A
AAS case, 241, 245
Abscissa, 54
Absolute value, 3, 49
in equations, 49
in inequalities, 50
properties, 49
Accuracy in computations, 241
Altitude, 85
Ambiguous case, 246
Amplitude, 188, 190
Angles, 197
complementary, 199
coterminal, 199, 202
degree measurement, 198
in a quadrant, 197
quadrantal, 197
radian measurement, 198
reference, 201, 206
standard position, 197
supplementary, 199
vertex, 197
Angular speed, 210
Arc length, 198, 203
Argument, 271
ASA case, 241, 245
Associative laws, 1
Asymptotes:
horizontal, 133, 135, 137
of a hyperbola, 339, 343
oblique, 135, 137
vertical, 132, 136
Axes, 54
rotation of, 349
Axioms for the real number system, 1
Axis, 54, 330
focal, 338
transverse, 338
B
Bearing, 241
Binomial coefficients, 381
Binomial theorem, 381, 384
proof, 384–385C
Cardioid, 266
Cartesian coordinate system, 54
Center:
of a circle, 57
of an ellipse, 337
of a hyperbola, 338Change-of-base formula, 168
Circle, 57, 62
center, 57, 62
radius, 57, 62
tangent line to, 85Closure laws, 1
Commutative laws, 1
Completing the square, 30, 34, 95
Complex fractions, 20, 23
Complex numbers, 3, 117, 270
nth roots, 272
operations, 22, 26 polar form, 272
standard form, 22 trigonometric form, 270
Complex plane, 270
Components:
of a vector, 254, 257 Compound interest, 154, 157, 172
continuous, 155, 158, 172 Conic sections, 356
degenerate cases, 356, 361
in polar coordinates, 360–361 Continuity, 389
Coordinate axes, 54
Cosine curve, 187
Cosines, 177, 187
law of, 241, 247 Coulomb’s law, 153
Cramer’s rule, 323
Critical point, 43
Cycloid, 267
D
Decibel, 169
Demand, 108
DeMoivre’s theorem, 271, 275
proof by mathematical induction, 372
Depreciation, linear, 84, 86
Depression:
angle of, 242
Derivative, 389
Descartes’ rule of signs, 118, 122
Determinants, 322
cofactors, 322
minors, 322
properties, 323, 325–326
Directrix:
of an ellipse, 359
of a hyperbola, 359
of a parabola, 330–331
Discriminant, 30, 356
Distance:
between two points, 54, 58
from point to line, 330, 332
Distributive laws, 1
Division:
of polynomials, 115
synthetic, 116
Dot product, 255
Double-angle formulas, 220, 224
E
e, 154
Earthquake intensity, 169, 173
Eccentricity, 339, 346
of a parabola, 359
Elevation:
angle of, 342
Ellipse, 337
equation of, 337, 339
Equations, 29
containing radicals, 31
equivalent, 29
exponential, 168
in quadratic form, 35
linear, 29
literal, 31, 35
logarithmic, 168
nonlinear, 29
quadratic, 30
second degree
classification, 356
systems of, 279
trigonometric, 213, 215
Exponential decay, 154, 156
Exponential growth, 154, 156
Exponents, 15
laws, 16
natural number, 15
negative integer, 15
properties, 154
zero, 15
F
Factor theorem, 116, 120
Factorial, 363
Focus:
of an ellipse, 337
of a hyperbola, 338
of a parabola, 330–331
FOIL multiplication, 2, 8
Formulas, 31
Function-inverse function relation, 105
Functions, 68
algebraic, 146
composite, 104, 108
constant, 69
decreasing, 69
domain, 68, 71
even, 69
exponential, 154
finding inverses, 106, 110
graph, 69
increasing, 69
input and output values, 68
inverse, 105, 110
largest possible domain, 68
linear, 79
linear, rule for, 79
logarithmic, 162
maximum, minimum values, 96
notation, 68
odd, 69
odd and even, 69, 112
one-to-one, 105, 109
periodic, 113, 177, 230
polynomial, 114
graphing, 118
product and quotient, 104
quadratic, 95
range, 68
rational, 132
graphing, 132, 135, 137
sum and difference, 104
Fundamental principle of fractions, 20
Fundamental theorem of algebra, 117
G
Gaussian elimination, 288
Gauss-Jordan elimination, 289
Graph, 55
of an equation, 55
H
Half-angle formulas, 221, 225
Half-angle identities, 221, 224
Heading, 241
Horizontal line test, 105
Hyperbola, 338
equation of, 339
I
Identities, 29, 178, 211
for negatives, 178
Pythagorean, 178, 184
quotient, 178
reciprocal, 178
trigonometric, 178, 211
verifying, 212–213
Identity laws, 1
Index:
of a radical, 21
of a sequence, 362
Inequalities, 41
equivalent, 42
linear, 42
nonlinear, 42–43
Inequality:
relations, 41
statements, 42
triangle, 49
Integers, 1
Intercepts, 55, 97
x-, 55
y-, 55
Intermediate value theorem, 118
Intervals, 41
Invariant, 358
Inverse laws, 2
Inverse square law, 150, 153
Irrational numbers, 1
K
Kepler’s third law, 152
L
Laws for negatives, 2
Laws for quotients, 2
Limits, 387
at infinity, 389
infinite, 388
one-sided, 388 Linear inequalities, 42
Lines:
horizontal and vertical, 79
normal, 85
parallel, 80
perpendicular, 80
point-slope form of equation, 80
slope-intercept form, 80
standard form of equation, 80
tangent, 85, 390
two-intercept form of equation, 81 Loci, 330
Logarithmic scales, 168
Logarithmic-exponential function relations, 162
Logarithms, 162
M
Major axis, 337
Mathematical induction (see Principle of mathematical induction)
Matrices, 287, 309
addition, 309
commutative law, 310
augmented, 288
equality, 309
identity, 314
inverses, 314
main diagonal elements, 309
multiplication, 313
non-commutativity, 314
non-singular, 314
orthonormal, 321
product with a scalar, 310
reduced row-echelon form, 288
row-echelon form, 288
row-equivalent, 287
singular, 314
square, 309
subtraction, 310
transpose, 312, 321
zero, 309 Median, 85
Midpoint formula, 58, 62
Minor axis, 337
Modulus, 271
Mollweide’s formula, 251
N
Natural numbers, 1
Newton’s law of cooling, 152
0
Ohm’s law, 150
Order of operations, 3
Ordering properties of real numbers, 2
Ordinate, 54
P
Parabolas, 95, 330
equations of, 331
vertex, 95, 330
Parametric equations, 262, 267
Partial fraction decomposition, 294
Period, 177
Perpendicular bisector, 63–64
Phase shift, 188
Phase-shift formula, 231, 237
Phase-shift identity, 231, 237
Polar and Cartesian coordinates, 262
transformation relationships, 264
Polar coordinate system, 261
Polar coordinates, 261
possible sets of, 261, 264
Polynomials, 7
addition, 7
complex zeros, 117
degree, 7, 10
division, 115, 119
factoring, 9–12
FOIL multiplication of, 8
multiplication, 8, 11
rational zeros, 117
subtraction, 8
zeros, 115
Population growth, 155
logistic, 155, 159
unlimited, 155, 159
Price/demand, 109, 112
Principle of mathematical induction, 368
extended, 369
Product-to-sum formulas, 221, 226
Projectile, 269
Proportion, 146
direct, 146, 149
inverse, 146, 149
Pythagorean theorem, 32
Q
Quadrants, 54
Quadratic formula, 30
R
Radical expressions, 21
rationalize denominator, 22, 25
rationalize numerator, 22, 25
simplest radical form, 21
Radicals, 21
properties, 21
rationalize denominator, 22, 25
rationalize numerator, 22, 25
simplest radical form, 21
Radioactive decay, 155, 160
Rate of change:
average, 70,
instantaneous, 390
Rational expressions, 20, 294
Rational numbers, 1
Real number line, 3
Reflection, 57
Remainder theorem, 116, 119
Resultant:
of forces, 259–260
Richter scale, 169, 173
Rose, 266
S
SAS case, 241, 248
Scalar quantity, 252
Scalars, 252
Scientific notation, 16, 18
Sequence, 362
arithmetic, 374
defined recursively, 362
Fibonacci, 364
geometric, 374
Series, 363
arithmetic, 374
infinite geometric, 375
Shifting
horizontal, 88
vertical, 87
Sine curve, 187
Sines, 177, 187
law of, 241, 244
Slope, 79
Sound intensity, 169, 173, 390
SSA case, 241, 246
SSS case, 241, 248
Stretching, 88
Sum and difference formulas, 220, 222
Summation notation, 363
Sum-to-product formulas, 221, 226
Symmetry, 55
tests for, 55
Systems of equations, 279
linear, 279
nonlinear, 302
Systems of linear equations, 279
classification, 280
equivalent, 279
solutions, 280
T
Tangent line:
to a circle, 85
to a graph, 390
Transformation equations:
in rotation, 349–350
reverse, 352
Transformations, elementary, 87
Triangle inequality, 49, 53
Triangles, 63, 240
oblique, 241
solving, 241
right, 200, 240
solving, 240
Trigonometric functions, 177
of acute angles, 200
calculator approximation, 208
cofunction formulas, 220
domain and range, 178–179
double-angle formulas, 220
even and odd properties, 186
graphs, 187
half-angle formulas, 221
half-angle identities, 221
inverse, 207, 212, 230
of important angles, 205–206
periodicity, 177
product-to-sum formulas, 221
redefined, 230
sum and difference formulas, 220
sum-to-product formulas, 221
unit circle definition, 177
U
Unit circle, 57, 176
V
Variation, 146
combined, 147
direct, 146, 149
inverse, 146, 149
joint, 147 Vector quantity, 252
Vectors, 252
addition, 253
algebraic, 254
angle between, 255, 258
dot product, 255
equivalent, 252
magnitude, 252, 255
multiplication by a scalar, 253
orthogonal, 260
subtraction, 253
unit, 259
zero, 253 Velocity, 390
Vertical line test, 69
W
Word problems, 32, 36
Z
Zero factor laws, 2
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