Numericall Graphicall. b reflecting the graph of across the -ais. For each value of, The parabola defined b can be obtained Self-Check a. Graph and on the same coordinate aes. b. Write an equation for in terms of. c. Compare the graphs of and. Eample clearl illustrates the reflection idea when we visuall compare these two graphs. According to David Bock, graphics research programmer for the National Center for Supercomputing Applications, This concept is used etensivel in threedimensional computer graphics and animation to create and position smmetrical objects and models. Eample Forming the Reflection of a Graph Use the graph of f () to graph f (). ƒ() (, ) (, ) (, ) (, ) ƒ() (, ) (, ) (, ) ƒ() (, ) Start b reflecting the ke points labeled on the graph of f (). Then use the shape of f () to sketch its reflection across the -ais.

40 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Self-Check Use the graph of f () to graph f (). (, ) (, ) (, ) Eample is used to eamine reflections when two functions are defined b a table of values. Eample Identifing a Reflection Across the -Ais Use the table of values for and to write an equation for. For each value of, is the additive inverse of. Answer: or Self-Check Use the table for f () to complete the table for f () Use Stretching and Shrinking Factors to Graph Functions What Does a Stretching or a Shrinking Factor Do to a Graph? Stretching or shrinking factors affect the vertical scale of the graph. The do not affect the horizontal scale, and the are not rigid translations. We start our inspection of scaling factors b eamining the functions and. Numericall 9 9 Graphicall 9 7

41 hal9_ch_-9.qd /7/9 : AM Page. Reflecting, Stretching, and Shrinking Graphs of Functions (-) Verball Each value of is times the corresponding -value. The graph of is the same basic V shape as but rises times as rapidl. Comparing the graph of to that of, we call a stretching factor because this factor verticall stretches the graph of. The following bo describes vertical scaling factors that can either stretch or shrink a graph. Then Eample eamines a scaling factor of that verticall shrinks the graph. Vertical Stretching and Shrinking Factors of f () For a function f () and a positive real number c: Algebraicall Graphicall Numericall If c : Original function: Scaled function: If c : Original function: Scaled function: f () cf () f () cf () To obtain the graph of c, For each value of, verticall stretch the graph of c. f () b a factor of c. To obtain the graph of c, verticall shrink the graph of f () b a factor of c. For each value of, c. The parabola defined b can be obtained b verticall shrinking the parabola defined b b a factor of At each -value, the height of is the. height of. Note that points on the -ais are not moved b scaling factors. (), thus (, ) is the verte of both parabolas. Eample Eamining a Shrinking of a Quadratic Function Compare the functions and numericall, graphicall, and verball. Numericall Graphicall 9 7 Verball For each value of,.

42 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Self-Check Compare and both numericall and graphicall. Both graphs in Eample have the characteristic shape of the square root function. The graph of can be obtained b verticall stretching b a factor of the graph of. For each function the domain is [, ) and the range is [, ). Eample Eamining a Stretching of a Square Root Function Compare the functions and numericall, graphicall, and verball. Numericall Graphicall Verball For each value of,. Self-Check Use the graph of f () to graph f (). Hint: Start b stretching the labeled points. (, ) (, ) f() (, ) Eample uses a table of values to identif a stretching factor. Eample Identifing a Stretching Factor Use the table of values for f () and f () to write an equation for in terms of f (). f () f () For each value of,. Thus f (). Answer: f ()

43 hal9_ch_-9.qd //9 : PM Page. Reflecting, Stretching, and Shrinking Graphs of Functions (-) Self-Check Use the table for f () to write an equation for in terms of f (). Ever graph in the famil of absolute value functions can be obtained from the V-shaped graph of. To obtain other members of this famil, we can use horizontal and vertical translations, stretching and shrinking factors, and reflections. In Eample 7, a function that both shrinks and reflects is eamined. The graph of can be obtained b verticall shrinking the graph of b a factor of and then reflecting this graph across the -ais. Both graphs have the characteristic V shape. The verte of both V-shaped graphs is (, ), but the open in opposite directions. Eample 7 Combining a Shrinking Factor and a Reflection Use the graph of to graph. Numericall Graphicall Verball Ever value can be obtained b multipling the corresponding value b. Self-Check 7 Compare the graphs of and. Eample illustrates how to analze a graph that involves horizontal and vertical translations and a reflection of a parabola.

44 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra Eample Combining Translations and a Reflection Use the graph of to write an equation for. Give the domain and the range of each function. 7 7 ƒ () The verte of is (, ), and the verte of f () is (, ). The parabola defined b f () is the same size as the parabola defined b. There is no stretching or shrinking in this eample. The graph of f () can be obtained from the graph of the parabola defined b b first shifting this graph to the right units, then reflecting this graph across the -ais, and finall shifting this graph up unit. Answer: ( ) For the domain is D and the range is R [, ). For the domain is D and the range is R (, ]. Self-Check Use a reflection and a translation of to write an equation for. ƒ () We can think of the shifts and reflections following the same order as the order of operations in the epression. For ( ), we first subtract inside the parentheses to produce a shift units to the right. Net we multipl b to produce a reflection across the -ais. Finall we add, to produce a shift unit up. The parabola f () has a verte of (, ). Eample 9 provides practice identifing the verte of a parabola from the defining equation. Eample 9 Determining the Verte of a Parabola Determine the verte of each parabola b using the fact that the verte of f () is at (, ). (a) f () 7 Verte of (, ) (b) f () ( 9) Verte of ( 9, ) The stretching factor of 7 does not move the verte of f (). Points on the -ais are not moved b scaling factors. This is a horizontal shift 9 units left of ever point on f (). The verte is shifted from (, ) to ( 9, ).

45 hal9_ch_-9.qd //9 9: PM Page. Reflecting, Stretching, and Shrinking Graphs of Functions (-) (c) f () ( ) Verte of (, ) The verte (, ) is shifted to (, ) b the horizontal shift. The reflection leaves this verte at (, ). The vertical translation shifts the verte to (, ). Self-Check 9 Determine b inspection the verte of the parabola defined b f () ( ). Self-Check Answers. a.. b. c. is a reflection of across the -ais. (, ) (, ) (, ).. For each value of,. The parabola defined b can be obtained b verticall stretching b a factor of the parabola defined b.. (, ) (, ) (, ). 7. f () The graph of can be obtained b verticall stretching the parabola defined b b a factor of and then reflecting this graph across the -ais.. 9. (, ). Using the Language and Smbolism of Mathematics. Vertical and horizontal shifts of a graph create another graph whose shape is the as that of the original graph and whose size is the as that of the original graph.. The reflection of the point (, ) across the -ais is the point.. If f () and c, then we call c a factor in the function cf().. If f () and c, then we call c a factor in the function cf().. If f () and c, then we call the graph of cf() a of the graph of f () across the -ais.. If we eamine the shapes of f () and f (), then we will observe that the two graphs have the basic shape but that f () is obtained b verticall f () b a factor of.. Quick Review In Eercises evaluate each epression for f ().. f (). f (). f ( ). f (). f ()

46 hal9_ch_-9.qd //9 : PM Page (-) Chapter A Preview of College Algebra. Eercises Objective Recognize and Use the Reflection of a Graph Eercises give some basic functions and reflections of these functions. Match each graph to the correct function.. f (). f (). f (). f (). f (). f () 7. f (). f () A. B. C. D. E. F. G. H. In Eercises 9 and, complete the given table of values. 9. f() f(). f() f() In Eercises and, use the graph of f () to graph f ()... Objective Use Stretching and Shrinking Factors to Graph Functions Eercises give functions that are obtained b either stretching or shrinking the graph of f (). Match each graph to the correct function.. f (). f (). f (). f () A. B. C. D.

47 hal9_ch_-9.qd //9 : PM Page 7. Reflecting, Stretching, and Shrinking Graphs of Functions (-7) 7 Eercises 7 give functions that are obtained b reflecting across the -ais or stretching or shrinking the graph of f (). Match each graph to the correct function. 7. f (). f () 9. f (). f () A. B. C. D. Eercises give functions that are obtained b either stretching or shrinking the graph of f (). Match each graph to the correct function.. f (). f (). f (). f () A. B. C. D. Eercises give functions that are obtained b either stretching or shrinking the graph of. Match each graph to the correct function.. f (). f () 7. f (). f () 9 A. B. C. D. Skill and Concept Development In Eercises 9, use the given graph of f () to graph each function. (Hint: First graph the three ke points on the new graph.) 9. f (). f (). f (). f (). f (). f ( ) In Eercises, use the given table of values for f () to complete each table.. f(). f () f() 9

48 hal9_ch_-9.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra 7. f(). In Eercises 9, use the given table to write an equation for in terms of f() f() f() In Eercises 7, determine the verte of each parabola b using the fact that the verte of f () is at (, ). 7. f (). f () 7 9. f () 7. f () ( 7) 7. f () ( ) 7. f () ( ) 9. f () ( ) 7. f () ( ) 9. Translate or reflect the graph of f () as described b each equation. a. Graph f () ( ), and describe how to obtain this graph from f (). b. Graph f () ( ), and describe how to obtain this graph from f () ( ). c. Graph f () ( ), and describe how to obtain this graph from f () ( ).. Translate or reflect the graph of f () as described b each equation. a. Graph f (), and describe how to obtain this graph from f (). b. Graph f (), and describe how to obtain this graph from f (). c. Graph f (), and describe how to obtain this graph from f (). Eercises 7 describe a translation, a reflection, a stretching, or a shrinking of f (). Match each description to the correct function. 7. A vertical stretching of f () A. f ( 7) b a factor of 7 B. 7f (). A vertical shrinking of f () C. b a factor of 7 f () 7 D. f () 9. A reflection of f () across E. f () 7 the -ais F. f ( 7). A horizontal shift of f () left 7 units. A horizontal shift of f () right 7 units. A vertical shift of f () up 7 units In Eercises, determine the range of each function, given that the range of f () is [, ).. f (). f. f (). () f () a n 7. f (). f () 9. Write the first five terms of each sequence. (Hint: See Eample in Section.7.) a. a n n b. c. a n n n d. a n n 7. Write the first five terms of each sequence. a. a n n b. a n n c. a n n d. a n n Connecting Concepts to Applications 7. Sequence of Retirement Bonuses As part of a bonus plan, a compan gives each secretar a number of shares in the compan at retirement. The number of shares given equals the number of ears the secretar has worked.

49 hal9_ch_-9.qd //9 : PM Page 9. Reflecting, Stretching, and Shrinking Graphs of Functions (-9) 9 a. Write a formula for A n, the sequence of the number of shares that would be given for a retirement after n ears. b. Write a formula for V n, the value of the shares that would be given for a retirement after n ears if the value of each share is $. 7. Comparing the Distance Traveled b Two Airplanes The formula D RT can be used to determine the distance D flown b an airplane fling at a rate R for time T. Complete the table if the rate of the second plane is double that of the first plane. Plane Plane Hours T Distance D Hours T Distance D 7. Comparing the Growth of Two Investments The formula I PRT can be used to determine the interest earned on an investment of P dollars for T ears at simple interest rate R. A second investment of the same amount is invested at a rate that is three-fourths that of the first investment. Complete the table for the second investment. Investment Investment Years T Interest I Years T Interest I,,,, 7. Comparing the Production of Two Assembl Lines The following graph shows the number of lightbulbs produced b the workers at an assembl line during a ver bus da shift. The night shift, which has fewer workers than the da shift, produces one-third as man bulbs as the da shift. Sketch the graph of the number of lightbulbs produced b the workers on the night shift. Lightbulbs, 9,, 7,,,,,,, Hours 7 9 Group discussion questions 7. Discover Question a. Graph all the points in this table. b. Sketch a parabola containing all these points. c. Use our knowledge of translations, reflections, stretchings, and shrinkings to write the equation of this parabola. d. Check our equation b using a graphing calculator. 7. Discover Question a. Graph all the points in this table. b. Sketch a parabola containing all these points. c. Use our knowledge of translations, reflections, stretchings, and shrinkings to write the equation of this parabola. d. Check our equation b using a graphing calculator. 77. Error Analsis A student graphing. obtained the following displa. Eplain how ou can tell b inspection that an error has been made. [,, ] b [,, ] 7. Error Analsis A student graphing obtained the following displa. Eplain how ou can tell b inspection that an error has been made. [,, ] b [ 7, 7, ]

50 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra. Cumulative Review. Write the equation of the horizontal line through (, ).. Write the equation of the vertical line through (, ).. Write in slope-intercept form the equation of the line through (, ) with slope m.. Write in slope-intercept form the equation of the line through (, ) and (7, ).. A line with slope m passes through (, ) and through another point with an -coordinate of. What is the -coordinate of this point? Section. Algebra of Functions Objectives:. Add, subtract, multipl, and divide two functions.. Form the composition of two functions. Problems, such as those in business, are often broken down into simpler components for analsis. For eample, to determine the profit made b producing and selling an item, both the revenue and the cost must be known. Separate divisions of a business might be asked to find the revenue function and the cost function; the profit function would then be found b properl combining these two functions. We shall eamine five was to combine functions: sum, difference, product, quotient, and composition of functions.. Add, Subtract, Multipl, and Divide Two Functions The sum of two functions f and g, denoted b f g, is defined as ( f g)() f () g() for all values of that are in the domain of both f and g. Note that if either f () or g() is undefined, then ( f g)() is also undefined. What Do I Get When I Add Two Functions? You get another function. This is consistent with earlier additions. When we add two real numbers, we get another real number. When we add two polnomials, we get another polnomial. Eample Determining the Sum of Two Functions Find the sum of f () and g() algebraicall, numericall, and graphicall. Algebraicall ( f g)() f () g() ( f g)() Numericall The new function called f g is determined b adding f() and g(). Substitute for f() and for g(). f() g() (f g)() This table contains onl a few values from the domain of input values from. These values illustrate that ( f g)() can be determined b adding f () g() for each input ;,,,, and.

51 hal9_ch_-99.qd //9 7: PM Page. Algebra of Functions (-) Graphicall g() (ƒ g)() 9 7 ƒ() The graph of f () is a parabola opening upward with its verte at (, ). The graph of g() is the horizontal line representing a constant output of. The new function f g adds all output values. Since g(), all output values of f are translated up units when g() is added to f (). Self-Check Given f () and g(), determine f g. When I Add Two Functions, How Can I Check M Answer? One wa is to compare the table of values for the original problem with the table of values for the sum that is determined. If two functions are equal, their input and output values are identical. If the domain of input values is the set of all real numbers, then we cannot list all the input-output pairs. However, a table of values can serve to check that two functions ield the same values. The difference of two functions f and g, denoted b f g, is defined as ( f g)() f () g() for all values of that are in the domain of both f and g. Eample Using the Difference of Two Functions to Model Profit Suppose that the weekl revenue function for u units of a product sold is R(u) u 7u dollars, and the cost function for u units is C(u) u. The fewest number of units that can be produced is and is the greatest number that can be marketed. Assuming profit can be determined b subtracting the cost from the revenue, find the profit function P and determine P(), the profit made b selling units. Word Equation Profit Revenue Cost Algebraic Equation P(u) R(u) C(u) P(u) (u 7u) (u ) P(u) u 7u u P(u) u u First write a word equation to model this problem. Substitute the given epressions for R(u) and C(u). The domain is u. (For some tpes of units, u ma have to be an integer.) This is the profit function.

52 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra The profit on units is P() () () Substitute for in the profit function P. P() Thus $ will be lost if onl units are sold. Self-Check Given f () and g() 7, determine ( f g)(). The operations of multiplication and division are defined similarl to addition ecept f that the quotient is not defined if g(). g f The domain of all these functions ecept is the set of values in both the domain of f g f and the domain of g. For, we also must have g(). g Operations on Functions Notation Definition Eamples: For f() and g() : Sum Difference Product Quotient f g f g f g f g ( f g)() f () g() ( f g)() for all real numbers ( f g)() for all real numbers ( f g)() for all real numbers ( f g)() f () g() ( f g)() f () g() a f f () b() g g() a f b() for all real numbers ecept g Eample Determining the Product and Quotient of Two Functions Given f () and g(), find the following. (a) ( f g)() (b) The domain of ( f g)() (c) a f g b() (d) The domain of a f g b() (a) ( f g)() f () g() ( f g)() ( )a b ( f g)() ( ) (b) f is defined for. g is defined for. f g is defined for. ( f g)() is defined onl for values for which both f and g are defined.

53 hal9_ch_-99.qd //9 7: PM Page. Algebra of Functions (-) (c) a f f () b() g g() a f g b() a f ( ) b() g a f b() g To simplif this comple rational epression, invert the divisor and multipl. (d) f is defined for. a f is defined for values for which both f and g are g b() g is defined for. defined and g(). g() for. f is defined for and. g Self-Check Given f () and g(), determine these two functions. a. (f g)() b. a f g b() Functions are defined not onl b their formulas, but also b the set of input values contained in the domains of the functions. We often allow the domain of a function to be implied b the formula. In this case, the domain is understood to be all real numbers for which the formula is defined and produces real number outputs. When we combine functions to produce new functions, we must take care that the formulas are used onl for input values that are allowed for the new function. This is illustrated b Eample. Two functions f and g are equal if the domain of f equals the domain of g and f () g() for each in their common domain. Eample Comparing Functions to Determine Whether The Are Equal Given g() and h() f (), determine whether:,, (a) f g (b) f h (a) f () g() for all real for g() ( ) for g() for f g because f is defined for and but g is not. Both and result in division b. f () and g() have the same values for all real numbers ecept and.

54 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra (b) f () h() for all real for all real h() ( ) for all real h() for all real f h because these functions produce the same output for each real number. There are no real numbers for which the denominator of h is. Self-Check Eplain wh f () is not equal to g(). As shown in the following figure, the graphs of the three functions f, g, and h in Eample are nearl identical. The onl difference is that the graph of g has holes at and because these values are not in its domain. ƒ() ( ) g() or g(), for ( ) h() or h(). Form the Composition of Two Functions Functions, especiall in formula form, are a powerful means of describing the relationship between two quantities. We can further amplif this power b chaining two functions together. This wa of combining two functions is called composition. What Smbol Do I Use to Denote the Composition of Two Functions? The smbol denotes the composition of two functions. The notation f g should not be confused with the smbol for the multiplication of two functions f g. Composite Function f g Algebraicall Verball Algebraic Eample ( f g)() f [g()] f g denotes the The domain of f g is the set composition of of -values from the domain function f with of g for which g() is in the function g. f g is read domain of f. f composed with g. For f () and g() : ( f g)() f [g()] f () Functions can be eamined b using mapping notation, ordered-pair notation, tables, graphs, and function notation. In each case, to evaluate ( f g)(), we first evaluate g() and then appl f to this result.

55 hal9_ch_-99.qd //9 7: PM Page. Algebra of Functions (-) Eample Determining the Composition of Two Functions Find f g for the given functions f and g. Verball Mapping Notation Answer g S S S f S 9 S 7 S g maps to ; then f maps to 9. g maps to ; then f maps to 7. g maps to ; then f maps to. f g g g f g f 9 f 7 f g. S 9 S 7 S Self-Check The function f maps to and the function g maps to. What do we know about f g? In Eample, f g g f. Although f g can equal g f in special cases, in general the order in which we perform composition of functions is important. Eample Determining the Composition of Two Functions Given f () and g(), evaluate these epressions. (a) ( f g)() (b) (g f)() (c) ( f g)() (d) (g f )() (a) ( f g)() f [g()] f [()] f () () (b) ( g f )() g[ f ()] g( ) g() () (c) ( f g)() f [g()] f () () 9 (d) ( g f )() g[ f ()] g( ) ( ) First appl the formula for g(). Evaluate g() (). Then appl the formula for f () to. Evaluate f (). First appl the formula for f(). Evaluate f (). Then appl the formula for g() to. Evaluate g() (). First appl the formula for g to. Then appl the formula for f to. First appl the formula for f to. Then appl the formula for g to. Self-Check Given f () and g(), evaluate these epressions. a. ( f g)() b. (g f )()

56 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra The functions f () and g() from Self-Check are inverses of each other. Note that ( f g)() g and (g f )() f f g For an input value, ( f g)() f [g()] f a b a b ( ) Likewise, (g f )(). Thus the composition of functions gives us another wa to characterize the inverse of a function. Inverse of a Function Algebraicall f The functions f and are inverses of each other if and onl if ( f f )() for each input value of f and ( f f )() for each input value of f Eample f () and f () are inverses because ( f f ) () f a b ( f f ) () a b ( f f )() and ( f f )() f ( ) ( f ( ) f )() ( f f )() This means that we can use tables to confirm that one function is the inverse of another. If we let f () and g(), then b letting ( ) we are representing f [g()]. Showing that has the same values as confirms that ( f g)(). Eample 7 illustrates how a problem can be broken down into pieces with individual functions as mathematical models. Then the overall relationship can be modeled b a composite function. Eample 7 Composing Cost and Production Functions The quantit of items a factor can produce weekl is a function of the number of hours it operates. For one compan this is given b q(t) t for t. The dollar cost of manufacturing these items is a function of the quantit produced; in this case, C(q) q q 7 for q. Evaluate and interpret the following epressions. (a) q() (b) C() (c) (C q)() (d) (C q)(t) (a) q(t) t q() () q() units can be produced in hours. Substitute into the formula for q(t).

57 hal9_ch_-99.qd //9 7: PM Page 7. Algebra of Functions (-7) 7 (b) C(q) q q 7 C() () () 7 C() 9, $9, is the cost of manufacturing units. (c) (C q)() C [q()] (C q)() C() (C q)() $9, $9, is the cost of hours of production. (d) (C q)(t) C [q(t)] (C q)(t) C[t] (C q)(t) (t) (t) 7 (C q)(t),t,t 7 This is the cost of t hours of production. Substitute into the formula for C(q). Substitute for q() from part (a). Substitute for C() from part (b). Substitute t for q(t). Then evaluate C for t. Self-Check 7 In Eample 7, assume that q(t) t and C(q) q q. a. Determine (C q)(t). b. Determine (C q)(). Eample illustrates the analsis of a geometric problem b using a composite function. Eample Composing Area and Length Functions A piece of wire m long is cut into two pieces. The length of the shorter piece is s m, and the length of the longer piece is L m. The longer piece is then bent into the shape of a square of area A m. s m L (a) Epress the length L as a function of s. (b) Epress the area A as a function of L. (c) Epress the area A as a function of s. of the Length of the (a) alength a longer piece b atotal length b shorter piece b L(s) s (b) Area Square of length of one side A(L) a L b (c) (A L)(s) A[L (s)] (A L)(s) A( s) s (A L)(s) a b First write a word equation that describes the relationship between the length of the longer piece and the length of the shorter piece. This equation epresses the length as a function of s where s. The length of one side of the square is one-fourth the perimeter. The composition A L epresses A as a function of s. Substitute s for L(s) from part (a). Evaluate A b using the formula from part (b).

58 hal9_ch_-99.qd //9 :7 PM Page (-) Chapter A Preview of College Algebra Self-Check A piece of wire that is cm long has a piece cm cut off. The remaining piece is bent into the shape of a rectangle whose length is twice its width. a. Epress the length L of the wire that remains as a function of. b. Epress the area A of the rectangle that is formed as a function of L. c. Epress the area A of the rectangle that is formed as a function of. An important skill in calculus is the abilit to take a given function and decompose it into simpler components. Eample 9 Decomposing Functions into Simpler Components Epress each of these functions in terms of f () and g(). (a) h() (b) h() (a) h() f () g[ f ()] (g f )() (b) h() g() f [g()] ( f g)() First substitute f() for. Then replace the square root function with the g function. First substitute g() for. Then rewrite the epression b using the definition of the f function. Self-Check 9 Epress h() in terms of f () and g(). Self-Check Answers. ( f g)(). f g maps to.. ( f g)(). a.. a.. a. ( f g)() b. b. b. a f g b() 7. a. (C q)(t),t,t. This is the cost of t hours or production. c.. The domain of g is but the domain of f is {}. b. (C q)(),. $, is the cost of hours of production. L() A(L) L ( ) A() 9. h() (g f )(). Using the Language and Smbolism of Mathematics. The sum of two functions f and g, denoted b f g, is defined as (f g)() for all values of that are in the domain of both f and g.. The difference of two functions f and g, denoted b f g, is defined as ( f g)() for all values of that are in the domain of both f and g.. The product of two functions f and g, denoted b f g, is defined as ( f g)() for all values of that are in the domain of both f and g. f. The quotient of two functions f and g, denoted b is defined as a f g, b() for all values g of that are in the domain of both f and g, provided g().. The composition of the function f with the function g, denoted b f g, is defined b ( f g)().

59 hal9_ch_-99.qd /7/9 : AM Page 9. Algebra of Functions (-9) 9. The domain of f g is the set of -values from the 7. The functions f and f are inverses of each other if and domain of g for which is in the domain onl if ( f f )() for each input of. value from the domain of and ( f f )() for each input value from the domain of.. Quick Review. Add ( 7 ) ( ).. Divide ( 7 ) ( ).. Subtract ( 7 ) ( ).. Multipl ( 7 )( ).. Find the inverse of the function f ().. Eercises Objective Add, Subtract, Multipl, and Divide Two Functions In Eercises and, evaluate each epression, given f () and g().. a. ( f g)(). a. ( f g)( ) b. ( f g)() b. ( f g)( ) c. ( f g)() c. ( f g)( ) d. a f d. a f g b() g b( ) In Eercises, use the given tables for f and g to form a table of values for each function. f() g() 7 9. f g. f g. g f. f g 7. g f f. g In Eercises 9, use f {(, ), (, ), (, 7)} and g {(, ), (, ), (, )} to form a set of ordered pairs for each function. 9. f g. f g. f g. f g In Eercises, use the given graphs for f and g to graph each function. (Hint: You ma want first to write each function as a set of ordered pairs.) ƒ g. f g. f g. f g f. g In Eercises 7, determine whether the functions f and g are equal. If f g, state the reason. 7. f (). g() f () g() 9. f {(, ), (, ), (, ), (, )} g(). f {( 7, 7), (, ), (, )} g(). f (). f () g() g() Objective Form the Composition of Two Functions In Eercises and, evaluate each epression, given f () and g().. a. ( f g)(). a. ( f g)( ) b. (g f )() b. (g f )( ) c. ( f f )() c. ( f f )( ) d. (g g)() d. (g g)( ). Use the given tables for f and g to complete the tables for f g. S g() S f() g() f[g()] S f[g()] S S 9 9 S 9 S S S 7 S S 7 S S S g f S. Use f {(, ), (, ), (, )} and g {(, ), (, ), (, )} to form a set of ordered pairs for a. f g b. g f

60 hal9_ch_-99.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra 7. Use the given graphs of f and g to graph f g. (Hint:. Use f () to determine You ma want first to write each function as a set of a. f () b. ( f f )() ordered pairs.) c. ( f f )() In Eercises, determine f g and g f.. f () 9. f () g() g(). f (). f () g() g(). f (). f () g() g() Skill and Concept Development In Eercises 7, determine f g f, f g, f g, and g, f g for the given functions. State the domain of the resulting function.. f (). f () g() g(). f () 7. f () 7 g() g() In Eercises and 9, use the graphs of f and g to graph f g.. 9. In Eercises and, determine both ( f f )() and ( f f )().. f (). f () f () f () f f g g g ƒ In Eercises, epress each function in terms of f () and g().. h(). h(),. h(). h() In Eercises 7, decompose each function into functions f and g such that h() ( f g)(). (Answers ma var.) 7. h(). h() h() ( ) ( ). h() Connecting Concepts to Applications. Profit as the Difference of Revenue Minus Cost The weekl revenue function for u units of a product sold is R(u) u u dollars, and the cost function for u units is C(u) u. Assume is the least number of units that can be produced and is the greatest number that can be marketed. Find the profit function P, and find P(), the profit made b selling units.. Determining a Formula for Total Cost and Average Cost The fied monthl cost F (rent, insurance, etc.) of a manufacturer is $,. The variable cost (labor, materials, etc.) for producing u units is given b V(u) u u for u,. The total cost of producing u units is C(u) F(u) V(u). Determine a. F() b. V() c. C() d. The average cost per unit when units are produced e. A formula for C(u) f. A formula for A(u), the average cost of producing u units. Determining a Formula for Total Cost and Average Cost A manufacturer produces circuit boards for the electronics industr. The fied cost F associated with this production is $, per week, and the variable cost V is $ per board. The circuit boards produce revenue of $ each. Determine a. V(b), the variable cost of producing b boards per week b. F(b), the fied cost of producing b boards per week c. C(b), the total cost of producing b boards per week d. A(b), the average cost of producing b boards per week e. R(b), the revenue from selling b boards per week f. P(b), the profit from selling b boards per week g. P(,) h. P(,) i. P(,)

61 hal9_ch_-99.qd /7/9 :7 AM Page. Algebra of Functions (-) j. The break-even value (the value producing a profit of $) b. Epress the area A of this circle as a function of the radius r. c. Determine (A r)(s) and interpret this result.. Composing an Area and a Width Function A border of uniform width is trimmed from all sides of the square poster board as shown.. Determining the Product of a Price Function and a Demand Function The number of items demanded b consumers is a function of the number of months that the product has been advertised. The price per item is varied each month as part of the marketing strateg. The number demanded during the mth month is N(m) m m, and the price per item during the mth month is P(m) m. The revenue for the mth month R(m) is the product of the price per item and the number of items demanded. Determine a. N(7) b. P(7) c. R(7) d. R(m). Composing Cost and Production Functions The number of sofas a factor can produce weekl is a function of the number of hours t it operates. This function is S(t) t for t. The cost of manufacturing s sofas is given b C(s) s s for s. Evaluate and interpret the following. a. S() b. C() c. (C S)() d. (C S)(t) e. (C S)() f. (C S)(). Composing Cost and Markup Functions The weekl cost C of making d doses of a vaccine is C(d).d. The compan charges % of cost to its wholesaler for this drug; that is, R(C).C. Evaluate and interpret the following. a. C(,) b. R(,9) c. (R C)(,) d. (R C)(d) 7. Composing an Area and a Radius Function A circular concrete pad was poured to serve as the base for a grain bin. This pad was inscribed in a square plot as shown. r s a. Epress the radius r of this circle as a function of the length s of a side of the square. a. Epress the area A of the border that is removed as a function of the remaining width w. b. Epress the remaining width w as a function of. c. Determine (A w) () and interpret this function. 9. Composing an Area and a Width Function A metal bo with an open top can be formed b cutting squares of sides cm from each corner of a square piece of sheet metal of width cm, as shown. cm w cm cm a. Epress the area of the base of this bo as a function of its width w. b. Epress w as a function of. c. Determine (A w)() and interpret this function. d. Epress the volume V of the bo as a function of. Group discussion questions. Discover Question For f () log and g(), determine (f g)(). What can ou observe from this result?. Discover Question a. Write the inverse of the function f {(, ), (, ), (, ), (, ), (, )}. Does f () f () for each input value of? b. Complete the function f {(, 7), (, ), (, ), (, ), (, ), (, ), (7, )} so that f () f (). c. Write the inverse of f () Does f () f (). for each input value of? cm w

62 hal9_ch_-99.qd /7/9 : AM Page (-) Chapter A Preview of College Algebra d. Have each member of our group write the equation of another function so that f () f (). Then eamine the graphs of all these functions for similar characteristics.. Discover Question a. For f () determine f. 7, b. Compare f and f. c. What does this impl about the graph of f ()?. Cumulative Review. The additive identit is.. The multiplicative identit is.. The additive inverse of is.. The multiplicative inverse of is.. The propert that justifies rewriting 7( ) ( ) as (7 )( ) is the of multiplication over addition. Section. Sequences, Series, and Summation Notation Objectives:. Calculate the terms of arithmetic and geometric sequences.. Use summation notation and evaluate the series associated with a finite sequence.. Evaluate an infinite geometric series. Objects and natural phenomena often form regular and interesting patterns. The stud of these objects and phenomena generall involves data collected sequentiall or in a sstematic manner. Here are some eamples of occurrences of sequences: Business: To enumerate the paments necessar to repa a loan Biolog: To describe the growth pattern of a living organism Calculator design: To specif the sequence of terms used to calculate functions such as the eponential function e and the trigonometric function cos Calculus: To give the areas of a sequence of rectangles used to approimate the area of a region We first eamined arithmetic sequences in Section. and geometric sequences in Section.. We now give a more formal definition of a sequence. A sequence is a function whose domain is a set of consecutive natural numbers. For eample, the sequence,,,,, 7 can be viewed as the function {(, ), (, ), (, ), (, ), (, ), (, 7)} with input values {,,,,, } and output values {,,,,, 7}. Alternative Notations for the Sequence,,,,, 7 Mapping Notation Function Notation Subscript Notation S S S S S S 7 f () f () f () f () f () f () 7 a a a a a a 7 A finite sequence has a last term. A sequence that continues without end is called an infinite sequence. A finite sequence with n terms can be denoted b a, a,..., a n, and an infinite sequence can be denoted b a, a,..., a n,... The notation consisting of three dots, which is used to represent the sequence a, a,..., a n, is called ellipsis notation. This notation is used to indicate that terms in the sequence are missing in the listing but the pattern shown is continued.

63 hal9_ch_-99.qd //9 7: PM Page. Sequences, Series, and Summation Notation (-). Calculate the Terms of Arithmetic and Geometric Sequences We now review the definitions and descriptions of arithmetic and geometric sequences. Arithmetic and Geometric Sequences Algebraicall Verball Numerical Eample Graphical Eample Arithmetic a n a n d or a n a n d Geometric An arithmetic sequence has a constant difference d from term to term. The graph forms a set of discrete points ling on a straight line.,,,,, 7 a n 7 7 (, ) (, ) (, ) (, ) (, ) (, 7) n a n a n r or a n ra n A geometric sequence has a constant ratio r from term to term. The graph forms a set of discrete points ling on an eponential curve.,,,,, a n 7, (, ) (, ) (, ) (, ) 7, n Eample Identifing Arithmetic and Geometric Sequences Determine whether each sequence is arithmetic, geometric, both, or neither. If the sequence is arithmetic, write the common difference d. If the sequence is geometric, write the common ratio r. (a) 7,,,,, p (b),,,,, p (c),,,,, p Arithmetic sequence with d Not a geometric sequence Not an arithmetic sequence Geometric sequence with r Arithmetic sequence with d Geometric sequence with r 7 ; ; ; d 7 9 ( ) 9 ; ; r ;

64 hal9_ch_-99.qd //9 : PM Page (-) Chapter A Preview of College Algebra (d),,,,,, p Not an arithmetic sequence Not a geometric sequence Self-Check Complete the finite sequence,,,, so that the sequence will be a. An arithmetic sequence b. A geometric sequence Since a n can represent an term of the sequence, it is called the general term. The general term of a sequence sometimes is defined in terms of one or more of the preceding terms. A sequence defined in this manner is said to be defined recursivel. Eamples of recursive definitions for parts (a) and (b) of Eample are (a) a n a n with a 7 7,,,,, p (b) a n a n with a,,,,, p Because recursive definitions relate terms to preceding terms, there must be an initial term or condition given so that the sequence can get started. This initial or seed value is frequentl used b computer programmers to construct looping structures within a program. Eample reeamines the sequence in Eample (d). This recursive definition requires two initial conditions. A Mathematical Note The sequence,,,,,,,... is known as a Fibonacci sequence, in honor of the Italian mathematician Leonardo Fibonacci (7 ). Fibonacci is known as the greatest mathematician of the th centur. His sequence appears in man surprising was in nature, including the arrangement of seeds in some flowers, the laout of leaves on the stems of some plants, and the spirals on some shells. There are so man applications that in 9 the Fibonacci Association was founded and began to publish The Fibonacci Quarterl. In its first ears, the association published nearl, pages of research. Eample Using a Recursive Definition Write the first five terms of the sequence defined recursivel b the formulas a, a, and a n a n a n. a a a a a a a a a a a a a a Answer:,,,, The first two terms are given; for n, a n is a, a n is a, and is a. Substitute for a and for a. For n, a n is a, is a, and is a. a n Substitute for a and for a. For n =, a n is a, is a, and is a. a n Substitute for a and for a. a n a n a n Self-Check Use the recursive definition a, a, and a n a n a n to write the net four terms of,,,,,,,,.

65 hal9_ch_-99.qd //9 7: PM Page. Sequences, Series, and Summation Notation (-) A spreadsheet is a ver good tool to use to investigate sequences. The abilit to define one cell in terms of others makes the use of recursive definitions ver powerful. Graphing calculators also have a more limited capabilit using a seq feature to create sequences. Recursive formulas are used etensivel in computer science and in mathematics. However, these formulas do require us to compute a sequence term b term. For eample, to compute the th term of a n a n, we first need the 99th term. Can I Determine the nth Term of an Arithmetic or Geometric Sequence Without Listing All of the Preceding Terms? Yes, we will now develop formulas for these terms. Arithmetic Sequence Geometric Sequence a n a n d a n ra n a a a a d a a r a a d (a d) d a d a a r (a r)r a r a a d (a d) d a d a a r (a r )r a r o o a n a n d a (n )d a n a n r a r n a n a (n )d a n a r n Start with the recursive formula for a n and note the pattern that develops. Arithmetic sequences add a common difference for each new term, and geometric sequences multipl b a common ratio for each new term. Eample Calculating a n for an Arithmetic and a Geometric Sequence Use the formulas for a n for an arithmetic sequence and a geometric sequence to calculate a for each sequence. (a) Determine a for an arithmetic sequence with a and d 7. (b) Determine a for a geometric sequence with a and r. (c) Determine a for a geometric sequence with a and r. a n a (n )d a ( )(7) a 9(7) a a 7 a n a r n a () a () 9 a (9,) a 7, a n a r n a ( ) a ( ) 9 a ( ) a, Substitute the given values into the formula for a n for an arithmetic sequence. Substitute the given values into the formula for a n for a geometric sequence. Use a calculator to evaluate this epression. Because r is negative, the terms in this geometric sequence will alternate in sign. All even-numbered terms will be negative. Self-Check Use the formulas a and a n a r n n a (n )d to determine a. a for an arithmetic sequence with a and d. b. a 9 for a geometric sequence with a and r.

66 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra The formula a n a (n )d relates a n, a, n, and d. Therefore, we can find an of these four variables when the other three are known. A similar statement can be made for the formula a n a r n. This is illustrated in Eample. Eample Determining the Common Difference and the Common Ratio Use the formulas for a n for an arithmetic sequence and a geometric sequence to calculate d and r. (a) Find d in an arithmetic sequence with a 7 and a 7 9. (b) Find r in a geometric sequence with a and a. a n a (n )d 9 7 (7 )d d d a n a r n ()(r) r r or r Substitute the given values into the formula for a n for an arithmetic sequence. Then solve for d. Substitute the given values into the formula for a n for a geometric sequence. Then solve this quadratic equation for both possible values of r. Check: The geometric sequences,,,,... and,,,,... both satisf the given conditions. Self-Check Use the formulas a and a n a r n n a (n )d to determine a. d for an arithmetic sequence with a and a. b. r for a geometric sequence with a and a,. Eample illustrates how to use the formulas for a n to calculate the number of terms in an arithmetic sequence and a geometric sequence. Eample Determining the Number of Terms in a Sequence Use the formulas for a n for an arithmetic sequence and a geometric sequence to calculate the number of terms in each sequence. (a),,, p, d ( ) 7 a n a (n )d (n )7 7(n ) n n This arithmetic sequence has terms. First note that this is an arithmetic sequence with a common difference d 7 and a first term of. Substitute a, a n, and d 7 into the formula for a n for an arithmetic sequence. Then solve for n.

67 hal9_ch_-99.qd //9 :7 PM Page 7. Sequences, Series, and Summation Notation (-7) 7 (b),,,,, p, r,.,, a n a r n (,)(.) n (.) n. log. n log. ( n ) log. log. log. n log. n n This geometric sequence has si terms. First note that this is a geometric sequence with a common ratio of r.. Substitute a,, a n, and r. into the formula for a n for a geometric sequence. Divide both sides of the equation b,. Take the common log of both members. Simplif b using the power rule for logarithms. Divide both sides of the equation b log.. Evaluate the right side with a calculator. You can check this answer b writing the first si terms of this geometric sequence. Self-Check Use the formulas a and a n a r n n a (n )d to determine a. The number of terms in the arithmetic sequence with a, a n, and d. b. The number of terms in the geometric sequence with a and r,, a n,.. Use Summation Notation and Evaluate the Series Associated with a Finite Sequence The sum of the terms of a sequence is called a series. If a, a,..., a n is a finite sequence, then the indicated sum a a p a n is the series associated with this sequence. For arithmetic and geometric series, there are formulas that serve as shortcuts to evaluating the series. Eample illustrates the meaning of a series. Then we eamine the shortcuts. Eample Evaluating an Arithmetic Series and a Geometric Series Find the value of the si-term series associated with these arithmetic and geometric sequences. (a) a n n a a a a a a () () () () () () 9 (b) a n n a a a a a a ,9 The series is the sum of the first si terms. Substitute the first si natural numbers into the formula a n n to determine the first si terms, and then add the terms of this arithmetic series. Calculate each of the si terms of this geometric sequence and then determine the sum. Self-Check Find the value of the five-term series associated with the sequence defined b a n n.

68 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra A convenient wa of denoting a series is to use summation notation, in which the Greek letter (sigma, which corresponds to the letter S for sum ) indicates the summation. Summation Notation Algebraicall Verball Algebraic Eample Last value of the inde Inde variable n a i a a... a n a n i Formula for general term Initial value of the inde The sum of a sub i from i equals to i equals n a a i a a a a a i Generall the inde variable is denoted b i, j, or k. The inde variable is alwas replaced with successive integers from the initial value through the last value. Eample 7 illustrates that the initial value can be a value other than. For eample, in a a i, i is i replaced with,, 7, and then to ield a a i a a a 7 a. i Eample 7 Evaluate each series. (a) a i i (b) Using Summation Notation to Evaluate an Arithmetic Series and a Geometric Series 7 a k k (a) (b) a i () () () () i 7 a k 7 k Replace i with,,, and then, and indicate the sum of these terms. Evaluate each term of this arithmetic series and then add these terms. Replace k with,,,, and then 7, and indicate the sum of these terms. Evaluate each term of this geometric series and then add these terms. Self-Check 7 Evaluate each series. k k a. a i b. a ( j j) c. a i j For a series with man terms, such as a i, it is useful to have a shortcut formula that i allows us to calculate the sum without actuall doing all the adding. We now develop formulas for an arithmetic series and a geometric series.

69 hal9_ch_-99.qd /7/9 : PM Page 9. Sequences, Series, and Summation Notation (-9) 9 Development of a Formula for an Arithmetic Series S n a (a d) p [a (n )d] [a (n )d] S n [a (n )d] [a (n )d] p (a d) a S n [a a (n )d] [a a (n )d] p [a a (n )d] [a a (n )d] S n (a a n ) (a a n ) p (a a n ) (a a n ) S n n(a a n ) S n n(a a n ) S n n[a (n )d] Development of a Formula for a Geometric Series S p a r n a r n n a a r a r rs a r a r a r p a r n a r n a r n n p a r n Sn rs n a S n ( r) a ( r n ) S n a ( r n ) r S n a a r n r S n a ra n r for r Sum the terms from a to a n. Sum the terms from a n to a. Add corresponding terms. Substitute a n for a (n )d. Note that (a a n ) is added n times. Solve for S n b dividing both sides b. Substitute a (n )d for a n for an alternative form of this formula. To obtain the second equation, multipl both sides of the first equation b r and shift terms to the right to align similar terms. Subtract the second equation from the first equation. Factor both sides. Divide both members b r. If r, r. Substitute a n for a r n to obtain an alternative form of this formula. Formulas for Arithmetic and Geometric Series S n a n a i i Arithmetic series Geometric series Algebraicall S n n (a a n ) or S n n [a (n )d] S n a ( r n ) r or S n a ra n r Algebraic Eample S a i S ( ) S () S S a k k S ( ) S ( ) S i

70 hal9_ch_-99.qd //9 7: PM Page 7 7 (-7) Chapter A Preview of College Algebra Eample eamines an application for both arithmetic and geometric series. Eample Using Arithmetic and Geometric Series to Model Applications (a) Rolls of carpet are stacked in a warehouse, with rolls on the first level, 9 on the second level, and so on. The top level has onl roll. How man rolls are in this stack? (See the figure.) (b) If ou could arrange to be paid $, at the end of Januar, $, at the end of Februar, $, at the end of March, and so on, what is the total amount ou would be paid for the ear? (a) The numbers of rolls on the various levels form an arithmetic sequence with a, n, and a. Algebraicall Numericall S n n(a a n ) S n a a p a n S a a p a ( ) S S 9 p S S Answer: There are rolls in this stack. (b) The paments at the end of each month form a geometric sequence with a,, r, n. Algebraicall S n a ( r n ) r S,( ),(,9) S S,9, S,9, Numericall Answer: The total amount for the ear would be $,9,. There is a constant difference of roll between consecutive levels in this stack. Thus the numbers of rolls form an arithmetic sequence, and the total number of rolls is determined b adding the terms of this arithmetic series. The total amount for the ear is determined b adding the terms of this geometric sequence. S n a a p a n Simplif and calculate S S a a p. a Although this answer ma seem unreasonable, it is S,,, the doubling pa scheme that makes the total unreasonable not the arithmetic.,,,,,,,,,,, S,9, Self-Check Use the formulas for arithmetic and geometric series to determine these sums. i i a. a i b. a i Compare the answers obtained algebraicall in Eample with the answers obtained numericall b actuall adding the terms in the series. Which method do ou prefer? Which method would ou prefer if there were, terms?

71 hal9_ch_-99.qd /7/9 :7 PM Page 7. Sequences, Series, and Summation Notation (-7) 7. Evaluate an Infinite Geometric Series If r, then the absolute values of the terms of the geometric sequence are decreasing. For eample, the geometric sequence is a decreasing, infinite,,,, p, n, p geometric sequence with r We alread have a formula for finding the sum of a finite. geometric sequence. Is the infinite sum a a i meaningful? Although we could never i b actuall add an infinite number of terms, we can adopt a new meaning for this sum. If the sum n a a i i S a r. approaches some limiting value S as n becomes large, then we will call this value the infinite sum. Smbolicall, a a i S if a a i approaches S as n increases. i i A general formula for the infinite sum can be obtained b eamining the formula for S n. S n a ( r n ) r n If then approaches as n becomes larger. Thus S n a ( r n ) r, r approaches r a ( ) a ; that is, S n approaches The infinite sum S is the limiting value, so r r. If r, the terms of a geometric series do not approach and a a i does not approach an limit as n becomes large. In this case, we do not assign a value to a. n n i a i i A Mathematical Note At the age of, Carl Friedrich Gauss (777 ) mentall computed the sum a i. This i was a problem that none of his fellow students were able to answer correctl b the end of the hour. Infinite Geometric Series S a a i i Algebraicall Algebraic Eample If r, S a r. S. p S... p with a. and r. If r, this sum does not eist. S a r S.. S..9 S

72 hal9_ch_-99.qd //9 9: PM Page 7 7 (-7) Chapter A Preview of College Algebra Eample 9 Write.777 as a fraction. Writing a Repeating Decimal in Fractional Form.777 p p This series is an infinite geometric series with a.7 and 7(.) 7(.) 7(.) p r.. a 7(.) i i S a r S.7. S.7.99 S Answer:.777 p This is the formula for the sum of an infinite geometric series. Substitute.7 for a and. for r. Simplif and epress S in fractional form. You can check this answer b dividing b. Self-Check 9 Write. p as a fraction. Self-Check Answers. a.,, 9,, b.,,,,.,,,,,,,,. a. a. a. n 7 b. n. a., b. a 9,,. 7 b.,97,. a. d b. r 7. a. b. 9.. p c. 9,. Using the Language and Smbolism of Mathematics. A sequence is a function whose domain is a set of consecutive numbers.. A sequence that has a last term is called a sequence.. A sequence that continues without end is called an sequence.. A sequence with a n a n d is an sequence.. A sequence with a n ra n is a sequence.. The sequence,,,,,,,... is an eample of a sequence. 7. Since a n can represent an term of a sequence, it is called the term.. A sequence where the general term is defined in terms of one or more of the preceding terms is said to be defined. 9. The formula for a n for an arithmetic sequence is a n.. The formula for a n for a geometric sequence is a n.. In the notation a, the inde variable i is. The initial value of the inde variable in this eample is, and the last value of the inde variable is.. The formula for an arithmetic series is S n.. The formula for a geometric series is S n.. The formula for an infinite geometric series is S if r. n a i

73 hal9_ch_-99.qd //9 7: PM Page 7. Sequences, Series, and Summation Notation (-7) 7. Quick Review. Write an algebraic epression for a sub equals.. Write an algebraic epression for a sub n equals four n plus eight. Determine the first five terms of each sequence.. a n n. a n n n. a n n. Eercises Objective Calculate the Terms of Arithmetic and Geometric Sequences In Eercises, determine whether each sequence is arithmetic, geometric, both, or neither. If the sequence is arithmetic, write the common difference d. If the sequence is geometric, write the common ratio r.. a.,,,,,9, 7,77 b.,,,,, c.,,,,, d.,,,,,. a.,,,,, b., 7,,, 9,. c. 9, 9, 9, 9, 9, 9 d. 9,, 7,,,. a. a n n b. a n n c. a n n d. a n 7. a. a and a n a n for n b. a and a n (a n ) for n c. a and a n a n for n d. a and a n for n In Eercises and, write the first si terms of each sequence.. a. a n n b. a n n n c. a n a n b d. a n ( ) n. a. a and a n a n for n b. a and a n a n for n c. a and a n a n for n d. a, a, and a n a n a n for n In Eercises 7 and, write the first si terms of each arithmetic sequence. 7. a.,,,,, b.,,,,, c. a 7, d d. a 9, d. a. a, a b. a n n c. a n n d. a. and a n a n. for n In Eercises 9 and, write the first five terms of each geometric sequence. 9. a.,,,, b.,,,, c. a, r d. a, r. a. a, a b. a n 9 a n b c. a n 9 a b n d. a and a n a n for n. Use the given information to calculate the indicated term of the arithmetic sequence. a. a, d, a b. a 7, a, a c. a, a, a d. a, a, a. Use the given information to calculate the indicated term of the geometric sequence. a. a, r, a b. a, a, a c. a, a, a d. a, a, a In Eercises, use the information given for the arithmetic sequence to find the quantities indicated..,,,..., ; n.,,..., 9; n. a, d, a. a 9, d, a 7. a, a, d. a 7, a, d

74 hal9_ch_-99.qd //9 7: PM Page 7 7 (-7) Chapter A Preview of College Algebra 9. a, a n, d, n. a, r, S. a, a n, d, n In Eercises, use the information given for the geometric sequences to find the quantities indicated..,, p, ; n.,, p, ; n. a, r, a. a, r, a. a 9, a, r. a 7, a 7, r 7. a n a n, a,, a 9. a n.a n, a 7,, a Objective Use Summation Notation and Evaluate the Series Associated with a Finite Sequence In Eercises 9 and, write the terms of each series and then add these terms. 9. a. a (i ) b. c. a d. a j k. a. k a b. a (i ) c. a d. a 7 j k In Eercises, use the information given to evaluate each arithmetic series.. a, a, S. a, a, S. a, a, S. a., a., S. a, d, S. a, d, S 7. a (i ). a (k ) i k k 9. a. a k i j k 7 j a (i ) i i j j In Eercises, use the information given to evaluate each geometric series.. a, r, S 7. a, r, S. a., r., S. a., r., S 7. a 79, r, S 7 7. a, a n.79, r., S n. p 7 9. a (.) i. a 7(.) k i. a, a n a n for n, S. a, a n a n for n, S Objective Evaluate an Infinite Geometric Series In Eercises, use the information given to evaluate the infinite geometric series.. a. a, r, r. a. a 7, r, r 7. a. a.9, r., r 9.. a a a a k j 7 b j k 9 b. 9 p. 9.7 p. a, a n for n a n. a, a n a for n 9 b a n In Eercises and, write each repeating decimal as a fraction.. a..... b..... c d a..... b..... c d..... Skill and Concept Development In Eercises 7 7, use the information given for the arithmetic sequences to find the quantities indicated. 7. a 77 9, d, S 77. S n, a, a n, n 9. S,, a 9, a 7. S 7 7, a, a 7 7. S, a, d 7. S, a 7, d k

75 hal9_ch_-99.qd //9 7: PM Page 7. Sequences, Series, and Summation Notation (-7) 7 In Eercises 7 7, use the information given for the geometric sequences to find the quantities indicated. 7. r, S,, a 7. a, r, S n,7, n 7. a, a n,7, S n,7, r a a i 7, a, r i a a k, r 9, a k a a i, r 7, a k participant, determine how man people will have been involved b the time the eighth generation has signed on but not et contacted anone.. Bouncing Ball A ball dropped from a height of m rebounds to si-tenths its previous height on each bounce. How far has it traveled when it reaches the ape of its eighth bounce? Give our answer to the nearest tenth of a meter. (Hint: You ma wish to consider the distances going up separatel from the distances going down.) Connecting Concepts to Applications 79. Stacks of Logs Logs are stacked so that each laer after the first has less log than the previous laer. If the bottom laer has logs and the top laer has logs, how man logs are in the stack? (See the figure.) m????. Rolls of Insulation Rolls of insulation are stacked so that each laer after the first has fewer rolls than the previous laer. How man laers will a lumberard need to use in order to stack rolls if rolls are placed on the bottom laer? (See the figure.). Vacuum Pump With each ccle a vacuum pump removes one-third of the air in a glass vessel. What percent of the air has been removed after eight ccles?. Arc of a Swing A child s swing moves through a -m arc. On each swing, it travels onl two-thirds the distance it traveled on the previous arc. How far does the swing travel before coming to rest?. Increased Productivit The productivit gain from installing a new robot welder on a machiner assembl line is estimated to be $, the first month of operation, $, the second month, and $, the third month. If this trend continues, what will be the total productivit gain for the first months of operation?. Seats in a Theater A theater has rows of seats, with seats in the back row. Each row has fewer seats than the row immediatel behind it. How man seats are in the theater?. Chain Letter A chain-letter scam requires that each participant persuade four other people to participate. If one person starts this venture as a first-generation 7. Multiplier Effect Cit planners estimate that a new manufacturing plant located in their area will contribute $, in salaries to the local econom. The estimate that those who earn the salaries will spend three-fourths of this mone within the communit. The merchants, service providers, and others who receive

76 hal9_ch_-99.qd //9 7: PM Page 7 7 (-7) Chapter A Preview of College Algebra this $, from the salar earners in turn will spend three-fourths of it in the communit, and so on. Taking into account the multiplier effect, find the total amount of spending within the local econom that will be generated b the salaries from this new plant. Group discussion questions. Challenge Question If people in a room shake hands with each other eactl once, how man handshakes will take place? 9. Discover Question The trichotom propert of real numbers states that eactl one of the following three statements must be true in each case. Determine which statement is true. a. i.. p ii.. p iii.. p b. i.. p ii.. p iii.. p c. i.. p ii.. p 9 9 iii.. p 9 d. What is the meaning of the three dots in ? e. Which of the following is true? i..999 p ii..999 p iii..999 p f. Are there an real numbers between.999 and? If so, give an eample. g. Multipl both sides of. p b. What does this prove? h. Is the statement, The number one-half on the number line can be represented b,, %,., and.99 a true statement? If not, wh is it false? 9. Risks and Choices One gimmick advertised in a newspaper was a sure-fire secret formula for becoming a millionaire. The secret discovered b those unwise enough to pa for it was as follows: On the first da of the month save, on the second da save, on the third, etc. How man das would be required to save a total of at least $,,? At this rate, what amount would be saved on the last da?. Cumulative Review. Write, in scientific notation.. Write. in scientific notation.. Write an inequalit for the interval [, ).. Write an inequalit for the interval (, ].. Write the interval notation for the inequalit 7. Section.7 Conic Sections Objective:. Graph the conic sections and write their equations in standard form. What Is Meant b the Term Conic Sections? Parabolas, circles, ellipses, and hperbolas can be formed b cutting a cone or a pair of cones with a plane. Therefore these figures collectivel are referred to as conic sections. Conic sections originall were studied from a geometric viewpoint. (See Fig..7..) Parabola Circle Ellipse Hperbola Figure.7. Conic sections.

77 hal9_ch_-99.qd //9 9: PM Page 77.7 Conic Sections (-77) 77 Two formulas that are useful for eamining relationships between points are the distance formula (Section.) and the midpoint formula. Note that the midpoint is found b taking the average of the -coordinates and the average of the -coordinates. Midpoint Formula: (, ) (, ), The distance and midpoint formulas are summarized in the following bo. Distance and Midpoint Formulas If (, ) and (, ) are two points, then: Algebraicall Numerical Eample Distance formula: Midpoint formula: (, ) a d ( ) ( ) For (, 7) and (, ) : Distance between the points:, b d ( ( )) ( 7) d d Midpoint between the points: (, ) a (, ) (, ), 7 ( ) b a, b. Graph the Conic Sections and Write Their Equations in Standard Form Using the distance formula, we now develop the equation of a circle with center (h, k) and radius r. A circle is the set of all points in a plane that are a constant distance from a fied point. (See Fig..7..) The fied point is called the center of the circle, and the distance from the center to the points on the circle is called the length of a radius. A diameter is a line segment from one point on a circle through the center to another point on the circle. The length of a diameter is twice the length of a radius. The distance r from an point (, ) on the circle to the center (h, k) is given b r ( h) ( k)

78 hal9_ch_-99.qd //9 7: PM Page 7 7 (-7) Chapter A Preview of College Algebra A Mathematical Note The term diameter is formed b combining the Greek words dia meaning across and metros meaning to measure. Euclid used diameter as an appropriate name for the chord across a circle through the circle s center. Squaring both sides of this equation gives an equation satisfied b the points on the circle: ( h) ( k) r Figure.7. To draw a circle, fi a loop of string with a tack and draw the circle as illustrated. Standard Form of the Equation of a Circle Algebraicall Graphicall Algebraic Eample The equation of a circle with center (h, k) and radius r is ( h) ( k) r k r (h, k) (, ) The equation of a circle with center (, ) and radius is ( ( )) ( ) ( ) ( ) h Eample Determining the Equation of a Circle Determine the equation of each graphed circle. (a) (, ) (, ) (b) (, ) Center (9, ) 7 9 (a) Center (h, k) (, ) Radius r ( ) [ ( )] Circle: r r ( h) ( k) r ( ) [ ( )] ( ) ( ) Determine the center b inspection, and use the distance formula to calculate the length of the radius. The segment (, ) to (, ) is a radius of this circle. Substitute the center and radius into the standard form for the equation of a circle.

79 hal9_ch_-99.qd //9 7: PM Page 79.7 Conic Sections (-79) 79 (b) Center Radius r (9 ) ( ) Circle: (h, k) a 9 (h, k) (, ), b r r ( h) ( k) r ( ) ( ) ( ) ( ) Use the midpoint formula to calculate the center of the circle with a diameter from (, ) to (9, ). Then use the center, one of the points on the circle, and the distance formula to calculate the length of the radius. Substitute the center and radius into the standard form for the equation of a circle. Self-Check A circle has a diameter from (, ) to (, ). a. Determine the center of this circle. b. Determine the length of this diameter. c. Determine the length of a radius. d. Determine the equation of this circle. The standard form of the circle ( ) ( ) can be epanded to give the general form. If we are given the equation of a circle in general form, we can use the process of completing the square to rewrite the equation in standard form so that the center and radius will be obvious. Eample Writing the Equation of a Circle in Standard Form Determine the center and the radius of the circle defined b the equation. ( ) ( ) ( ) ( ) ( ) ( ) [ ( )] ( ) Answer: The circle has center (, ) and radius. Divide both sides of the equation b. Regroup terms. Complete the square. Write in standard form. Self-Check Write 99 in standard form and identif the center and the radius. Equations of the form c, for c, define circles centered at the origin. The following bo also describes the cases where c or c.

80 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra Equations of the Form c As shown in the figure,. is a circle with center (, ) and radius.. is a circle with center (, ) and radius.. is a circle with center (, ) and radius.. is the degenerate case of a circle a single point (, ).. has no real solutions and thus no points to graph. Both and are nonnegative, so also must be nonnegative. r r r B the vertical line test, circles are not functions. Thus graphing calculators do not graph circles with as a function of. Nonetheless, there are was to get the graph of a circle to appear on a graphing calculator displa. One wa is to split a circle into two semicircles an upper semicircle and a lower semicircle, both of which are functions. This is illustrated with the circle. Solving for, we obtain. The upper semicircle is defined b Y, and the lower semicircle is defined b Y. An ellipse is the set of all points in a plane, the sum of whose distances from two fied points is constant. (See Fig..7..) The two fied points F ( c, ) and F (c, ) are called foci. The major ais of the ellipse passes through the foci. The minor ais is shorter than the major ais and is perpendicular to it at the center. The ends of the major ais are called the vertices, and the ends of the minor ais are called the covertices. If the ellipse is centered at the origin, then the equation of the ellipse as shown in Fig..7. is a b where a b. (, ) b Coverte Verte a d F c Major ais d Minor ais F c Verte a b Coverte Figure.7. d + d = a. Figure.7. To draw an ellipse, loop a piece of string around two tacks and draw the ellipse as illustrated.

81 hal9_ch_-99.qd //9 7: PM Page.7 Conic Sections (-) Eample Graph. 9 Graphing an Ellipse Centered at the Origin Plot the -intercepts ( 7, ) and (7, ). Plot the -intercepts (, ) and (, ). Then use the known shape to complete the ellipse. ( 7, ) 9 7 (, ) (, ) 7 9 (7, ) 9 From the given equation, 9 this is an ellipse centered at the origin with a 9 and b. Thus a 7, b, and the intercepts are ( 7, ), (7, ), (, ), and (, ). Self-Check Sketch the graph of. 9 We now use translations (Section.) to eamine ellipses that are not centered at the ( h) ( k) origin. The graph of is identical to the graph of, a b a b ecept that it has been translated so that the center is at (h, k) instead of (, ). Standard Form of the Equation of an Ellipse The equation of an ellipse with center (h, k), major ais of length a, and minor ais of length b is: Algebraicall Horizontal major ais: ( h) ( k) a b (h, k b) b (h, k) (h a, k) (h a, k) a (h, k b) Algebraic Eample ( ) 9 ( ) (, ) 7 (, ) 7 9 (, ) (9, ) a 7 b (, 7)

82 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra Vertical major ais: ( h) ( k) b a (h, k a) a (h, k) (h b, k) (h b, k) b ( ) ( ) (, ) (, ) (, ) b a 9 7 (, ) (h, k a) 9 7 (, ) Note: a b. Eample Writing the Equation of an Ellipse in Standard Form Determine the equation of the graphed ellipse. (, ) (, ) Minor ais Major ais (7, ) 7 (, ) ( h, k) a 7 ( h, k) (, ), b a (7 ( )) ( ) a a a a The center of the ellipse is at the midpoint of the major ais. The length of the major ais is a, and the length of the minor ais is b. Use the distance formula to determine these values. b ( ) ( ( )) b b b b ( h) a ( ) ( ) ( k) b ( ) ( ) 9 The horizontal ais is the major ais; thus we use the standard form of an ellipse with a horizontal major ais. Substitute a, b, h, and k into the standard form. Self-Check Write the equation in standard form and graph the ellipse.

83 hal9_ch_-99.qd //9 7: PM Page.7 Conic Sections (-) A hperbola is the set of all points in a plane whose distances from two fied points have a constant difference. (See Fig..7..) The fied points are the foci of the hperbola. If the hperbola is centered at the origin and opens horizontall as in Fig..7., then the equation of the hperbola is a. b This hperbola is asmptotic to the lines b and b As becomes larger, the hperbola gets closer and closer to these a a. lines. The asmptotes pass through the corners of the rectangle formed b (a, b), ( a, b), ( a, b), and (a, b). This rectangle, which is shown in Fig..7., is called the fundamental rectangle and is used to sketch quickl the linear asmptotes. (, ) d d ( a, b) b (a, b) F c a a F c a a ( a, b) b (a, b) Figure.7. d d a. Figure.7. a. b Eample Graph. 9 Graphing a Hperbola Centered at the Origin Plot the -intercepts (, ) and (, ). Sketch the fundamental rectangle with corners (, ), (, ), (, ), and (, ). Draw the asmptotes through the corners of this rectangle. Then sketch the hperbola through the -intercepts that opens horizontall, using the asmptotes as guidelines. From the given equation, this 9 is a hperbola centered at the origin with a and b 9. Thus a and b. 9 (, ) (, ) 7 (, ) 7 (, ) Self-Check Graph.

84 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra We now use translations to eamine hperbolas whose center is (h, k). The following bo covers both hperbolas that open horizontall and those that open verticall. Standard Form of the Equation of a Hperbola The equation of a hperbola with center (h, k) is Algebraicall Opening horizontall: ( h) ( k) a b Algebraic Eample ( ) ( ) 9 7 b (h a, k) (h, k) (h a, k) a Vertices: (h a, k) and (h a, k). The fundamental rectangle has base a and height b. Opening verticall: ( k) ( h) a b ( ) 9 ( ) 7 (h, k a) a (h, k) b (h, k a) Vertices: (h, k a) and (h, k a). The fundamental rectangle has base b and height a.

85 hal9_ch_-99.qd //9 7: PM Page.7 Conic Sections (-) Eample Writing the Equation of a Hperbola in Standard Form Determine the equation of the hperbola graphed in the figure. (h, k) (, ) ( k) ( h) a b a [ ( )] [ ( )] a a a b ( ) [ ( )] b ( ) b b The center of the hperbola was determined b inspection. Select the form for a hperbola that opens verticall. Use points on the fundamental rectangle to compute the values of a and b. [ ( )] [ ( )] ( ) ( ) 9 Substitute (, ) for (h, k) and for a and for b into the standard form for this hperbola. Self-Check Write the equation of the hperbola graphed in the figure. 9 We have alread graphed parabolas in several sections of this book (including Section.). For completeness of this introduction to conic sections, we now note that a parabola can be formed b cutting a cone with a plane. A parabola is the set of all points in a plane the same distance from a fied line L (the directri) and from a fied point F (the focus). See Fig Note that the ais of smmetr passes through the verte, and it is perpendicular to the directri. If the verte of the parabola in Fig..7.7 is at the origin, then the equation of this parabola can be written in the form If the verte is translated to p. the point (h, k), then the equation becomes k The equation p ( h). can also be rewritten in the form a ( ) b ( ). This form reveals the horizontal and vertical translations that we covered in Section.. F (focus) (, p) p p Ais of smmetr d Verte (, ) d L (directri) p Figure.7.7 d d.

86 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra Eample 7 Graphing a Parabola Graph ( ). 9 7 ( ) 7 9 (, ) This parabola can be obtained b translating the graph of right units and down units. The verte of this parabola is (, ), and the -intercept is (, 7). Self-Check 7 Graph ( ). Eample involves a circle and a parabola and uses the graphs of these conic sections to solve the corresponding sstem of equations and inequalities. Eample Use graphs to solve. (a) (a) Solving a Nonlinear Sstem of Equations and a Nonlinear Sstem of Inequalities (, ) (b) (, ) (, ) (, ), (, ), and (, ) all satisf both equations. is a circle centered at the origin with radius r. is a parabola opening upward with its verte at (, ). Do all these points check?

87 hal9_ch_-99.qd //9 7: AM Page 7.7 Conic Sections (-7) 7 (b) (, ) (, ) (, ) The points satisfing lie on and inside the circle. Use (, ) as a test value. The points satisfing lie on and above the parabola. Use (, ) as a test value. The shaded points satisf both inequalities. Self-Check Sketch the graph of the solution of e. f Self-Check Answers. a. (, ) b. c. d.. ( ) ( ( )) a with b. center (, ) and radius r. (, ) 7 (, ) (, ) (, ) 7.. ( ) ( ) 9 7 (, ) (, ) (, ) (, ) (, ) 7 9 ( ). ( ) ( )

88 hal9_ch_-99.qd //9 7: PM Page (-) Chapter A Preview of College Algebra.7 Using the Language and Smbolism of Mathematics. Parabolas, circles, ellipses, and hperbolas are collectivel referred to as.. A is the set of all points in a plane that are a constant distance from a fied point.. An is the set of all points in a plane the sum of whose distances from two fied points is constant.. A is the set of all points in a plane whose distances from two fied points have a constant difference.. A is the set of all points in a plane the same distance from a fied line and a fied point.. A line segment from the center of a circle to a point on the circle is a. 7. A line segment from one point on a circle through the center to another point on the circle is a.. The ends of the major ais of an ellipse are called. 9. The ends of the minor ais of an ellipse are called.. The hperbola is to the lines a b b and b a a..7 Quick Review. The verte of the parabola defined b a b c. Epand 9( ) ( ) and simplif the has an -coordinate of and a -coordinate result. of f a b. a b. Solve 7 b completing the square.. Complete the square to rewrite the function. The verte of the parabola defined b f () is. f () in the form f () a( h) k..7 Eercises Objective Graph the Conic Sections and Write Their Equations in Standard Form In Eercises, calculate the distance between each pair of points and the midpoint between these points.. (, ) and (, ). (, ) and (, ). (, ) and (, ). (a, b) and (a, b ). Calculate the length of the radius of a circle with center at (, ) and with the point (7, ) on the circle.. Calculate the length of the diameter of a circle with endpoints on the circle at (, ) and (, ). 7. A circle has a diameter with endpoints at (, ) and (, ). Determine the center of this circle. In Eercises, match each graph with the corresponding equation A. B. C. D. E. 9 7

89 hal9_ch_-99.qd //9 7: PM Page 9.7 Conic Sections (-9) 9 In Eercises 7, write in standard form the equation of the circle satisfing the given conditions.. Center (, ), radius. Center (, ), radius. Center (, ), radius. Center (, ), radius. 7. Center a, radius b, In Eercises, determine the center and the length of the radius of the circle defined b each equation.. 9. ( ) ( ). ( ) ( ).... In Eercises, determine the center and the lengths of the major and minor aes of the ellipse defined b each equation, and then graph the ellipse.. ( ) ( ). 9 9 ( ) ( ) 7.. 9,7 In Eercises 9, write in standard form the equation of the ellipse satisfing the given conditions. 9. Center (, ), -intercepts ( 9, ) and (9, ), -intercepts (, ) and (, ). Center (, ), -intercepts (, ) and (, ), -intercepts (, ) and (, ). Center (, ), horizontal major ais of length, vertical minor ais of length. Center (, ), horizontal major ais of length, vertical minor ais of length. Center (, ), vertical major ais of length, horizontal minor ais of length. Center (, ), vertical major ais of length, horizontal minor ais of length. Center (, ), a, b, major ais vertical. Center (, ), a, b, major ais horizontal In Eercises 7, determine the center, the values of a and b, and the direction in which the hperbola opens, and then sketch the hperbola ( ) ( ). 9 9 ( ) ( ) In Eercises, write the standard form of the equation of the hperbola satisfing the given conditions.. The center is (, ), the hperbola opens verticall, and the fundamental rectangle has height and width.. The center is (, ), the hperbola opens horizontall, and the fundamental rectangle has height and width.. The hperbola has vertices (, ) and (, ), and the height of the fundamental rectangle is.. The hperbola has vertices (, ) and (, ), and the width of the fundamental rectangle is. In Eercises 7, graph each parabola. 7. a.. a. b. b. c. ( ) c. ( ) 9. a.. a. b. b. c. c. In Eercises, use the given graphs to solve each sstem of equations and each sstem of inequalities.. a. b.. a. b. ( ) ( ) ( ) ( ) (, 7) (, ) 9 7 (, ) (, 7)

90 hal9_ch_-99.qd /7/9 : PM Page 9 9 (-9) Chapter A Preview of College Algebra. a. b.. a. b. (, ) e e (, ) (, ) Group discussion questions. Discover Question A line and a circle can intersect at,, or points. Illustrate each possibilit with a sketch.. Discover Question A circle and an ellipse can intersect at,,,, or points. Illustrate each possibilit with a sketch. 7. Discover Question A circle and a hperbola can intersect at,,,, or points. Illustrate each possibilit with a sketch.. Challenge Question Use a graphing calculator to graph each pair of functions. a. Y, Y Y (Hint: Compare to.) b. Split into Y and Y, which represent upper and lower semiellipses. Then graph Y and Y..7 Cumulative Review Solve each equation.. ( ) 7( ).. ( ). 7. log( ) Chapter Ke Concepts. Methods for Solving Sstems of Linear Equations Use the addition method and the substitution method to eliminate variables. Use the augmented matri method.. Augmented Matri A matri is a rectangular arra of numbers consisting of rows and columns. A row consists of entries arranged horizontall. A column consists of entries arranged verticall. The dimension of a matri is given b first stating the number of rows and then the number of columns. An augmented matri for a sstem of linear equations consists of the coefficients and constants in the equations. The augmented matri for is a b c a b c a b c d d. d a b c z d a b c z d a b c z d. Elementar Row Operations on Augmented Matrices An two rows in the matri ma be interchanged. An row in the matri ma be multiplied b a nonzero constant. An row in the matri ma be replaced b the sum of itself and a constant multiple of another row.. Reduced Row Echelon Form of a Matri The first nonzero entr in a row is. All other entries in the column containing the leading are s.

91 hal9_ch_-99.qd /7/9 : PM Page 9 Ke Concepts (-9) 9 All nonzero rows are above an rows containing onl s. The first nonzero entr in a row is to the left of the first nonzero entr in the following row.. Equation of a Plane The graph of a linear equation of the form A B Cz D is a plane in threedimensional space.. Basic Functions Eamined in This Book Each tpe of function has an equation of a standard form and a graph with a characteristic shape. Linear functions: First-degree polnomial functions (The shape of the graph is a straight line.) Quadratic functions: Second-degree polnomial functions (The shape of the graph is a parabola.) Cubic functions: Third-degree polnomial functions Absolute value functions (The graph is V-shaped.) Square root functions Cube root functions Eponential functions Logarithmic functions Rational functions 7. Vertical and Horizontal Translations of f() If c is a positive real number, f () c shifts the graph of f () up c units. f () c shifts the graph of f () down c units. f ( c) shifts the graph of f () left c units. f ( c) shifts the graph of f () right c units.. Stretching, Shrinking, and Reflecting f() f () reflects the graph of f () across the -ais. cf () stretches the graph of f () verticall b a factor of c for c. cf () shrinks the graph of f () verticall b a factor of c for c. 9. Operations on Functions If f and g are functions, then for all input values in both the domain of f and g: Sum: ( f g)() f () g() Difference: ( f g)() f () g() Product: ( f g)() f ()g() Quotient: a f f () b() for g() g(). Composite Function f g If f and g are functions, then ( f g)() f [g()] for all input values in the domain of g for which g() is an input value in the domain of f. If f and f are inverses of each other, then ( f f )() for each input value of f and ( f f )() for each input value of f.. Sequences A sequence is a function whose domain is a set of consecutive natural numbers. A finite sequence has a last term. An infinite sequence continues without end. A sequence is arithmetic if a n a n d; d is called the common difference. The points of an arithmetic sequence lie on a line. A sequence is geometric if a n ra n ; r is called the common ratio. The points of a geometric sequence lie on an eponential curve.. General Term of a Sequence A formula for a n is a formula for the general term of a sequence. A formula defining a n in terms of one or more of the preceding terms is called a recursive definition for a n. The Fibonacci sequence defined b a, a, and a n a n a n is an eample of a recursive definition. A formula for the general term of an arithmetic sequence is a n a (n )d. A formula for the general term of a geometric sequence is a n a r n.. Series A series is a sum of the terms of a sequence. Summation notation is used to denote a series: n a a i a a p a n a n i The sum of the first n terms of an arithmetic sequence is S n n (a a n ). The sum of the first n terms of a geometric sequence is S n a ( r n ). r If r, the infinite geometric series S a is S a If r, this sum does not eist. r. a i i. Distance and Midpoint Formulas For two points (, ) and (, ): Distance: d ( ) ( ) Midpoint: (, ) a, b. Standard Forms for Conic Sections Circle: ( h) ( k) r Ellipse: ( h) ( k) with horizontal major ais a b ( h) ( k) with vertical major ais b a Hperbola: ( h) ( k) opens horizontall a b ( k) ( h) opens verticall a b Parabola: k ( h) p

92 hal9_ch_-99.qd //9 7: PM Page 9 9 (-9) Chapter A Preview of College Algebra Chapter Review Eercises Solving Sstems of Linear Equations. Write an augmented matri for each sstem of linear equations. a. 7 b. z z z. Write the solution for the sstem of linear equations represented b each augmented matri. a. c b. d. Write the general solution and three particular solutions for the sstem of linear equations represented b this augmented matri:. In Eercises 7, use the given elementar row operations to complete each matri.... r c r c d d r c r c d d r c r r c d d r 7. r r r r r In Eercises, solve each sstem of linear equations z. z z z z. z z z. Graph the plane defined b the linear equation.z.. Weights of Pallets of Bricks and Blocks A truck has an empt weight of, lb. On the first trip, the truck delivered two pallets of bricks and three pallets of concrete blocks. The gross weight on this deliver was, lb. On the second trip, the truck delivered four pallets of bricks and one pallet of concrete blocks. The gross weight on this deliver was 9, lb. Use this information to determine the weight of a pallet of bricks and the weight of a pallet of concrete blocks. 7. Price of Calculators A department store chain with three stores retails calculators of tpes A, B, and C. The table shows the number sold of each tpe of calculator and the total income from these sales at each store. Find the price of each tpe of calculator. Total Sales A B C ($) Store Store 9 Store 7 Translations of Graphs and Functions. Match each function with its graph. a. f () b. f () c. f () A. B. C. 9. Match each function with its graph. a. f () b. f () c. f () A. B. 7 9

93 hal9_ch_-99.qd //9 7: PM Page 9 Review Eercises (-9) 9 C.. Match each function with its graph. a. f () b. f () c. f () A. B. C.. Use the given graph of f () to graph each function. a. f ( ) b. f () c. f ( ). Determine the verte of each parabola b using the fact that the verte of f () is at (, ). a. f () 7 b. f () ( 9) c. f () ( ). Use the given table of values to complete each table. a. f() b. 7 9 f( ) 9 ƒ() f() 9. The graph of each of these functions is a translation of the graph of f (). Match each function to the correct description. a. f () 7 A. A translation 7 units right b. f ( 7) B. A translation 7 units up c. f () 7 C. A translation 7 units left d. f ( 7) D. A translation 7 units down. Use the given table and match each function with its table. f() a. f () b. f ( ) c. f ( ) d. f ( ) A. B. 7 C. D. Stretching, Shrinking, and Reflecting Graphs of Functions. Use the graph of f () shown and match each function with its graph. a. f () b. f () c. f () d. f () A. B. 7 (, )

94 hal9_ch_-99.qd //9 7: PM Page 9 9 (-9) Chapter A Preview of College Algebra C. D. 7. Use the graph of f () shown and match each function with its graph. a. f () b. f () c. d. f () f () A. B. C. D Use the given table of values for to complete each table. 7 7 f () 7 7 ƒ() f() a. b. c. d. 9. Match each function with the description that compares its graph to the graph of f (). a. f () b. f () A. A reflection of f () across the -ais c. f ( ) B. A horizontal shift of d. f () f () units left C. A vertical shift of f () units up D. A vertical stretching of f () b a factor Algebra of Functions In Eercises, evaluate each epression, given f () 7 and g().. a. f () b. g () c. ( f g)() d. ( f g)(). a. (f g)() b. a f g b() c.. a. b. c. d. a g f b() f() f() d. (g f )() ( f g)() (g f )() ( f f )() (g g)(). Use the given tables for f and g to form a table of values for each function. f() 7 g() 9 9 a. f g b. f g c. f g d. f () f() f g

95 hal9_ch_-99.qd //9 7:7 PM Page 9 Review Eercises (-9) 9. Use the given graphs for f and g to form a set of ordered pairs for each function. a. f g b. f g c. f g d.. Use the given tables for f and g to complete a table for f g.. Use the given graphs of f and g to graph f g. g() f() 9 7 In Eercises 7 and, determine each function, given f () and g(). Give the domain for each function. 7. a. f g b. f g c. f g f. a. f g b. c. g f g 9. Given and g() f () state the reason, that f g.. Given f (), determine f. Then determine (f f )().. Determining a Formula for Total Cost and Average Cost The fied weekl costs for a sandwich shop are $; F (). The variable dollar cost for sandwiches is V().. Determine a. F() b. V() c. C(), the total cost of making sandwiches in a week d. A(), the average cost per sandwich when sandwiches are made in a week f g() ƒ() f g g. Composing Cost and Production Functions The number of desks a factor can produce weekl is a function of the number of hours t it operates. This function is N(t) t for t. The cost of manufacturing N desks is given b C(N) a N b N. a. Evaluate and interpret N(). b. Evaluate and interpret C(). c. Evaluate and interpret (C N)(). d. Determine (C N)(t). e. Eplain the logic of the domain t. Sequences and Series. Write the first si terms of an arithmetic sequence that satisfies the given conditions. a. a, d b. a, d c. a, a d. a, a. Write the first si terms of a geometric sequence that satisfies the given conditions. a. a, r b. a, r c. a, a d. a, a. Write the first five terms of each sequence. a. a b. a n n n n 7 c. a n a n b d. a and a n a n 7 for n. An arithmetic sequence has a and d. Find a. 7. A geometric sequence has a and r. Find a.. An arithmetic sequence has a, a n, and d. Find n. 9. A geometric sequence has a,,, a n 7, and r. Find n.. Write the terms of each series, and then add these terms. a. a (i ) b. i j j k c. a d. a 7 a (k k) i