26_169841 bindex.qxd 2/29/08 2:58 PM Page 365

Index

law, for limits, 345 Symbols and Numerics additive identity, 14 > (greater than), 21 additive inverse property, 14 ≥ (greater than or equal to), 21 advanced identities, described, 195 < (less than) symbol, 21 Algebra, Fundamental Theorem of, 82–83, 95 ≤ (less than or equal to), 21 algebra, labeling and expressing ellipses, θ (theta) 269–270 graphing calculator, entering as a variable algebra I and II, compared to pre-calculus, 10 in, 251–252 Algebra I and II Workbook For Dummies as the input of a trig function, 147 (Sterling), 240, 356 length of radius, 243 Algebra II For Dummies (Sterling), 240 as plotting point, 244 algebraic expressions, simplifying, 178 theta prime, as the name given to the algebraic method, to find limit, 342–345 reference angle, 141 amplitude in trig functions, 125 changing, 160–163, 168–169 2 x 2 matrix, inverse of, 310 defined, 160 2(π) radian, 150 period, compared to, 165 3 x 3 system of equations, and Cramer’s positive values of, 161 rule, 314 sine functions, 161–163 30-60 triangle, 132–133, 135 transformations, combining, 167–170 45er triangle, 131–132, 138 analytical method for finding a limit, 360, convenience of, 120 341–342 AND statement, 25 angle(s) • A • arc as, 145 central, 145–146 AAS (Angle Angle Side) triangle, solving, common, 128–129 218, 220–221 correct placement, 134–136 absolute value, 13, 22–24, 36, 48 cosine of, 122–123 absolute-value functions, 36 in degrees, 197 acceleration, moving objects with, 338 doubling the trig value of, 205–207 acute triangle, 222 COPYRIGHTEDdrawing, MATERIAL 120, 130–131 addition formulas for finding, 200–202, 229 of, 13 knowing two measures, 219–221 commutative property of, 13 negative, 130–131, 246 of complex numbers, 241 reference angles, 141–144 of degrees, 197 sides, using to find, 228–230 of exponents, to multiply, 362 sine of, 121–122, 139, 197, 227 of fractions, 11, 179 as sums or differences, 196–197, 200–202 functions, 58–59 tangent of, 123–124, 203 of matrices, 297–298 on unit circle, 129, 140–144 of radians, 197 26_169841 bindex.qxd 2/29/08 2:58 PM Page 366

366 Pre-Calculus For Dummies

Angle Angle Side (AAS) triangle, solving, base, 103, 106 218, 220–221 base numbers, fractions as, 99 Angle Side Angle (ASA) triangle, solving, binomial coefficients, 329–336 218–220 binomial equation, factoring, 69 angle theta, deriving from the tangent, 248 binomial expansion, 329, 332–336 Annual Percentage Rate (APR), 321–322 binomial theorem answer matrix, 301 breaking down into parts, 330 answers, checking, 284, 355 defined, 329–330 antilogarithm, 106 expanding by using, 332–336 APR (Annual Percentage Rate), 321–322 using, 329 arc length, calculating, 145–146 binomials, 67, 74, 329 arccos (inverse cosine), 125 biology, exponential concepts in, 98 Archimedes Spiral, graphing, 250 British Method, 30, 69, 71–73, 241 arcs, making and measuring, 145–146 arcsin (inverse sine), 125, 223 arctan (inverse tangent), 125–126 • C • area, calculating triangle, 232–233 calculator, and , 231. Argand coordinate plane, 238 See also graphing calculator arithmetic sequences, 319–321, 325–326 calculus Arone, Wendy (author), 228 compared to pre-calculus, 338–339 ASA (Angle Side Angle) triangle, solving, described, 337 218–220 habits helping, 351–357 associative property of habits to break before, 359–363 addition/, 13 cancellation, 362–363 asymptote Cardiod, graphing, 250 changing for logs, 109 Cartesian coordinates, 15, 247, 278–282 of cosecant graph, 158 center of cotangent graphs, 152 of ellipse, 269–270 of a graph, 49 of hyperbola, 274–276 graphing, 53–54 center-radius form, 257 horizontal, 51, 53–56, 98, 101–102 central angle, 145–146 of hyperbola, 275–277 change of base formula, 105 oblique, 51–52 checking, answers, 284, 355 of secant graph, 156 circle(s) of tangent graphs, 152 circumference formula, 145 vertical, 49–50, 53–57, 153, 155 described, 254–255, 257 attack, planning, 352–353 eccentricity value for, 281 augmented form, of a matrix, 305 graphing, 250, 257–259 axes, of ellipses, 271 unique characteristics of, 256 axis of symmetry, 260, 263–266, 274 closed interval, 25 co-function identities, 185–187, 202, 204 co-terminal angles, 120 • B • co-vertices of ellipse, 269–272 back-solving, executing, 304–305 coefficient matrix, 301, 311–313 back-substitution process, described, 293 coefficients, 39 bad math habits, breaking, 359–363 coefficients of binomials, 331–332 26_169841 bindex.qxd 2/29/08 2:58 PM Page 367

Index 367

combination formula, using, 334 intersection of, 285 combinations, calculator button for, 331 no more than four solutions for two, 290 combined functions, 58–62 parabolas as, 260 common denominator, for angles in parametric form, 278–280 radians, 198–199 on polar plane, 280–282 common difference, 319–321 conics, expressed outside of Cartesian common logarithms, 103, 111 coordinates, 278–282 common ratio, finding, 326 conjugate, multiplying by, 30, 191–192, commutative property 204, 344 of addition, 13 conjugate axis, of a hyperbola, 274, 276 of multiplication, 13, 212 consecutive terms, 319–320, 322–323 comparisons, of expressions, 16 consistent and independent system, 285 completing the square, described, 78–80 constants complex conjugates, 82, 241 cannot be added to variables, 360 complex coordinate plane, 238, 242 in conic equations, 256 complex numbers described, 68, 294 adding, 241 multiplying equations by, 288 defined, 12, 238 scalar, 241, 297–298 dividing, 241–242 continuity, in functions, 346–348 expansion with, 335–336 continuous graph, 346 expressing answers as, 95 conversion factor, for radians, 247–248 form of, 83 coordinate pair (x, y), 15, 246 graphing, 238, 242–243 coordinate plane manipulating arithmetically, 240 described, 33 multiplying, 240–241 graphing on, 16 operations involving, 240–242 points existing on, 126 with real and imaginary parts, 240 shifting the waves on, 166–167 subtracting, 241 unit circle lying on, 126 system of, 83, 239–242 coordinates, of points on unit circle, 135 types of, 240 cosecant complex roots, 83, 240 of angle, 139 complicated polar coordinates, 246 defined, 124 composed function, 60–62 evaluating on the unit circle, 138 composition of functions, breaking down, 60 graphing, 156, 158–159, 173–176 computer program, finding matrix cosine inverses, 310 of angle, 122–123, 139 concepts, understanding, 356 changing functions in equation to, 180 cones, cutting with a plane, 254–255 defined, 122 congruence postulates, for triangles, 228 double angle of, 207–208 congruent angles, same values for different half-angle formula, 210 trig functions, 134 point-in-the-plane definition, 137 conic sections reciprocal of, 124 characteristics unique to each type of, 256 sine, changing to, 208 described, 253 sum and difference formulas, 200–202 equations of, 256–257 cosine ⋅ cosine product-to-sum graphing, 278–282 formula, 212 identifying, 254–257 cosine functions, 148–151, 170, 174 26_169841 bindex.qxd 2/29/08 2:58 PM Page 368

368 Pre-Calculus For Dummies

cosine graphs difference cosine parent graph, characteristics cosine, formula for, 200 of, 151 of cubes, 74, 76 described, 150–151 sine, formula for, 197 springs, compared to, 160–170 of squares, 74–75 tangent and cotangent, compared to, 152 tangent, formula for, 203–205 cotangent, 125, 138–139, 152 transporting from sums or differences to, cotangent functions, graphing, 152, 154–155 213–214 cotangent graphs, transforming, 170–173 using formulas, 198–199 counting numbers, defined, 11 difference-to-product identities, 213 Cramer, Gabriel (mathematician), 311 dimensions, of a matrix, 297 Cramer’s rule, 311–314 directrix for parabolas, 260, 263–266 critical point, 35, 92–93 Dirty Monkeys Smell Bad , 86 cross product, 299 discontinuity, dealing with, 347–348 cube root, 27, 29, 37–38 discontinuous function, 48–49, 337 cubic functions, 37, 42–43 discriminant, 82, 240 curve, area under, 338 disjoint sets, 25 distance, formula for, 17, 127 distributing, improper, 363 • D • , 14, 197 decay constant, 113 dividend, defined, 85–86 decomposing, partial fractions, 293–295 division degree mode, setting calculator to, 123 of functions, 59 degrees of polynomials, 51 adding and subtracting, 197 rational roots, 86–88 calculating in, 197–198 synthetic, 88–90, 96 of numerator and denominator, 52–53 division algorithm, 86 of polynomials, 51, 68, 81 division bar, as grouping symbol, 360 working with angles in, 196 division law, for limits, 345 denominators divisor, defined, 85–86 conjugate of, 30 DNE value, 343 containing a binomial with a radical, 30 domain equal degree with numerators, 55–56 combined function, 61 with greater degree, 53–55 cosecant graph, 176 multiplying by conjugates of, 192 cosine function, 150, 170 rationalizing, 28–31, 204 cotangent function, 173 splitting up, 360 exponential function, 99 in trig proofs, 189–192 of relation, 15 dependent system, 285 secant function, 175 depreciation, 321–322, 324, 326 sine function, 148 depressed polynomial, 84, 90, 96 tangent function, 172 derivative, finding, 346 x value as, 147 Descartes, Rene (mathematician), 15 double-angle formula, 205–209 Descartes’ Rule of Signs theorem, double root, 89, 95 81, 83–84, 95 drawing determinants of coefficient matrices, while problem solving, 352 311–314 to represent right triangles, 201 for trig problems, 130 26_169841 bindex.qxd 2/29/08 2:58 PM Page 369

Index 369

even functions, 34, 151, 183 • E • even multiplicity, at a root, 95 eccentricity value, conic section, 280–281 even-odd identities, 183–185 element, of a cone, 253 exact/approximate mode, for graphing elementary row operations, 303, 306–307 calculator, 18 elements, in matrices, 284 excluded values, for a domain, 61 elimination expansion problems, normal, 332–333 Gaussian, 306–309 exponential decay, 98–99 method of, to solve linear systems, exponential equations 285, 287–288, 292 changing logs to, 106 ellipse real-world applications, 113–115 in astronomical terms, 253 solving, 110–112 axes and foci of, 271 taking the log of both sides, 111 co-vertices of, 269–272 turning logs into, 112 described, 255, 268 variable on one side, solving, 110 expressing with algebra, 269–270 variables on both sides, solving, 110 foci on, 268–271, 273 exponential functions graphing polar functions of, 281–282 changing to logs, 103 in non-standard form, 272–273 as continuous at every point, 346 parts of, 269–273 defined, 98 unique characteristics of, 256 domain of, 99 vertical, 269, 271 graphing, 99–102, 108 vertices and co-vertices, 271–272 investigating the inverse of, 102–109 empty set, as the solution, 113 rules applying to, 98–100 end behavior, of a graph, 92 transforming, 100–102 equalities exponential growth, 98–99, 113–115 co-function identities, 186–187 exponents even-odd identities to prove, 184–185 adding to multiply, 362 periodicity identities to prove, 188 defining and relating, 26–27 properties of, 191 described, 26, 330 Pythagorean identities to prove, 183 in exponential functions, 98 reciprocal identities to prove, 180–181 logarithms as, 103 equations rewriting radicals as, 27–28 of the asymptotes, hyperbola, 277 solving equations with, 109–113 of conic sections, 254, 256–257 expressions, simplifying, 184 exponents and logs, solving, 109–113 for the four conic sections, 256–257 • F • more than one log, solving, 112 needed for each system variable, 283 factor theorem, 91 parameter, solving, 279 factored equation, finding the roots of, 78 putting in standard form, 272 factorial, 331 rewriting to find identities, 206 factorial button, on calculator, 331 simplifying with periodicity identities, factoring 187–188 to cancel out terms, 59 solving, overview, 22–24 defined, 69 systems with more than two, 291–293 a difference of squares, 74 types applying to ellipses, 270 to find a limit algebraically, 342–343 26_169841 bindex.qxd 2/29/08 2:58 PM Page 370

370 Pre-Calculus For Dummies

factoring (continued) dividing, 59 four or more terms, 77 with domains not all real numbers, 61 polynomial expressions, 69–77 even, 34 in trig proofs, 189 finding the limits of, 339–345 using to get a solution, 143 graphs of, 163–167 factors inverse, 63–65 checking for greatest common, 143 multiplying, 59 defined, 69 odd, 34 multiple raised to a power inside operating on, 58–62 parentheses, 99 piece-wise, 47–49 using solutions to find, 91 quadratic, 34–35 families, on the unit circle, 140 square root, 35–36 Fibonacci Sequence, 318–319 studying changes to, 338 finite sum, 326 subtracting, 58–59, 361 focal distance, parabola, 264–265 transforming point by point, 46–47 focus (foci) in trigonometry, 121 of ellipse, 268–271, 273 Fundamental Theorem of Algebra, 82–83, 95 of hyperbola, 275 of parabola, 260, 263–266 FOIL Method, 30, 69, 71–73, 241 • G • formulas, 316, 320, 354 Gauss coordinate plane, 238, 242 45er triangle, 131–132, 138 Gaussian elimination, 306–309 fractals, 239 GCF (greatest common factor), 69–73, 291 fraction(s) general expression, for a sequence, 317 adding, 11, 179 geometric sequences, 321–329 breaking up, 193, 294 geometric solids, volume for, 338 creating working with reciprocal geometric summation, 328 identities, 190–191 geometry, compared to pre-calculus, 10 dividing by other fractions, 179 Geometry For Dummies (Arone), 228 multiplying a period, 165 Geometry Workbook For Dummies in proofs, 189 (Ryan), 356 reducing, 199 graphing calculator starting off with, 191 described, 18–20, 93 subtracting, 11, 179, 185 matrix inverses on, 310 in trig proofs, 189–190 order of operations, 231 fractional exponents, 27 parentheses for numerator and function(s) denominator, 232 absolute-value, 36 polar function plotting, 251 adding, 58–59 setting to parametric mode, 280 breaking down, 60 vertex on, 267 calculating limits to, 339 graphs and graphing continuity in, 346–348 absolute-value functions, 36 cube root, 37–38 circles, 257–259 cubic, 37 combined functions, 58 defined, 33 complex numbers, 242–243 determining if continuous, 347 cosine functions, 148–151 26_169841 bindex.qxd 2/29/08 2:58 PM Page 371

Index 371

cube root functions, 37–38 moving, 101–102 cubic functions, 37 of parent exponential functions, 98 denominator with greater degree, 53–54 horizontal displacement, of a circle, 257 described, 14–18 horizontal ellipse, 269, 271 end behavior of, 92 horizontal equation, for an ellipse, 270 equalities and inequalities, 16, 295 horizontal hyperbola exponential functions, 99–102, 108 asymptote, 275, 277 gathering information from, 16–18 described, 273–275 horizontal shifts, 41 equation for, 274 hyperbolas, 275–276 slopes of asymptotes, 275 inverse functions, 63–64, 106–107 transverse axis and conjugate axis, 274 limit, method for finding, 340–341 vertices, 275 logarithms, 106–109 horizontal parabola parent graph. See parent graphs described, 260 plotting, 94–95 determining, 263 polar coordinates with negative values, equation for, 261 246–247 forms of, 262 polynomials, 68, 91–96. See also graphing, 266 polynomial functions, graphing parts of, 265–266 quadratic functions, 34–35 transformations of, 262 rational functions, 52–57 horizontal reflections, 43 secant, 156–157, 173–176 horizontal shifts sine functions, 148–151 combining transformations, 167–170 to solve system of inequalities, 295 of cotangent functions, 172–173 square root functions, 35–36 described, 41, 166 transformed logs, 107–109 parabolas, 262–263 trig functions, 147 representing, 259 vertical shifts, 42–43 revealing, 168–169 greater than (>), 21 vertical shifts, compared to, 42 greater than or equal to (≥), 21 horizontal transformation, 38, 40, greatest common factor (GCF), 69–73, 291 101–102, 160 grouping, to factor four or more terms, 77 hyperbola growth constant, 113 asymptotes, 277 guess-and-check method, of factoring, 71 defined, 273 described, 255–256, 273 eccentricity value for, 281 • H • graphing, 275–276 habits to address, 351–357, 359–363 standard form for, 274 half-angle formulas, 210–211 types of, 273–275 half-angle identities, 210 hypotenuse Heron’s Formula, 233 in a 30-60 triangle, 133 horizontal asymptote of a 45er triangle, 132 described, 55 of every triangle in a unit circle, 134 drawing, 53–54 identifying, 121 finding, 51 of a right triangle, 128 graphing, 56 26_169841 bindex.qxd 2/29/08 2:58 PM Page 372

372 Pre-Calculus For Dummies

Internet, practice math problems on, 356 • I • interval, finding the solutions on, 206–207 identities interval notation, 10, 24–26 advanced, 195 intervals, testing values in, 54–55 basic, 177–188 inverse cosine (arccos), 125 co-function, 185–187 inverse functions even-odd, 183–185 described, 63–65, 125–126 looking for, 193 double-angle solving, 209 periodicity, 187–188 finding, 106, 108 primer on, 178 graphing, 63–64, 106–107 proving, 204–205 notation for, 63 Pythagorean, 181–183 solving for, 65 reciprocal, 179–181 swapping domain and range, 108–109 rules for working with, 179 trigonometric, 121, 209 identity matrix, 303–304, 310 verifying, 65 imaginary axis, 242 writing, 309 imaginary numbers inverse logarithms, 106 as complex numbers, 240 inverse matrices, 309–311 defined, 12, 238 inverse sine (arcsin), 125, 223 importance to the real world, 239 inverse tangent (arctan), 125–126 multiplying complex numbers by, 241 inverting, functions, 64–65 real numbers, compared to, 238–239 irrational numbers, defined, 12 some roots as, 95–96 irrational roots, 84 imaginary part, in the denominator, 241 imaginary roots, accounting for, 82–84 • K • imaginary unit, defined, 12 inconsistent system, 285 key words, underlining in questions, 352 index, described, 31 kth partial sum, 324–325 inequalities breaking in two, 24 defined, 21 • L • expression solutions with interval Law of Cosines, 217, 228–232 notation, 24–26 Law of Sines, 217–227 graphing, 16 Law of Tangents, 217 solving, 21–26 leading coefficient test, 92 systems of, 284, 295–296 leading coefficients, 51, 68, 92, 302 infinite geometric sum, 326 least common denominator (LCD), infinite solutions and absolute value, 23 191, 344–345 infinite sum, finding, 326–329 least common multiple (LCM), 292 infinity, described, 12 left limit, of a function, 339 initial side, of an angle, 120 Lemniscate, graphing, 251 instantaneous rate of change, 338 length, measuring an arc as, 145 integers, defined, 11 less than or equal to (≤), 21 integral, derivation of, 324 less than (<) symbol, 21 integration, using, 338 Limacon, graphing, 251 intercepts, defined, 52 limit laws, 345–346 26_169841 bindex.qxd 2/29/08 2:58 PM Page 373

Index 373

limit process, 53 matrix (matrices) limits adding, 297–298 algebraic, finding, 342–345 applying basic operations to, 297–298 analytic, finding, 341–342 basics, 296–301 described, 337 described, 296 for functions, finding, 339–345 inverse of, 309–310 graphic, finding, 340–341 multiplying by each other, 298–301 understanding and communicating about, multiplying by its inverse, 309–311 339–340 operating on, 297 line segment, from a point perpendicular reduced row echelon form of, 302–305 to a line, 260 simplifying, 301–305 linear equations, 17, 289, 291 solving difficult systems, 305–314 linear function, graphing, 48 subtracting, 297–298 linear inequalities, systems of, 295 using to solve systems, 284 linear systems, 285–288, 291–292 matrix form, writing any system as, 301–302 lines, slope of, 17–18 matrix multiplication, visual representation logarithm equations, 112–113 of, 299 logarithmic functions, 106–107, 346 maximum and minimum points, calculating logarithms (logs) for a cosine graph, 151 base of, changing, 105, 112 maximum area, for rectangles, 260 better handle on, 103 maximum function, graph of, 267–268 calculating from numbers, 106 maximum of vertical parabola, 266–268 combining to one base, 112 maximum point, calculating for a sine defined, 102–103 function graph, 149 exponential functions, changing to, 103 measurement problems, can’t be graphing, 106–109 negative, 355 investigating the inverse of exponential methods functions, 102–109 for solving linear systems, 285–288 mistakes in working with, 104–105 for solving system equations, 283 no negative numbers allowed inside, 113 midpoint, finding, 17 properties and identities of, 104–105 minimum, of a vertical parabola, 266–268 solving equations with, 109–113 minimum point, calculating for a sine types of, 103 function graph, 149 long division, 51, 86–88 minor axis of ellipse, 269–271 long leg, in a 30-60 triangle, 132–133 minus signs, losing track of, 361 lower limit, of a sum, 325 mnemonic devices lowest common denominator (LCD), Dirty Monkeys Smell Bad, 86 191, 344–345 PEMDAS, 13 SOHCAHTOA, 121 modes, of graphing calculators, 19 • M • monomials major axis of ellipse, 269–271 binomial theorem written as the sum of martini, of parabolas, 264 two, 330 mathematical statements, in visual form, defined, 67 14–18 multiplying with exponents, 99, 362 raising to a power pre-expansion, 333–335 26_169841 bindex.qxd 2/29/08 2:58 PM Page 374

374 Pre-Calculus For Dummies

multiplication associative property of, 13 • O • commutative property of, 13, 212 objects, moving with acceleration, 338 of functions, 59 oblique asymptotes, 51–52, 57 of matrices by each other, 298–301 oblique triangle, 217, 232–233 multiplication law, for limits, 345 obtuse triangle, 222 multiplicative identity, 14 odd functions, 34, 150, 183 multiplicative inverse property, 14 odd multiplicity, at a root, 95 multiplicative property, of zero, 14 open interval, 25 multiplicity of the solution, 80, 89 operating, on functions, 58–62 multiplicity of two, root with, 89 operations applying to matrices, 297–298 • N • combining with matrices, 298 order of, 359 natural logarithms, 103, 113 performing with complex numbers, natural numbers, defined, 11 240–242 negative angles, 130–131, 246 OR statements, graphing, 25–26 negative base, exponential function, 99 order, of a matrix, 297 negative coefficients, in parabolas, 261 order of operations negative constant, multiplying period, 165 defined, 12–13 negative exponents, 99 for exponential functions, 99 negative infinity, in interval notation, 26 following when using the Law of negative measure, of an angle, 120 Cosines, 231 negative numbers keeping in mind, 290 multiplying only the input, 44 for matrices, 297 multiplying or dividing an equality by, 22 origin, 35, 257–261 multiplying whole functions, 43 outputs, 49–52, 54 square roots of, 12 oval. See ellipse trig functions multiplied by, 161 negative radius, 246–247 negative real roots, determining, 83 • P • negative roots, 81–83 parabolas negative slopes, 18 characteristics of, 261 negative values, graphing, 246–247 described, 254–255, 259–260 negative variable, in trig function, 184 eccentricity value for, 281 non-consecutive terms, geometric as graph of any quadratic function, 35 sequence, 323 graphing, 264–265 non-real roots, polynomial functions as, 95 parametric curve for, 279–280 nonlinear equations, 285, 288–291 unique characteristics of, 256 nonlinear systems, 288–291, 295 parameter, in parametric form, 278 nonremovable discontinuity, 347–348 parametric equations, described, 278 numbers parametric form, graphing conic sections calculating from logs, 106 in, 278–280 fundamental operations on, 12–13 parametric mode, for a graphing properties of, 13–14 calculator, 280 types of, 11–12 parent functions, of logs, 106–107 numerator, 29, 55–57, 343–344 26_169841 bindex.qxd 2/29/08 2:58 PM Page 375

Index 375

parent graphs mapping with x-y coordinates, 248 of cosine functions, 150 multiple names for every point, 249 described, 34, 147 negative values, graphing, 246–247 for exponential functions, 100–102 plotting, 243–246 sine and cosine of, 148–151 strange and remarkable equations, of sine functions, 148–150 graphing, 250–251 transforming, 38–47, 100–102 and x-y coordinates, 248–249 of trig function, altering, 159–160 polar equations, picturing, 250–252 parentheses, in interval notation, 25 polar form, 10, 278 partial fractions, decomposing, 293–295 pole, concentric circles around, 243 partial sums, 324, 327–328 polynomial expressions, factoring, 69–77 Pascal, Blaise (mathematician), 331 polynomial functions, graphing Pascal’s triangle, 331 factoring, 69–77 paths, of objects moving in space, 253 function of degrees and roots, 68–69 pattern, in a sequence, 316 goals, 67 PEMDAS mnemonic, 13, 359 quadratic equation, 78–80 perfect square root factors, finding, 31 roots of factored equation, 78 perfect squares, factoring, 74–75 solutions to find factors, 91 period unfactorable polynomials with degrees altering, 163–165, 168–169, 171–172 higher than two, 80–90 for cotangent function, 172 polynomials described, 160 breaking into sets of two, 77 for function graphs, 163–165 counting the number of roots, 81 positive values of compared to fraction defined, 67 values, 165 degree of, 68 shifts of, factoring out of expressions, 168 dividing two, 86–88 of sine graph, 150 factoring, 70–71 for tangent and cotangent, 171 factoring special types of, 73–76 for transformation combinations, 167–170 graphing, 68, 91–96 periodic graph, 150 long division of, 51 periodicity identities, 187–188, 204 testing roots by dividing, 85–86 pictures, drawing positive angles, for values of theta, 246 while problem solving, 352 positive constant, multiplying a period, 165 to represent right triangles, 201 positive measure, of an angle, 120 for trig problems, 130 positive real roots, determining, 83 picturing, an ellipse, 268 positive roots, 27, 81–82 piece-wise functions, 47–49 positive slopes, 18 plane, intersecting cone, 254–255 power law, for limits, 345 planning, problem solution, 352–353 power-reducing formulas, 214–215 plug-and-chug method, 16, 58, 279 power rule, 104–105, 111–112 point-in-the-plane definition, 126, 137 powers, 26 points, transforming a function from a set prime, polynomial as, 70 of random, 46–47 prime factors, breaking down terms into, 70 polar coordinate plane, 243–246, 280–282 principal root, 27 polar coordinates problems changing to and from, 247–250 drawing while problem solving, 352 described, 243 figuring out, 351–352 26_169841 bindex.qxd 2/29/08 2:58 PM Page 376

376 Pre-Calculus For Dummies

problems (continued) quadratic term, defined, 68 with no solution, 354–355 questions, asking teachers, 357 practicing, 356 quitting, knowing when to, 354–355 word problems, 122–123, 286, 352 quotient, defined, 85–86 product rule, 104–105 quotient rule, 104–105 product-to-sum formulas, 212 products, 211–214 proofs • R • described, 177 r, as a plotting point, 244 fractions in, 189 radian(s) identities, 178 adding, 197 for tangents, applying to, 204–205 angle measures in, 247 techniques for difficult trig, 189–194 calculating in, 198–199 working on both sides of, 193–194 described, 119–120 pure imaginary roots, 83 expressing solutions to trig equations Pythagorean identities, 181–183 in, 140 Pythagorean Theorem formula for, 146 described, 121 graphing in, 147 length of the hypotenuse, finding, 127 mode, for polar coordinates, 251 missing values, finding, 201 showing clear relationships on the unit for polar coordinates, 250 circle, 140–141 radius of a triangle, finding, 248 subtracting, 197 working with angles in, 196 radicals • Q • defining and relating, 26–27 quadrant I, 134–136 oversimplifying, 361 quadrants rewriting as exponents, 27–28 in the coordinate plane, 136 simplifying, 31 determining where solutions lie, 142 radicand, 61 rules, on the unit circle, 141 radius, 126, 257 of tangents, 153 range in the unit circle, 129 affected by amplitude, 161 quadratic equation, 50, 78–80, 291 of combined function, 61 quadratic expression, 68, 70 of composed function, 62 quadratic formula for cosine function, 150 described, 78 for cotangent function, 155 ending up with a negative number under defined, 15 a square root sign, 95 of function shifted vertically, 170 solving an equation, 289 for sine function, 148–149 solving the depressed polynomial, 96 for tangent function, 153 using, 79 of transformed cosecant graph, 176 quadratic functions of transformed secant function, 175 described, 34–35 y value as, 147 graphing, 48 rate of change, finding, 338 parabolas as, 260 ratio, defined, 120 transforming, 38 rational exponents, 27–28 26_169841 bindex.qxd 2/29/08 2:58 PM Page 377

Index 377

rational expressions, simplifying, 362 for secant, sines and cosines, 187, 207 rational functions secant to cosine, changing, 207 calculating outputs for, 49–52 simplifying expressions with, 180 cotangent graphs as, 152 using to prove equalities, 180–181 denominator with the greater degree, rectangles, maximum area for, 260 53–55 recursive sequences, described, 318–319 described, 49 reduced row echelon form, 302–305, domains not all real numbers, 61 308–309 graphing, 52–57 reference angle, 141–144 mathematical definition of, 49 reflection, 38, 43–44 numerator and denominator with equal reflexive property, of numbers, 13 degrees, 55–56 relation, on a coordinate plane, 15 numerator with the greater degree, 57 remainder, 85–86 subtracting, 361 remainder theorem, 90 tangent graphs as, 152 removable discontinuity, 347–348 types of, 53 repeating decimals, expressing as infinite undefined values, 339 sums, 329 rational numbers, 11, 172–173 right limit, of a function, 339 Rational Root Theorem, 84–85, 95 right triangles, ratios in, 120–121 rational roots, 84–88 root(s) rationalized fraction, simplifying, 204 counting a polynomial’s total number rationalizing of, 81 denominators, 28–31 defined, 26, 67 numerators, 343–344 of a factored equation, 78 ratios to find factors, 91 consecutive paired terms, 321–324 finding every, 84 in right triangles, 120–121 to frame a graph, 94 real axis, 242 to graph polynomial functions. See real numbers polynomial functions, graphing all roots as, 92–95 imaginary, 82–84 basic operations on, 12–13 imaginary, some numbers are, 95–96 compared to imaginary, 238–239 multiplicity of two, 89 as complex numbers, 240 real, all numbers are, 92 defined, 12 testing by dividing polynomials, 85–86 real roots, 81, 84–90 root sign, representing radicals, 27 reciprocal(s) Rose, graphing, 251 forgetting, 360–361 row echelon form, matrix in, 308 rules of, using to simplify, 193 rows, 307 of sine, 124 Ryan, Mark (author), 356 of trig functions, 124, 127, 179 of zero, as undefined, 158 reciprocal functions, 124–125, 173–174 • S • reciprocal identities SAS (Side Angle Side) triangle, solving, described, 179–181 230–233 fractions working with, 190–191 scalar constant, 241, 297–298 listing of, 179 26_169841 bindex.qxd 2/29/08 2:58 PM Page 378

378 Pre-Calculus For Dummies

secant sine graphs of an angle, finding, 139 after transformations, 162–163 defined, 124–125 cosine graphs, compared to, 150 evaluating on the unit circle, 138 cotangent, compared to, 152 graphing, 156–157, 173–176 described, 148–150 secant graphs, transforming, 173–176 springs, compared to, 160–170 secant parent graph, picturing, 156–157 tangent, compared to, 152 segment, finding the midpoint of, 17 transforming a cosecant graph, 175 semiperimeter variable, 233 sinusoid, 149 sequence(s) sinusoidal axis, 160–161 arithmetic, 319–321 slope, of a line, 16–18 defined, 316 slope-intercept form, 16, 277 described, 315 snail rule, 103–104 general formula for any, 316 SOHCAHTOA mnemonic, 121 geometric, 321–324 solids of revolution, volume of, 338 kth partial sum of, 324–325 solution(s), of a given equation, 67 recursive, 318–319 solution angle, finding, 141 series, 315, 324–329 solutions. See also root(s); specific sets, of numbers, 11–12 problems shifts, described, 166 absolute values, 22–24 short leg, in a 30-60 triangle, 133 arriving at one, 225–227 shortest leg, in a 30-60 triangle, 132 finding none, 227 shrink, vertical transformation as a, 39 preparing for two, 222–225 side, finding a missing in a triangle, 228 that don’t make sense, 355 Side Angle Side (SAS) triangle, solving, with triangles, 142, 217–218 230–233 using to find factors, 91 Side Side Angle (SSA) triangle, solving, solved variable, substituting, 286–287 218, 221–227 special angles, cutting in half, 210 Side Side Side (SSS) triangle, solving, square brackets, in interval notation, 25 228–230, 233 square matrices, with inverses, 309 sides, formulas for finding missing, 228 square root signs, of points on the unit circle, 136 on both sides of a trig function, 144 simple polar coordinates, 246 graph of, 35 simplifying, algebraic expressions, 178 of a negative number, 237 sin2x (double-angle formula), 205–209 rationalizing expressions with, 29 sine simplifying, 31 of angle, 121–122, 139, 197, 227 taking, 27 changing functions in an equation to, 180 in trig proofs, 189 cosines, changing to, 208 square root functions, 35–36, 61 defined, 121, 137 squaring, 208, 360 of a doubled angle, 205–207 SSA (Side Side Angle) triangle, solving, half-angle formula, 210 218, 221–227 sine ⋅ cosine product-to-sum formula, 212 SSS (Side Side Side) triangle, solving, sine ⋅ sine product-to-sum formula, 228–230, 233 212–213 standard form, 70, 274 sine functions, 148–151, 164, 175, 223 standard position, for an angle, 120 26_169841 bindex.qxd 2/29/08 2:58 PM Page 379

Index 379

Sterling, Mary Jane (author), 240, 356 stretch, vertical transformation as a, 39 • T • substitution tangent to solve systems, 284 of angle, 123–124, 139 solving linear systems, 285–287, 289 defined, 123, 152 solving nonlinear systems, 285 double-angle formula, 208–209 subtraction evaluating on the unit circle, 138 complex numbers, 241 expressing in terms of x and y on the unit degrees, 197 circle, 202 fractions, 11, 179, 185 half-angle formula, 210 functions, 58–59, 361 reciprocal of, 125 matrices, 297–298 returning a value of, 248 radians, 197 as the slope of the radius, 140 rational functions, 361 of a sum or a difference of angles, 202 subtraction law, for limits, 345 sum or difference formula for, 202–203 sum(s) tangent functions, 152–154, 172 arithmetic sequences, 325–327 tangent graphs, transforming, 170–173 cosine sum formulas, 200 tangent line, slope of, 338 of cubes, 74, 76 tangent parent graph, 154–155 expressing products as, 211–213 teachers, asking questions of, 357 geometric sequences, 326–329 term of the sequence, 316 tangent sum formulas, 202–205 terminal side of angle, 120, 128 transporting to products, 213–214 terms using sum formulas, 197–198 calculating a sequence’s, 316–317 sum and difference formulas combining the wrong, 360 applying, in proof, 199–200, 202, 204–205 forming expressions from, 317–318 cosine, 200, 202 identifying for a geometric sequence, finding missing trig ratios to use, 201–202 322–323 sine, 197 two to define a sequence, 320–321 sum-to-product identities, 213 test values, plotting outputs of, 54 summation notation, reviewing, 324–325 theta (θ) symbols, expressing inequalities, 21 conic sections in polar coordinate form, symmetric property, of numbers, 13 280–282 symmetrical functions, 34 graphing calculator, entering as a variable symmetry, of parabolas, 260 in, 251–252 synthetic division, 88–90, 96 as the input of a trig function, 147 system(s) length of radius, 243 described, 283–284 as plotting point, 244 of inequalities, 295–296 theta prime, as the name given to the with one nonlinear equation, 289–290 reference angle, 141 solving two-equation algebraically, 285 in trig functions, 125 solving with more than two equations, things, finding a number of, 355 291–293 30-60 triangle, 132–133, 135 with two nonlinear equations, 290–291 3 x 3 system of equations, and Cramer’s writing in matrix form, 301–302 rule, 314 system-solving options, described, 284 26_169841 bindex.qxd 2/29/08 2:58 PM Page 380

380 Pre-Calculus For Dummies

360, convenience of, 120 trig functions TI-89 graphing calculator, 18 calculator, evaluating without, 134 transformation(s) of common angles, 210–211 combining, 44–46, 167–170 described, 121 cosecant function, 176 eliminating exponents in, 214–215 cosecant graph, 175–176 evaluating on the unit circle, 138–140 cotangent function, 170–173 finding values for, 137–140 exponential functions, 100–102 graphing, 147, 159, 161 graphs, domain and range of, 167 isolating, 141–142 horizontal, 38, 40, 101–102, 160 multiplied by a negative number, 161 logs, graphing, 107–109 square root of both sides, 144 parabolas, 262 of sums and differences, 196–205 parent graphs, 38 unit circle, evaluating on, 136–140 reciprocal functions, 174 values of, same for a specific angle, 134 secant function, 175 trig graphs, transforming secant graph, 174 complicated functions, 159–160 sine function, 176 cosecant graphs, 173–176 sine graphs, 162–163, 175 cosine graphs, 160–170 tangent, 170–173 cotangent graphs, 170–173 of trig functions, 159 secant graphs, 173–176 of trig graphs. See trig graphs, sine graphs, 160–170 transforming tangent graphs, 170–173 types of, 38 trig identities vertical. See vertical transformation advanced, 195 transitive property, of numbers, 13 basic, 177–188 translations, 38, 41–43, 166 co-function, 185–187 transverse axis, hyperbola, 274–276 even-odd, 183–185 triangle(s) looking for, 193 30-60 triangle, 132–133, 135 periodicity, 187–188 Angle Angle Side (AAS), solving, 218, primer on, 178 220–221 proving, 204–205 Angle Side Angle (ASA), solving, 218–220 Pythagorean, 181–183 congruence postulates, 228–230 reciprocal, 179–181 filling in by calculating area, 232–233 rules for working with, 179 45er triangle, 131–132, 138 trig proofs, 189–190 fusing with the unit circle, 134–144 trig ratios, 120–126 hypotenuse, 121, 128, 132–134 trig value, of twice an angle, 205 Law of Cosines, 228–232 trigonometry, compared to pre-calculus, 10 Law of Sines, 218–227 Trigonometry Workbook For Dummies with no solution, 227 (Sterling), 356 ratios, digesting special, 131–133 trinomial equation, FOIL method, 69 Side Angle Side (SAS), solving, 230–233 trinomials, 67, 71–73, 143–144 Side Side Angle (SSA), solving, 218, 2 x 2 matrix, inverse of, 310 221–227 two-equation systems, finding solutions Side Side Side (SSS), solving, 228–230, 233 algebraically, 285–291 solving, 217–232 2(π) radian, 150 two different, solving, 222 26_169841 bindex.qxd 2/29/08 2:58 PM Page 381

Index 381

vertical asymptote • U • of the cotangent parent graph, 155 undefined fraction, 152 described, 49, 55 undefined slope, 18 drawing, 53–54, 57 undefined values finding for a tangent function, 153 for a domain, 61 graphing, 56 for rational functions, 49 searching for, 50 reciprocal of, 158 vertical displacement, of a circle, 257 unfactorable polynomials, solving with a vertical ellipse, 269, 271 degree higher than two, 80–90 vertical equation, for an ellipse, 270 union symbol, in interval notation, 26 vertical hyperbola unit circle asymptotes, 275, 277 angles in radians, 198–199 described, 273–275 angles of, combining to get a requested equation for, 274 angle, 197 slopes of asymptotes, 275 angles on, 129, 140–144 transverse axis and conjugate axis, 274 defined, 126 vertices, 275 described, 119 vertical parabola entire, 136–137 determining, 263 families on, 140 equation for, 261 fusing with triangles, 134–144 graphing, 260 getting a good grasp on, 128–131 information from, 262 solving for angles on, 140–144 min and max on, 266–268 trig-function values on, 136–140 vertical reflections, 44 unknown variables, solving for, 287–288 vertical shifts upper limit, of a sum, 325 described, 166–167, 170 graphs, 42–43 of horizontal parabolas, 262–263 • V • representing, 259 for tangent, 171 variable matrix, 301 of vertical parabolas, 262 variables vertical transformation cannot be added to constants, 360 described, 38–40 with a coefficient of 1, 286 parabolas, 262–263 crossing out from terms, 362 of parent exponential functions, 101–102 eliminating one from a system, 287 parent graph of any trig graph, 160 isolating in a trig function, 125–126 range, changing, 167 lining up underneath each other, 287 vertical translation, 167 solving for one in a system, 286 vertices. See vertex (vertices) squaring in the equation of a parabola, 261 vectors, multiplication of dot products of, 299–300 • W • vertex (vertices) of ellipse, 271–272 waves, sine graphs moving in, 148 finding by completing the square, 267 Web sites, practice math problems, 356 of function, 35 whole numbers, defined, 11 of hyperbola, 275 word problems, 122–123, 286, 352 of parabolas, 260–266 work, showing each step of, 354 26_169841 bindex.qxd 2/29/08 2:58 PM Page 382

382 Pre-Calculus For Dummies • X • • Y • x-rcos(θ), 248 y-axis, 15 x-axis, 15, 40 y-intercept, 16 x-intercepts y-intercepts for cosine graph, 150–151 locating, 52 for a cotangent function, 155 plotting, 53–54, 56–57 locating, 52 of polynomials, 92 plotting, 53–54, 56–57 for sine function graph, 149 for tangent function, 153 Z zeroes as, 91 zero, multiplicative property of, 14 x-value, between two roots, 68 zero product property x-y coordinate plane, 238 described, 14, 78, 143 x-y coordinates, and polar coordinates, in polynomial expression, 69–70 248–250 using, 291 zero slope, 18 26_169841 bindex.qxd 2/29/08 2:58 PM Page 383

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