Stewart J. College Algebra: Concepts and Contexts

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I4 INDEX

Exponential model(s) (continued)

in terms of instantaneous growth

rate, 358

of world population, 289, 296, 369–70

Exponential notation, T14–T15

for nth root of a, T20

for nth root of a

, T21

Exponential patterns, 316–19

finding exponential functions fitting

data, 317–19

recognizing exponential data, 316–17

Extraneous lines in graph, avoiding,

T82–T83

Extraneous solutions, T50

Extrapolation, 191–92

Extreme numbers, 312–14

Factor(s), T1

canceling common, T39

factoring out common, T33–T34

Factored form of rational function, 539

Factoring

algebraic expressions, T33–T38

completely, T36

distributive property used in, T4–T5

factoring out common factors,

T33–T34

graphing polynomial functions by,

505–8

by grouping, T37

solving quadratic equations by,

448–50, T56–T57

special factoring formulas, T35–T36

trinomials, T34–T35

Fair division of assets, 242–45

Falling sky diver, 61

False equation, 582

Family of linear equations, graphing,

177–78, 180–81

Family of logarithmic functions,

graphing, 329–30, 338–39

Farms in United States, average rate of

change in number of, 144–45

Fechner, Gustav, 344

Femur length, height and, 197

Fencing a garden, 94–96, 443

Financial problem, using linear system to

model, 585–86

First differences, 28–30

Fish, length-at-age data for, 521–22

Fish population, limited, 277–78

Fitt’s Law, 350

Fixed cost, 26

Focusing distance for camera lenses,

536, 537–39

FOIL, multiplying binomials using, T28

Force/mass/acceleration, Newton’s

formula of, 107

“Forgetting” curves, experimenting

with, 525

Formula(s), 101–12

for average (mean), 104–5

compound interest, 265

definition of, 101

distance, T68–T69

equivalent, 105

finding, 102–3

midpoint, T69–T70

quadratic, 451, 454, 455, T59–T61

reading and using, 105–8

simple interest, 203

for surface area, 103

variables with subscripts in, 104–5

Fractional positive powers, power

function with, 497–98

Fractions

properties of, T40

solving an equation involving, T50

Frequencies of musical notes, 410–12

Fudging the data, 128

Function(s), 35–52

absolute value, 70

average rate of change of, 142–53

composition of, 377–79

constant, 65

cube root, 498

decreasing, 78–79

definition of, 36–37

dependent variable in, 37–38, 54

difference of, 484–87

domain of, 56–57, 77–78

equations that represent, 38–40, 42

evaluating, 54–56

exponential. See Exponential

function(s)

four ways to represent, 43

of functions, 377–79

graphical addition and subtraction of,

485–87, 505

graph of, 41–43, 64–88

graph of, finding values of its inverse

from, 380

graph of, reading, 74–76

graph of, using graphing calculator,

67–68

identity, 66

increasing, 78–79

independent variable in, 37–38, 54

inverse, 379–86, 498

linear. See Linear function(s)

local maximum and minimum values

of, 79–81

logarithmic. See Logarithmic

function(s)

logistic, 297

modeling with, 89–100

natural exponential, 351–52

natural logarithm, 353

one-to-one, 383–85

piecewise defined, 57–58, 68–70

polynomial, 504–16, 520–22

power. See Power functions

product of, 488–89

quadratic. See Quadratic function(s)

quotient of, 488

range of, 77–78

rational, 536–45, 560–63

reciprocal, 527–30

as relation, 36–37

root, 497–98

as rule, 52–53

square root, 66–67, 497–98

squaring, 428–29, 432

sum of, 484–87

surge, 563–66

value of f at x, 53, 55–56, 75–76

vertical shifts, 414–15, 417–18

Function form, 39

Function notation, 52–54

Galileo Galilei, 153

Galileo’s Law, 493, 498

Garden

fencing a, 94–96, 443

irrigating, 91–92

Gas mileage

formula for, 102

maximum, for car, 442

Gasoline price, year and, 38

Gas-powered and hybrid-electric cars,

cost comparison of, 212–13

Gaussian elimination, 581–82

General form of equation of line, 172–73

General form of quadratic function,

429–30

completing the square for, 440

General Social Survey (GSS), 635

Geometry, constructing model involving,

205–6

Geometry of space, inverse square laws

and, 531–32

Germany, population of, 375

Global warming, 141

Grade of road, 157, 158

Graph(s)

INDEX I5

of absolute value function, 70

of circle, T76, T83–T84

of exponential functions, 286–95,

406–9

of family of linear equations, 177–78,

180–81

finding domain and range of function

from, 77–78

finding exponential function from,

290–91

finding linear functions from, 160–61

finding local maximum and minimum

values from, 79–81

finding values of function from, 75–76

of function, 41–43, 64–88

of function, finding values of its

inverse from, 380

of function, reading, 74–76

of general linear equation, 172–73

horizontal shifts of, 416–18

of increasing and decreasing functions,

78–79

of inequalities, T8

intercepts, finding, T75. See also

x-intercepts; y-intercepts

intersection points on, 68, 76

of intervals, T9, T10

of inverse square function, 530

of linear function, 155

of logarithmic functions, 328–30,

338–39

misleading, 128–30

of model, getting information from,

93–96

of natural exponential functions, 352

of parallel lines, 179, 180

of piecewise defined functions, 68–70

of polynomial function, 505–8

of power functions, 495

of quadratic functions, 432–34,

449–50

rates of change and shapes of, 230–33

of rational functions, 536–37, 539–41

reading, 16–17, 138–39, 473–75

of reciprocal function, 527–28

reflecting, 419–22, 495

of root functions, 497, 498

scatter plot, 14, 519

semi-log, 407–9, 518, 519

shifting, 414–18, 495

solving a polynomial equation

graphically, 511

of squaring function, 432

stretching and shrinking, vertically,

418–19, 495

of system of two linear equations in

two variables, 571–72

transformations of function and,

414–27, 495

two-variable data in coordinate plane,

14–16, T67–T68

of two-variable equations, T72–T75

verbal description from, 75

vertical shifts, 414–15, 417–18

Graphical addition of functions,

485–86, 505

Graphical division, 536

Graphical method of finding intersection

points of linear functions, 211

Graphical method of solving equations,

T85–T88

equation in an interval, T88

inequalities, T88–T89

quadratic equations, T87

Graphical subtraction of functions,

486–87

Graphing calculator, T80–T85

avoiding extraneous lines, T82–T83

choosing viewing rectangle, T80–T82

CubicReg command on, 521

exponential function, 296

family of exponential functions, 289

graphing circle on, T83–T84

graphing functions with, 67–68

logistic growth function, 278, 297–98

LOG key, 338

multiplying matrices on, 614–15

PwrReg command, 517, 519

QuadReg command on, 462, 463

two graphs on same screen, T82

Xmin and Xmax command, T81

Ymin and Ymax command, T81

Gravitation, Newton’s Law of, 535

Gravitational force

between moon and astronaut in space

ship, 73

Newton’s formula for, 104, 107

Greater than symbol (>), T7

Grimshaw, John, 34

Grouping, factoring by, T37

Growth

attenuated, 398

exponential. See Exponential growth

linear model of, 156–57

logistic, 272, 276–78, 297–98

Growth factor, 249–51, 252

changing time period and, 262–65

Growth rate, 251–52

annual percentage yield, 266

changing time period and, 262–65

comparing, 264

instantaneous, 355–64

percentage rate of change,

272–75, 297

Half-life, 254, 264

Hanselman sextuplets, weights of, 8

Hare and tortoise race, modeling, 210–11

Health-care expenditures, U.S., 301

Heat transfer coefficient, 370

Height

age and, 5–6

of box, 106–7

femur length and, 197

Hidden variable, 239–42

Highway engineering, 80–81

Home sales in United States, 492

Hooker, Steven, 194

Hooke’s Law, 188–89

Horizontal asymptote, 287, 288, 528,

540, 541

Horizontal lines, 171–72

Horizontal Line Test, 384–85

Horizontal shifts of graphs, 416–18

combining vertical shifts and, 417–18

House, median price of, 4–5

“Housing bubble,” bursting of U.S.,

259–60

Howler monkeys, modeling population

of, 637–39

Hybrid car sales, 22, 302

Hydrogen ion concentration, pH scale

and, 344

“Iceman” of Neolithic Age, radiocarbon

dating of, 372

Identity function, 66

Identity matrix, 619

Image of x under f, 53

Income, average and median, 3–4

Inconsistent system, 571, 572, 582–84,

596–98

Increasing functions, 78–79

Independent variable, 37–38, 54

India

cell phone usage in, 363

population of, 270, 392

Inequalities, T62–T66

graphing, T8

linear, solving, T63

nonlinear, solving, T63–T65

operations on, T62

order on real line and, T7–T8

reversing direction of, T62

solving, graphically, T88–T89

I6 INDEX

Infant mortality, U.S., 190–91, 192, 195

Infinite intervals, T9

Initial population, 249

Initial value, 26, 156, 250, 356

Inner product, 612

Input(s), 12–14

evenly spaced, 28

in function, 36

Installation of flooring, average rate

of, 143

Instantaneous rates of growth or decay,

355–64

expressing exponential models in

terms of, 358

Integer exponents, T14–T19

Integers, T1

Intelligibility, noise and, 199

Intensity

of earthquakes, 346–47

sound, decibel scale of, 344–45

Intercepts. See x-intercepts; y-intercepts

Interest

compound, 265–66, 369

continuously compounded, 354–55

simple, 203–4

Internet usage in China, 363

Interpolation, 192

Intersection

of intervals, T10–T11

of sets, T8–T9

Intersection points, 68, 76

equilibrium point, 214

of linear functions, 211–15

Interval(s), T9–T11

closed, T9

frequency, on musical scale, 410–12

graphing, T9, T10

infinite, T9

open, T9

solving equation in, T88

unions and intersections of, finding,

T10–T11

verbal description of, T10

Inverse function(s), 379–86

definition of, 380

of distance function, 498

of exponential and logarithmic

functions, 385–86

finding values of, graphically, 380

properties of, 382

steps in finding, 380

Inverse of matrix, 620–25

Inverse proportionality, 528–30

Inverse square function, graph of, 530

Inverse square laws, 530–32

laws of nature as, 531–32

Investment, growth of, 265–66, 369

Irrational numbers, T1

e as, 351

Jet takeoff, sound intensity of, 345

Kepler’s Law for Periods of Planets, 524

Kepler’s Third Law, 493

King Hiero’s crown, amount of gold in,

568, 573

Kirchhoff’s Laws, 589

Labor force, U.S., 492

La Condamine, Charles-Marie de, 134

Lang Lang, 410

Law of Laminar Flow, 50

Law of the Lever, 209, 568

Law of the Pendulum, 503

Laws of Logarithms, 334–41

defined, 335

expanding and combining logarithmic

expressions using, 335–36

Lead emissions, U.S., 524

Leading entry of matrix row, 592

Leading term of polynomial, 504

end behavior of polynomial function

and, 509–10

Leading variable in linear system, 595

Least common denominator (LCD), T41

Length-at-age data for fish, 521–22

Less than symbol (<), T7

Lever, Law of the, 209, 568

Life expectancy

demographics and, 23

in United States, 198

“Like terms,” combining, T26, T30

Limited resources, growth with. See

Logistic growth

Line(s)

equations of, 165–77. See also Linear

equation(s)

extraneous, T82–T83

horizontal, 171–72

parallel, 178–80, 182

perpendicular, 180–82

regression (line of best fit), 189,

190–94, 195, 519

slope of, 157, 158–59

through two given points, finding

equation for, 169

vertical, 171–72

Linear data. See Linear patterns

Linear equation(s), 165–77, 201–19

defined, T49

general form of, 172–73

graphing family of, 177–78, 180–81

models that lead to, 202–9

point-slope form of, 167–71

slope-intercept form of, 166–67

solving, T49–T51

systems of. See Linear system(s)

in two variables, 166

varying coefficients in, 177–82

Linear function(s), 153–65

average rate of change of, 155–56

comparing exponential functions and,

275–76

comparing squaring functions and,

428–29

definition of, 154

finding, from a graph, 160–61

fitting patterns in data, finding, 234–37

graphing, 155

graphing quotients of, 536–39

identifying, 154

initial value of, 156

intersection points of 211–15

modeling supply and demand, 214–15

modeling with, appropriateness of,

518, 519

rate of change and, 155–57, 159–61

slope and, 157–61

Linear growth versus exponential growth,

275–76

Linear inequality, T63

Linearizing data, 408–9

Linear model(s), 26–35, 275

for cost, 89–90

definition of, 26

for demand, 170–71

for depth-pressure data, 29–30

getting information from, 30

of growth, 156–57

making, 165–77

for radar, 166–67

for temperature and elevation, 28–29

for volume, 168–69

Linear patterns, 233–37

finding functions to fit data, 234–37

properties of linear data, 234–35

Linear regression, 189–201

analyzing data using, 239

correlation coefficient, 195

line of best fit, 189, 190–94, 519

line of best fit, used for prediction,

191–94

Linear scale, 342, 343

Linear system(s)

augmented matrix of, 590–91,

593, 594

elementary row operations to

solve, 592

INDEX I7

modeling with, 584–86

number of solutions of, 571

operations leading to equivalent, 581

in reduced row-echelon form,

solutions of, 595

in several variables, 580–89

solving, using reduced row-echelon

form, 594–95

solving, using row-echelon form, 593

in triangular form, 580, 581, 582

in two variables, 568–80

written as single matrix equation,

621–25

Line of best fit (regression line), 189,

190–91

correlation coefficient, 195

using, for prediction, 191–94

Line segment, midpoint of, T69–T70

Living alone, number of Americans, 200

Local maximum value, 79–81

Local minimum value, 79–81

Logarithm(s)

base a, 326–27

basic properties of, 328

change of base, 337–39

common (base 10), 324–26

comparing, 336–37

expanding and combining expressions,

335–37

exponential equations solved using,

401–2

laws of, 334–41

natural, 353–54

Logarithmic equations, 367–68

steps in solving, 367

Logarithmic form, 327

Logarithmic function(s), 328–34

with base a, 328

definition of, 328

graphing, 328–30

graphing, using change of base

formula, 338–39

inverse functions of, 385–86

natural, 353

transformations of, 421–22

Logarithmic meter stick, 404

Logarithmic scales, 342–50, 407

decibel scale, 344–45

distance between semitones on, 410

linear scale compared to, 342–43

pH scale, 343–44

Richter scale, 345–47

Logarithm ruler, 403–4, 407

Logistic functions, 297

Logistic growth, 272, 276–78

modeling, 297–98

Logistic growth equation, 277

Log-log plot, 518, 519

Loma Prieta earthquake (1989), 43

Long division, T42–T43

Magnitude

of earthquakes, 346, 349

orders of, 405

Main diagonal, 619

Mass of earth, 108

Matrix, matrices

addition of, 605–6, 611–12

augmented, of linear system, 590–91,

593, 594

coefficient, 621

cross-tab, 603, 636

defined, 590

dimension of, 590

entries of, 590

identity, 619

inverse of, 620–25

multiplying, 612–16

multiplying, times a column matrix,

606–7

organizing categorical data in, 603–5,

636–37

proportion, 636

reduced row-echelon form of, 594–95

row-echelon form of, 592–94

rows and columns, 590

scalar multiplication of, 606, 611–12

singular, 623

solving systems of equations using,

590–602

subtraction of, 611–12

transition, 637–39

Matrix equations, 621–25

modeling with 624–25

solving, 622–25

Maximum area, modeling, 94–96

Maximum gas mileage for car, 442

Maximum height for model rocket, 442

Maximum revenue from ticket sales,

443–44

Maximum value of function, local, 79–81

Maximum value of quadratic function,

440–41

Maximum volume, finding, 510–11

Mean (average), 2–3

formula for, 104–5

Measurement

bias in, 128–33, 130–33

data collection, 134–36

Median, 3–5

Median income, 3–4

Median price of house, 4–5

Medication, exponential decay model for,

252–54

Meter stick, 404

Midpoint formula, T69–T70

Millionaire measuring tape, 403

Minimum value of function, local, 79–81

Minimum value of quadratic function,

440–41

Minneapolis, population of, 362

Misleading graphs, 128–30

Mixture problem, 575–76

Mixtures and concentration, model for,

204–5

Model, 26, 89. See also Linear model(s);

Modeling

constructing, 202–6

Modeling, 89

absorption and metabolism of alcohol,

P3–P4, 261, 272, 295, 376

definition of, 26

determining appropriate model for

data, 518–20

direct proportionality, 182–84

with exponential functions, 249–55,

261–66, 274–75, 290, 295–98,

356–58

with functions, 89–100

getting information from graph of

model, 93–96

with linear systems, 584–86

logistic growth, 297–98

with matrix equations, 624–25

models that lead to linear equations,

202–9

with polynomial functions, 510–11,

520–22

with power functions, 499, 517–20,

554–57

prediction based on model, P4

with quadratic functions, 442–44,

453–56, 461–66, 476–77

radioactive decay, 320–22

of real-world relationships, 88–100

species-area relation, 517–18

species survival, 637–39

supply and demand, 214–15, 218–19

with surge functions, 563, 565–66

with systems of equations, guidelines

for, 574

using algebra to make model, P3

volume of box, 93–94

Model rocket

maximum height for, 442

quadratic function modeling height of,

455–56

Monomial, T26

I8 INDEX

Mosquito prevalence in Saga City,

Japan, 199

Mount Kilimanjaro, temperatures on, 34

Moving object, average speed of,

145–46, 148

Multiplication. See also Product

of algebraic expressions, T27–T29

of exponents, T16

of matrices, 612–16

of matrices, getting information by,

615–16

of matrix times a column matrix,

606–7

of polynomials, T30

of rational expressions, T40

Music, 409–12

frequencies of notes in musical scale,

410–12

intervals, frequencies, and dissonance,

411–12

National debt, U.S., 132–33, 270, 314

Natural exponential function, 351–52

Natural logarithm function, 353

Natural logarithms, 353–54

properties of, 353

Natural numbers, T1

Negative exponents, T15–T16

Negative number, square root of, T20

Net change, 38, 42, 54–56

in value of function, 55–56

Newton, Isaac, 104, 107

Newton’s Law of Cooling, 370–71, 375,

399, 427

Newton’s Law of Gravitation, 535

Nietzsche, Friedrich, 409

Noise and intelligibility, 199

Nonlinear inequalities, solving, T63–T65

algebraically, T64–T65

steps in, T64

Norman window, maximizing area

admitting light in, 447

Northridge earthquake in Los Angeles

(1994), 85

Notation

arrow, 509, 528

exponential, T14–T15, T20, T21

function, 52–54

scientific, 312–14

set-builder, T8

Number(s)

extreme, 312–14

irrational, 351, T1

natural, T1

rational, T1

real, T1–T7

triangular, 479–80

using letters to stand for, T3

Numerical data, 602

Numerical representation of function, 43

Nutritional analysis, 598

Olympic pole vault records, 193–94,

198–99

Olympic swimming records, 200

One-to-one function, 383–85

test of, 384–85

One-variable data, 2–5

analyzing, 136–37

Open interval, T9

Operations

on equations, T47

on inequalities, T62

order of, T2

on real numbers, T1–T2

Ordered pair, 12, T67

as solution to equation, T72

Order of operations, T2

Order on real line, T7–T8

Orders of magnitude, 405

Origami, 401–2

Outliers, 4

Output, 12–14

in function, 36

Pag, Andy, 34

Pan evaporation, 24

Paper consumption, function model for,

90–91

Parabola, 432. See also Quadratic

function(s)

opening downward, 434, 440

opening upward, 433, 440

vertex of, 432, 440

Parallel lines, 178–80

finding equations of lines parallel to

given line, 182

Parameter of system of solutions, 584

Patterns

exponential, 316–19

linear, 233–37

quadratic, 478–81

Paulos, John Allen, 239

Pendulum, Law of the, 503

Percentage rate of change, 272–75, 297

as constant, 273, 274

Perfect square, T57

recognizing, T36

Perimeter, constructing model for, 205–6

Periods of Planets, Kepler’s Law for, 524

Perpendicular lines, 180–82

finding equations of lines

perpendicular to given line, 182

Philadelphia, population of, 86

Phoenix Mars Lander, 152–53

pH scale, 343–44

Piecewise defined functions, 57–58

graphing, 68–70

Pitch of roof, 157, 158

Pizza, cutting for maximum number of

pieces, 481

Point-slope form of equation of line,

167–71

Pole vaulting, 192–94, 198–99

Polynomial(s)

combining, T29–T31

defined, T29–T30

degree of, 504

leading term of, 504, 509–10

Polynomial equation, solving

graphically, 511

Polynomial function(s), 504–16

defined, 504

end behavior of 508–10

graphing, 505–8

modeling with 510–11, 520–22

ratio of. See Rational functions

Population

of Atlanta, 150–51

bacterial growth, 248–49, 263, 264

of Belgium, 302–3

of California, 282

carrying capacity and, 298

changing time period for growth,

262–63

comparing growth rates of, 264

doomsday, 316

of Ethiopia, 374–75

of Germany, 375

growth factor, 249–51, 252

growth rate of, 251–52

of India, 270, 392

initial, 249

limited fish, 277–78

linear vs. exponential growth, 275–76

logistic growth, 277–78, 297–98

of Minneapolis, 362

of Philadelphia, 86

transition matrix to predict, 637–39

U.S., census data on, 225, 300–301

world, 282

world, doubling of, 369–70

world, exponential models for, 289,

296, 369–70

world, limited, 285

INDEX I9

Power equations, T51–T53

Power functions, 493–503

combining. See Polynomial function(s)

comparing exponential functions and,

494–95

defined, 493

direct proportionality to, 496–97

with fractional positive powers, 497–98

inverse proportionality to, 528–30

inverse square laws, 530–32

modeling with, 499, 517–20, 554–57

with negative powers, 527–35

with positive integer powers, 493

reciprocal function, 527–30

transformations of, 495

Powers of Ten (film), 402

Precipitation

average annual, 16–17

Olympic Peninsula in state of

Washington, 10

Prediction

based on model, P4

using line of best fit for, 191–94

Preschoolers, average age of, 3

Pressure, depth and, 6–7, 14–15, 26

model for, 29–30

Pressure, volume, and temperature

formula, 101, 105

Principal square root, T20

Principle of Substitution, T27, T28

Product

of functions, 488–89

inner, 612

Product Formulas, T28–T29

Profit, finding, 487

Profit sharing, 245

Proportionality, 557–60

analyzing problem of road capacity

using, 560–63

constant of, 183, 496, 529, 557

direct, 182–84, 496–97

inverse, 528–30

in natural world, 559–60

of surface area and volume, 558–59

Proportionality symbol, 557–60

Proportion matrix, 636

Proportions, categorical data and, 604–5

Quadrants, T67

Quadratic equation(s), 448–60, T56–T61

defined, 448, T56

discriminant of, 452–53

solving, algebraic method, T86

solving, by completing the square,

T58–T59

solving, by factoring, 448–50,

T56–T57

solving, graphical method, T87

solving, using quadratic formula, 451,

454, 455, T59–T61

Quadratic formula, 451, 454, 455,

T59–T61

defined, T60

Quadratic function(s), 428, 429–47

of best fit, 461–66, 476–77

defined, 429

finding, from its graph, 434, 450

in general form, 429–30, 440

graphs of, 432–34, 449–50

identifying, 430

maximum and minimum values of,

440–41

modeling with, 442–44, 453–56,

461–66, 476–77

in standard form, 431–34

Quadratic patterns, 478–81

Quotient(s). See also Division

of functions, 488

of linear functions, graphing, 536–39

in long division, T43

Rabbit population, exponential growth

model for, 252

Radar, linear model for, 166–67

Radical(s)

conjugate, T42

rational exponents and, T20–T24

Radioactive elements, decay of, 254–55

of different amounts, 290

half-lives, 254, 264

modeling with coins, 320–21

modeling with dice, 321–22

Radiocarbon dating, 371–72

Radium, exponential decay model for,

254–55, 264–65

Radius of ball, effect of doubling,

496–97

Rainfall and crop yield, 465

Range

of function, 77–78

of relation, 13, 14

Rate of change, 156

average, 142–53, 155–56, 272–75

changing, 229–33

constant, 153–65

finding linear model from, 157

linear functions and, 155–57, 159–61

percentage, 272–75, 297

shapes of graphs and, 230–33

slope and 159–61, 167

Rational exponents, T20–T24

1/n

, T20

m/n

, T21

rules of exponents and, T21, T22

simplifying by writing radicals as,

T22–T23

Rational expressions, T39–T45

adding, T41–T42

defined, T39

dividing, T40–T41

long division of, T43

multiplying, T40

simplifying, T39, T42

subtracting, T41–T42

Rational functions, 536–45

analyzing problem of road capacity

using, 560–63

in compound fraction form, 539

defined, 536

in factored form, 539

graph of, 536–37, 539–41

quotients of two linear functions,

536–39

in standard form, 539

Rationalizing the denominator, T42

Rational numbers, T1

Real number line (real line), T7

distance between points on, T12

graphing intervals on, T9, T10

order on, T7–T8

Real numbers, T1–T7

operations on, T1–T2

properties of, T2–T5

Reciprocal function, 527–30

graph of 527–28

Reduced row-echelon form of matrix,

594–95

Reflecting graphs, 419–22, 495

Regression, linear. See Linear regression

Regression line (line of best fit), 189,

190–94, 519

correlation coefficient, 195

using, for prediction, 191–94

Relation(s), 12–14

definition of, 12

domain of, 13, 14

function as, 36–37

range of, 13, 14

Relativity Theory, 50

Remainder in long division, T43

Resistors in parallel, 544–45

Richter, Charles, 345

Richter scale, 345–47

Rise, 158

Root(s). See Solution(s)

I10 INDEX

Root function, 497–98

Row(s)

elementary row operations, 591–92

of matrix, 590

Row-echelon form of matrix, 592–94

reduced, 594–95

Run, 158

Safe following distance, 560

carrying capacity of road and 561–62

San Francisco earthquake (1906), 346

SAT scores, change in rate of change

of, 233

Savings, exponential growth of, 315–16

Scalar multiplication, 606

of matrix, 606, 611–12

Scaling factor, 554–55

Scatter plot, 14, 519

Scientific notation, 312–14

Score(s)

finding average, 104–5

needed to get specific average, 106

Second differences, 478–79, 480, 481

Sedimentation rate, 165

Semi-log graphs (semi-log plot), 407–9,

518, 519

Semitone, 410

Set(s), T8

in coordinate plane, graphing, T68

empty, T8

union and intersection of, T8–T9

Set-builder notation, T8

Shape and size

power functions modeling

relationships between, 554–57

proportionality symbol used to relate,

557–60

Shifting graph, 414–18, 495

Shrinking graphs, 418–19

Sichuan, China, earthquake in (2008),

346–47

Similar objects, 554–57

areas of, 555–56

scaling factor of, 554–55

volumes of, 556

Simon, Erika and Helmut, 372

Simple interest, 203–4

Simplifying rational expressions,

T39, T42

Simpson, O.J., 370

Singular matrix, 623

Slope-intercept form of equation of line,

166–67

Slope of line, 157, 158–59

definition of, 158

linear functions and, 157–61

parallel lines, 178

perpendicular lines, 181

point-slope form, 167–71

rate of change and, 159–61, 167

slope-intercept form, 166–67

Smoking in United States, 492

Snowfall, Sierra Nevada mountain

range, 10

Solar cookers, 471

Solar panels, electrical capacity of,

182–84

Solution(s)

of equation, T47

equation with no, T50–T51

extraneous, T50

of linear system in reduced row-

echelon form, 595

of linear system in two variables,

number of, 571

of system of equations, 569

of two-variable equation, T72

of two-variable equation, table of,

T73, T74

Solving the equation, T47–T56

algebraic method, T85, T86

with distributive property, T48

graphical method, T85–T88

linear equations, T49–T51

for one variable in terms of others,

T53–T54

power equations, T51–T53

Sorensen, Søren Peter Lauritz, 343

Sound intensity, decibel scale of, 344–45

Special factoring formulas, T35–T36

Special Product Formulas, T28–T29

Species-area relation, 516

in bat cave, 499, 517–18

modeling, 517–18

Species survival, predicting, 637–39

Speed(s)

average, 145–46, 148

combining, 415

doubling, 418

Sphere

surface area of, 60

volume of, 48, 496–97

Spring constant, 188–89

Square root

of negative number, T20

principal, T20

Square root function, 66–67, 497–98

Squaring function, 428–29

graph of, 432

Staircase, slope of, 158

Standard form

of equation of circle, T76

of quadratic function, 431–34

of rational function, 539

Starting time, horizontal shift of graph

and changing, 416

Steepness. See Slope of line

Stopping distance of car, 503

Straight-line depreciation, 176

Streptococcus A bacterium, exponential

growth of, 248–49

Stretching graphs, 418–19, 495

Subscripts, variables with, 104–5

Substitution, principle of, T27, T28

Substitution method, 569–70

Subtraction. See also Difference(s)

of algebraic expressions, T26–T27

graphical, of functions, 486–87

of matrix, 611–12

of polynomials, T30

of rational expressions, T41–T42

Sumatra, earthquake in (2004), 346–47

Sum of functions, 484–87

Super-size portions, 226

Supply and demand, modeling, 214–15,

218–19

Supply equation, 214, 218

Surface area

formula for, 103

species-area relation in bat cave, 499,

517–18

of sphere, 60

volume and, 558–59

Surge functions, 563–66

experimenting with, 563–65

modeling alcohol data with, 563,

565–66

Survey

analyzing, 136–37

taking, 135–36, 635

Swimming pool fill time, 201–2

Symbolic representation of function, 43

System of equations, 567–602

applications, 573–76

defined, 568

dependent, 571, 582–84, 596–98

elimination method of solving, 570–71

equivalent, operations leading to, 581

graphical interpretation of, 571–72

guidelines for modeling with, 574

inconsistent, 571, 572, 582–84, 596–98

linear, in several variables, 580–89

linear, in two variables, 568–80

linear, written as single matrix

equation, 621–25

INDEX I11

matrices to solve, using, 590–602

solution of, 569

substitution method of solving,

569–70

Table of solutions, T73, T74

Tacoma Narrows bridge collapse, 237

Temperature

average rates of change of, 230–33

chirping rate of crickets and, 198, 208

of cooling coffee, 41

elevation and, 28–29

time and, 6, 14–15

Term, T1

Ticket sales

maximum revenue from, 443–44

quadratic function modeling revenue

from, 454–55

Time and temperature, 6, 14–15

Time periods in exponential models

changing, 262–65, 266

exponential decay models, 253

exponential growth model, 250

Time periods in logistic growth

model, 277

Tire inflation, modeling proper, 461, 463

Tire Pressure Monitoring (TPM)

system, 461

Tones in music, 410

Torricelli, Evangelista, 476

Torricelli’s Law, 61, 392–93, 466,

476–77

testing theory, 477

Traffic management, 560–63

Trains traveling on same track, 179–80

Transformations of function, effects on

graph, 414–27, 495

exponential functions, 420–21

horizontal shifts, 416–18

logarithmic functions, 421–22

power functions, 495

reading graph, 473–75

reflecting graphs, 419–22, 495

vertical shifts, 414–15, 417–18

vertical stretching and shrinking,

418–19, 495

Transition matrix, 637–39

Triangular form, linear system in, 580,

581, 582

Triangular numbers, 479–80

Trinomial, T26

factoring, T34–T35

Tuition fees, rate of change in, 229–30

Twain, Mark, 128

Two-variable data, 5–7

analyzing, 137

dependent and independent variables,

37–38, 54

graphing, in coordinate plane, 14–16

as relation, 13

Two-variable equations, T71–T80

of circle, T76–T77

defined, T71

graphs of, T72–T75

reading, T72

solution of, T72

Union

of intervals, T10–T11

of sets, T8–T9

Unit cost, 26, 27

U.S. Census Bureau, 134

U.S. national debt, 132–33, 270, 314

U.S. population, national census data on,

225, 300–301

U.S. Senate, women in, 22

Universal gravitational constant, 104

Value of function, 53

finding, from a graph, 75–76

local maximum and minimum, 79–81

net change in, 55–56

Variable(s), T25

dependent, 37–38, 54

hidden, 239–42

independent, 37–38, 54

leading, in linear system, 595

solving equation for one variable in

terms of others, T53–T54

with subscripts, 104–5

Verbal description of interval, T10

Verbal representation of function, 43, 54

verbal description from graph, 75

Vertex of parabola, 432, 440

Vertical asymptote, 329, 528, 540, 541

Vertical lines, 171–72

Vertical line test, 41, 42–43

Vertical shifts of graphs, 414–15

combining horizontal shifts and,

417–18

Vertical stretching and shrinking of

graphs, 418–19, 495

Viewing rectangle, choosing, T80–T82

Volume(s)

of box, modeling, 93–94

of can, 60

linear model for, 168–69

maximum, finding, 510–11

proportionality to cube of “length”

measurement, 557

of similar objects, 556

of sphere, 48

of sphere, radius and, 496–97

surface area and 558–59

Voter survey, 42

Water rates, usage and, 58, 69

Weight

of astronaut, net change, 55–56

of whole world, 107–8

Well-Tempered Clavier, The (Bach), 411

Wheat, supply and demand for, 214–15

Windmill, watts of power generated by, 40

Wing loading, 112

Women

in science and engineering, 201

in U.S. Senate, 22

World population, 282

different models for, 289

doubling of, 369–70

exponential models for, 289, 296

limited, 285

x-axis, T67

x-coordinate, T67

Xeriscaping, 57

x-intercepts, T75

factored form of rational function to

find, 539

graphing a polynomial function to

find, 505–8

using discriminant to find number of,

452–53

y-axis, T67

y-coordinate, T67

y-intercepts, 158, 539, T75

Zero exponents, T15–T16

Zero-product property, 448, T56

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■

Lines

Slope of a line through the points and :

Point-slope equation of a line through the points and with slope m:

Slope-intercept equation of a line with slope m and y-intercept b:

Two lines with slopes and are:

■

Parallel if they have the same slope

■

Perpendicular if the slopes are

negative reciprocals

■

Exponents

Properties of exponents

■

Logarithms

Laws of logarithms

3. log

= C log

log

= log

A - log

log

AB = log

A + log

-1

1ab 2

= a

m -n

= a

m +n

= m

y = b + mx

y - y

= m1x - x

, y

m =

- y

- x

, y

21x

, y