I-2 Index. Coterminal Angles, 738 Counting numbers, 8 Cramer, Gabriel, 309 Cramer s rule, 306 Cube root, 427, 434 Cylinder, right circular, 117
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1 Index Absolute value, 18 equations, 154, 162 inequalities, 159, 162 Absolute error, 158 Addition, 4 associative property, 19 commutative property, 18 of complex numbers, 481 of fractions, 21 of functions, 248 of polynomials, 50 of radical expressions, 457, 458 of rational expressions, 381 of real numbers, 28, 29 Addition property of equality, 101 of inequalities, 143 Additive identity, 23 Additive inverse, 16, 23 Agnesi, Maria Gaetana, 492 Alternating sequence, 630 Amplitude, 746 Analytic geometry, 183, 293 Angles, 129 acute, 129 base, 132 central, 715 complementary, 130 coterminal, 738 degree measure, 129 of depression, 842 of elevation, 842 negative, 714 obtuse, 129 positive, 714 reference, 734 right, 129 sides, 129 standard position, 724 straight, 129 supplementary, 130 vertex, 129 Antilogarithm, 605 Area, 116 Arithmetic sequence, 80, 644 general term, 645 partial sum, 646 Associative properties, 19 Asymptote(s), 689, 691 horizontal, 412 vertical, 399 Augmented matrix, B 1 Axes of a coordinate system, 84, 180 Axis, 180 Bar chart, 3, 178 Base, 4 of a logarithm, 588, 614 Bearing, 845 Brahmagupta s formula, 456 Break-even point, 343 Binomial, 50 multiplication of, 54 square of, 56 Binomial coefficient, 662 Binomial formula, 664 Blueprint for problem solving, 126, 312 Boundary line, 215, 216 Braces, in set-builder notation, 7 Brackets, in interval notation, 144 Calculators, graphing, see Graphing calculators Calculators, scientific, 45, 46 finding logarithms, 604 finding natural logarithms, 607 finding roots, 430 change of base, 614 rational exponents, 430, 442 trigonometric functions, 732 Carbon-14 dating, 438 Cartesian coordinate system, 180 Central Angle, 715 Change of base (logarithms), 614 I-1
2 I-2 Index Characteristic, 604 Circle, 677, 679 circumference, 350 tangent to a line, 686 Closed interval, 148 Coefficient, 49 leading, 50 Combining similar terms, 21 Common denominator, 21, 382 Common difference, 80, 644 Common logarithms, 603 Common ratio, 81, 652 Commutative properties, 18, 19 Comparison symbols, 3 Completing the square, 494, 497, 499 Complex fractions, 389 Complex numbers, 479, 480 addition, 481 conjugates, 483 division, 483 equality, 480 i, 478 imaginary part, 479 multiplication, 481 real part, 479 standard form, 479 subtraction, 481 Composite number, 10 Composition of functions, 250 Compound inequality, 147 Compound interest, 571, 573 Conclusion, C 1 Conditional statement, C 1, C 3 notation, C 2 Conic sections, 677 Conjugates, 464, 483 Constant term, 101, 107, 203 Constant of variation, 255 Contrapositive, C 2, C 3 Continued inequalities, 148 Converse, C 2 Conversion factor, 409 Coordinates, 8 Cosecant function, 725, 838 graph of, 755 Cosine function, 725, 838 graph of, 746 Cost, 49, 52, 114, 247, 343, 505 average, 360 Cotangent function, 725, 838 graph of, 756 Coterminal Angles, 738 Counting numbers, 8 Cramer, Gabriel, 309 Cramer s rule, 306 Cube root, 427, 434 Cylinder, right circular, 117 Decibel, 597 Decreasing sequence, 121, 630 Degree measure, 129, 714 converting to radians, 717 Degree of a polynomial, 50 Demand, 371 Dependent equations, 277, 281, 291, 293 Dependent variable, 236 Depreciation, 625 Descartes, René, 180, 273 Determinant, 298, 301 expansion by minors, 300, 301 sign array, , , 299, 301 Difference, 4, 29 Difference of two cubes, 73 Difference of two squares, 57, 71 Difference quotient, 362, 363 Direct variation, 255 Discriminant, 517 Distance formula, 677, 678 Distributive property, 16, 20 Division, 4 of complex numbers, 483 of fractions, 33 of function, 248 of polynomials, 360, A 1 of radical expressions, 464 of rational expressions, 374 of real numbers, 31 synthetic, A 1 with rational exponents, 440 with zero, 34 Domain, 224, 225, 226 of a rational function, 353 Double-angle formulas, 817 Doubling time, 611 e, 572 Element of a set, 6
3 Index I-3 Ellipse, 676, 677, 687, 688, 692, 694, 711, 712 major axis of, 698, 712 minor axis of, 698, 712 reflective property, 676 Empty set, 6 Equality, properties, 101, 102 squaring property, 469 Equations absolute value of, 154, 162 equivalent, 101 exponential, 612 linear, 101, 182, 203, 206, 207 with logarithms, 589 quadratic, 107 quadratic in form, 526 with radicals, 468 rate, 350 with rational expressions, 394 solution set, 101, 276, 289 systems of, 276, 280, 283, 289, 306, 307 trigonometric, 823 Equilateral triangle, 132 Euclid s Elements, 432 Eves, Howard, 432 Expanded form, 4, 44 Expansion by minors, 300 Exponents, 4 negative integer, 40 properties of, 39 43, 429 rational, 429, 430 Exponential decay, 466 Exponential depreciation, 625 Exponential equation, 612 Exponential form, 4, 589 Exponential function(s), 567, 568 graphing, 569 natural, 572 Extraneous solution(s), 396, 469, 528 Factor, 10 prime, 10 Factorial, 15, 662 Factoring, 61, 70, 75 the difference of two cubes, 73 the difference of two squares, 71 by grouping, 62 the sum of two cubes, 73 trinomials, with rational exponents, 441 Family of curves, 196 Fibonacci, Leonardo, 2, 675 Fibonacci number, 86 Fibonacci sequence, 2, Finite set, 6 FOIL method, 54 Formulas, 112 Fractions addition, 21 complex, 389 division, 33 multiplication, 17 reducing, 11 Function, 223, 225, 583 composition, 250 difference of, 248 domain of, 224, 225, 353 exponential, 567, 568 input, 223 inverse, 578, 581, 583 logarithmic, 588 map, 224 notation, 236, 237, 452, 581 one-to-one, 580 output, 223 product of, 248 quadratic, 493 quotient of, 248 range, 224, 225 rational, 345, 352, 353, 411 sum of, 248 trigonometric, 725, 838 Gauss, Karl, 97 General term of a binomial expansion, 666 of an arithmetic sequence, 645 of a geometric sequence, 652 of a sequence, 81, 120, 629, 632 Geometric sequence, 652 general term, 652 partial sum, 654 Geometric series, infinite, 655, 656 Geometry, facts from angles, 129 area, 116 base angles, 132 complementary angles, 130 degree measure, 129 equilateral triangle, 132
4 I-4 Index Geometry, facts from (Continued) Euclid s Elements, 432 Eves, Howard, 432 golden ratio, 433, 529 golden rectangle, 432 hypotenuse, 133 isosceles triangle, 132 perimeter, 116 Pythagorean theorem, 133, 432, 451 right circular cylinder, 117 right triangle, 133 special lines, 185, 209 spiral of roots, 445, 451, 453, 491 supplementary angles, triangle, triangle, 500 Geometry of equations in three variables, 293 Golden ratio, 433, 462, 529 Golden rectangle, 426, 432, 461, 465 Graph of a circle, 680, 681 of direct variation, 256 of an ellipse, 686, 687, 688, 693, 694, 712 of exponential functions, 569 of a function, 226, 238 of a hyperbola, 689, 691, 695 of inequalities, 143, 144, 148, 216, 217, 327, of inverse variation, 257, 259 of a line, 182 of a logarithmic function, 590 of a parabola, 536, 537, 542 of a point, 180, 293 of a rational function, 354, 399, 411 of trigonometric functions, 744, 746, 753, 755, 756 Graphing calculators change of base, 614 complex solutions, cost, profit, revenue, 247, 505, 513 cube roots, 434 degree mode, 746 exploring asymptotes, 400, 412 extraneous solutions, 532 evaluating formulas, 121 family of curves, 196 finding logarithms, 616 finding terms of a sequence, 630, 631 graphing functions, 240 Graphing calculators (Continued) graphing lines, 186, 209 linear inequalities, 219 quadratic functions, 512, 531 radian mode, 746 recursive formula, 453, 632, 674 solving equations, solving quadratic inequalities, 551 solving systems that intersect in exactly one point, 284 solving for y, 186 spiral of roots, 453 spreadsheets, 136 summing series, 639 tables, 228, 241, 400, 412, 630 trace, 186, 285, 352, 512, 521, 523, 531, 554 trigonometric functions, 732 window, 186, 187 zoom, 186, 285, 352, 512, 523, 531, 554 Greatest common factor, 61 Half-angle formulas, 819 Heron s formula, 456 Horizontal asymptote, 412 Horizontal axis, 84, 178 Horizontal line, 185, 209 Horizontal line test, 581 Hypatia of Alexandria, 712 Hyperbola, 677, 689, 692, 694, 695 asymptotes, , 695 Hypotenuse, 133 Hypothesis, C 1 i, 478 Identity, 106 trigonometric, 796 Imaginary numbers, 480 Inconsistent equations, 276, 281, 292, 293 Increasing sequence, 120, 629 Independent variable, 236 Index of a radical expression, 427 of summation, 638 Inductive reasoning, 79, 80 Inequalities absolute value, 159, 162 compound, 147 continued, 148
5 Index I-5 Inequalities (Continued) graphing, 144, 148, 217, linear, 143, 215, 217 quadratic, 549 rational, 555 second-degree, 700 systems, 705 Inequality, properties, 143, 145 Infinite set, 6 Integers, 8, 9 Intercepts, 183, 536 Intersection of sets, 7, 147 Interval notation, 144, 148 graphing, 144, 148 Inverse function, 578, 581, 583 notation, 581 Inverse reasoning, 585 Inverse statement, C 2 Inverse trigonometric function, 780 Inverse variation, 255, Irrational numbers, 9 Isosceles triangle, 132 Jacobs, Harold, 82 Joint variation, 260 Kovalevskaya, Sofia, 98 Largest value, 539 Law of cosines, 862 Law of levers, 264 Law of sine s, 851 ambiguous case, 857 Leading coefficient, 50 Least common denominator (LCD), 21, 382 Like terms, 50 Light intensity, 272 Line graph, 83, 84, 178 Linear depreciation, 625 Linear equation in one variable, 101 solving, 105 in three variables, 289, B 2 geometry of, 293 in two variables, 182, 203, 206, 207 Linear inequalities in one variable, 143 in two variables, 215, 217, 327 Linear systems, 276, 389, B 1 solution by addition method, 277, 280 solution by graphing, 276 solution by substitution method, 282, 283 solution using augmented matrix, B 1 Linear term, 107 Lithotripter, 676 Logarithm(s), 588 base, 588 change of base, 614 characteristic, 604 common, 603 identities, 591 mantissa, 604 natural, 607 properties, 597, 598, 614 Logarithmic form, 589 Logarithmic function(s), 588 graphing, 590 Magnitude (absolute value) of a number, 18 Mantissa, 604 Mathematical model, 82 Matrix augmented, B 1 row operations, B 1 Member, of a set, 6 Midpoint, 214 Minor, 300 Modular arithmetic, 26 Monomial, 49 Multiplication, 4, 10 associative property, 19 commutative property, 19 of complex numbers, 481 of fractions, 17 of functions, 248 of polynomials, of radical expressions, 462 of rational expressions, 373 of real numbers, 30, 31 with rational exponents, 439 Multiplication property of equality, 102 of inequality, 145 Multiplicative identity, 23 Multiplicative inverse, 17, 23 Multiplicity, 519
6 I-6 Index Natural exponential function, 572 Natural logarithms, 607 Natural numbers, 8 Negative exponent, 40 Negative number, square root of, 428, 478 Nonlinear systems, 702 nth partial sum, 646, 654 nth root, 427 Null set, 6 Numbers complex, 479, 480 composite, 10 counting, 8 integers, 8 irrational, 9 natural, 8 prime, 10 rational, 8, 347 real, 8 whole, 8 Numerical coefficient, 49 One-to-one function, 580 Open interval, 148 Operation symbols, 4 Opposites, 16 Order of operations, 5 Ordered pair, 180 Ordered triple, 289 Origin, 7, 180 line through, 185, 209 Paired data, 84, 178 Parabola, 493, 536, 537, 542, 677 concave down, 538 concave up, 537 intercepts, 536, 537 vertex, 536, 537, 542 Parallel lines, 195 Parentheses, in interval notation, 144 Pascal, Blaise, 273 Pascal s triangle, 662 Percent, 117 Perfect square, square root of, 450 Perimeter, 116 Period, 745 Perpendicular lines, 195 Petters, Arlie O., 565 ph, 602, 606 Phase shift, 759 Pi ( ), 113, 350, 572 Point-slope form of a line, 206 Polynomial, 49 addition, 50 binomial, 50 coefficient, 49 degree, 50 division, 360, A 1 factoring, 61, 70, 75 long division, 364 monomial, 49 multiplication, prime, 63 subtraction, 51 trinomial, 50 Prime factor, 10 Prime number, 10 Prime polynomial, 63 Product, 4 Profit, 49, 52, 114, 247, 343, 505 Pythagorean theorem, 133, 432, 451 Quadrant, 180 Quadratic equation, 107 solving by completing the square, 494, 497, 499 solving by factoring, 109 standard form, 107 Quadratic formula, 505, 506 Quadratic inequalities, 549 Quadratic term, 107 Quotient, 4, 31 Radian(s), 715 converting to degrees, 718 Radian measure, 715 Radical(s), 427 equations with, 468 similar, 457 Radical expressions, 427 addition, 457, 458 division, 464 multiplication, 462 properties of, 446, 450, 451 simplified form, 446 subtraction, 457, 458 Radical sign, 427 Radicand, 427
7 Index I-7 Range, 224, 225, 226 Rate equation, 350 Ratio, 350 Rational exponent, 429, 430 Rational expression(s), 347, 422 addition, 381 division, 374 equations involving, 394 multiplication, 373 properties of, 347 reducing, 347 subtraction, 381, 441 Rational function, 345, 352, 353 graph of, 399, 411 Rational numbers, 8, 9, 347 Rationalizing the denominator, 448 Real numbers, 7, 9 arithmetic, 28 properties, 18 Real number line, 7 Reciprocal, 17 Recorde, Robert, 175 Rectangular coordinate system, 180 Recursion formula, 632 Recursive process, 452 Reference angle, 734 Relation, 226, 583 Revenue, 49, 52, , 247, 343, 505, 543 Richter scale, 588, 592 Right triangle, 133 hypotenuse, 133 legs, 133 Rise of a line, 192, 193 Roots, 427, 429 Run of a line, 192, 193 Scatter diagram, 83, 176, 225 Scientific notation, 44 Secant function, 725, 838 graph of, 756 Second-degree inequalities, 700 Sequence, 79, 628 alternating, 630 arithmetic, 80, 644 decreasing, 121, 630 Fibonacci, general term, 120, 629, 632, 645, 652 geometric, 81, 652 Sequence (Continued) increasing, 120, 629 notation, 628 recursive, 632 Series, 637 infinite geometric, 655, 656 Set, 6 empty, 6 finite, 6 infinite, 6 intersection of, 7, 147 null, 6 operations, 6 union, 6, 147 Set-builder notation, 7 Sierpinski triangle, 661 Sigma notation, 638 Sign array of a determinant, 301 Similar radicals, 457 Similar terms, 21, 50 Simplified form for radicals, 446 Sine function, 725, 838 graph of, 744 Slope, 192, 193, 203, 209 Slope-intercept form of a line, 203 Smallest value, 538 Solution set, 101, 276, 289 Solutions of multiplicity 1, 519 of multiplicity 2, 519 Speed average, 350, 423 Spiral of roots, 445, 451, 453, 491, 874 Spreadsheets, 136, 179 Square root, 424, 426, 427 of a negative number, 428, 478 of a perfect square, 450 Squaring property of equality, 469 Standard form of a complex number, 479 of a linear equation, 182, 207 of a quadratic equation, 107 Standard position, 724 Study skills, 2, 100, 177, 275, 346, 425 Subscript, 119 Subset, 6 Subtraction, 4 of complex numbers, 481 of functions, 248 of polynomials, 51 of radical expressions, 457, 458
8 I-8 Index Subtraction (Continued) of rational expressions, 381, 441 of real numbers, 29 Sum, 4 Sum and difference formulas, 809 Sum of two cubes, 73 Summation notation, 638 index, 638 Synthetic division, A 1 Systems of inequalities, 327, 705 of linear equations, 276, 280, 283, 289, 306, 307, B 1, B 2 nonlinear, 702 Tangent function, 725, 838 graph of, 753 Term, 21 constant, 101, 107 of an equation, 101 of a polynomial, 49 of a sequence, 119, 628 variable, 101, 107 Topographic map, 837 contour lines, 837 Trace feature on a graphing calculator, 186, 285, 352, 512, 521, 523, 531, 554 Tree diagram, 89 Triangle(s) equilateral,132 isosceles, 132 right, 133 special, 500 Trigonometric equations, 823 Trigonometric functions, 725, 838 graphs, 744, 746, 753, 755, 756 inverse, 780 signs of, 728 Trigonometric identities, 796 proving, 804 Pythagorean, 798 ratio, 797 reciprocal, 796 Trinomial, 50 factoring, perfect square, 70 Union of sets, 6, 147 Unit analysis, 405, 409 Unit circle, 810 Variable term, 101, 107 Variation constant of, 255 direct, 255 inverse, 255, joint, 260 proportionality constant, 255 Venn, John, 6, 14 Venn diagram, 6, 14 Vertex of an angle, 129 of an ellipse, 688 of a hyperbola, 692 of a parabola, 536, 542 Vertical asymptote, 399 Vertical axis, 84, 178 Vertical line, 185, 209 Vertical line test, 229 Vertical translations, 766 Wallis, John, 144 Whole numbers, 8 Witch of Agnesi, 492 x-axis, 180 x-coordinate, 180 x-intercept, 183 x y -coordinate system, 692 y-axis, 180 y-coordinate, 180 y-intercept, 183, 203 Zero-factor property, 107 Zeno of Elea, 344 paradoxies, 344 Zoom feature on a graphing calculator, 186, 285, 352, 512, 523, 531, 554
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