Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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Linear equation (continued)
in two variables, 772–773
applications, 778–784
elimination method, 775–777
substitution method, 774–775
Linear function, 162, 252, 290–292
catalog, 170
Linear inequalities, 308–309
Linear models
applications, 125–127
linear regression, 123–124
mathematical model, 120–121
residual, 122
Linear rate of change, 64
Linear rational functions, 292
Linear Regression Theorem, 123
Linear speed, 438
Line graphs, 41, 87
Local extrema, 274
Local maxima and minima, 165
rational graph, 293
Logarithmic equations, 403
Logarithmic functions
algebraic solutions, 399–408
to base b, 393
common and natural logarithmic
functions, 375–385
models, 409–418
to other bases, 392–399
properties of logarithms, 385–392
radical equations, 350–357
Logarithmic models, 409
Logarithm Laws, 395
Logarithms
power law, 395
product, 395
product law, 395
quotient law, 395
Logistic function, 238
Logistic model, 364–365
Log x, 376
Lola, 588
Lower bound, 266
M
Magnitude, 640, 641–642
Mahoney, Matt, 815
Mahoney, Sharon, 110, 114–115
Market for a product, 137
Mathematical induction, 864–873
Mathematical model, 120
Matrix
equality, 807
identity, 810
inverse, 810
inverses and solutions of a system, 813
invertible, 810
multiplication, 807–808
from a system of equations, 812
Matrix methods
linear equations, large systems of, 795–798
for square systems, 806–807
applications, 814–819
identity matrix and inverses, 810–812
inverse matrices, 812–814
multiplication, 807–810
Maximum/minimum finder, 84
I-8 Subject Index
Midpoint Formula, 43, 69
Modulus of complex number, 626–627
Moon, periodic cycle of, 571–572
Motion, energy of, 927
Multiplication
matrix methods for square systems,
807–810
polar form, 629
Multiplicity, 273
of roots, 330
m n matrix, 807
N
n!, 857
Natural exponential function, number e and,
363–364
Natural logarithmic function, 378
Natural logarithms, 377–378
properties, 380
Natural numbers, 2
Nautical mile, 436
Negative angle identities, 463, 516, 576
Negative angles, 429
Negative correlation, 124
Negative numbers
square roots, 324
square roots of, 8
Negative of a vector, 644
Negatives, 6
N factorial, 857
Nonlinear equations in two variables,
784–786
second-degree equations, 787–788
Nonnegative numbers, 6
Nonvertical asymptotes, 304–307
Norm. See magnitude
Norman arch, 340
Nth root, 342–343
complex numbers, 633–635, 636–638
formula, 635
Nth term
of arithmetic sequence, 838–839
of geometric sequence, 845
The number e
continuous compounding and, 372–373
natural exponential function and,
363–364
Number line, 4
Number of roots, 257, 330
Numerical equation solving, 96–97
advantages and disadvantages, 99
O
Oblique asymptote, 304
Oblique triangles, solution of, 598–602
One-sided limits, 886
One-step equation solvers, 96
One-step solver, 99
One-to-one function, 217
Open interval, 5
Open interval, continuous on, 908
Optimization problems, solving, 114–118
Order, 4–5
Order of operations, 3
Orthogonal vectors, 656
Outputs, rule of a function, 144
P
Parabola, 240–241, 670
applications, 709–712
equations, 700–704
graphing techniques, 707–708
latus rectum, 703
parametric equations, 716–717
vertical and horizontal shifts, 705–707
Parabolic arch, 340
Parallelogram rule, 631
Parallel vectors, 655
Parameter, 176, 728
Parametric equations, 176
circles, 713–714
ellipses, 714–715
hyperbolas, 715–716
parabolas, 716–717
plane curves, 727–742
Parametric graphing, 175–178
Partial fraction decomposition, 801
Partial sums, 834–835, 840–842
of a geometric sequence, 847–850
Pascal’s triangle, 859
Period, 478
of tangent, 472
Periodic graphs and simple harmonic
motion, 477–490
Periodicity identities, 462, 516, 576
Phase shift, 482
Piecewise-defined function, 156–157
graphs, 164
Pistons and flywheels, 512
Plane curves, 727–742
Point, 670
Point-in-the-plane description, 447, 499,
575, 586
Point of diminishing returns, 282
Point-slope form of the equation
of a line, 58
Points of inflection, 275
Polar axis, 743
Polar coordinates, 743–753
Polar form, 627–628
division rules, 629
multiplication rules, 629–630
Polar graphs, 748–751
Polar/rectangular coordinate conversion,
744, 746
Pole, 743
Polynomial and rational inequalities
applications, 315
basic principles for solving, 308
linear inequalities, 308–309
polynomial inequalities, 310–311
quadratic and factorable inequalities,
311–313
rational inequalities, 313–314
Polynomial equation of degree n, 26
Polynomial function, 250–252
arithmetic, 252–254
graphs, 270
applications, 277–278
bending, 274–275
complete, 275–277
continuity, 271
local extrema, 274
shape when |x| is large, 271–272
x-intercepts, 273–274
Polynomial function (continued)
limits, 891–893
constant, 891–892
identity function, 891–892
real roots
bounds, 265–267
Factor Theorem, 264–265
rational roots, 263–264
summary, 267–268
remainders and roots, 254–257
synthetic division, 259–262
Polynomial in x, 251
Polynomial limits, 892
Polynomial models, 283–285
Polynomial solver, 99
Positive angles, 429
Positive correlation, 124
Positive numbers, 6
Potential energy, 927
Power Law for Logarithms, 388, 395
Power models, 409
Power Principle, 350–353
Power-reducing identities, 537
Powers, computing with DeMoivre’s
Theorem, 632–633
Principal nth root, 343
Principal square root, 8
Principle of Mathematical Induction, 866
Product, 195–196
of functions, 196
logarithm of, 395
to sum identities, 540
Product function, 200
Product Law for Logarithms, 385, 395
Projectile motion, 736
Projection of u on v, 657
Properties
of limits, 890
of limits at infinity, 919
of logarithms, 379–381, 394
Pythagorean identities, 460, 500–501,
516, 576
Q
Quadrants, 69
Quadratic and factorable inequalities,
311–313
Quadratic equations, 20–26
Quadratic Formula, 23, 69, 560–562
Quadratic functions, 240–244,
245–246, 252
Quadratic regression, 283–285
Quartic regression, 283–285
Quotient Law for Logarithms, 387, 395
Quotients of functions, 195–196
R
Radian/degree conversion, 433–434
Radian measure, 430
Radians, 575
Radical equations, 98, 350–357
Radical notation, 346–347
Radicals and rational exponents,
342–350
Radius (r) of circle, 46
Rational exponents, 344–346
Subject Index I-9
Rational functions, 288
applications, 299–300
big-little principle, 288
domain, 288
finding accurate graphs, 296–299
graphs, properties of, 293–296
linear, 290–292
oblique asymptote, 304
technology, 292–293
vertical asymptote, 305–306
Rational inequalities, 313–314
Rationalizing denominators and numerators,
347–348
Rational limits, 893
Rational numbers, 3
Rational Root Test, 263
Real axis, 626
Real numbers, roots of, 342–343
Real number system
absolute value, 9–11
arithmetic, 3–4
decimal representations, 16–17
distance on the number line, 11–13
integers, 2
irrational numbers, 3, 17
natural numbers, 2
negative numbers and negatives of
numbers, 6
number line and order, 4–6
rational numbers, 3
relationships among types of, 3
scientific notation, 7–8
square roots, 8–9
Real root, 255
Real solutions, 19, 24
Reciprocal identities, 516
Recursively defined sequence, 829
Reduced row echelon form, 799
Reducing equations, 560–562
Reflections in the x- and y-axis, 184–185
Regular polygon, 619
Remainders, polynomial division, 254
Remainder Theorem, 255
Repeating and nonrepeating decimals, 17
Residual, 122
Restricted cosine function, 548
Restricted tangent function, 550–553
Richter scale, 390
Right, continuous from, 907
Right angle, 584
Right triangle, 584
applications, 588
description, 576, 585
solving, 579–582
Root, 255
of multiplicity k, 273
of unity, 636
Rose, 750
Rotation equations, 723
Rotation of axes, 722–727
Rotations and second-degree equations,
718–720
Round-Trip Theorem, 222
Row echelon form matrix, 797
Row operations, 795–796
Rows, 807
Rows, augmented matrix, 795
Rows and columns, 807
Rules for parentheses, 4
Rutherford, Ernest, 700
S
Scalar multiplication, 642–643
properties, 645
Scatter plots, 69, 87
Schwarz inequality, 656
Schwarzschild radius, 928
Scientific notation, 7
Secant function, 497
graphs of, 501–502
period of, 500
Secant line, 208–209
Second-degree equation. See also quadratic
equations
graphs (discriminant of equation),
720–721
nonlinear equations in two variables,
787–788
Semicircular arch, 340
Sequences, 826–831
arithmetic, 837–840
constant, 828–829
term of the, 826–828
Shortage, 138
Shorthand language, 152
Simple harmonic motion, 485
Sine functions, 442–457
graphs, 466–472
Sinusoidal graphs, 491
Slope, 54
of parallel lines, 60
of perpendicular lines, 61
properties of, 56
theorem for perpendicular
lines, 534
Slope-intercept form of the equation of
a line, 57
Solution algorithms for basic trigonometric
equations, 557–559
Solution methods, 99
Solution of a system, 772
number of, 773, 802
Solve an equation, 19
graphically, 93, 95
Special values, 445, 456
Speed of light, 927
Spiral, Archimedean, 750
Square functions, 162–163
catalog, 170
Square root, 342
negative numbers, 324
of negative numbers, 8
real number system, 8–9
of squares, 10
of zero, 97
Square root function, 163
catalog, 170
Square window, calculator, 83
Standard form, complex number, 322
Standard notation for triangles, 597
Standard position, 586
Standard viewing window,
calculator, 82
Step function, 163
Stored energy, 927
Substitution method
linear equations in two variables,
774–775
reducing equations, 560–562
Summation index, 832
Summation notation, 831–834
Sum of an infinite geometric series, 853
Sum of the covergent series, 853
Sum/product identities, 540–542
Sums, 195–196
Sum to product identities, 541
Sun, periodic cycle of, 571–572
Supplemental angle identity, 609
Supply curve, 137
Surplus, 138
Surveying, 666–667
Symmetries
coordinate and algebraic tests, 192
even and odd functions, 192–193
origin symmetry, 191–192
x-axis, 190–191
y-axis, 189–190
Synthetic division, 254, 259–262
System of equations
linear equations, large systems of, 792–793
applications, 801–806
elementary operations, 793–795
Gauss-Jordan method, 798–800
matrix methods, 795–798
linear equations in two variables, 772–773
applications, 778–784
elimination method, 775–777
substitution method, 774–775
matrix methods for square systems,
806–807
applications, 814–819
identity matrix and inverses, 810–812
inverse matrices, 812–814
multiplication, 807–810
nonlinear equations in two variables,
784–786
applications, 789–791
second-degree equations, 787–788
T
Tangent, half-angle identities, 539–540
Tangent function, 442–457, 443, 452, 455
graphs, 466–472
Technology tips. See calculator use
Terminal point, vectors in the plane, 640
Terminal side, 428, 586
Term of the sequence, 826–828
Theorems
Angle Theorem, 655
Binomial Theorem, 857–862
games of chance, 877–878
Conjugate Roots Theorem, 331
DeMoivre’s Theorem, 632–633
Equation Theorem, 911
Factor Theorem, 256
Fundamental Theorem of Algebra, 328
Infinite Limit Theorem, 919–921
Intermediate Value Theorem, 910–911
Limit Theorem, 894
Linear Regression Theorem, 123
Remainder Theorem, 255
Round-Trip Theorem, 222
I-10 Subject Index
Theory of equations
factorization over the complex
numbers, 329
Fundamental Theorem of Algebra,
328–329
number of roots, 330
polynomials with real coefficients,
330–332
Time interval, 205
Trace, 81
Triangle inequality, 11
Triangles
area formulas, 617
oblique, solution of, 598–602
standard notation for, 597
Triangle trigonometry
angles, functions of, 574–588
applications, 588–597
area of a triangle, 617–620
law of cosines, 597–606
law of sines, 606–616
Triangular form, 794
Triangular form system, 794
Trigonometric form, 627–628
Trigonometric functions
algebra and identities, 457–465
alternate descriptions, 498–499
angles and measurement, 428–435, 574
arc length and angular speed, 435–441
basic graphs, 466–477
cotangent, secant, and cosecant functions,
497–498
cotangent and tangent functions,
reciprocal identities of, 499
damped and compressed trigonometric
graphs, 493–495
periodic graphs and simple harmonic
motion, 477–490
period of secant, cosecant, and
cotangent, 500
point-in-the-plane description, 499
Pythagorean identities, 500–501
secant function, 501–502
sine, cosine, and tangent functions,
442–457
sinusoidal function, 490–493
Trigonometric identities and equations
addition and subtraction identities, 523–531
basic equations, 555–560
basic identities and proofs, 514–523
cofunction identities, 527–528
double-angle identities, 535–537
graphical solution method, 562–563
half-angle identities, 538–540
inverse trigonometric functions, 545–555
lines and angles, 532–535
power-reducing identities, 537
reducing equations with substitution,
factoring, quadratic formula, and
identities, 560–562
sum/product identities, 540–542
Trivial solution, 800
Two-sided limits, 887
U
Unit circle, 48
Unit vectors, 645–646
Unity, roots of, 636
Upper bound, 266
V
Variable costs, 69
Vector addition, 643–644
properties, 645
Vectors in the plane,
639–653
angles, 654–656
applications, 659–660
arithmetic, 642–645
direction angle, 647
direction angles, 647–649
dot product, 653–654
equivalent, 640–641
initial point, 640
length, 640
magnitude, 640
magnitude (or norm), 641–642
projections and components,
657–659
terminal point, 640
unit vectors, 645–646
Vector subtraction, 644
Vertex, 241, 428
of hyperbola, 687
of parabola, 700
Vertical asymptote, 289, 294, 297,
305–306, 915
rational graph, 294
Vertical lines, 58
Vertical Line Test, 169
Vertical shifts, 180, 677, 705
Viewing window, 80
W
Work, 650
Work, equation for, 660
X
X, change in, 54
X-axis, 39
reflections, 184–185
symmetries, 190–191
X-intercept, 44–45, 92, 273
Y
Y, change in, 54
Y-axis, 39
reflections, 184–185
symmetries, 189–190
Y-intercept, 44–45
Z
Zero, 255
factoring method, need for one side to
equal, 21
Zero polynomial, 251
Zero products, 20
Zoom, 82
Trigonometry
If t is a real number and P is the point where the terminal side of an angle of t radians in standard position
meets the unit circle, then
Point-in-the-Plane Description
For any real number t and point (x, y) on the terminal side of an angle of t radians in standard position:
Periodic Graphs
If and then each of and has
amplitude period
Right Triangle Trigonometry Special Values
Special Right Triangles
Law of Cosines Law of Sines
Area Formulas
Herron’s Formula: where Area
1
2
ab sin Cs
1
2
1a b c2Area 1s1s a21s b21s c2,
c
2
a
2
b
2
2ab cos C
b
2
a
2
c
2
2ac cos B
a
2
b
2
c
2
2bc cos A
tan u
opposite
adjacent
cos u
adjacent
hypotenuse
sin u
opposite
hypotenuse
phase shift c/b.2p/b,
0
A
0
,
g1t2 A cos1bt c2f 1t2 A sin1bt c2b 7 0,A 0
cot t
x
y
1y 02sec t
r
x
1x 02 csc t
r
y
1y 02
tan t
y
x
1x 02cos t
x
r
sin t
y
r
cot t
cos t
sin t
sec t
1
cos t
csc t
1
sin t
tan t
sin t
cos t
sin t y-coordinate of Pcos t x-coordinate of P
Degrees Radians
001 0
1
1 0 undefined
p
2
90°
13
1
2
13
2
p
3
60°
12
2
12
2
p
4
45°
13
3
13
2
1
2
p
6
30°
0°
tan Ucos Usin U
U
Adjacent
Opposite
u
Hypotenuse
t
r
(x, y)
1
45°
45°
2
11
3
30°
2
60°
B
a
C
b
A
c
a
sin A
b
sin B
c
sin C
Trigonometric Identities
Reciprocal Identities Pythagorean Identities Negative Angle Identities
Periodicity Identities Cofunction Identities
Addition and Subtraction Identities
Double Angle Identities Half-Angle Identities
Product Identities Factoring Identities
cos x cos y 2 sin
a
x y
2
b
sin
a
x y
2
b
cos x sin y
1
2
1sin1x y2 sin1x y22
cos x cos y 2 cos
a
x y
2
b
cos
a
x y
2
b
cos x cos y
1
2
1cos1x y2 cos1x y22
sin x sin y 2 cos
a
x y
2
b
sin
a
x y
2
b
sin x sin y
1
2
1cos1x y2 cos1x y22
sin x sin y 2 sin
a
x y
2
b
cos
a
x y
2
b
sin x cos y
1
2
1sin1x y2 sin1x y22
cos 2x 2 cos
2
x 1
tan
x
2
sin x
1 cos x
cos
x
2
±
B
1 cos x
2
tan 2x
2 tan x
1 tan
2
x
cos 2x cos
2
x sin
2
x
tan
x
2
1 cos x
sin x
sin
x
2
±
B
1 cos x
2
cos 2x 1 2 sin
2
xsin 2x 2 sin x cos x
cos1x y2 cos x cos y sin x sin y
tan1x y2
tan x tan y
1 tan x tan y
cos1x y2 cos x cos y sin x sin y
sin1x y2 sin x cos y cos x sin y
tan1x y2
tan x tan y
1 tan x tan y
sin1x y2 sin x cos y cos x sin y
csc x seca
p
2
xbsec x csca
p
2
xbcot1x p2 cot xtan1x p2 tan x
cot x tana
p
2
xbtan x cota
p
2
xbsec1x 2p2 sec xcsc1x 2p2 csc x
cos x sina
p
2
xbsin x cosa
p
2
xbcos1x 2p2 cos xsin1x 2p2 sin x
tan1x2 tan x 1 cot
2
x csc
2
xcot x
1
tan x
tan x
1
cot x
cos1x2 cos x 1 tan
2
x sec
2
xcot x
cos x
sin x
tan x
sin x
cos x
sin1x2 sin xsin
2
x cos
2
x 1csc x
1
sin x
sec x
1
cos x