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ALGEBRA QUADRATIC FORMULA SPECIAL PRODUCT FORMULAS SPECIAL FACTORING FORMULAS If a 0, the roots of 共x ⫹ y兲共x ⫺ y兲 ⫽ x2 ⫺ y2 x2 ⫺ y2 ⫽ 共x ⫹ y兲共x ⫺ y兲 ax2 ⫹ bx ⫹ c ⫽ 0 are 共x ⫹ y兲2 ⫽ x2 ⫹ 2xy ⫹ y2 x2 ⫹ 2xy ⫹ y2 ⫽ 共x ⫹ y兲2 x ⫽ ⫺b ⫾ 2b2 ⫺ 4ac 共x ⫺ y兲2 ⫽ x2 ⫺ 2xy ⫹ y2 x2 ⫺ 2xy ⫹ y2 ⫽ 共x ⫺ y兲2 2a 共x ⫹ y兲3 ⫽ x3 ⫹ 3x2y ⫹ 3xy2 ⫹ y3 x3 ⫺ y3 ⫽ 共x ⫺ y兲共x2 ⫹ xy ⫹ y2兲共x ⫺ y兲3 ⫽ x3 ⫺ 3x2y ⫹ 3xy2 ⫺ y3 x3 ⫹ y3 ⫽ 共x ⫹ y兲共x2 ⫺ xy ⫹ y2兲 EXPONENTS AND RADICALS BINOMIAL THEOREM INEQUALITIES 共aamma兲n ⫽ am⫹nn a1m/n/n ⫽⫽22nn aam 共x ⫹ y兲n ⫽ xn ⫹ 冉1n冊xn⫺1y ⫹ 冉2n冊xn⫺2y2 ⫹ If a ⬎ b,atnhde nb a⬎⫹c,cth⬎enb a⫹⬎cc 冉 a共 aamnba⫽兲冊n ⫽nam⫽⫺anbabnn a冑2nmn/an ab⫽⫽⫽共22n2nnaaab兲m2n b where ⭈ ⭈冉⭈nk⫹冊 ⫽冉nkk冊!共xnn⫺n⫺k!ykk⫹兲! ⭈ ⭈ ⭈ ⫹ yn, If a ⬎ b and c ⬍⬎ 0, then ac ⬍⬎ bc a⫺n ⫽ a1n 兹m2n a ⫽ 2mn a ABSOLUTE VALUE 共d ⬎ 0兲 SEQUENCES EXPONENTIALS AND LOGARITHMS 兩x兩 ⬍ d if and only if nth term of an arithmetic sequence with first y ⫽ loga x means ay ⫽ x ⫺d ⬍ x ⬍ d term a1 and common difference d loga xy ⫽ loga x ⫹ loga y 兩x兩 ⬎ d if and only if either an ⫽ a1 ⫹ 共n ⫺ 1兲d x x ⬎ d or x ⬍ ⫺d Sum Sn of the first n terms of an arithmetic loga y ⫽ loga x ⫺ loga y MEANS sequence Sn ⫽ 2n 共a1 ⫹ an兲 laologgaxx⫽r ⫽x r loga x Arithm Ae⫽tica m1 ⫹eana 2A⫹of⭈n⭈ n⭈u⫹mabners or Sn ⫽ 2n 关2a1 ⫹ 共n ⫺ 1兲d兴 loga 1ax⫽⫽0x n nth term of a geometric sequence with first term a1 and common ratio r loga a ⫽ 1 Geometric mean G of n numbers G ⫽ 共a1a2 ⭈ ⭈ ⭈ an兲1/n, ak ⬎ 0 Sum Sn of the firsat nn⫽tearm1rsn ⫺o1f a geometric longxx⫽⫽lologegx10 x sequence loga u logb u ⫽ a1共1 ⫺ rn兲 loga b Sn ⫽ 1 ⫺ r FORMULAS FROM GEOMETRY area A perimeter P circumference C volume V curved surface area S altitude h radius r RIGHT TRIANGLE TRIANGLE EQUILATERAL TRIANGLE c a c a s s h h b b s Pythagorean Theorem: c2 ⫽ a2 ⫹ b2 A ⫽ 21 bh P ⫽ a ⫹ b ⫹ c h ⫽ 23 s A ⫽ 23 s2 2 4 RECTANGLE PARALLELOGRAM TRAPEZOID a w h h l b b 1 A ⫽ lw P ⫽ 2l ⫹ 2w A ⫽ bh A ⫽ 共a ⫹ b兲h 2 CIRCLE CIRCULAR SECTOR CIRCULAR RING r u s r r R A ⫽␲ r 2 C ⫽ ␲2 r A ⫽ 1 r␪2 s ⫽␪r A ⫽␲ 共R2 ⫺ r 2兲 2 RECTANGULAR BOX SPHERE RIGHT CIRCULAR CYLINDER h h r w l r ␲ ␲ 4␲ 3 ␲ 2 2 V ⫽ lwh S ⫽ 2共hl ⫹ lw ⫹ hw兲 V ⫽ r S ⫽ 4 r V ⫽ r h S ⫽ 2 rh 3 RIGHT CIRCULAR CONE FRUSTUM OF A CONE PRISM r h h h r R V ⫽ 1 3␲r 2h S ⫽ ␲ r2r 2 ⫹ h2 V ⫽ 13␲ h共r 2 ⫹ rR ⫹ R2兲 V ⫽ Bh with B the area of the base ANALYTIC GEOMETRY DISTANCE FORMULA EQUATION OF A CIRCLE 2 2 2 2 2 d共P1, P2兲 ⫽ 2共x2 ⫺ x1兲 ⫹ 共y2 ⫺ y1兲 y 共x ⫺ h兲 ⫹ 共y ⫺ k兲 ⫽ r y r (h, k) P1(x1, y1) P2(x2, y2) x x SLOPE m OF A LINE GRAPH OF A QUADRATIC FUNCTION 2 2 y y2 ⫺ y1 y ⫽ ax , a ⬎ 0 y ⫽ ax ⫹ bx ⫹ c, a ⬎ 0 m ⫽ l x2 ⫺ x1 y y (x1, y1) (x 2, y2) c x x b x ⫺ 2a POINT-SLOPE FORM OF A LINE CONSTANTS ␲ ⬇ 3.14159 y y ⫺ y1 ⫽ m共x ⫺ x1兲 l e ⬇ 2.71828 (x1, y1) CONVERSIONS x 1 centimeter ⬇ 0.3937 inch SLOPE-INTERCEPT FORM OF A LINE 1 meter ⬇ 3.2808 feet 1 kilometer ⬇ 0.6214 mile y y ⫽ mx ⫹ b 1 gram ⬇ 0.0353 ounce l (0, b) 1 kilogram ⬇ 2.2046 pounds 1 liter ⬇ 0.2642 gallon x 1 milliliter ⬇ 0.0381 fluid ounce INTERCEPT FORM OF A LINE 1 joule ⬇ 0.7376 foot-pound 1 newton ⬇ 0.2248 pound x y y ⫹ ⫽ 1 共a 0, b 0兲 l 1 lumen ⬇ 0.0015 watt a b (0, b) 1 acre ⫽ 43,560 square feet (a, 0) x CLASSIC TWELFTH EDITION ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY E A R L W. 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Swokowski CONTENTS Preface viii C H A P T E R 1 Fundamental Concepts of Algebra 1 1.1 Real Numbers 2 1.2 Exponents and Radicals 16 1.3 Algebraic Expressions 27 1.4 Fractional Expressions 40 Chapter 1 Review Exercises 49 Chapter 1 Discussion Exercises 51 C H A P T E R 2 Equations and Inequalities 53 2.1 Equations 54 2.2 Applied Problems 61 2.3 Quadratic Equations 73 2.4 Complex Numbers 87 2.5 Other Types of Equations 94 2.6 Inequalities 102 2.7 More on Inequalities 111 Chapter 2 Review Exercises 119 Chapter 2 Discussion Exercises 122 C H A P T E R 3 Functions and Graphs 123 3.1 Rectangular Coordinate Systems 124 3.2 Graphs of Equations 130 3.3 Lines 140 3.4 Definition of Function 155 3.5 Graphs of Functions 171 3.6 Quadratic Functions 185 3.7 Operations on Functions 197 Chapter 3 Review Exercises 205 Chapter 3 Discussion Exercises 211 iv C o n t e n t s v C H A P T E R 4 Polynomial and Rational Functions 213 4.1 Polynomial Functions of Degree Greater Than 2 214 4.2 Properties of Division 222 4.3 Zeros of Polynomials 229 4.4 Complex and Rational Zeros of Polynomials 241 4.5 Rational Functions 248 4.6 Variation 265 Chapter 4 Review Exercises 272 Chapter 4 Discussion Exercises 275 C H A P T E R 5 Inverse, Exponential, and Logarithmic Functions 277 5.1 Inverse Functions 278 5.2 Exponential Functions 287 5.3 The Natural Exponential Function 299 5.4 Logarithmic Functions 308 5.5 Properties of Logarithms 323 5.6 Exponential and Logarithmic Equations 330 Chapter 5 Review Exercises 342 Chapter 5 Discussion Exercises 345 C H A P T E R 6 The Trigonometric Functions 347 6.1 Angles 348 6.2 Trigonometric Functions of Angles 358 6.3 Trigonometric Functions of Real Numbers 375 6.4 Values of the Trigonometric Functions 393 6.5 Trigonometric Graphs 400 6.6 Additional Trigonometric Graphs 412 6.7 Applied Problems 420 Chapter 6 Review Exercises 433 Chapter 6 Discussion Exercises 439 vi C O N T E N T S C H A P T E R 7 Analytic Trigonometry 441 7.1 Verifying Trigonometric Identities 442 7.2 Trigonometric Equations 447 7.3 The Addition and Subtraction Formulas 457 7.4 Multiple-Angle Formulas 467 7.5 Product-to-Sum and Sum-to-Product Formulas 477 7.6 The Inverse Trigonometric Functions 482 Chapter 7 Review Exercises 496 Chapter 7 Discussion Exercises 499 C H A P T E R 8 Applications of Trigonometry 501 8.1 The Law of Sines 502 8.2 The Law of Cosines 512 8.3 Vectors 522 8.4 The Dot Product 536 8.5 Trigonometric Form for Complex Numbers 546 8.6 De Moivre’s Theorem and nth Roots of Complex Numbers 552 Chapter 8 Review Exercises 557 Chapter 8 Discussion Exercises 560 C H A P T E R 9 Systems of Equations and Inequalities 563 9.1 Systems of Equations 564 9.2 Systems of Linear Equations in Two Variables 573 9.3 Systems of Inequalities 582 9.4 Linear Programming 590 9.5 Systems of Linear Equations in More Than Two Variables 598 9.6 The Algebra of Matrices 614 9.7 The Inverse of a Matrix 623 9.8 Determinants 628 9.9 Properties of Determinants 634 9.10 Partial Fractions 642 Chapter 9 Review Exercises 648 Chapter 9 Discussion Exercises 651