A Absolute deviation, 38 Absolute value, Properties of, 28 Absolute value equation(s) defined, 28 solving, 27 31, 51 algebraically, 27 graphically, 27 numerically, 27 with two absolute values, 30, 31 writing the equation, 29 Absolute value function(s) defined, 9 graph of parent function, 9, 10, 12, 13 transformations of, 12, 13, 17, 50 51 Absolute value inequality(ies) defined, 36 solving, 35 38, 51 algebraically, 35 graphically, 35 numerically, 35 Addition additive inverse of complex numbers, 160 of complex numbers, 157 of polynomials, 217 221, 278 vertically and horizontally, 218 of radicals and roots, 298, 299 of rational expressions, 435 437, 453 of rational numbers, 409 sum of two cubes, factoring, 233 of two functions, 321, 322 Algebra tiles, completing the square, 163 Analyzing Mathematical Relationships, Throughout. See for example: factoring to solve equation, 165 horizontal shrink, 371 identifying functions, 10 inverse variation, 414 percent rates of change from natural base exponential functions, 357 quadratic form, 233 translation, 419 vertical stretch, 101, 305, 418 and (intersection), 36 Another Way absolute value equation, 29 change-of-base formula, 382 equation with two radicals, solving, 316 function operations, division, 323 intersection points, 195 polynomials, factoring, 234 quadratic equations, solving, 164, 175, Quadratic Formula, 187 quadratic functions, 167 rational equations, solving, 446 rational expressions, multiplying, 429 rational functions inverse variation and constant of variation, 413 rewriting and graphing, 420 Rational Root Theorem, 243 three-variable system, 69 Area of rectangular prism, 207 of sphere, 332 Asymptote(s) defined, for exponential functions, 348 for rational functions, 418 Augmented matrix defined, 76 solving a system using technology, 77 78 writing for systems, 76 77 Average rate of change, 129 Axis of symmetry defined, of parabola, 108 graphing a parabola, 307 and standard equations of parabolas, 121 122 B Binomials cube of, 217, 219 as factor in polynomial, 234 multiplying three, 219 square of, 219 using Pascal s Triangle to expand, 221 Bounded intervals, defined, on real number line, 4 C Change-of-base formula, 381 382 Circle compared to other conic sections, 424 graphing with center at origin, 307 standard form of, 186 Classifying data, 394, 413 Coefficient of determination, 131 Coefficient matrices, writing, 75 Combinations of transformations, 13, 21 Common denominators, adding or subtracting rational expressions, 436 Common Errors augmented matrices, 77 imaginary solutions, 175 nth roots, 292 polynomials long division, 226 squaring and cubing, 219 subtraction, 218 properties of logarithms, 380 Quadratic Formula, 174 in quadratic functions, 109, 111 rational exponents, 291 and radicals, 299 rational expressions simplifying, 428 subtracting, 438 substitution, 186 Common logarithm(s) changing a base, 381 382 defined, 363 evaluating, 363 Completing the square, 163 167, 201 compared to other methods for solving quadratic equations, defined, 164 solving quadratic equations by completing the square, 165 166 using square roots, 164 writing quadratic functions in vertex form, 166 Complex conjugates, 251 252 Complex Conjugates Theorem, 251 252 Complex fraction(s) defined, 439 simplifying, 409 Complex number(s), 155 159, 200 additive inverse of, 160 complex solutions and zeros, 159 A59
defined, 156 equality of two, 157 imaginary numbers, 155 156 imaginary unit i, 155 157 operations with, 157 158 adding, 157 multiplying, 158 subtracting, 157 relationships in set of complex numbers, 155, 156 set of, 155 Compositions of functions, 332 Compound inequalities, 36, 38 Compound interest, 351, 358 Concept Summary methods for solving quadratic equations, solving inequalities, 38 zeros, factors, solutions, and intercepts, 264 Conic sections, 424 Conjugate(s) complex, 251 252 defined, 298 Irrational Conjugates Theorem, 245 Consistent linear system, 59 Constant function graphs of parent function, 9, 10 standard form of, 210 Constant term in function, 210 Constant of variation, 412 Continuous functions, 208 Continuously compounded interest, 358 Correlation coefficient, 45 Critical values, 194 Cross multiplying, to solve rational equation, 444 Cube of a Binomial, special product pattern, 217, 219 Cube root function, parent function for, 304 305 Cubic equations and imaginary solutions, 249 and repeated solutions, 241 solving by factoring, 242 Cubic functions cubing binomials, 217 inverse of, 331 standard form of, 210 transforming graph of, 257 writing for set of points, 272 Cubic regression, 273, 274 D Data classification, 394, 413 Decay factor, 348, See also Exponential decay function(s) Deductive reasoning, 98 Degree of function, 210 and end behavior, 211 relationship to number of solutions, 250 Denominators, See also Least common denominator (LCD) like and unlike, adding or subtracting rational expressions, 436 439 rationalizing, 147, 297 Descartes, René, 252 Descartes s Rule of Signs, 252 253 Difference of Two Cubes, 232, 233 Differences of outputs, 130, 271 273, 394 Dimensions of a matrix, 75, 76 Direct variation, 411 413 Directrix, of parabola, 120, 137 Discriminant, in Quadratic Formula, 176 Distance Formula, 97 writing equation of parabola, 120 Division of polynomials, 225 228, 279 long division, 226 Remainder Theorem, 228 synthetic division, 227 of rational expressions, 427, 430 431, 453 by polynomial, 431 of two functions, 322, 323 Domain of function, finding, 345 of rational expression, 428 Doppler effect, 424 E Electrical circuits, 158 Elements of a matrix, 76 Elements of a set, 4, 6 Elimination Gaussian, 70 solving a system of three linear equations, 67 70, 91 solving nonlinear system by, 185 Ellipse, compared to other conic sections, 424 End behavior of function s graph, 211 Endpoints, 4 Equality for Exponential Equations, Property of, 386 Equality for Logarithmic Equations, Property of, 387 Equivalent row-echelon form, 70 Equivalent systems, 67 operations that produce, 70 Euler, Leonhard, 356 Euler number, 356, See also Natural base e Even functions, 267 Exponential decay function(s), 347 351, 402 defined, 348 graphing, 349 graphing natural base functions, 357 Exponential decay models, 349 351 Exponential equation(s) defined, 386 Property of Equality for Exponential Equations, 386 solving, 385 387, 404 Exponential form, 362 Exponential function(s) defined, 348 graph of parent function, 9 growth and decay functions, 347 351, 402 inverse function of, 361 inverse properties of, 364 modeling with, 393 397, 404 natural base e, 355 358, 402 transformations of, 369 371, 373, 403 writing recursive rules for, 396 writing using two points, 395 Exponential growth function(s), 347 351, 402 defined, 348 graphing, 349 graphing natural base functions, 357 Exponential growth models, 349 351 Exponential inequalities, solving, Exponential Property of Inequality, Exponential models exponential regression, 397 writing and finding, 393 397 Exponential Property of Inequality, Exponential regression, 397 Exponents properties of integer exponents, 287, 295 properties of rational exponents, 291, 295, 296 using, 345 Expressions evaluating, 1 with rational exponents, 291, 295 simplifying algebraic, 207 A60
simplifying radical, 297 299 writing in radical form, 289 writing in rational exponent form, 289 Extraneous solution(s) defined, 31 solving an equation with, 315 solving rational equation with, 446 F Factor, equivalent statements, 264 Factor by grouping, 233 Factor Theorem, 234 235, 252 Factored completely, 232 Factoring compared to other methods for solving quadratic equations, polynomials, 143, 148, 231 235, 279 determining whether binomial is a factor, 234 Difference of Two Cubes, 232, 233 by grouping, 233 in quadratic form, 233 Sum of Two Cubes, 232, 233 solving polynomial equations by, 242 solving quadratic equations by, 146, 148 special products, 143 Finite differences, 272 273 Finite length, of bounded interval, 4 First differences, 130, 271 273 Flawed reasoning, 98 Focus, of parabola, 119 123, 137 defined, 120 FOIL method, 103, 158 Formulas change-of-base, 381 Distance Formula, 97 Quadratic Formula, 174 volume of cube, 262 of rectangular prism, 207 of right triangle pyramid, 262 Function families, identifying, 9 10 Functions compositions of, 332 continuous, 208 cubic (See Cubic functions) exponential (See Exponential functions) inverses (See Inverses of functions) linear (See Linear functions) logarithmic (See Logarithmic functions) logistic, 393 operations on, 321 324, 340 parent (See Parent functions) polynomial (See Polynomial functions) quadratic (See Quadratic functions) quartic (See Quartic functions) rational (See Rational functions) Fundamental Theorem of Algebra, 249 253, 281 G Galileo Galilei, 153 Gauss, Carl Friedrich, 70, 250 Gaussian elimination defined, 70 solving systems by, 70 Gaussian function, 393 Golden rectangles, 450 Graph of the system, 193 Graph of a system of linear inequalities, 83 Graphing, See also Graphs compared to other methods for solving quadratic equations, exponential functions, 349, 361 linear inequalities, 81 logarithmic functions, 361, 365 polynomial functions, 209 213, 264, 278 analyzing, 263 267, 282 quadratic functions in standard form, 109 transformations of, 11 13, 99 103, 136 using symmetry, 108, 121 122 using x-intercepts, 111 quadratic inequality in one variable, 194 in two variables, 192 radical functions, 303 307, 339 parabolas and circles, 307 square root and cube root, 304 transforming, 305 rational functions, 417 421, 438, 452 hyperbola, 418 translation of, 419 solving nonlinear system by, 184, 187 quadratic equations by, 146, 187 quadratic inequality by, 194 sum of two functions, 321 system of linear inequalities, 83 Graphing calculator cubic regression feature, 273, 274 exponential regression, 397 function operations, 324 graphing system of linear inequalities, 58 intersect feature, 187, limitations of, 144 linear regression feature, 45 logarithmic regression, 397 maximum feature, 131, 266 minimum feature, 266 quadratic regression feature, 127, 131 regression feature, 271 square viewing window, 2, 144 standard viewing window, 2 table feature, 208, 350, 429 trace feature, 274, 350, 421 using Location Principle, 265 variables displayed, 195 writing radical functions, 306 zero feature, 266 Graphs, See also Graphing of cubic and quartic functions, transforming, 257 of linear parent functions, 9 10, 50 of polynomial functions, analyzing, 263 267, 282 even and odd functions, 267 turning points, 263, 266 using Location Principle, 265 of radical functions, 303 307 transformations of radical functions, 339 Growth factor, 348, See also Exponential growth function(s) H Hooke s Law, 335 Horizontal axis of symmetry, of parabolas, 121 122 Horizontal Line Test, 330 Horizontal shrinks of polynomial functions, 258, 259, 306 Horizontal stretches A61
of polynomial functions, 258 Horizontal translations of linear functions, 18 of polynomial functions, 258 of quadratic functions, 100 Hyperbolas, 418 420 I Imaginary number(s) defined, 156 solutions of cubic and quartic equations, 249 solving equations with imaginary solutions, 175 Imaginary unit i defined, 156 equality of two complex numbers, 157 finding square roots of negative numbers, 156 Inconsistent linear system, 59 of radical, 290 Inequalities, solving, 38 Inequality Exponential Property of, Logarithmic Property of, Infinite length, of unbounded interval, 4 5 Infinity end behavior of function, 211 and unboundedness of intervals, 5 Intercept form, See also x-intercepts graphing quadratic function in, 111 writing quadratic equations, 128 129 Interval notation using, 4 5, 50 writing, 5 Inverse functions, 329 Inverse properties of exponential functions and logarithmic functions, 364 Inverse variation, 411 414, 452 classifying equations and data, 412 413 defined, 412 writing equations, 413 Inverses of functions, 327 332, 340 of cubic function, 331 determining whether functions are inverses, 332 formula for input of function, 328 graphing, 327 horizontal line test, 330 of linear function, 329 of nonlinear functions, 330 331 of quadratic function, 330 of radical function, 331 of rational functions, 447 Irrational Conjugates Theorem, 245 K Kepler s third law, 294 L Leading coefficient, 210 and end behavior, 211 Least common denominator (LCD), 436 437 to solve rational equation, 445 Least common multiple (LCM), 436 437 Like radicals, 298 Line of best fit, 45 Line of fit, 44 Line of reflection, 11, 329 Linear equation in three variables, 60 Linear equations solving absolute value equations, 27 31 writing from a graph or table, 42 43 Linear functions graphs of parent function, 9 10, 11, 50 inverse of, 329 modeling with, 41 45, 52 standard form of, 210 transformations of, 11, 18 21, 50 51 Linear inequalities graphing systems of, 83 solving absolute value inequalities, 35 38, 51 solving systems of, 81 84, 92 in two variables, graphing, 57 writing systems of, 84 Linear regression, 45 Linear systems consistent and inconsistent, 59 in row-echelon form, 67 solving using elimination, 67 70, 91 using substitution, 59 63, 90 using technology, 75 78, 92 writing augmented matrices for, 76 77 Literal equations, rewriting, 57, 287 Local maximum, 266 Local minimum, 266 Location Principle, 265 Logarithm(s) change-of-base formula, 381 382 and logarithmic functions, 361 365, 402 403 properties of, 379 382, 403 rewriting logarithmic expressions, 381 Logarithm of 1, 363 Logarithm of b with base b, 363 Logarithm of y with base b, 362 Logarithmic equation(s) defined, 387 Property of Equality for Logarithmic Equations, 387 rewriting, 362 solving, 385 388 Logarithmic form, 362 Logarithmic functions finding logarithmic model, 397 graphing, 365 inverse properties of, 364 and logarithms, 361 365, 402 403 natural base e, 402 parent graphs for, 365 transformations of, 369, 372 373, 403 Logarithmic inequalities, solving, Logarithmic Property of Inequality, Logarithmic models finding, 397 logarithmic regression, 397 Logarithmic Property of Inequality, Logic, deductive reasoning, 98 Logistic function, 393 Long division of polynomials, 226 M Matrix augmented, 76 78 defined, 75, 76 Maximum value(s) defined, of parabola, 110 finding with zeros of function, 149 local maximum, 266 Measure, converting units of, 410 Membership in set, symbol for, 6 Minimum value(s) defined, of parabola, 110 finding with zeros of function, 149 local minimum, 266 Modeling with functions a dropped object, 150 A62
with exponential functions, 393 397, 404 a launched object, 178 with linear functions, 41 45, 52 with logarithmic functions, 393 397 with polynomial functions, 271 274, 282 with quadratic functions, 127 131, 138 Modeling with Mathematics, Throughout. See for example: exponential and logarithmic functions, annual interest, 358 linear functions absolute deviation, 38 combining transformations, 21 function identification, 13 writing linear equation from a table, 43 polynomial functions, and volume of pyramid, 260 quadratic equations and baseball, 167 perimeter and area of land, 195 quadratic functions parabola and golf shot, 112 path of water spray, 103 rational functions 3-D printers, 421 inverse variation, 414 Monomials, finding a common factor, 232 Multiplication of complex numbers, 158 of polynomials, 217, 219 220, 278 cube of binomials, 219 square of binomials, 219 sum and difference product, 219 vertically and horizontally, 219 Properties Power of Product, 220, 296 Product of Powers, 219, 296 Zero-Product, 148 rational equations, solving by cross multiplying, 444 of rational expressions, 427, 429 430, 453 by polynomial, 430 of two functions, 322, 323 Multiplicity, repeated solutions, 242 Multi-step problem inverse of function, 332 quadratic functions, 149 N Natural base e, 355 358, 402 approximating, 355 defined, 356 graphing natural base functions, 355, 357 Natural base exponential function, 355, 357 translating, 371 Natural logarithm(s) changing a base, 381 382 defined, 363 evaluating, 363 Negative Exponent Property, 296 Negative infinity end behavior of function, 211 and unboundedness of intervals, 5 Negative real zeros, 252 253 Newton s Law of Cooling, 387 Nonlinear systems of equations, solving, 183 187, 202 by elimination, 185, 202 by graphing, 184 by substitution, 185, 186 nth root(s) defined, of a, 290 and rational exponents, 289 292, 338 real nth roots of a, 290 solving equations, 292 Number line, 27 bounded and unbounded intervals on real number line, 4, 5 Numbers of solutions of nonlinear systems, 184 circle and line, 186 relationship to degree of polynomial, 250 using Quadratic Formula, 176 O Odd functions, 267 Operations on functions, 321 324, 340 addition, 321, 322 division, 322, 323 multiplication, 322, 323 subtraction, 322, 323 Operations that produce equivalent systems, 70 Ordered triple, 60 P Parabola(s) compared to other conic sections, 424 defined, 100 directrix, 120 122 Distance Formula to write equation of, 120 equation of translation of, 123 finding maximum and minimum values, 110 focus of, 119 123, 137 graphing with horizontal axis of symmetry, 307 latus rectum, 126 properties of graph of, 109, 111 satellite dishes and spotlights, 119 standard equations of, 121 122 and symmetry, 107 108 Parabolic reflectors, 119, 123 Parent function(s) for absolute value function, 9, 10, 12, 13 for constant function, 9, 10 defined, 10 for exponential decay functions, 348 for exponential growth functions, 348 identifying, 9 10 for linear functions, 9 10, 11 for logarithmic functions, 365 for quadratic functions, 9, 10, 11 13, 50 for simple rational functions, 418 for square root and cube root functions, 304 306 Pascal, Blaise, 221 Pascal s Triangle defined, 221 patterns for cubing binomials, 217, 221 using to expand binomials, 221 Patterns Pascal s Triangle and cubing binomials, 217, 221 special factoring patterns of polynomials, 232 special product patterns of polynomials, 219, 220 Pendulum period, 310 Perfect square trinomial, 165 Performance Tasks Accident Reconstruction, 135 Algebra in Genetics: The Hardy- Weinberg Law, 199 For the Birds Wildlife Management, 277 Circuit Design, 451 Fun and Games, 89 Measuring Natural Disasters, 401 Secret of the Hanging Baskets, 49 Turning the Tables, 337 Point-slope form, writing an equation of a line, 42 A63
Polynomial(s) adding and subtracting, 217 221, 278 defined, 210 dividing, 225 228, 279 rational expression by, 431 factoring, 143, 148, 231 235, 279 multiplying, 217, 219 220, 278 rational expression by, 430 set of closed under addition and subtraction, 218 closed under multiplication, 219 not closed under division, 227 Polynomial equations Fundamental Theorem of Algebra, 249 253, 281 solving, 241 245, 280 by factoring, 242 Polynomial function(s) analyzing graphs of, 263 267, 282 defined, 210 end behavior of, 211 finding zeros of, 242, 244, 251 graphing, 209 213, 264, 278 identifying and evaluating, 210 211 modeling with, 271 274, 282 standard form of, 210 summary of common types of, 210 transformations of, 257 260, 281 turning points of, 263, 266 Polynomial identity, proving, 220 Polynomial long division, 226 Positive infinity end behavior of function, 211 and unboundedness of intervals, 5 Positive real zeros, 252 253 Power function, 304 Power of Power Property, 290, 296 Power of Product Property, 220, 296 Power Property of Logarithms, 380 Power of Quotient Property, 296 Principal (investment), 351 Principal root, 290 Prisms, rectangular, 207 Product of Powers Property, 219, 296 Product Property of Logarithms, 380 Product Property of Radicals, 297, 305 Property(ies) of Equality for Exponential Equations, 386 of Equality for Logarithmic Equations, 387 Exponential Property of Inequality, of exponents, 287, 295 of finite differences, 273 of graph of parabola in intercept form, 111 in standard form, 109 inverse properties of exponential and logarithmic functions, 364 Logarithmic Property of Inequality, of logarithms, 379 380, 403 Power Property, 380 Product Property, 380 Quotient Property, 380 Negative Exponent, 296 Power of Power, 290, 296 Power of Product, 220, 296 Power of Quotient, 296 Product of Powers, 219, 296 Properties of Absolute Value, 28 Quotient of Powers, 296 of radicals Product Property of Radicals, 297, 305 Quotient Property of Radicals, 297 of rational exponents, 291, 295, 296, 338 Zero Exponent, 296 Zero-Product, 148 Pure imaginary numbers, 156 Q Quadratic equation in one variable, 146 Quadratic equations, See also System(s) of nonlinear equations solving, 145 150, 200 by completing the square, 165 166 complex solutions and zeros, 159 by factoring, 146, 148 by graphing, 146, 187 using Quadratic Formula, 174 175 using square roots, 146, 147 summary of methods for solving, writing with three points, 130 Quadratic form, factoring polynomials in, 233 Quadratic Formula, 173 178, 201 analyzing the discriminant, 176 compared to other methods for solving quadratic equations, defined, 174 deriving, 173 solving equations using, 174 175 Quadratic function(s) characteristics of, 107 112, 136 defined, 100 focus of a parabola, 119 123 graphing in standard form, 109 using symmetry, 108 using x-intercepts, 111 graphs of parent function, 9, 10, 11 13, 50 identifying graphs of, 99 inverse of, 330 modeling with, 127 131, 138 in standard form, 108 transformations of, 11 13, 99 103, 136 writing in vertex form, 166 Quadratic inequality(ies), 191 195, 202 graphing in one variable, 194 graphing in two variables, 192 solving, 191 by graphing, 194 Quadratic inequality in one variable, 194 Quadratic inequality in two variables, 192 Quadratic regression, 127, 131 Quartic equations, and imaginary solutions, 249 Quartic functions finding zeros of, 242 standard form of, 210 transforming graphs of, 257 Quotient polynomial, 226 Quotient of Powers Property, 296 Quotient Property of Logarithms, 380 Quotient Property of Radicals, 297 R Radical equation(s) defined, 314 solving, 313 317, 339 with extraneous solutions, 315 with rational exponent, 316 317 steps, 314 with two radicals, 316 Radical expressions, simplifying, 297 299 Radical form compared to rational exponent form, 291 writing expressions in, 289 Radical function(s) defined, 304 A64
graphing, 303 307, 339 inverse of, 331 transformations of, 305 Radical inequalities, solving, 317, 339 Radicals products and quotients of, 295 properties of, 297, 338 Range of function, finding, 345 Rational equations, solving, 443 447, 454 by cross multiplying, 444 with extraneous solution, 446 by using the least common denominator, 445 Rational exponents and nth roots, 289 292, 338 properties of, 291, 295, 296, 338 Rational expression(s) adding and subtracting, 435 438, 453 defined, 428 dividing, 427, 430 431, 453 by polynomial, 431 multiplying, 427, 429 430, 453 by polynomial, 430 rewriting rational functions, 438 simplifying, 428 simplifying complex fractions, 439 Rational function(s) defined, 418 graphing, 417 421, 438, 452 inverse of, 447 inverse variation, 411 414, 452 Rational numbers, adding and subtracting, 409 Rational Root Theorem, 243 244 Rationalizing the denominator, 147, 297 Reading Factor Theorem, 234 inverse functions, 329 local maximum and local minimum, 266 positive infinity, 211 Real nth roots of a, 290 Real number line absolute value equation and, 27 bounded intervals on, 4 unbounded intervals on, 5 Real numbers, relationship in set of complex numbers, 155, 156 Real-life problems, Throughout. See for example: exponential and logarithmic functions car value, 350 continuously compounded interest, 358 Newton s Law of Cooling, 387 sound loudness in decibels, 382 linear systems seats in amphitheater, 63 system of inequalities, shoe sale, 85 polynomial functions baseball distance, 271 biomass energy, 274 Descartes s Rule of Signs, and tachometer, 253 electric vehicles, 213 roller coaster, 235 quadratic equations complex numbers and electrical circuits, 158 dropped object compared to thrown object, 178 monthly magazine, 149 quadratic inequality, rope climbing, 193 quadratic functions golf shot, 112 parabolic reflectors, 123 radical functions annual depreciation rate, 292 function operations, rhino, 324 hurricane wind velocity, 315 rational functions 3-D printers, 447 annual per capita income, 431 Reasoning absolute value equation, 29 deductive, 98 vertical stretch and shrink, 12 Rectangular prism, 207 Recursive rules for exponential functions, 396 Reflection(s) defined, 11 graph of inverse function, 329 graphing and describing quadratic functions, 11 of linear functions, in x-axis and y-axis, 19 of polynomial functions, 258, 259 of quadratic functions, in x-axis and y-axis, 101 Regression features of graphing calculator cubic, 273, 274 exponential, 397 linear, 45 logarithmic, 397 quadratic, 127, 131 Relative maximum, 266 Relative minimum, 266 Remainder polynomial, 226 Remainder Theorem, 228 Factor Theorem, special case, 234 235, 252 Remember average rate of change, 129 compound inequality with and, 36 differences of functions, 394 exponential functions, 395 extraneous solutions, 31 FOIL method to multiply binomials, 103 function notation, 11 inverses of functions, 364 matrix rows and columns, 78 Power of Product Property, 220 Product of Powers Property, 219 proportional relationship, 42 rational equations, 447 recursive rule and equation, 396 slope-intercept form, 11 vertical stretches and shrinks, 259 x-intercept of quadratic function, 111 Repeated solution(s) of cubic equations, 241 defined, 242 Richter scale for earthquakes, 378, 401 Root of an equation, 146 Roots, approximating and evaluating, 288 Row operations, 70 Row-echelon form, 67, 70 Rule of Signs, Descartes s, 252 253 S Second differences, 130, 271 273 Set, defined, 3, 4 Set notation, writing subsets in, 3 Set of real numbers and subsets, 4 Set-builder notation defined, 3, 6, 50 using, 6 Shrinks graphing and describing vertical, 12 of polynomial functions, 258 306 A65
Simplest form of a radical, 297 Simplified form of rational expression, 428 Simplifying complex fractions, 439 Slope-intercept form, writing equation of a line, 42 Solution of a system of linear inequalities, 82 Solution of a system of three linear equations, 60 Solutions equivalent statements, 264 extraneous, 31 number from linear equation in three variables, 60 Special patterns of polynomials factoring, 232 product, 219 Sphere, surface area and radius of, 332 Spreadsheet, using, 346 Square of a Binomial, special product pattern, 219 Square root function, parent function for, 304 306 Square roots compared to other methods for solving quadratic equations, of negative numbers, 156 simplifying, 143 solving quadratic equations using, 146, 147, 164 Stair-step pattern, 67, 70 Standard equations, of parabola with vertex at (h, k), 122 at origin, 121 Standard form of circle, 186 of common polynomial functions, 210 defined, for quadratic functions, 108 graphing, quadratic functions, 109 of linear functions, 210 Straight line depreciation, 41 Stretches graphing and describing vertical, 12 of polynomial functions, 258 Study Skills Analyzing Your Errors Application Errors, 311 Study Errors, 425 Creating a Positive Study Environment, 171 Forming a Weekly Study Group, 377 Keeping Your Mind Focused, 239 Taking Control of Your Class Time, 25 Ten Steps for Test Taking, 73 Using the Features of Your Textbooks to Prepare for Quizzes and Tests, 117 Subset defined, 3, 4 writing in set notation, 3 Substitution solving a system of three linear equations, 61 63, 90 solving nonlinear system by, 185, 186 Subtraction of complex numbers, 157 difference of two cubes, factoring, 232 233 of polynomials, 217 221, 278 vertically and horizontally, 218 of radicals and roots, 298, 299 of rational expressions, 435 436, 438, 453 of rational numbers, 409 of two functions, 322, 323 Sum and Difference Product of polynomials, 219 Sum of Two Cubes, 232, 233 Syllogism, 98 Symmetric about the origin, 267 about the y-axis, 267 Synthetic division defined, 227 of polynomials, and remainder, 227 228 System of linear equations, See Linear systems System(s) of linear inequalities, 81 85, 92 defined, 82 graphing, 57, 58, 83 writing, 84 System(s) of nonlinear equations defined, 184 solving, 183 187, 202 by elimination, 185 by graphing, 184 by substitution, 185, 186 System of quadratic inequalities, graphing, 193 System of three linear equations defined, 60 solving using elimination, 67 70 using substitution, 61 63 using technology, 75 78 Systems of equations, See Linear systems T Technology, See also Graphing calculator solving systems of equations using augmented matrices, 77 78 Third differences, 273 Three-variable system solving using elimination, 67 70 using substitution, 61 63 using technology, 75 78 visualizing, 60 Transformation(s) of absolute value functions, 17, 19, 50 51 combinations of, 13, 21 defined, 11 describing, 11 12 of exponential functions, 369 371, 373, 403 of figures, 1 of linear functions, 11, 18 21, 50 51 of logarithmic functions, 369, 372 373, 403 of parent quadratic function, 10, 11 13, 50 of polynomial functions, 257 260, 281 of quadratic functions, 11 13, 99 103, 136, 339 Translation(s) defined, 11 of linear functions, 11, 18 19 of natural base exponential function, 371 of polynomial functions, 258, 259 of quadratic functions, 100 of simple rational functions, 419 Turning points of polynomial functions approximating, 263 finding, 266 local maximum and local minimum, 266 A66
U Unbounded intervals, defined, on real number line, 5 Units of measure, converting, 410 V Variation, See Inverse variation Vertex defined, of parabola, 102 and standard equations of parabolas, 121 122 Vertex form defined, of quadratic function, 102 writing quadratic equations, 128 writing quadratic functions in, 166 Vertical axis of symmetry, of parabolas, 121 122 Vertical shrinks defined, 12 of polynomial functions, 258, 259 Vertical stretches defined, 12 of polynomial functions, 258 Vertical translations of linear functions, 18 of polynomial functions, 258 of quadratic functions, 100 W Writing, Throughout. See for example: exponential functions, 395 396 an exponential model, 350 inverse variation equations, 413 polynomial functions for data sets, 272 quadratic equations, 128 129 to model data, 130 131 radical functions, 306 recursive rules for exponential functions, 396 system of linear inequalities, 84 transformations of exponential and logarithmic functions, 373 of polynomial functions, 259 260 of quadratic functions, 102 103 X x-axis reflections of linear functions, 19 x-intercepts equivalent statements, 264 finding, of the graph of a linear equation, 97 graphing quadratic functions using, 111 of polynomial graphs, identifying, 209 using to graph polynomial functions, 264 writing quadratic equations using, 128 129 as zeros of function, 148 Y y-axis reflections of linear functions, 19 Z Zero Exponent Property, 296 Zero(s) of a function defined, 148 and Descartes s Rule of Signs, 252 253 equivalent statements, 264 finding number of solutions, 250 of polynomial function, finding, 242, 244, 251 using Location Principle, 265 of quadratic function, finding, 148, 159 using to write polynomial function, 245 Zero-Product Property, 148 A67