5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-1 Index Absolute value defined, 46 functions and, 126 59 Absolute value equations, solutions, 46 49 Absolute value inequalities, solutions, 263 267 Acceleration, 292 Addition of complex numbers, 217 218 of functions, 151 of matrices, 571 572 of polynomials, 13 of radical expressions, 7 of rational expressions, 34 35 Addition method systems of equations, 517 519, 521, 618 619, 621 systems of nonlinear equations, 712, 714 715 Additive identity, 187 Additive identity for matrices, 573 Additive inverse, 43, 187 Additive inverse for matrices, 589 590 Amortization Formula, 445 450 defined, 447 Annual Compound Interest Formula, defined, 432 Annuity Formula, 441 445 defined, 442 Area of square formula, 170 171 Arithmetic General Formula, defined, 739 Arithmetic sequences, 738 740 defined, 738 Arithmetic series, 755 Asymptotes defined, 348 finding, 354 355 horizontal asymptotes, 349 352 of hyperbolas, 689, 692 linear asymptotes, 353 oblique asymptotes, 352 356 quadratic asymptotes, 355 vertical asymptotes, 348 349 Augmented matrices, 617, 618, 619, 620 Average rate of change, 176 180 Average rate of change formula, 176 177 Axis of symmetry defined, 662 of parabolas, 662 663, 664 Base Change of Base Formula, 407 408 logarithmic functions and, 404 407 Big Picture, concepts of, 41 44, 45 Binomials division by, 16 17 FOIL method and, 22 multiplying binomial by trinomial, 14 special binomials, 24 25, 226 Binomial squared formula, 14 Binomial squares, 701 Central box, of hyperbolas, 689 Certain events, 777 Change of Base Formula, 407 408 defined, 407 Circles as conic sections, 660, 678 679 defined, 679 general form of, 705 identifying, 699 solutions, 713 714 as special ellipses, 678 679 Coefficient matrix, 603, 605, 617 Coefficients correlation coefficient, 548 549 general form of a conic and, 699 Cofactors, 597 Collaborative activities exponential functions, 392 factoring techniques, 29 30 finding equations, 528 529 function review, 489 493 graphing techniques, 78 identifying conics, 710 inverse functions, 202 linear equations, 51, 512 513 loans and mortgages, 455 457 logarithmic and exponential equations, 429 430 multiplicative inverses of matrices, 601 603 operations on functions, 168 169 parabolas versus ellipses, 686 687 polynomial functions, 299 properties of matrices, 588 589 property of logarithms, 415 416 rational functions, 361 365 sequences, 749 750 solving inequalities graphically, 275 solving quadratic equations, 239 240 theoretical versus empirical probability, 789 792 Combinations counting theory and, 766 769 defined, 766 Combinations Formula, defined, 766 Common differences arithmetic sequences and, 739, 742 common ratios versus, 380 383 polynomials and, 289 290, 291, 292 293, 294 slope of line and, 497 tables and, 291, 292 293, 380 381, 382, 383, 384 Common logarithms, 407 Common ratios common differences versus, 380 383 geometric sequences and, 740, 741, 742 tables and, 381, 382, 383, 384 Commutative property of multiplication, 577, 591 Complement formulas for, 781 probability and, 780 783 Completing the square, 229 231, 701 702 Complex Conjugate Zeros Theorem, 340 defined, 338 Complex fractions defined, 35 simplification of, 35 37 Complex numbers addition and subtraction of, 217 218 defined, 217 definition of i and, 216 217 division of, 219 220 fractals and, 221 223 multiplication of, 218 219 Composed functions defined, 158 domain and range of, 159 163 real-valued solutions and, 159 Composition, of functions, 158 163, 172 176 Compound Interest Formula, defined, 433 Compression, 317 318, 666 defined, 318 Concave down graphs, 137 Concave up graphs, 137 Conic sections circles, 660, 678 679 686 687, 710 distance formula and, 660 661 ellipses, 660, 672 679 general form of conics, 699 706 hyperbolas, 660, 688 694 identifying conics, 698 706, 710 parabolas, 660, 661 669 types of, 660 Conjugate axis, of hyperbolas, 689 Conjugates, 219 Consistent systems of equations, 515, 517 Constant denominators, 251 Constant matrix, 603, 617 Constant of variation, A-14, A-15 Constant polynomials, 289 290, 294, 302 Constants, exponential and logarithmic systems, 521 522 Constraints, A-7, A-8 Continuous Compound Interest Formula, defined, 434 Continuous polynomials, 300, 301 Contradiction, 45 Correlation coefficient, 548 549 Counting principle counting theory and, 761 763, 764, 766 defined, 761 Counting theory combinations and, 766 769 counting principle, 761 763, 764, 766 permutations and, 763 766 Covertices of ellipses, 673 of hyperbolas, 689 Cramer s Rule, 607 610 Cube roots, 58 Cubic function models, 546, 549, 552 Curve of best fit, 530, 532, 549 550 Cyclic behavior, 217 Data analysis appropriate models and, 545 548 determination of good models, 548 554 linear models and, 496 504 modeling with graphing calculator and, 530 538 systems of two-variable equations and, 514 524 Decay applications of, 458 465 384 386, 394 395 Decibels, 472 473 Decreasing functions, 136, 137, 138 Defined area, 221 Degenerate cases, 705 Degree of a polynomial, defined, 13 Degrees, Demand function, 504 Denominators constant denominators, 251 least common denominator, 34, 251 rationalization of, 6, 8 9 I-1
5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-2 I-2 Index Denominators (continued) single variable term in, 251 252 variables in, 252 253 Dependent systems of equations, 516 517, 519 Dependent variables defined, 107 functions and, 108, 109, 170 172 Descending order, Determinants Cramer s Rule and, 607 610 defined, 596 matrices and, 596 598 Difference of two cubes formula, 24 Difference of two squares formula, 14, 24 Difference quotient, 176 180 Dimension fractional dimension, 222 of matrices, 568 569, 571 573, 579 Directrix, of parabolas, 661, 662, 664 Direct variation, A-14 A-15 defined, A-14 Distance formula defined, 661 parabolas and, 662 Pythagorean Theorem and, 660 661 Distinguishable permutations, 768 769 Distributive property, 218 Dividends, defined, 16 Division of complex numbers, 219 220 of functions, 151 of polynomials, 15 17, 324 331, 617 of rational expressions, 33 Divisors, defined, 16 Domain of combined functions, 153, 154 of composed functions, 159 163 defined, 120 of functions, 122 125, 193 194 of square roots, 123, 125 126 Domain of all real numbers, 123, 124 Dominating terms, 350 e defined, 397 397 398 natural logarithms and, 407 Equations. See also Systems of equations absolute value equations, 46 49 Big Picture and, 41 43 528 529 exponential equations, 246 247, 417 420, 429 430 flow charts for solving, 423 425 fractional exponent equations, 243 248 linear equations, 41 46, 51, 499 500, 512 524 logarithmic equations, 420 423, 429 430 matrix equations, 603 613 quadratic equations, 225 235, 239 240 quadratic-like equations, 253 258 square root equations, 240 244, 248 without solution or with many solutions, 45 46 Even functions, 308 Even-power polynomials, 305 Events certain events, 777 defined, 776 impossible events, 777 independent events, 778 mutually exclusive events, 779 Expansion about a row or column, 597 Exponential decay, 385 386 Exponential equations 429 430 factoring and, 246 248 greatest common factor and, 246 247 solutions, 417 420 Exponential expressions, 2 3 Exponential function models, 532, 535, 538, 547, 549, 551, 552 553 Exponential functions basic characteristics of, 393 397, 521 392 construction of, 386 387 defined, 395 finding, 522 523, 524 geometric sequences and, 740 growth and decay, 384 386 inverse of, 398 400 nature of, 380 384 number e and, 397 398 Exponential growth, 384 385 Exponential notation, 2 3 defined, 2 Exponents exponential notation, 2 3 fractional exponents, 4 6, 243 248 logarithms and, Extraneous solutions, 242 Factorials defined, 733 simplification of, 767 Factoring 29 30 combination of factoring techniques, 26 exponential equations and, 246 248 by greatest common factor, 20 21, 226 by grouping, 25 26, 226 polynomials, 19 26 quadratic equations and, 226 227 special binomials, 24 25 trinomials, 21 23, 226 Factor Theorem, 329 331 defined, 330 Feasible region, A-8 Finite sequences, 732 Flipping translations, 316 317 Focus, of parabola, 661, 662 663 FOIL method, 14, 21 23, 218 Form of horizontal line, 499 Form of vertical line, 499 Formulas Amortization Formula, 445 450 Annual Compound Interest Formula, 432 Annuity Formula, 441 445 area of square, 170 171 Arithmetic General Formula, 739 average rate of change formula, 176 177 binomial squared formula, 14 Change of Base Formula, 407 408 Combinations Formula, 766 combining functions and, 157 complement, 781 composed functions and, 162 163 Compound Interest Formula, 433 Continuous Compound Interest Formula, 434 difference of two cubes formula, 24 difference of two squares formula, 14, 24 difference quotient formula, 176 177 for ellipses, 673 674 Fahrenheit-Celsius conversion formula, 104 105, 121, 170, 195, 196 197 as functions, 106 107, 122 123, 125 Geometric General Formula, 741 for hyperbolas, 689 for inverse, 598 of lines, 497 501 mutually exclusive events, 779 non-mutually exclusive events, 779 for nth partial sum of arithmetic series, 755 for nth partial sum of geometric series, 757 for parabolas, 661 662, 665 for permutations, 764 ph Formula, 471 472 quadratic formula, 231 235, 257 258 for sequences, 741 745 series, 754 slope formula, 176 177, 290 291, 496 497 for sum of infinite geometric series, 758 sum of two cubes formula, 24 Fractals, 221 223 Fractional dimension, 222 Fractional exponents defined, 4 radicals and, 6 simplifying expressions with, 5 solutions of fractional exponent equations, 243 248 Fractions complex fractions, 35 37 functions in fractional form, 126 Functions. See also Polynomials application of composition, 172 176 application of function operations, 170 180 168 169, 489 493 combining functions, 150 163, 176 180 composition of, 158 163, 172 176 compound interest and, 431 437 defined, 102, 120 dependent variables and, 108, 109, 170 172 difference quotient and average rate of change, 176 180 exponential functions, 380 387, 393 400, 521, 522 523, 524 formulas as, 106 107, 122 123, 125 fractals and, 221 222 function notation, 107 110, 124 62 76 increasing and decreasing functions, 136 138 interval notation, 127 inverse functions, 186 198, 202, 398 linear functions, 125, 288, 290 291, 293, 294, 496 503 logarithmic functions, 403 411, 521, 522 523, 524 maximums and minimums of, 135 145
5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-3 Index I-3 non-quadratic functions, 320 321 odd functions, 308 309 one-to-one functions, 187 188, 192 193 operations on, 151 152 piecewise-defined functions, 84 87, 127 129 rational functions, 347 356, 361 365 relations and, 100 107, 120 130, 665 translating functions, 314 322, 665 Gaussian elimination applications and, 633 augmented matrices and, 617 Gaussian method, 617 620 616, 621 628 General Form of a Conic, 699 706 defined, 699 observations about, 704 705 Geometric General Formula, defined, 741 Geometric sequences, 740 745 defined, 740 Geometric series, 757 758 Geometry area as function of perimeter, 172 173 area of circle, 180, 186, 214, A-18 area of rectangular poster, 185 area of right triangle, 184 area of square, 110, 170 171 circumference of circle, 180, 210 dimensions of box, 347 expansion of cubed figure, 201 volume of box, 135, 148 149, 182, 237, 238, 239 volume of cone, 210 volume of conical water tank, 184 volume of cube, 117, 118, 180, 186 volume of cylinder, 182 volume of sphere, 117, 118, 180, A-18 volume of water in water tank, 182 Graphing calculators. See also TI-83/84; TI-86 absolute value and, 59 absolute value inequalities and, 266 267 approximating zeros using, 341 342 basic commands, 52 60 brightness of screen (contrast), 53 calculator keys, 52 combinations of translations and, 666 668 complements and, 782 783 conics in general form and, 706 determinants and, 598 769 ellipses and, 676 677 ENTER key, 53, 76 and exponential functions, 394 finding a graph, 69 76 FRAC key, 59 functions and, 62 76 Gaussian elimination and, 616, 621 628 graphing with, 62 76 Home Screen, 54 hyperbolas and, 689 690 inverse of matrices and, 593 596 local maximums and minimums and, 140 143 logarithmic functions and, 407 matrices and, 569 571 minus sign key, 55 56 MODE key, 53 modeling with, 530 538 MODE menu, 53 MORE key, 53 negative sign key, 55 56 ON key, 52, 76 order of operations and, 52, 55 performing operations, 54 piecewised-defined functions and, 84 87, 127 129 polynomials and, 301 quadratic equations and, 226 227 raise to the power key, 57 range of a function and, 125 126 rational functions and, 353 354 2nd key, 52, 53 sequences and, 734 736 specific points and, 66 68 square roots, 54, 57 58 systems of nonlinear equations and, 717 718 and systems of two-variable equations, 514 515 table of values for sequences, 737 TRACE command, 64, 65 typing in a function, 63 WINDOW menu, 65 66 window size and, 63, 65 66 Xscl, 72 Yscl, 72 Y screen, 62 zeros/roots and, 80 84 ZOOM Decimal, 65, 70 ZOOM Standard, 64, 70 Graph of an equation, defined, 514, 521 Graphs and graphing 78 combining functions and, 155 156, 157 composing functions and, 162 163 concave down graphs, 137 concave up graphs, 137 of ellipses, 675, 678 exponential decay and, 395 62 76 hole in graph, 354 of inverse functions, 196 197 of linear equations with two variables, 514 515, 521 local maximums and minimums and, 139 140 of logarithmic functions, 403 406 of polynomials, 300, 302 303, 306 307 solutions in three variables, 611 613 Greatest common factor (GCF) factoring by, 20 21, 226 solutions to exponential equations and, 246 247 Grouping, factoring by, 25 26, 226 Growth applications of, 458 465 384 386 Horizontal asymptotes, 349 352 defined, 349 Horizontal line, form of, 499 Horizontal line test, 188 Horizontal translations, 665 defined, 316 Hyperbolas as conic sections, 660, 688 694 defined, 688 formulas for, 689 general form of, 705 689 690 identifying, 699 reflective property of, 694 sketching, 690 694 solutions, 714 715 transforming hyperbola into standard form, 703 704 vertical hyperbolas, 692 693 I, defined, 216 217 Identity, 43, 46 Identity matrices, 573 Imaginary numbers, 216 217 Impossible events, 777 Inconsistent systems of equations, 515, 623 Increasing functions, 136 137, 138 Independent events, defined, 778 Independent systems of equations, 515 Independent variables defined, 107 functions and, 108, 109 Index defined, 4 of summation, 752 753 Inequalities absolute value inequalities, 263 267 275 linear inequalities, 262 263, A-1 A-3 polynomial inequalities, 267 269 rational inequalities, 269 271 reversing of, 262 solutions, 261 271 systems of linear inequalities, A-4 A-6 Infinite sequences, 732, 751 Infinite series defined, 751 value for, 752 Infinite solutions, 46, 519, 623 624, 712 Infinite sums, 758 Inputs, 100 102, 123 Integers, 753 754 Interval notation, 127 Inverse of exponential functions, 398 400 formula for, 598 Inverse functions 202 defined, 187 188 difficult inverses, 197 example of, 186 187 finding inverse logically, 191 193, 398 finding inverse of one-to-one function, 192 finding inverses of non-one-toone functions, 193 195 graphs of, 196 197 observations of, 189 191 Inverse matrix method, 603, 606 607, 633 Inverse method, systems of equations, 605 607 Inverse properties of logarithms, 408 Inverse relations, 191 Inverse variation, A-15 A-16 defined, A-15 Irrational Conjugate Zeros Theorem, 338 Iterations, 223 Julia Set, 222 Least common denominator (LCD) rational equations and, 251 rational expressions and, 34 Least squares best fit, 530 Like terms, defined, 7 Limits, 349 Linear asymptotes, 353 Linear equations 51, 512 513 solutions, 41 46, 499 500
5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-4 I-4 Index Linear equations (continued) systems of two-variable equations, 514 524 Linear function models, 532, 533, 537, 546, 549 Linear functions arithmetic sequences and, 739 common differences and, 293, 294 formulas of lines and, 497 501 parallel and perpendicular lines and, 501 503 range and, 125 slope formula and, 496 497 as type of polynomial function, 288, 290 291 Linear inequalities defined, A-1 solutions, 262 263, A-1 A-3 Linear models, data analysis, 496 504 Linear polynomials, 288, 290 291, 293, 294, 302, 304 Linear programming, optimization, A-7 A-11 Lines formulas of, 497 501 parallel and perpendicular lines, 501 503 slope formula, 496 497 Line solution, 613 Local maximums, 8, 139 Local minimums, 9 Logarithmic equations 429 430 solutions, 420 423 Logarithmic function models, 532 Logarithmic functions basics of, 403 408, 521 finding, 522 523, 524 properties of logarithms and, 408 411 Logarithms applications of, 471 474 415 416 defined, 399 estimating, 399 400 evaluating, 400 exponents and, properties of, 408 411, 419 Logistics function models, 550 Log of 1 property of logarithms, Long division of polynomials, 324 325, 326, 617 Major axis, of ellipses, 673 Mandelbrot Set, 221 Mathematical induction, 754 Matrices addition of, 571 572 additive identity for, 573 additive inverse for, 589 590 applications of, 579 581, 632 639 basics of, 568 570 588 589, 601 603 defined, 568 determinants and, 596 598 Gaussian elimination, 616 628, 633 identities and inverses, 589 598 inverse matrix method, 603, 606 607, 633 matrix multiplication, 575 581 matrix operations, 568 581 multiplicative identity for, 590 592 multiplicative inverses for, 592 596 scalar multiplication and, 573 575 subtraction of, 572 573 systems of equations and, 603 613 word problems and, 634 638 Matrix equations Cramer s Rule, 607 610 defined, 603 graphic representations of solutions in three variables, 611 613 inverse method, 605 607 solutions, 603 605 Minor, 597 Minor axis, of ellipses, 673 Monomials division of monomial by monomial, 15 Multiplication of complex numbers, 218 219 defined, 2 factoring as undoing of, 20 of functions, 151 of matrices, 575 581 of polynomials, 14 of radical expressions, 8 of rational expressions, 32 33 scalar multiplication, 573 575 Multiplicative identity, 187 Multiplicative identity for matrices, 590 592 Multiplicative inverse, 187 Multiplicative inverses for matrices, 592 596 Multiplicity, 229, 308 Mutually exclusive events, defined, 779 Natural logarithms, 407 Natural log function models, 534 535, 537, 547 Negative rule for exponents, 2, 3 Non-quadratic functions, translations of, 320 321 No real roots, 234 No solution, 45, 48, 515, 518, 612, 712, 715 716 nth Partial Sum defined, 751, 752 finding, 754 756 Objective function, A-7 Oblique asymptotes, 352 356 defined, 352 Odd functions, 308 309 Odd-power polynomials, 304 305 One solution, 45, 515, 612, 712 One-to-one functions, 187 188, 192 193 Optimization, linear programming, A-7 A-11 Order of combinations, 766 of matrices, 568 of permutations, 765 766 Ordered pairs, 120 122 Order of operations, graphing calculators, 52, 55 Outcomes, defined, 776 Outputs, 100 102, 123 Parabolas combinations of translations, 319 320, 666 668 compression and stretching, 317 318, 666 as conic sections, 660, 661 669 defined, 661 formulas for, 661 662, 665 general form of, 705 horizontal translations of, 316, 665 identifying, 699 multiplication by negative, 316 317 quadratic polynomials and, 314, 661 sideways parabolas, 664 665, 669 sketching and labeling, 668 669 solutions, 712 714 in standard position, 663 665 transforming parabola into standard form, 702 703 vertical translations of, 314 316, 665 Parallel lines defined, 501 finding, 501 503 Partial sums, series, 751, 752, 754 758 Perfect square trinomials, 701 702 Permutations counting theory and, 763 766 defined, 763 768 769 Permutations formula, defined, 764 Perpendicular lines defined, 502 finding, 502 503 PH Formula, 471 472 Piecewised-defined functions, graphing calculators, 84 87, 127 129 Plane solution, 613 Point of symmetry (center) defined, 672 of ellipses, 672 673 Point-slope form of a line, 498 Polynomial functions, 225 Polynomial inequalities, solutions, 267 269 Polynomials addition of, 13 characteristics of, 300 309 299 constant polynomials, 289 290, 294, 302 continuity and, 300, 301 cubic polynomials, 302, 304 defined, 12 division of, 15 17, 324 331, 617 end behavior and, 304 305 factoring, 19 26 fifth-power polynomials, 302 finding polynomial through three points, 637 638 introduction to, 288 294 linear polynomials, 288, 290 291, 293, 294, 302, 304 long division and, 324 325, 326, 617 multiplication of, 14 operations on, 12 17 quadratic polynomials, 288, 291 292, 293, 294, 302, 304, 314, 637 638 quartic polynomials, 302, 304 rational functions and, 347 356 remainder theorem, 328 329 shapes of polynomial graphs, 302 303 simplification of, 13, 14 sketching by hand, 307 subtraction of, 13 symmetry and, 308 309 translating functions and, 314 322 x-intercepts and, 305 307 zeros/roots of, 334 343 Power function models, 548 Power property of logarithms, Power rule for exponents, 2, 3 Power Rule for Logarithms, 416 Principal square root, 5, 46 47 Probability basics of, 776 780 789 792 defined, 776 use of complement, 780 783 Product property for logarithms, Product rule for exponents, 2, 3 Product Rule for Logarithms, 415 Product rule for radicals, 6
5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-5 Index I-5 Pythagorean Theorem defined, 660 distance formula and, 660 661 Quadratic asymptotes, 355 Quadratic equations 239 240 completing the square, 229 231 factoring method, 226 227 quadratic formula, 231 235 recognition of quadratics, 225 226 square root method, 227 229 Quadratic formula defined, 232 solving quadratic equations and, 231 235 solving quadratic-like equations and, 257 258 Quadratic function models, 532, 533 534, 537, 546, 549, 551, 552 Quadratic inequalities, solutions, 267 268 Quadratic-like equations, solutions, 253 258 Quadratic polynomials end behavior and, 304 finding through three points, 637 638 parabola as, 314, 661 shape characteristics, 302 translating functions and, 314 as type of polynomial function, 288, 291 292, 293, 294 Quadratic trinomials, factoring, 21 Quartic function models, 547, 548, 552 Quartic polynomials, 302, 304 Quotient property of logarithms, Quotient rule for exponents, 2, 3 Quotient Rule for Logarithms, 416 Quotient rule for radicals, 6 Quotients, defined, 16 Radical expressions addition of, 7 multiplication of, 8 simplification of, 6 subtraction of, 7 Radicals defined, 4 fractional exponents and, 6 operations on, 7 8 rationalizing denominators and, 8 9 Radicands, defined, 4 Range of a function, 122, 125 126, 193 of combined functions, 153 154 of composed functions, 159 163 defined, 120 of square roots, 123, 125 126 Rate of change, 176 180, 291, 292, 294 Rational equations, solutions, 251 253 Rational expressions addition of, 34 35 defined, 31 division of, 33 multiplication of, 32 33 operations on, 31 35 simplification of, 31 subtraction of, 34 35 Rational functions 361 365 examples of, 347 348 353 354 horizontal asymptotes and, 349 352 oblique asymptotes and, 352 356 vertical asymptotes and, 348 349 x-intercepts for, 356 Rational inequalities, solutions, 269 271 Rationalizing denominators, 6, 8 9 Rational Zeros Theorem, defined, 336 Real-value inputs, 123, 152 Real-value outputs, 123 Recursive sequences, 737 738 defined, 737 Reduced row echelon form of matrix, 593 Reflective property of ellipses, 679 of hyperbolas, 694 of parabolas, 663 Regression analysis, 530 Relations defined, 100, 102, 120 functions and, 100 107, 120 130, 665 inverse relations, 191 Relative maximums, of functions, 82 84 Relative minimums, of functions, 82 84 Remainder Theorem, 328 329 defined, 329 Restricted domain, 123, 124, 193 194 Roots. See Zeros/roots Roots of a polynomial, defined, 335 Rotated conics, 701 Rotation translation, 665, 666 defined, 317 Sample spaces, defined, 776 Scalar multiplication, matrices, 573 575 Scientific notation, 3 4 defined, 3 Self-similar, fractals as, 222 Sequence of partial sums defined, 751 series, 751 752 Sequences arithmetic sequences, 738 740 basics of, 732 737 749 750 defined, 732, 752 finding nth term of, 742 745 geometric sequences, 740 745 recursive sequences, 737 738 series and, 751 758 Series basics of, 751 752 defined, 751, 752 partial sums, 754 758 summation notation and, 752 754 Sierpinski s Triangle, 222 Simplification of complex fractions, 35 37 of complex numbers, 218 of factorials, 767 of fractional exponents, 5 of polynomials, 13, 14 of radical expressions, 6 of rational expressions, 31 of square roots, 216 217, 218 Sine function, 553 554 Singular matrix, 596 Slope formula, 176 177, 290 291, 496 497 Slope-intercept form of a line, 497 498 Smooth polynomials, 300, 301 Solutions absolute value equations, 46 49 absolute value inequalities, 263 267 composed functions and realvalued solutions, 159 counting principle, 762 equations, 41 43 exponential equations and, 417 420 extraneous solutions, 242 flow charts for solving equations, 423 425 fractional exponent equations, 243 248 functions for dependent variables, 171 172 inequalities, 261 271 linear equations, 41 46, 499 500 linear inequalities, 262 263, A-1 A-3 logarithmic equations and, 420 423 parabolas, 712 714 polynomial inequalities, 267 269 quadratic equations, 225 235 quadratic-like equations, 253 258 rational equations, 251 253 rational inequalities, 269 271 sequences, 733 734, 742 745 square root equations, 240 244, 248 systems of equations and, 514 521 systems of linear inequalities, A-4 A-6 systems of nonlinear equations, 711 718 Special binomials, 24 25, 226 Square matrices, 592, 595, 598 Square root method, 227 229 Square roots absolute value equations and, 46 47 domain and range and, 123, 125 126 graphing calculators, 54, 57 58 imaginary numbers and, 216 217 principal square root, 5, 46 47 simplification of, 216 217, 218 solutions of square root equations, 240 244, 248 Standard Form of a Circle, defined, 679 Standard form of a line, 498 499 Standard Form of an Ellipse, defined, 676 Standard Form of an Hyperbola, defined, 693 Standard Form of a Non-Function (Sideways) Parabola, defined, 669 Standard Form of the Parabola Function, defined, 668 Standard position ellipses, 674, 700 hyperbolas, 689 parabolas, 662, 663 665 Stretching, 317 318, 666 defined, 318 Substitution method systems of equations, 516 517, 521 systems of nonlinear equations, 711, 713 Subtraction of complex numbers, 217 218 of functions, 151, 157 of matrices, 572 573 of polynomials, 13 of radical expressions, 7 of rational expressions, 34 35 Summation notation defined, 752 series, 752 754 Sum of an Infinite Geometric Series, defined, 758
5639_Holtfrerich_Index 6/2/05 11:45 AM Page I-6 I-6 Index Sum of two cubes formula, 24 Symmetry axis of, 662 polynomials, 308 309 Synthetic division of polynomials, 325 328, 617 Systems of equations addition method, 517 519, 521, 618 619, 621 applications with matrices, 632 639 Cramer s Rule, 607 610 defined, 514 exponential and logarithmic equations, 521 524 Gaussian elimination and, 616 628 graphical solutions, 514 515, 521 inverse method, 605 607 matrices and, 603 613 substitution method, 516 517, 521 translating word problems and, 520 521 two-variable equations, 514 524, 712 Systems of linear inequalities, solutions, A-4 A-6 Systems of logarithmic equations, 716 717 Systems of nonlinear equations, solutions, 711 718 Tables combining functions and, 155, 157 common differences and, 291, 292 293, 380 381, 382, 383, 384 common ratios and, 381, 382, 383, 384 composing functions and, 161 162, 163 compound interest and, 431 432 exponential characteristics and, 393 exponential decay and, 385 386, 395 396 397 exponential growth and, 384 385 increasing and decreasing functions and, 138 local maximums and minimums, 139 140 of logarithmic functions, 404, 406 number e and, 397 table of values for sequences, 737 Terms of arithmetic sequences, 738 739 defined, 7 of recursive sequences, 737 738 of sequences, 732 734, 742 745 Test Point Method, 264 265, 267 270 TI-83/84 absolute value, 59 769 Gaussian elimination method and, 621 622, 624 626 inverse of matrix and, 594 596 MATH menu, 58 matrices and, 569 570 minus sign key, 55 56 modeling with, 531 538, 545 548, 550 negative sign key, 55 56 piecewise-defined functions and, 84 87 QUIT command, 54 relative maximums and relative minimums, 82 84 sequences and, 734 736 TRACE command, 64, 65, 73 window size and, 73 74 Yscl, 72 zeros/roots and, 80 84 ZOOM Decimal, 65, 75 ZOOM OUT, 71 ZOOM Standard, 64, 70, 73 TI-86 absolute value, 59 769 EXIT key, 54 Gaussian elimination method and, 621 622, 624 626 inverse of matrix and, 594 596 MATH menu, 58 matrices and, 569 570 minus sign key, 55 56 modeling with, 531 538, 545 548, 550 negative sign key, 55 56 piecewise-defined functions and, 84 87 relative maximums and relative minimums, 82 84 sequences and, 734 736 TRACE command, 64, 65, 73 window size and, 73 74 Yscl, 72 zeros/roots and, 80 84 ZOOM Decimal, 65, 66, 75 ZOOM OUT, 71 ZOOM Standard, 64, 70, 73 Translated conics, 700 Transverse axis, of hyperbolas, 689 Trinomials completing the square and, 701 factoring, 21 23, 226 multiplying binomial by trinomial, 14 Unknown constants, 521 Variable matrix, 603 Variables defined, 107 in denominators, 252 253 dependent variables, 107, 108, 109, 170 172 graphic representations of solutions in three variables, 611 613 independent variables, 107, 108, 109 systems of two-variable equations, 514 524, 712 Variation direct variation, A-14 A-15 inverse variation, A-15 A-16 other variation statements, A- 17 Velocity, 292 Vertical asymptotes, 348 349 defined, 348 Vertical line, form of, 499 Vertical line test, 106, 188, 676 Vertical translations, 314 316, 665 defined, 315 Vertices of ellipses, 673 of hyperbolas, 689 of parabolas, 661, 664, 665, 667 of quadratic polynomials, 314 Word problems matrices and, 634 638 systems of equations and, 520 521 x-intercepts 395 polynomials and, 305 307 for rational functions, 356 x-intercept Theorem, 333 334 y-intercepts, exponential functions, 395 Zero of a polynomial, defined, 335 Zero rule, 2 Zeros/roots 80 84 polynomials and, 334 343 roots of polynomials, 335 336, 338 339, 342 343 zeros of polynomials, 336 338, 340 342 Zeros Theorem, defined, 339