- ! (factorial), 342–343
- ∑ (epsilon), 330
- ≥ (greater than or equal to), 25
- ≤ (less than or equal to), 25
- < (less than), 25
- > (greater than), 25
- || (absolute value), 30
- ^ (caret), 91
A
- absolute value inequalities, 53–54
- absolute values, 30–34
- defined, 30
- graphs of, 89
- inequalities, 31–33
- solving equations, 31
- abundant numbers, 368
- addition (adding). See also summing sequences
- ancient symbols, 29
- associative property of,
- casting out 9s, 357–358
- commutative property of,
- complex numbers, 276–277
- matrices, 290–291
- order of operations, 11–12
- additive identity, applying, 10
- additive inverses
- defined, 11
- determining, 299
- algebra
- basics of,
- properties of, –11
- associative property,
- commutative property,
- distributive property, –10
- identities, 10
- inverses, 11
-
Algebra For Dummies, , 30
-
Algebra I For Dummies, 39
-
Algebra I Workbook For Dummies, 150, 163
- Algebra Rules icon,
- alternating sequential patterns, 309–310
- amicable numbers, 367
- ancient symbols, 29
- arithmetic sequences
- overview, 313
- recursive rule for, 317
- summing, 319–320
- associative property,
- asymptotes
- defined, 160
- graphing
- oblique asymptotes, 163
- overview, 160
- vertical and horizontal asymptotes, 161–163
- horizontal, 161
- of hyperbolas, 223–224
- oblique, 163
- vertical, 160
- axis of symmetry, of quadratic functions, 124–125
B
- bases, of exponential functions, 179–182
- basketballs, shooting, quadratic function example, 128–130
- binomials
- defined,
- factoring into the product of two, 71
- synthetic division by, 153–154
- blocks, stacking, 323
- bouncing a ball, 324–326
C
- calculators, graphing
- entering equations into, 90–93
- entering exponents, 91
- entering fractions, 90–91
- finding irrational roots of polynomials, 145
- negating or subtracting in, 92
- calculators, graphing (continued)
- radicals in, 91–92
- sketching a parabola, 126
- solving problems with, 89–93
- window of, 92–93
-
Calculus For Dummies (Ryan), 112
- candles, quadratic function example, 127–128
- caret (^), 91
- Cartesian coordinate system, 74
- circles
- with center at the origin, 217
- circumference of, 217
- defined, 215
- graphs of, 89
- parabolas intersecting with
- finding common solutions, 258–259
- multiple intersections, 256–258
- standard form for the equation of, 215
- standardizing, 215–216
- unit, 217
- circumference of circles, 217
- coefficients
- defined,
- quadratic functions, 116
- coexisting lines, systems of linear equations and
- identifying, 237–238
- recognizing, 235–236
- columns of matrices, 289
- combinations
- defined, 348
- mixing up sets with, 348–350
- tree diagrams for, 352
- combining terms with fractional exponents, 68
- common denominator
- finding, 361
- least (LCD)
- finding, 56–57
- solving rational equations, 56–59
- commutative property,
- complements of sets, 335
- completing the square
- conic sections, 43–46
- twice over, 45–46
- complex numbers
- adding, 276
- applications of, 276
- examples of, 276
- multiplying, 277
- multiplying by the conjugate to perform division, 277–279
- operations on, 276–277
- standard form, 275
- subtracting, 277
- complex solutions
- solving polynomials with, 282–285
- solving quadratic equations with, 280–282
- complex zeros, 283–285
- composition of functions, 110
- compound inequality, 28
- compound interest formula, 186–188
- compounding, continuous, 188–189
- conic sections. See also parabolas
- completing the square, 43–46
- definition and features, 206
- identifying from their equations, 227–228
- overview, 205–206
- conjugate, multiplying by to perform division, 277–279
- conjugate axis, 222
- conjugate pairs, complex zeros in, 283
- constant, defined,
- constant terms
- polynomials with, Rational Root Theorem and, 146–147
- polynomials without, Rational Root Theorem and, 147
- continuous compounding, 188–189
- coordinate plane, 74–75
- coordinates, defined, 75
- counting off the slope, 83
- Cramer, Gabriel, 229, 241
- Cramer’s Rule, 238–241
- crossing polynomials, 261–262
- cross-products, solving rational equations with, 61
- cube roots, solving cubes by taking, 48
- cubics, 87
- curves
- cubic, 87
- disconnecting, on graphing calculators, 93
- exponential, 88–89
- logarithmic, 88–89
- quadratic, 86
- quartic, 87
- radical, 88–89
- rational, 88–89
D
- Dantzig, George, 296
- decomposing fractions, 248–250
- deficient numbers, 368
- derivatives, difference quotient and, 111
- Descartes, Rene, 74
- diagrams
- tree
- for combinations, 352
- for permutations, 351
- Venn
- adding sets to, 339–341
- applying, 337–338
- overview, 336
- with set operations, 338–339
- difference between terms, in sequential patterns, 311
- difference of cubes, quadratic equations, 47–48
- difference of two perfect squares, factoring two terms, 16
- difference of two squares, factoring, quadratic equations, 38–39
- difference quotient, 111–112
- Diophantes, 29
- directrix, defined, 206
- disconnecting curves, on graphing calculators, 93
- discontinuities
- defined, 108, 160
- rational functions and, 160
- removable, rational functions and
- determining an existent limit without tables, 169–170
- determining which functions have limits, 170
- evaluating limits at discontinuities, 168–169
- evaluating the removal restrictions, 165
- overview, 164–166
- showing on a graph, 165–166
- distributive property, –10
- divide/average method, for finding square roots, 279
- division (dividing)
- complex numbers, 278–279
- exponents, 13–14
- matrices, by using inverses, 304
- order of operations, 11–12
- synthetic
- by binomials, 153–154
- Remainder Theorem, 154–155
- to test for roots, 150–153
- divisors, determining, 362
- domains
- of functions, 99–100
- of rational functions, 158
- double roots, trinomials, 40
E
- effective rates, 187–188
- elimination, solving systems of two linear equations by using, 233–236
- ellipses
- defined, 218
- determining the shape of, 219–220
- finding the foci of, 220–221
- overview, 218
- sketching graph of, 221–222
- standard equation for, 218–219
- three dots following a short list of terms, 308
- empty (null) sets, 331
- epsilon (∑), 330
- equations, entering into graphing calculators, 89–93
- even functions
- applying to graphs, 103–104
- overview, 102–104
- exponential curves, 88–89
- exponential expressions, evaluating, 178
- exponential functions (exponential equations)
- base of, 179–182
- evaluating, 178
- financial applications, 186
- compound interest formula, 186–188
- effective rates, 187–188
- overview, 186
- target sums, 186–187
- graphing
- identifying a rise or fall, 197
- overview, 196
- grouping according to their bases, 179–180
- grouping according to their exponents, 180
- intersecting, 263–265
- inverses of, 200–201
- rewriting log equations as, 195–196
- solving exponential equations
- making bases match, 182–184
- recognizing and using quadratic patterns, 184–185
- exponential rules, 13
- exponents
- entering into graphing calculators, 91
- fractional
- combining negative exponents and, 71–72
- combining terms with, 69
- factoring, 69
- factoring out variables with, 70–71
- introduction to, 14
- overview, 68
- solving equations with, 70–72
- multiplying and dividing, 13–14
- negative
- combining fractional exponents and, 71–72
- factoring out to solve equations, 66–68
- flipping to positive exponents, 65–66
- rational equations with, 65–68
- use of, 15
- expression, defined,
F
- factor (noun), defined,
- factor (verb), defined,
- factorials
- overview, 342–343
- in sequences, 309
- symbol for (!), 342–343
- factoring
- binomials, 37–39
- four or more terms by grouping, 19
- fractional exponents, 69
- by grouping, quadratic equations, 40
- introduction to, 15
- for polynomial roots
- overview, 143
- patterns and groupings, 143–144
- unfactorable equations, 144–145
- into the product of two binomials, 71
- removal of discontinuities by, 164–165
- solving cubes by, 47
- three terms, 17
- trinomials, 39–40
- two terms, 16–17
- variables with fractional exponents, 70–71
- factors, Rational Root Theorem and changing from roots to, 147–148
- first difference, in sequential patterns, 311
- Fit button, on graphing calculators, 93
- flattening curves, complex roots and, 284
- focus of parabolas, defined, 206
- FOILing
- multiplying complex numbers, 277
- overview, 17, 18
- four-color problem, 349
- fractional exponents
- combining negative exponents and, 71–72
- combining terms with, 69
- factoring, 69
- factoring out variables with, 70–71
- introduction to, 14
- overview, 68
- solving equations with, 70–72
- fractions. See also rational functions
- decomposing, 248–250
- determining divisors, 362
- entering into graphing calculators, 90–91
- linear equations and, 23
- rational equations, 56
- solving quadratic equations, 52
- vulgar, 166
- zeros in the denominators, 99
- functions
- composition of, 110
- definition of, 98
- difference quotient, 111–112
- domain of, 99–100
- evaluating, 98–99
- even or odd, 102–104
- inverse, 112–114
- logarithmic. See logarithmic functions
- notation, 98
- one-to-one, 104–106
- piecewise, 106–110
- quadratic. See quadratic functions
- range of, 100–102
- rational. See rational functions
G
- Gallimard, 29
- GCF (greatest common factor)
- factoring out
- for quadratic binomials, 38
- to solve rational equations, 66–67
- factoring three terms, 17
- factoring two terms, 16, 17
- solving exponential functions and, 185
- geometric sequences
- general formula or rule for, 315–316
- recursive rule for, 317
- summing, 320–322
- graphing calculators
- entering equations into, 90–93
- entering exponents, 91
- entering fractions, 90–91
- finding irrational roots of polynomials, 145
- negating or subtracting in, 92
- radicals in, 91–92
- sketching a parabola, 126
- solving problems with, 89–93
- window of, 92–93
- graphs (graphing; sketching)
- absolute values, 89
- applying even and odd functions to, 103–104
- asymptotes
- oblique asymptotes, 163
- overview, 160
- vertical and horizontal asymptotes, 161–163
- basic forms of, 86–89
- with calculators. See graphing calculators
- Cartesian coordinate system, 74
- circles, 89
- connecting the points in order, 75–76
- coordinate plane, 74–75
- cubics and quartics, 87
- ellipses, 221–222
- horizontal line test, 105, 106
- hyperbolas, 224–226
- inequalities, 251, 268–269
- intercepts, 76–77
- lines, 81–85
- overview, 73–74
- piecewise functions, 107–108
- radicals and rationals, 88–89
- rational functions, 173–175
- removable discontinuities, 165–166
- solutions of linear systems, 230–233
- symmetry, 76, 78–80
- vertical line test, 105
- greater than (>), 25
- greater than or equal to (≥), 25
- greatest common divisor (GCD), finding, 362
- greatest common factor (GCF)
- factoring out
- for quadratic binomials, 38
- to solve rational equations, 66–67
- factoring three terms, 17
- factoring two terms, 16, 17
- solving exponential functions and, 185
- grouping
- factoring by, 19, 40
- finding quadratic factors in a, 41
H
- happy numbers, 368
- hexagonal numbers, 366–367
- horizontal asymptotes, 161
- hyperbolas
- asymptotes of, 223–224
- axes of, 222–223
- basic equations for, 223
- defined, 222–223
- foci of, 222
- graphing, 224–226
I
- icons used in this book, –4
- identities, defined, 10
- identity matrices
- characteristics of, 289–290
- determining multiplicative inverses and, 299–304
- imaginary numbers. See also complex numbers
- defined, 273
- simplifying radicals, 279–280
- income tax, piecewise functions, 109–110
- inequalities
- absolute value, 31–33, 53–54
- expressing ranges in, 100
- expressing the domain of functions with, 99
- graphing, 251, 268–269
- intersecting to share a portion of the plane, 269
- linear, 25–29
- moving in opposite directions, 32–33
- overview, 268
- quadratic, 49–53
- rational, 52
- sandwiching the values in, 32
- inequality notation, interval notation and, 27–28
- infinity
- adding all the terms of geometric sequences to, 321–322
- rational functions and, 170–173
- rational limits at, 172–173
- intercepts. See also x-intercepts; y-intercepts
- graphing log functions with, 199
- of polynomials
- counting, 135–138
- interpreting relative value and absolute value, 135
- overview, 134
- solving for, 138–139
- in quadratics, 118–119
- of rational functions, 159
- streamlining the graphing process with, 76–80
- interval notation
- expressing the domain of functions with, 99
- overview, 27–28
- intervals of polynomials, positive and negative
- overview, 139–140
- sign-line method, 140
- inverse functions, 112–114
- inverse matrices
- dividing matrices by using, 304
- identifying, for any size square matrix, 301–303
- multiplicative inverses, 299–304
- quick-and-slick rule for, 303–304
- inverses
- additive
- defined, 11
- determining, 299
- defined, 11
- of exponential functions, 200–201
- multiplicative
- defined, 11
- determining, 299–304
- rational functions as, 267–268
- irrational numbers
- quadratic equations, 42–43
- rational numbers distinguished from, 146
- irrational roots, unfactorable equations, 144–145
- irrational solutions, straightening out, solving quadratic equations, 42–43
L
- lead coefficient of the standard form, 116, 117
- least common denominator (LCD)
- finding, 56–57
- solving rational equations, 56–59
- Leibniz, 29
- less than (<), 25
- less than or equal to (≤), 25
- linear, defined,
- linear equations
- absolute value and, 30–34
- clearing out fractions, 23
- isolating different unknowns, 24–25
- overview, 21–22
- solving basic, 22–23
- systems of
- Cramer’s Rule for solving, 238–241
- decomposing fractions, 248–250
- defined, 229
- elimination method for solving, 233–236
- graphing, 230–233
- matrices for finding solutions for, 305–306
- with more than three linear equations, 244–247
- overview, 229
- parallel lines, 232–233
- pinpointing the intersection, 231–232
- real-world applications, 247–248
- recognizing solutions indicating parallel or coexisting lines, 235–236
- standard form for, 230
- substitution method for solving, 235–236
- with three linear equations, 241–247
- two equations representing the same line, 232
- linear inequalities
- compound inequality, 28–29
- overview, 25–26
- solving, 26–27
- lines
- basic forms of graphs, 86
- coexisting
- identifying, 237–238
- recognizing, 235–236
- combined with parabolas
- finding two solutions, 253
- no-answer answer, 254–255
- overview, 252
- point(s) where a line and parabola cross paths, 253–254
- settling for one solution, 253–254
- finding the slope of, 81
- graphs of, overview, 81–84
- identifying parallel and perpendicular, 85
- intersections of a rational functions and, 265–266
- mixing polynomials and, 260–261
- parallel. See parallel lines
- slope of
- counting off, 83
- finding, 81–82
- formulating the value of, 81–82
- identifying characteristics of, 81
- slope-intercept form, equation for
- changing from standard form to, 84
- overview, 83–84
- standard form, equation for
- changing from slope-intercept form to, 84
- overview, 82
- logarithmic curves, 88–89
- logarithmic functions (logarithmic equations)
- expanding with log notation, 192
- graphing
- with intercepts, 199
- overview, 196–197
- sketching exponential graphs, 197–198
- overview, 189
- properties of, 190–191
- rewriting for compactness, 192–193
- rewriting log equations as exponentials, 195–196
- setting log equal to log, 194–195
- solving, 193–196
- uses of, 191–192
M
- matrices
- adding and subtracting, 290–291
- applications of
- determining how many of each item was sold, 294–295
- determining how to increase sales, 296–297
- determining sales by salesperson, 295–296
- overview, 293–294
- determining dimensions of, 292
- finding solutions for systems of equations and, 305–306
- identity, 289–290
- inverse
- dividing matrices by using, 304
- identifying, for any size square matrix, 301–303
- multiplicative inverses, 299–304
- quick-and-slick rule for, 303–304
- multiplying
- by scalars, 291
- two matrices, 291–292
- overview, 287
- quick-and-slick rule for, 303–304
- row operations, 297–298
- rows and columns of, 289
- simplex method, 296
- square, 289
- zero, 289
- monomial, defined,
- mortgage, paying off your, 25
- multiples, sequential patterns, 312–313
- multiplication (multiplying)
- ancient symbols, 29
- associative property of,9
- commutative property of,
- complex numbers, 277
- distributing,
- exponents, 13–14
- matrices
- by scalars, 291
- two matrices, 291–292
- multiplication (multiplying) (continued)
- order of operations, 11–12
- property of zero, 12
- time-saving tricks
- casting out 9s, 358–359
- finding the next perfect square, 356
- multiplying by 5, 360–361
- multiplying by 11, 359–360
- multiplying two-digit numbers, 362–363
- recognizing the pattern in multiples of 9 and 11, 357
- squaring numbers that end in 5, 355–356
- multiplication principle, applied to sets, 344–345
- multiplicative identity, use of, 10
- multiplicative inverses
- defined, 11
- determining, 299–304
N
- narcissistic numbers, 368–369
- negative button, in graphing calculators, 92
- negative exponents
- combining fractional exponents and, 71–72
- factoring out to solve equations, 66–68
- flipping to positive exponents, 65–66
- rational equations with, 65–68
- use of, 15
- negative numbers
- origin of, 13
- solving linear inequalities, 26–27
- negative roots of polynomials, 149–150
- negatives
- factoring out to solve rational equations, 66–68
- order of operations on graphing calculators, 92
- nonlinear equations
- combined parabolas and lines, 252
- crossing parabolas with lines, 252
- notation
- of exponents, 13
- function, 98
- interval, overview, 27–28
- for inverse functions, 112
- for limits of rational functions, 167
- sequence, 308
- set
- overview, 329–330
- set builder notation, 330
- subsets, 331–333
- universal and empty (null) sets, 331
- numbers
- abundant, 368
- amicable, 367
- complex
- adding, 276
- applications of, 276
- examples of, 276
- multiplying, 277
- multiplying by the conjugate to perform division, 277–279
- multiplying numbers by FOILing, 277
- operations on, 276–277
- standard form, 275
- subtracting, 277
- deficient, 368
- happy, 368
- hexagonal, 366–367
- irrational
- quadratic equations, 42–43
- rational numbers distinguished from, 146
- ISBN, 30
- narcissistic, 368–369
- palindromic, rational equations and, 64
- perfect, 367
- prime, 369
- square, 366
- triangular, 365–366
O
- oblique asymptotes, 163
- odd functions
- applying to graphs, 103–104
- overview, 102–104
- one-sided limits, rational functions, 171–173
- one-to-one functions
- defining, 104–105
- eliminating violators, 105–106
- overview, 104
- order of operations
- basic description of, 11–12
- on graphing calculators, 92
- origin, symmetry with respect to the, 80
P
- palindromic numbers, rational equations and, 64
- parabolas. See also quadratic functions
- axis of symmetry of, 207
- circles intersecting with
- finding common solutions, 258–259
- multiple intersections, 256–258
- combined with lines
- finding two solutions, 253
- no-answer answer, 254–255
- overview, 252
- point(s) where a line and parabola cross paths, 253–254
- settling for one solution, 253–254
- completing the square on the equation of a, 43
- converting equations to the standard form, 214–215
- defined, 115, 206–207
- general form of parabola equations, 210–211
- graphing, 118–119
- overview, 86
- sketching graphs of
- overview, 211
- real-world applications, 212–214
- steps to follow, 212
- parallel lines
- identifying, 85
- systems of linear equations and
- identifying, 237–238
- overview, 232–233
- recognizing, 235–236
- patterns
- factoring for polynomial roots and, 143–144
- sequential
- difference between terms, 311
- multiples and powers, 312–313
- overview, 309–310
- perfect numbers, 367
- perfect square trinomials, finding, 17–18
- perfect squares
- factoring two terms, 16
- finding the next, 356
- permutations
- of sets, 345–348
- tree diagrams for, 351
- perpendicular lines, identifying, 85
- piecewise functions
- applying, 108–110
- example of, 107
- overview, 106–107
- polynomial equations, 46–49
- polynomial roots
- factoring for roots, 143–145
- negative roots, 149–150
- overview, 143
- positive roots, 149
- Rule of Signs, 149–150
- synthesizing root findings, 150–155
- synthetic division to test for roots, 150–153
- polynomials, 133–155
- comparing graphs that have differing sign behaviors, 142
- with complex solutions, 282–285
- defined, , 133
- determining positive and negative intervals, 139–142
- intercepts of
- counting, 135–138
- interpreting relative value and absolute value, 135
- overview, 134
- solving for, 138–139
- intersecting, 261
- mixing lines and, 260–261
- Rational Root Theorem. See Rational Root Theorem
- Remainder Theorem, 154–155
- roots of
- factoring for roots, 143–145
- negative roots, 149–150
- overview, 143
- positive roots, 149
- Rule of Signs, 148–150
- synthesizing root findings, 150–155
- synthetic division to test for roots, 150–153
- standard form, 134
- turning points of
- counting, 135–138
- interpreting relative value and absolute value, 135
- positive integers
- sum of odd-numbered, 332
- sum of the first 10 squares of, 332
- positive roots of polynomials, 149
- powers
- of i, 274–275
- raising to
- exponents and, 14–15
- order of operations, 11–12
- solving equations with fractional exponents, 70–72
- sequential patterns, 312–313
- prime numbers, 369
-
Probability For Dummies (Rumsey), 344
- proportions
- definition and features of,60
- solving rational equations with, 60–61
- Pythagoras, 262
Q
- quadratic equations
- absolute value inequalities, 53–54
- completing the square, 43–46
- defined,
- doubling up on a trinomial solution, 39–40
- factoring binomials, 37–39
- factoring by grouping, 40
- factoring difference of squares, 38–39
- factoring out the greatest common factor (GCF), 38
- finding rational solutions, 42
- finding two solutions in trinomials, 39
- graphs of, 86
- increasing the number of factors, 53
- overview, 35–36
- polynomial equations, 46–49
- quadratic formula, 41–43
- rational inequalities, 52
- solving exponential functions as, 185
- solving quadratic inequalities, 49–53
- solving with complex solutions, 280–282
- square root rule, 36–37
- straightening out irrational solutions, 42–43
- quadratic formula, 41–43
- quadratic functions, 115–131. See also parabolas
- applying to the real world
- launching a water balloon, 130–131
- selling candles, 127–128
- shooting basketballs, 128–130
- axis of symmetry of, 124–125
- b and c coefficients, 117–118
- intercepts in, 118–119
- lead coefficient of, 116, 117
- standard form of
- overview, 116
- starting with a in, 116–117
- vertices of. See vertex
- x-intercepts of, 120–123
- y-intercept of, 119–120
- quadratic inequalities
- keeping inequalities strictly quadratic, 50–51
- solving, 49–53
- quadratic trinomials
- defined, 16
- factoring, 16–17
- solving exponential functions, 185
- quadratic-like trinomials
- defined, 48
- overview, 48–49
- solving, 67–68
- quartics, 87
- quick-and-slick rule for matrices, 303–304
R
- radical curves, 88–89
- radicals
- calming, in rational equations, 63–64
- in graphing calculator, 91–92
- as grouping symbol, 12
- replacing with fractional exponents, 13
- ridding rational equations of, 61–64
- to show roots, 13
- simplifying, 279–280
- raising to powers
- exponents and, 14–15
- order of operations, 12
- solving equations with fractional exponents, 70–72
- range, of functions, 100–102
- rational curves, 88–89
- rational equations, 56–61
- calming two radicals in, 63–64
- factoring out negatives to solve, 66–68
- least common denominator (LCD) and solving, 56–59
- with negative exponents, 65–68
- overview, 56–57
- ridding of radicals, 61–64
- solving by working with fractional exponents, 70–72
- solving with cross-products, 61
- solving with proportions, 60–61
- squaring both sides of, 62–63
- systematically solving, 56–59
- rational functions. See also fractions
- asymptotes of. See asymptotes
- definition and features of, 157
- domains of, 158
- graphs of, 173–175
- intercepts of, 159
- intersections of lines and, 265–266
- as inverses, 267–268
- limits of
- at infinity, 170–173
- notation for, 167
- overview, 167–168
- with removable discontinuities
- determining an existent limit without tables, 169–170
- determining which functions have limits, 170
- evaluating limits at discontinuities, 168–169
- evaluating the removal restrictions, 165
- overview, 164–166
- showing on a graph, 165–166
- rounding up, 265–268
- sketching rational graphs from clues, 173–175
- rational inequalities, 52
- rational numbers, defined, 146
- Rational Root Theorem
- changing from roots to factors, 147–148
- constant terms
- polynomials with, 146–147
- polynomials without, 147
- list of numbers that may be roots of a particular polynomial, 146
- rational solutions, quadratic equations, 42–43
- real numbers, defined, 148, 273
- real roots, of polynomials, 148
- rectangles, graphing hyperbolas and, 225
- recursively defining functions, sequences and, 317–318
- relative maximum or minimum values, of polynomials, 135
- Remainder Theorem, 154–155
- Remember icon,
- removable discontinuities
- determining an existent limit without tables, 169–170
- determining which functions have limits, 170
- evaluating limits at discontinuities, 168–169
- evaluating the removal restrictions, 165
- overview, 164–166
- showing on a graph, 165–166
- roots
- exponents and, 14
- finding, order of operations, 11
- fractional exponents and, 14
- of polynomials
- factoring for roots, 143–145
- negative roots, 149–150
- overview, 143
- positive roots, 149
- Rule of Signs, 148–150
- synthesizing root findings, 150–155
- synthetic division to test for roots, 150–153
- roster notation, for set elements, 330
- row operations, on matrices, 297–298
- rows, of matrices, 289
- Rule of Signs, 148
S
- scalars, multiplying matrices by, 291
- second difference, in sequential patterns, 311–312
- sequences
- sequential patterns
- difference between terms, 311
- multiples and powers, 312–313
- overview, 309–310
- series, 318–322
- set builder notation, 330
- set notation
- overview, 329–330
- set builder notation, 330
- subsets, 331–333
- universal and empty (null) sets, 331
- sets
- complements of, 335
- counting, 344
- counting the elements in, 335–336
- empty (null), 331
- factorials and, 342–344
- intersections of two, 334–335
- listing elements with a roster, 330
- multiplication principle applied to, 344–345
- operating on, 333
- permutations of, 345–348
- subsets, 331–333
- union of two, 333–334
- universal, 331
- Venn diagrams
- adding sets to, 339–341
- applying, 337–338
- overview, 336
- with set operations, 338–339
- sign-line method
- polynomials, 140–141
- solving quadratic equations, 50, 52, 53
- simplex method, 296
- simplify,
- sketching. See graphs
- slant asymptotes, 163
- slope of a line
- counting off, 83
- finding, 81–82
- formulating the value of, 81–82
- identifying characteristics of, 81
- slope-intercept form of equation for a line, 83–84
- changing from standard form to, 84
- solve,
- square matrices, 289
- identifying an inverse for any size, 301–303
- square numbers, 366
- square root rule, quadratic equations, 36–37
- square roots, 279
- squares
- completing the
- conic sections, 43–46
- twice over, 45–46
- difference of two, factoring quadratic equations, 38
- difference of two perfect, factoring two terms, 16
- perfect
- factoring two terms, 16
- finding the next, 356
- squaring, rational equations, 62–63
- squaring numbers that end in 5, 355–356
- standard form
- of equation for lines
- changing from slope-intercept form to, 84
- overview, 82–83
- of polynomials, 134
- systems of linear equations, 230
- subsets
- counting the number of, 332–333
- notation, 332
- overview, 331
- substitution method
- line/parabola systems of equations, 253
- for solving systems of linear equations, 235–236
- subtract button, in graphing calculators, 92
- subtracting (subtraction)
- ancient symbols, 29
- complex numbers, 277
- matrices, 290–291
- order of operations, 11–12
- sum or difference of cubes, 47–48
- summation notation, 318–319
- summing sequences
- arithmetically, 319–320
- geometrically, 320–322
- adding all the terms to infinity, 321–322
- adding the first n terms, 320–321
- real-world applications for sums of sequences, 323–326
- bouncing a ball, 324–326
- negotiating your allowance, 323–324
- special formulas, 326–327
- stacking blocks, 323
- special formulas, 326–327
- symmetry
- axis of, of a quadratic function, 124–125
- of graphs, 76, 78–80
- with respect to the origin, 80
- with respect to the x-axis, 78–79
- with respect to the y-axis, 78
- synthetic division
- by binomials, 153–154
- Remainder Theorem, 154–155
- to test for roots, 150–153
- system of equations, defined, 229
- systems of linear equations
- Cramer’s Rule for solving, 238–241
- decomposing fractions, 248–250
- defined, 229
- elimination method for solving, 233–236
- graphing, 230–233
- matrices for finding solutions for, 305–306
- with more than three linear equations, 244–247
- overview, 229
- parallel lines, 232–233
- pinpointing the intersection, 231–232
- real-world applications, 247–248
- recognizing solutions indicating parallel or coexisting lines, 235–236
- standard form for, 230
- substitution method for solving, 235–236
- with three linear equations, 241–247
- two equations representing the same line, 232
T
- Tartaglia, 29
- Technical Stuff icon,
- term,
- third difference, in sequential patterns, 312
- three linear equations, systems of, 241–247
- Tip icon,
- tree diagrams
- for combinations, 352
- for permutations, 351
- triangular numbers, 365–366
- trinomials
- definition of,
- factoring, 17, 39–40
- finding two solutions in, 39
- perfect square, finding, 17–18
- quadratic
- defined, 16
- factoring, 16–17
- solving exponential functions, 185
- quadratic-like
- defined, 48
- overview, 48–49
- solving, 67–68
- turning points, of polynomials
- counting, 135–138
- interpreting relative value and absolute value, 135
- two-digit numbers, multiplying, 362–363
U
- unfactorable equations, polynomials, 144–145
- unFOILing, 17, 18, 184
- unit circle, 217
- universal sets, 331
- utility companies, piecewise functions, 108–110
V
- variables
- defined,
- substitution method and, 236–237
- Venn diagrams
- adding sets to, 339–341
- applying, 337–338
- overview, 336
- with set operations, 338–339
- vertex (vertices)
- defined, 118, 207
- finding, in quadratic functions
- lining up along the axis of symmetry, 124–125
- overview, 123–124
- sketching a graph, 125–127
- at the origin
- opening to the right or left, 207–209
- opening upward or downward, 209–210
- overview, 207
- vertical asymptotes, 160
- vertical line test, 105
- vertical lines, slope as non-existent, 81
- vulgar fractions, 166
W
- Warning icon,
- water balloons, launching, quadratic function example, 130–131
- words
- expressing ranges in, 100
- expressing the domain of functions with, 99
X
-
x-axis
- defined, 75
- symmetry with respect to, 78–79
-
x-intercepts
- finding, 77
- on graphing calculator window, 93
- overview, 77
- of polynomials
- overview, 134–135
- solving for, 138–139
- unfactorable equations, 144–145
- of quadratic functions, 120–123
- of rational functions, 159
Y
-
y-axis
- defined, 75
- symmetry with respect to, 78
-
y-intercepts
- finding, 77
- overview, 77
- of quadratic functions, 119–120
- of rational functions, 159
- solving for, 138–139
Z
- zero(s)
- complex, 283–285
- finding y-intercepts of rational functions, 159
- multiplication property of, 12
- unfactorable equations, 144–145
- zero factorial, 309
- zero matrices, 289
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