Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

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I-4 Index

Factoring

chart of methods of, 46

difference of two squares, 43

explanation of, 39, 46, 82

by grouping, 40–41

nested, 53

perfect square trinomials, 43–44

polynomial equations, 46–49

quadratic forms, 45–46

quadratic polynomials, 41–43

sum or difference of two perfect cubes,

44–45

trial-and-error method for, 42–43

Factors

greatest common, 39–40

Factor theorem

explanation of, 387, 471

to ﬁnd factors of polynomials, 387–389

ﬁnding zeroes using, 389

proof of, A–34

Families

of curves, 711

of equations, 25

of functions, 204, 236

of polynomials, 16

Feasible region, 621

Finite sequences, 762, 838

Finite series, 765–766

Focal chord

of ellipse, 723

explanation of, 723

of hyperbolas, 738, 743

of parabola, 748, 753

Foci

applications of, 724–725

of ellipse, 711, 720–725

explanation of, 710

of hyperbola, 712, 737–740

of parabola, 710, 711, 746–749

Foci formula, 723, 738

Focus-directrix form of equation of parabola,

746–749

F-O-I-L method, 19, 41, 285, 399

Folium of Descartes, 467

Formulas

area, 33, 37, 455, 491, 627, 660, 727,

A–14 – A–16

average rate of change, 361

binomial coefﬁcients, 831–832

binomial cubes, 291

binomial probability, 836

body mass index, 37

change-of-base, 522–523

Clark’s rule, A–12

compounded interest, 541

distance, 95–96, 708, 755

Euler’s polyhedron, 37

explanation of, 155

exponential growth, 495, 545

factorial, 814

foci, 723, 738

inverse of matrices, 704

lift capacity, 37

midpoint, 95, 708, 755

perimeter, 33, 660, 727, A–14 – A–16

perpendicular distance from point to line,

601, 709, 713, 755

Pick’s theorem, 132

pitch diameter, A–12

population density, 381, 441

Pythagorean Theorem, 75–76, 83

radius, 490

required interest rate, 269

right parabolic segment, 752

simple interest, 539

slope, 107, 108, 110, 138

Stirling’s formula, 814

sum of cubes of ﬁrst n natural numbers, 793

sum of ﬁrst n natural numbers, 781

sum of squares of ﬁrst n natural

numbers, 781

surface area of cylinder, 162

surface area of rectangular box with square

ends, 323

velocity, 79, 80, 236

vertex, 315–316

vertex/intercept, 323

volume, 34, 218, 491, 627

Fractions

compound, 58–59

partial, 682–688, 702

Function families, 204

Function form, of linear equations, 137, 138

Function notation, 125–127

Functions

absolute value, 204, 225, 257

algebra of, 340–346, 373

applications of, 189, 345–346

base, 378

composite, 356–357

composition of, 352–359, 374

constant, 137

continuous, 247

cube root, 205

cubing, 205

domain and range of, 122–125, 128

evaluation of, 126–127

even, 190–191

explanation of, 119–120, 181–182

exponential, 492–499, 566–567

on graphing calculator, 127, 244–245

graphs of, 120–122, 127–128, 190–197,

204–214, 273, 343–345, 485–487

identiﬁcation of, 120

identity, 137, 204

increasing or decreasing, 194–195

inverse, 481–487, 527, 566

linear, 138, 169–171

logarithmic, 504–508, 567

maximum and minimum value of, 196,

621–622

nonlinear, 331–332

objective, 620

odd, 191–192

one-on-one, 480–484, 486, 566

piecewise-deﬁned, 247–255, 276

polynomial, 192–193, 394–405, 411–420

positive and negative, 192–193

power, 236–242, 275, 296

products and quotients of, 341–343

quadratic, 293–294, 313–321, 372

range of, 123–124

rational, 232–236, 239–242, 275,

430–439, 473

as relations, 119–122

root, 236, 238–239

sand dune, 257

smooth, 247

square root, 204

squaring, 204

step, 254–255

sums and differences of, 340–341

vertical line test for, 120–122

zeroes of, 192–194

Fundamental principle of counting (FPC),

806–807, 820

Fundamental properties of logarithms

explanation of, 517

to solve equations, 516–519

Fundamental property of rational

expressions, 53–54

Fundamental theorem of algebra,

394–397, 399

Galileo Galilei, 236, 363

Gauss, Carl Friedrich, 394, 642

Gaussian elimination, 640, 642, A–30

Gauss-Jordan elimination, 642–645, A–30

General form, of equation of circle, 98

General functions, transformations of,

211–214

General linear equations, 147

Geometric sequences

applications of, 789–791

explanation of, 782–783, 839

ﬁnding nth partial sum of, 787–788

ﬁnding nth term of, 783–787

sum of inﬁnite, 788–789

Geometric series, 782

Geometry

analytical, 708–712

perimeter and area formulas, A–14 – A–16

plane, 708–709

unit conversion factors, A–18 – A–21

verifying theorem from basic, 708

volume, A–16 – A–18

Global maximum, 196

Goodness of ﬁt, 167, 376–377

Graphical solutions

intersect method for, 150–152

for linear inequalities, 154–155

x-intercept/zeroes method for, 152–154

Graphing calculator features

intersect, 150–151

QuadReg command on, 329, 330

repeat graph, 758

split screen viewing, 399

TABLE feature, 92–93, 402, 511–512, 544

window size, 603–604

Graphing calculators

absolute value on, 221, 224, 225, A–6

asymptotes on, 240

circles on, 99–100, 715

combinations on, 810

complex numbers on, 285, 286

composition of functions on, 355, 358–359

compound interest on, 542

division on, A–7

elongation and eccentricity on, 758–759

evaluating expressions on, 4

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Index I-5

exponential functions on, 496–499

functions on, 127, 344–345

hyperbolas on, 735, 736, 739

imaginary and complex numbers on, 285

intersection-of-graphs method on, 160,

519, 545–546

inverse functions on, 484–487

linear equations on, 114

linear inequalities on, 617, 619

linear programming on, 633–634

logarithms on, 505, 506, 508, 511–512,

520, 528–535

logistic equations on, 533, 534

matrices on, 643–645, 653, 656–658, 665,

667–669, 672–673

maximums and minimums on, 196–197,

317, 319–321

order of operations on, A–9

parallel lines on, 140

partial sums on, 766–767, 769

permutations on, 808

piecewise-deﬁned functions on, 250

polynomial inequalities on, 460

power functions on, 237

probability on, 821, 823, 835

quadratic equations on, 298, 299, 301, 302

rational functions on, 234, 447, 452–454

regression on, 171–173, 178, 242,

376–377, 552, 554–557

relations on, 92–94

repeating decimals on, A–2

sequences on, 763–765, 775–776,

785–786, 789–790

summation on, 767, 769

system of linear equations on, 577, 578,

580–586, 599

system of nonlinear equations on, 605–608

transformations on, 213, 265

translations on, 206–208

variation on, 264, 279

x-intercept on, 153–154, 293–294, 296

zeroes method on, 154

Graphs

of circles, 95–99

continuous, 91

of ellipses, 717–724

end-behavior of, 195, 413

of exponential functions, 493–498

of functions, 120–122, 127–128, 190–197,

204–214, 273, 343–345

of hyperbolas, 731–735, 737–740

of linear equations, 105–114, 136–142

of lines, 138–139

of logarithmic functions, 506–507

one-dimensional, 591

of parabolas, 91, 92, 193, 745–749

of piecewise-deﬁned functions, 248–254

of polynomial functions, 411–420, 472

of power functions, 237–239

quadratic, 378

of quadratic functions, 313–321

of rational functions, 232–236, 435–438,

446–450, 472–473, 485–487

of reﬂections, 208–210

of relations, 89–94

of semicircles, 92

of sequences, 776, 786

to solve inequalities, 193, 194

symmetry and, 190–192

of system of linear equations, 577

of system of nonlinear equations, 604–605

transformations of, 211–214

of translations, 206, 207

two-dimension, 591

of variation, 262–265

vertically stretching/compressing basic,

210–211

Gravity, 236, 244, 269, 271, 363

Greater than symbol, A–4

Greatest common factors (GCF), 39–40

Grid lines, 89

Grouping, factoring by, 40–41

Grouping symbols, 341

Growth rate, 545

Half planes, 615–616

Horizontal asymptotes, 234, 433–434

Horizontal boundary lines, 123–124, 134

Horizontal change, 107

Horizontal hyperbolas, 732

Horizontal lines, 109–110

Horizontal line test, 480, 486

Horizontal parabolas, 92, 745–747

Horizontal reﬂections, 209–210

Horizontal translations, 207–208

Hyperbolas

applying properties of, 739, 740

branches of, 731

central, 731–732, A–33

directrix of, 712

equation of, 731–736, 756, A–32 – A–33

equation of semi-, 742

explanation of, 731, 737–738, 756

focal chord of, 738, 743

foci of, 712, 737–740

on graphing calculator, 735, 736, 739

graphs of, 731–735, 737–740

horizontal, 732

with rational/irrational values, 759

standard form of equation of, 733–735

vertical, 733, 735–736

Identities

additive, 5–6, 81

explanation of, 27

multiplicative, 5–6, 81

Identity function, 137, 204

Imaginary numbers

calculations with, 285, 291

explanation of, 282

on graphing calculator, 285, 286

historical background of, 291

Imaginary unit (i), 282, 287

Implied domain, 124–125

Inclusion, of endpoint, 122

Inconsistent system of equations

matrices and, 644–645

in three variables, 596–598

in two variables, 581, 582

Independent variable, 88

Index, A–8

Index of summation, 766, 838

Induction. See Mathematical induction

Induction hypothesis, 800

Inequalities. See also Linear inequalities

absolute value, 223–227, 231, 275

additive property of, 27–28

applications of, 32, 464–465

compound, 29–31

graphs to solve, 193, 194

interval tests to solve, 461–464

joint, 31

linear, 27–29, 154–155

multiplicative property of, 27–28

polynomial, 459–462, 474

push principle to solve, 476–477

quadratic, 303–308, 371

rational, 461–464, 474

writing mathematical models

using, A–4

zeroes and, 475–476

Inequality symbols, A–4

Inﬁnite geometric series, 788–789

Inﬁnite sequences, 762, 838

Inﬁnity symbol, 123

Input values, 3

Integers, 81, A–2

Intercept method

explanation of, 106, 136, 137

to graph hyperbolas, 713–732

Interest

compound, 540–541, 569

continuously compounded, 541–542,

545, 569

simple, 539, 569

Interest rate, 269, 539, 540

Intermediate value theorem (IVT), 397–399

Interpolation, 556–557, 570

Intersection, 29

Intersection-of-graphs method

on graphing calculator, 160, 519, 545–546

to solve equations, 150–154, 183, 519, 534

to solve linear inequalities, 154–155

use of, 160, 544

Interval notation, 122, 123

Intervals

where function is increasing or decreasing,

194–195

where function is positive or negative,

192–194

Interval test method

explanation of, 304, 305

to solve function inequalities, 461–464

to solve quadratic inequalities, 304–305

Inverse

additive, 6, 81

of functions, 482, 566

graphs of function and its, 485–487

matrix, 665–667, 696–697, 704–705

multiplicative, 6, 81

Inverse functions

algebraic method to ﬁnd, 482–485

applications of, 487

explanation of, 481–482, 566

on graphing calculator, 484–487

graphs of, 485–487

use of, 527

Inverse variation, 264–265, 277

Irrational numbers, 81, A–3

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I-6 Index

Joint inequality, 31

Joint variation

explanation of, 266, 277

on graphing calculator, 279

Latus rectum, 753

Leading coefﬁcients, 17, 41

Least common denominator (LCD), 26,

56–58

Less than symbol, A–4

Lift capacity formula, 37

Like terms, 7

Limited value, 788

Linear association, 166

Linear depreciation, 143

Linear equations. See also Equations

applications of, 31–32, 113–114, 142–144

explanation of, 105, 181

forms of, 186–187

in function form, 137, 138

general, 147

on graphing calculators, 114

graphs of, 105–106, 136–142

horizontal and vertical lines and, 109–111

intercept/intercept form of, 147

intercept method to graph, 106, 136, 137

methods to solve, 82, 136

in one variable, 25–26, 82

parallel and perpendicular lines and,

111–113

in point-slope form, 141–142

properties of equality to solve, 25–26

in slope-intercept form, 137, 138

slope of line and rates of change and,

106–109

standard form of, 46

in two variables, 105, 106

Linear factorization theorem

explanation of, 395, 396

proof of, 8, A–35

Linear function models, 169–171, 184

Linear functions, 138

Linear inequalities. See also Inequalities

applications of, 615

explanation of, 27

on graphing calculator, 617, 619

intersection-of-graphs method to solve,

154–155

method to solve, 27–29

system of, 618–620

in two variables, 615–616, 618

Linear programming

applications of, 622–625

explanation of, 620–621, 631–632

on graphing calculators, 633–634

solutions to problems in, 621–622

Linear regression, 171–173

Linear systems. See System of linear

equations; System of linear equations

in three variables; System of linear

equations in two variables

Line of best ﬁt, 171–173

Lines

auxiliary, A–15

equations of, 138–141

horizontal, 109–110

parallel, 111–112

perpendicular, 112–113

slope-intercept form and graph of,

138–139

slope of, 106–108

vertical, 109–110

Literal equations, 155–156

Local maximum, 196

Logarithmic equations

explanation of, 569

forms of, 532

on graphing calculator, 532–533

method to solve, 517–519, 527–530

system of, 608

Logarithmic form, 504–505, 519

Logarithmic functions

domain of, 507–508

explanation of, 504, 567

graphs of, 506–507

Logarithmic regression model, 554–555

Logarithmic scales, 508–509

Logarithms

applications for, 508–512, 534–535

base-10, 522

base-e, 522

change-of-base formula and, 522–523

common, 504–506

explanation of, 567

fundamental properties of, 516–519

on graphing calculator, 505, 506, 508,

511–512, 520, 528–535

natural, 504–506

product, quotient, and power properties of,

9, 519–521, 527, 568, A–36

uniqueness property of, 528–529

Logistic equations

explanation of, 533

on graphing calculator, 533, 534

investigation of, 572

method to solve, 533–534

regression models and, 555–556

Logistic growth, 533–534, 536

Logistic growth model, 555

Long division, 382

Lorentz transformation, 52

Lower bound, 404

Mapping notation, 88, 89, 354

Market equilibrium, 585–586

Mathematical induction

applied to sums, 796–798

explanation of, 796, 840

general principle of, 799–800

to prove statement, 798–799

Mathematical models, 2–3, A–4

Matrices

addition and subtraction of, 652–653, 700

applications of, 645–646, 692–697,

701, 702

associated minor, 669–670

augmented, 638–642, 704–705, A–30

coefﬁcient, 638–639

determinants and, 668–672, A–31

equality of, 650–651

explanation of, 638, 700

on graphing calculator, 643–645, 665,

667–669, 672, 673

identity, 663–665

inconsistent and dependent systems and,

644–645

inverse of, 665–667, 696–697, 704–705

multiplication and, 653–658, 663, 700

in reduced row-echelon form, 394, 642

in reduced row-echelon form and, A–30

singular, 668–672

to solve system of equations, 640–643

square, 638, 664

in triangularized form, 640, 678

Matrix equations

applications of, 672–673

explanation of, 663, 701

on graphing calculator, 672–673

to solve systems of equations,

667–668, 701

Matrix multiplication

explanation of, 654–656

on graphing calculator, 656–658

identity matrices and, 663–665

product matrix and, 663

properties of, 657

Matrix of constraints, 638

Maximum values

explanation of, 196

of functions, 196, 621–622

on graphing calculator, 196–197, 317,

319–321

Members of set, A–1

Message encryption, 694–697

Metric units, A–19 – A–20

Midinterval points, 418, 420

Midpoint formula, 95, 708, 755

Midpoint of line segment, 94–95

Minimum values

explanation of, 196, 621

of functions, 196

on graphing calculator, 196–197, 317,

319, 320

Minors, 669–670

Mixture problems, 159–160, 582–583

Modeling

mathematical, 2–3

step function for, 255

system of linear equations in two variables,

582–586

Monomials, 16, 18. See also Polynomials

Multiplication

associative property of, 5, 81

commutative property of, 5, 81

of complex numbers, 285–287

matrix, 653–658, 663–665, 700

of polynomials, 18–21

of radical expressions, 70–72

of rational expressions, 55

scalar, 653–654

using exponential properties, 11

Multiplicative identity, 5–6, 81

Multiplicative inverse, 6, 81

Multiplicative property

of absolute value, 222

of equality, 25

of inequality, 27–28

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Index I-7

Multiplicity

vertical asymptotes and, 432–433

zeroes of, 396

Mutually exclusive events, 822

Nappe, 710

Natural exponential functions, 496

Natural logarithms

explanation of, 504

method to ﬁnd, 505–506

Natural numbers, 81, 781, A–1

Negative association, 165

Negative exponents, 13, 15, 81

Negative numbers, A–2

Negative slope, 108

Nested factoring, 53

Newton’s law of cooling, 498–499

Nondistinguishable

permutations, 809

Nonexclusive events, 821–823

Nonlinear association, 166

Nonlinear asymptotes, 448–450

Nonlinear divisor, 385

Nonlinear functions, 331–332

Nonlinear systems

applications of, 610–611

conic sections and, 749–750

of equations, 604–608, 631

of inequalities, 609–610, 631

Nonrepeating and nonterminating

decimals, A–3

Notation/symbols

for composition of functions, 353

delta, 107

explanation of, A–1

exponential, 11, A–7

factorial, 765

function, 125–127

grouping, 341

inequality, A–4

inﬁnity, 123

intersection, 29

interval, 122, 123

mapping, 88, 89, 354

proper subset of, A–1

radical, A–8

rate of change, 495

scientiﬁc, 15–16

set, 81, 122, A–1, A–2

subscript, 796

summation or sigma, 766, 767, 838

union, 29

nth term

of arithmetic sequence, 774–777

explanation of, 762

of geometric sequence, 783–787

Null set, A–1

Number line, 122, A–2

Numbers. See also Integers

complex, 282–288, 291,

370–371, 408

imaginary, 282, 285, 286, 291

irrational, 81, A–3

natural, 81, 781, A–1

negative, A–2

positive, A–2

rational, 53, 54, 81, A–2

real, 5–7, 81, A–3, A–4

sets of, 81, A–1

whole, 81, A–1 – A–2

Objective functions, 620

Objective variables, 155–156, 622

Oblique asymptotes, 448–450

Odd functions, 191–192

One-dimensional graphs, 591

One-to-one functions. See also Exponential

functions

explanation of, 480, 486, 566

identiﬁcation of, 480–481

inverse of, 482, 566

restricting domain to create, 483–484

Ordered pairs, 88–91

Ordered triples, 591–593

Order of operations, A–8 – A–9

Order property of real numbers, A–4

Origin, symmetry to, 191–192

Output values, 3

Parabolas

applications of analytic, 750

deﬁnition of, 747

explanation of, 91, 710–711, 744–745,

756–757

focal chord of, 748, 753

focus-directrix form of equation of,

746–749, 756

graphs of, 91, 92, 193, 745–749

horizontal, 92, 745–747

right parabolic segment of, 752

vertex of, 196, 205, 316

vertical, 91, 745, 747

with vertical axis, 745–746

Parallel lines

equations for, 139–141

explanation of, 111

slope of, 111, 112, 139

Parametric equations, 467

Parent functions, transformations of, 212

Parent graph, 205

Pareto’s law, 525

Partial fractions

explanation of, 682

rational expressions and, 682–688, 701

Partial sums

of arithmetic sequence, 777–778

explanation of, 766

of geometric sequence, 787–788

on graphing calculator, 766–767, 769

of series, 766

Pascal’s triangle, 829–831

Perfect cubes, 44–45, 394, A–7

Perfect squares, 43, A–7

Perfect square trinomials

explanation of, 20

factoring, 43–44

Perimeter

explanation of, 32, A–14

formulas for, 33, 660, 727, A–14 – A–16

Permutations

distinguishable, 807–808

on graphing calculator, 808

nondistinguishable, 809

Perpendicular distance, 601, 709, 713, 755

Perpendicular lines

equations for, 139–141

explanation of, 112

slope of, 112, 113, 139

Pick’s theorem, 132

Piecewise-deﬁned functions

applications of, 254–255

continuous, 247, 251

discontinuous, 252, 253

domain of, 248–249

equation of, 249, 253–254

explanation of, 247, 276

on graphing calculator, 250

graphs of, 249–251

Pitch diameter, A–12

Placeholder substitution, 45

Plane, 591, A–14

Plane geometry, relationships from, 708–709

Point of inﬂection, 124

Point-slope form

explanation of, 141

to ﬁnd function model, 143–144

linear equations in, 141–142

Pointwise deﬁned relations, 88, 89

Poiseuille’s law, 52

Polygons, 37, A–14

Polyhedron formula, 37

Polynomial equations

with complex coefﬁcients, 312

explanation of, 46–48

Polynomial form

converting between standard form

and, 759

of equation of circle, 715

of equation of ellipse, 718

of equation of hyperbola, 735

Polynomial functions

applications of, 405

complex, 394

dominant term of, 413

graphs of, 411–420

zeroes of, 192–193, 394–405, 471

Polynomial graphs

end-behavior of, 412–415, 419

guidelines for, 419–420

identiﬁcation of, 411–412

turning points and, 411

zeroes of multiplicity and, 415–418

Polynomial inequalities

explanation of, 459, 474

on graphing calculator, 476

interval tests to solve, 461–462

method to solve, 459–461

push principle to solve, 476

Polynomial modeling, 421–422

Polynomials. See also Binomials;

Monomials; Trinomials

addition and subtraction of, 17–18

applications of, 421–422

complex, 312

degree of, 16

division of, 383, 384

evaluation of, 386–387

explanation of, 16, 82

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I-8 Index

Polynomials—Cont.

factoring, 39–49

factors of, 387–389, 395

families of, 16

identifying and classifying, 16–17

multiplication of, 18–21

prime, 42, 46

quadratic, 41–43

quartic, 425

rational zeroes of, 400–402

real, 397–399

Population density, 381, 441

Positive association, 165

Positive numbers, A–2

Positive slope, 108

Power functions

applications of, 239–242

explanation of, 236, 275, 496

on graphing calculators, 237

graphs of, 236–239

transformations of, 238, 239

variation and, 280

Power property

of equality, 72, 74

of exponents, 11, 12, 15

of logarithms, 520–521, 527, 568, A–36

Present value equation, 540

Pressure, 10

Prime polynomials, 42, 46

Principal square roots, 282, 408, A–8

Prisms, A–21

Probability

binomial, 834–836

deﬁning events and, 816

elementary, 817

explanation of, 816, 841

on graphing calculator, 821, 823, 835

of mutually exclusive events, 822

of nonexclusive events, 821–823

properties of, 818–820

quick-counting and, 820–821

Problem-solving guide, 157–160

Product property

of exponents, 11, 12, 81

of logarithms, 520–521, 527, 568, A–36

of radicals, 68–69, 292

Product to power property, 12, 15, 81

Projectile height, 311, 318–319, 324

Projectile velocity, 333

Proof by induction, 796

Proper subset of whole numbers, A–1

Property of negative exponents, 13

Push principle, 476–477

Pythagorean theorem, 75–76, 83

Quadrants, 89

Quadratic equations

applications of, 306–307

checking solutions to, 327–328

completing the square to solve, 297–299

explanation of, 46–47, 292, 371

on graphing calculator, 298, 299, 301, 302

quadratic formula to solve, 300, 301

square root property of equality and,

295–297

standard form of, 46

zero product property and, 47–48

Quadratic factors, 396

Quadratic forms

explanation of, 45–46

u-substitution to factor expressions in,

45–46, 49, 75

Quadratic formula

discriminant of, 301–302

explanation of, 299–300

to solve quadratic equations, 300, 301

Quadratic functions

completing the square to graph, 313–315

equation of, 316

explanation of, 293–295, 372

extreme values and, 316–321

ﬁnding equation of, 316

vertex formula to graph, 315–316

zeroes of, 293–294

Quadratic graphs, 378

Quadratic inequalities

applications of, 30–308

domain and, 305–306

explanation of, 303, 371

interval test method for, 304–305

method to solve, 303–306

Quadratic models, 328–331, 372

Quadratic polynomials, 41–43

Quadratic regression, 331–332, 372

Quadrilaterals, A–14

Quartic polynomials, 425

Quick-counting techniques, 820–821

Quotient property

of exponents, 13, 15, 81

of logarithms, 520–521, 527, 568, A–36

of radicals, 69

Quotients

difference, 337, 359–364, 374

to power property, 12, 15

Radical equations

explanation of, 72–73

methods to solve, 73–75, 83

Radical expressions. See also Cube roots;

Square roots

addition and subtraction of, 70

evaluation of, 65

explanation of, 65

methods to simplify, 65–66, 68–69

multiplication and division of, 70–72

rational exponents and, 67–68

Radicals

explanation of, 65, 83

product property of, 68–69, 292

quotient property of, 69

Radical symbol, A–8

Radicand, 65, 66, A–8

Radioactive elements, 546

Radius

of circle, 96–98, 103, 491, 710, 714

of sphere, 490

Range

of functions, 123–124

of relations, 88, 134–135

Rate of change

applications of, 332

average, 332–334, 359–361, 373

difference quotient and, 359–361

explanation of, 181

nonlinear functions and, 331–332

notation for, 495

rational functions and, 445

slope as, 106–107, 109, 136, 142

Rational equations, 59–60

Rational exponents

explanation of, 67

power property of equality and, 74

radical expressions and, 67–68

simplifying expressions with, 68

solving equations with, 74, 83

Rational expressions

addition and subtraction of, 56–58

decomposition for, 682, 685–688

explanation of, 53, 82

fundamental property of, 53–54

multiplication and division of, 55–56

partial fractions and, 682–688

in simplest form, 53–55

simplifying compound fractions and, 58–59

Rational functions

applications of, 239–241, 439, 451–454

domain of, 431

end-behavior of, 233, 234

equations of, 236, 438

explanation of, 232, 275, 430, 473

on graphing calculator, 234, 452–454

graphs of, 235–236, 435–438, 446–450,

472–473

horizontal asymptotes of, 234, 433–435

with oblique or nonlinear asymptotes,

448–450

reciprocal function and, 232, 233, 235

removable discontinuities and, 446–447

vertical asymptotes of, 235, 430–433

writing equation of, 236

Rational inequalities

analysis to solve, 462–463

explanation of, 474

interval test method to solve, 461–464

Rationalizing the denominator, 71

Rational numbers, 53–54, 81, A–2

Rational zeroes theorem, 399–402, 429

Raw data, 165

Real numbers

absolute number of, A–5 – A–6

explanation of, 81, A–3

properties of, 5–7, 81, A–4

Real polynomials, 397–399

Reciprocal function, 232, 235

Reciprocal quadratic function, 235

Reciprocal square function, 233

Rectangles

central, 733

explanation of, 33, A–14

perimeter and area formulas for, 33, 660,

A–14

Rectangular coordinate system, 89

Rectangular solid, 35, 601, A–17

Recursive sequences, 764–765

Reduced row-echelon form, 394, 642, A–30

Reference intensity, 509

Reﬂections

horizontal, 209–210

vertical, 208–209

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Index I-9

Regression

applications of, 556–557

explanation of, 165

forms of, 552–553

on graphing calculator, 171–173, 178, 242,

376–377, 552, 554–557

quadratic, 331–332, 372

Regression equations

applications of, 556–557

explanation of, 171–172, 552, 570

Regression line, 171

Regression models

exponential, 553–554

logarithmic, 554–555

logistic equations and, 555–556

selection of, 178, 328, 570

Regular polygon, area of, 37

Relations

domain and range of, 88, 134–135

explanation of, 88, 180

functions as, 119–122

on graphing calculator, 92–94

graphs of, 89–94

methods to represent, 88, 89

pointwise deﬁned, 88, 89

Relative maximum, 196

Remainder theorem

application of, 390

to evaluate polynomials, 386–387

explanation of, 386, 470, 471

proof of, A–34

Removable discontinuities

explanation of, 446

rational functions and, 446–447

Repeated zeros, 475–476

Repeating and nonterminating

decimals, A–2

Required interest rate, 269

Residuals, 376–377, 552

Right angles, A–14

Right circular cones, 35, A–17

Right circular cylinders, 35, A–17

Right parabolic segment, 752

Right prisms, A–21

Right pyramids, 35

Right square pyramid, A–17

Right triangles, 75

Root functions

explanation of, 236

transformations of, 238–239

Roots

cube, 66, A–8

of equation, 25

extraneous, 60, 527

square, 65–66, 282, 408, A–8

Root tests for quartic polynomials, 425

Row-echelon form, 642

Row operations, elementary, 640–641

Run, 107

Sample outcome, 805

Sample space, 805, 816, 817

Sand dune function, 257

Scalar multiplication, 653–654

Scatterplots

explanation of, 165

linear/nonlinear association and, 166–167

positive/negative association and, 165–166

Scientiﬁc notation, 15–16, 81

Secant lines, 140, 146

Semicircles, graph of, 92

Semimajor axis, 717

Semiminor axis, 717

Sequences. See also Series; speciﬁc types of

sequences

applications of, 768–769, 779, 789–791

arithmetic, 773–779, 839

explanation of, 762, 838

ﬁnding terms of, 762–764

ﬁnite, 762

geometric, 782–791, 839

on graphing calculator, 763–765, 775–776,

785–786, 789–790

graphs of, 776, 786

graphs of arithmetic, 776

inﬁnite, 762

recursive, 764–765

Series. See also Sequences

explanation of, 762, 765, 838

ﬁnite, 765–766

geometric, 782

properties of, 769

Set notation, 81, 122, A–1, A–2

Sets of numbers, 81, A–1 – A–3

Sigma notation, 766, 767, 838

Similar triangles, A–20

Simple interest, 539, 569

Simplest form

radical expressions in, 71

rational expressions in, 53–55

Singular matrices

determinants and, 669–672

explanation of, 669

Sinking fund, 544

Slope

of line, 106–108

positive and negative, 108

as rate of change, 106–107, 109, 136, 142

Slope formula, 107, 108, 110, 138

Slope-intercept form

explanation of, 137

graph of line and, 138–139

linear equations in, 137, 138

Smooth functions, 247

Solution region, 616

Solution sets, 27

Spheres, 35, 218, A–17

Spherical shells, 52

Square matrices, 638, 664

Square root function, 204

Square root property of equality, 295–297

Square roots. See also Radical expressions

of complex numbers, 408

principal, 282, 408, A–8

simpliﬁcation of, 65–66

Squares. See also Completing the square

binomial, 20–21, 43, 759, 830

difference of two, 43

explanation of, 33

factoring difference of two, 43

perfect, 43, A–7

perimeter and area formulas for, 33, A–16

trinomial, 730

Square systems, 597

Square units, A–14

Squaring function, 204

Standard form

of complex numbers, 283–284

converting between polynomial form

and, 759

of equation of circle, 96, 716

of equation of ellipse, 717

of equation of hyperbola, 733–734

of linear equations, 46

of polynomial expressions, 17

of quadratic equations, 46

of system of equations, 579

Statistics, 816

Step functions, 254–255

Stevens, Stanley, 571

Stevens’law, 571

Stirling’s formula, 814

Stretches, vertical, 210–211

Subscripted variables, A–14

Subscript notation, 796

Substitution

applying power property after, 12

back, 578

to check complex root, 286, 287

to check solutions, 26

explanation of, 578

placeholder, 45

to solve system of equations, 578–580

Subtraction

of complex numbers, 284–285

of matrices, 652–653, 700

of polynomials, 17–18

of radical expressions, 70

of rational expressions, 56–58

Summation

applications of, 803–804

explanation of, 766

index of, 766, 838

properties of, 767–768

Summation notation, 766, 767

Surface area. See also Area

of cone, 80

of cylinder, 52, 311, 349, 455

of rectangular box, 323

Symbols. See Notation/symbols

Symmetry

axis of, 205

to origin, 191–192

polynomial graphs and, 419

y-axis, 190–191

Synthetic division, 383–384, 470, A–28 – A–29

System of linear equations

applications of, 645–646

augmented matrix of, 638–640

determinants and Cramer’s Rule to solve,

679–682

equivalent, 579, 593–594

explanation of, 576

on graphing calculators, 643–644

inconsistent and dependent, 581–582,

596–598, 644–645

matrices to solve, 640–643, 663–673, 701,

A–30

square, 597

verifying solutions to system of, 576

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I-10 Index

System of linear equations in three variables

applications of, 598–599

coincident dependent, 592

elimination to solve, 593–596, 634–635

explanation of, 591–592, 630–631

inconsistent and dependent, 596–598

linearly dependent, 592

solutions to, 592–593

System of linear equations in two variables

applications of, 582–586

elimination to solve, 579–581, 634–635

explanation of, 576, 630

on graphing calculator, 577, 578, 580–586

graphs to solve, 577

inconsistent and dependent, 581–582

modeling and, 582–586

substitution to solve, 578–579

System of linear inequalities

applications of, 619–620

explanation of, 615

on graphing calculator, 619

method to solve, 618–620

System of logarithmic equations, 608

System of nonlinear equations

conic sections and, 749–750

elimination to solve, 607–608

explanation of, 604, 631

on graphing calculator, 605–608

graphs of, 604–605

possible solutions for, 604–605

substitution to solve, 605–607

System of nonlinear inequalities,

609–610, 631

System of two equations in two

variables, 576

Temperature conversions, 132

Terminating decimals, A–2

Toolbox functions

direct variation and, 261–264

explanation of, 204–206, 274

horizontal translations and, 207–208

vertical translations and, 206–207

Transcendental functions, 503

Transformations

of general functions, 211–214

to graph exponential functions, 495

on graphing calculator, 213, 265

to graph logarithmic functions, 506

Lorentz, 52

of parent graph, 205

of power functions, 238, 239

of reciprocal functions, 235

review of, 274

of root functions, 238

Translations

on graphing calculator, 206–208

horizontal, 207–208

vertical, 206–207

Transverse axis, 732

Trapezoids, 33–34, A–14

Tree diagrams, 804–805

Trial-and-error method, for factoring, 42–43

Trials, 805

Triangles

explanation of, 33

Pascal’s, 829–831

perimeter and area formulas for, 33, 455,

688, A–14

right, 75

similar, A–20

Triangularizating augmented matrix,

640–642, 678

Trigonometric graphs, 200

Trinomials. See also Polynomials

explanation of, 16

factoring, 41–43

perfect square, 20, 43–44

in quadratic form, 45

Trinomial squares, 730

Two-dimension graphs, 591

Unbounded region, 621

Uniform motion, 158–159, 583–584

Union, 29

Uniqueness property

to solve exponential equations, 496–499

to solve logarithms, 528–529

Unique solutions, 592

Unit conversion factors, A–18 – A–21

Upper and lower bounds property, 402, 404

Upper bound, 404

U.S. Customary Units, A–19

u-substitution, 45–46, 49, 75

Variables

dependent, 88

descriptive, 2

independent, 88

object, 155–156

subscripted, A–14

Variation

constant of, 261

direct, 261–264, 277

on graphing calculators, 264, 279

graphs of, 262–265

inverse, 264–265, 277

joint or combined, 266, 277

power functions and, 280

Variation equations, 261, 263, 265

Velocity, 333, 363, 364

Velocity formula, 79, 80, 236

Vertex

of cone, 710

of hyperbola, 732

of parabola, 196, 205, 316

Vertex formula, 315–316

Vertex/intercept formula, 323

Vertical asymptotes, 235, 430–433

Vertical axis, 591

Vertical boundary lines, 122, 134

Vertical change, 106–107

Vertical format, 3

Vertical hyperbolas, 732, 735–736

Vertical lines, 109–110

Vertical line test, 120–122

Vertical parabolas, 91, 745, 747

Vertical reﬂections, 208–209

Vertical shifts. See Vertical translations

Vertical stretches, 210–211

Vertical translations, 206–207

Volume

of box, 52

of cone, 35, 491

of cylinder, 176

of cylindrical shell, 52

explanation of, 34, A–16, A–18 – A–19

formulas for, 34, 491, A–17

method to compute, 35

of open box, 392

of right prism, A–21

of sphere, 218

of triangular pyramid, 691

Whole numbers, 81, A–1 – A–2

Written information

translated into equations, 31–32

translated into mathematical model, 2–3

x-intercepts

explanation of, 106

on graphing calculators, 153–154,

293–294, 296

of quadratic functions, 293–294

x-intercept/zeroes method, 152–154

xy-plane, 89

y-axis, 190–191

y-intercepts, 106

Zeroes

approximation of real, 429

complex, 475–476

division with, 384–385, A–6

factor theorem to ﬁnd, 389

of functions, 192–194

intermediate value theorem to ﬁnd, 398

of multiplicity, 396, 415–416, 475

of polynomial functions, 192–193,

394–405, 415–416, 471

of quadratic functions, 293–294

quotient of, 384–385, A–6

rational, 400–402, 429

repeated, 475–476

Zeroes/x-intercept method, 152–154, 298

Zero exponent property, 13, 15

Zero exponents, 13, 81

Zero product property, 47–48

cob19545_ndx_I1-I16.qxd 12/23/10 9:13 PM Page I-10

Photo Credits

C-1

Chapter R

Allison/Getty Images/RF.

Chapter 1

Smak/Alamy/RF.

Chapter 2

McGraw-Hill Companies, Inc./Ken Cavanagh

Carey/Getty Images/RF; p. 259: Courtesy John

Chapter 3

Images/RF; p. 330: Courtesy NASA; p. 338:

Fish & Wildlife Service/Tracy Brooks/RF;

McGraw-Hill Companies, Inc./Barry Barker,

Chapter 4

Images/RF; p. 393, p. 409, p. 444, p. 468:

Chapter 5

Publishing/age Fotostock/RF; p. 525 bottom-left:

Inc./Andrew Resek, photographer; p. 525

Education/Carlyn Iverson, photographer; p. 537:

Images/RF.

Chapter 6

McGraw-Hill Companies, Inc./Jill Braaten,

Getty Images/RF.

Chapter 7

Chapter 8

Wiesenhofer/PhotoLink/Getty Images/RF; p. 753

Chapter 9

CORBIS.

Appendices I

Appendices II

Images/RF.

cob19545_cre_C1-C2.qxd 12/23/10 9:15 PM Page C-1

This page intentionally left blank

ISBN: 0-07-351954-5

Author: John W. Coburn

Title: College Algebra, 3e

Front endsheets

Color: 5

Pages: 2, 3

▼

13 ⬇ 1.732112 ⬇ 1.4142e ⬇ 2.7183␲ ⬇ 3.1416

Special Constants

▼

Special Products

1a  b2

 a

 3a

b  3ab

 b

1a  b2

 a

 3a

b  3ab

 b

1a  b2

 a

 2ab  b

1a  b2

 a

 2ab  b

1a  b21a  b2 a

 b

1x  a21x  b2 x

 1a  b2x  ab

▼

Special Factorizations

 b

 1a  b21a

 ab  b

 b

 1a  b21a

 ab  b

 2ab  b

 1a  b2

 2ab  b

 1a  b2

 b

 1a  b21a  b2x

 1a  b2x  ab  1x  a21x  b2

▼

Formulas from Plane Geometry: P S perimeter, C S circumference, A S area

Rectangle Square Regular Polygon

A 

P  ns

A  s

P  4s

A  lw

P  2l  2w

Parallelogram Trapezoid Triangle

A 

bhA 

1a  b2

A  bh

Triangle

Sum of angles

Right Triangle

Pythagorean Theorem

Circle

C  2␲r  ␲d

A  ␲r

 b

 c

A  B  C  180°

Ellipse Right Parabolic Segment

A 

C ⬇ ␲ 221a

 b

A  ␲ab

▼

Formulas from Solid Geometry: S S surface area, V S volume

Rectangular Solid Cube Right Circular Cylinder

S  2␲r1r  h2

V  ␲r

S  6s

V  s

S  21LW  LH  WH2

V  LWH

Right Circular Cone Right Square Pyramid Sphere

S  4␲r

V 

␲r

S  b

 b2b

 4h

V 

S  ␲r1r  s2

V 

␲r

▼

Distance between P

and P

Slope of Line Containing P

and P

m 

¢y

¢x



 y

 x

d  21x

 x

 1y

 y

Formulas from Analytical Geometry: P

S (x

, y

), P

S (x

, y

)

Equation of Line Containing P

and P

Point-Slope Form

Equation of Line Containing P

and P

Slope-Intercept Form (slope m, y-intercept b)

where b  y

 mx

y  mx  b,y  y

 m1x  x

Parallel Lines

Slopes Are Equal:

Perpendicular Lines

Slopes Have a Product of m

11:m

 m

Intersecting Lines

Slopes Are Unequal:

Dependent (Coincident) Lines

Slopes and y-Intercepts Are Equal: b

 b

 m

 m

Interest Compounded n Times per Year Interest Compounded Continuously

A  Pe

A  Pa1 

Accumulated Value of an Annuity Payments Required to Accumulate Amount A

P 

11  R2

 1

A 

冤11  R2

 1冥

Arithmetic Sequences Geometric Sequences

冟

6 1S



1  r

;



 a

1  r

 a

, ... , a

 a

n 1

 a

r,a



冤2a

 1n  12d冥



 a

 a

 2d, ... , a

 a

 1n  12da

 a

 d,a

▼

Logarithms and Logarithmic Properties

log

 P

log

Mlog

b log

M  log

Nlog

1MN 2 log

M  log

log

x 

log

 xlog

 x

log

1  0log

b  1y  log

x 3 b

 x

▼

Applications of Exponentials and Logarithms

Sequences and Series:

r S common ratiod S common difference,S

S sum of n terms,a

S nth term,a

S 1st term,

Binomial Theorem

b

k!1n  k2!

0!  1n!  n1n  121n  22

###

132122112;

1a  b2

 a

 a

n 1

 a

n 2



###

 a

n  1

b a

n 1

 a

b a

t S time in yearsR S interest rate per time period a

br S interest rate per year

n S compounding periods/yearP S periodic paymentP S initial deposit,A S amount accumulated

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