Algebra II Vocabulary Word Wall Cards



Algebra, Functions, and Data AnalysisVocabulary Word Wall CardsMathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development.?The cards should be used as an instructional tool for teachers and then as a reference for all students. Table of ContentsExpressions and OperationsReal NumbersComplex Numbers Complex Number (examples)Absolute ValueOrder of OperationsExpressionVariableCoefficientTermScientific NotationExponential FormNegative ExponentZero ExponentProduct of Powers PropertyPower of a Power PropertyPower of a Product PropertyQuotient of Powers PropertyPower of a Quotient Property HYPERLINK \l "polynomial" PolynomialDegree of PolynomialLeading CoefficientAdd Polynomials (group like terms)Add Polynomials (align like terms)Subtract Polynomials (group like terms)Subtract Polynomials (align like terms)Multiply BinomialsMultiply PolynomialsMultiply Binomials (model)Multiply Binomials (graphic organizer)Multiply Binomials (squaring a binomial)Multiply Binomials (sum and difference) HYPERLINK \l "factors_monomial" Factors of a MonomialFactoring (greatest common factor)Factoring (perfect square trinomials)Factoring (difference of squares)Difference of Squares (model) HYPERLINK \l "sum_diff_cubes" Factoring (sum and difference of cubes)Factor by GroupingDivide Polynomials (monomial divisor)Divide Polynomials (binomial divisor)Prime PolynomialSquare RootCube Rootnth RootSimplify Radical ExpressionsAdd and Subtract Radical ExpressionsProduct Property of RadicalsQuotient Property of RadicalsEquations and InequalitiesZero Product PropertySolutions or RootsZerosx-InterceptsCoordinate Plane HYPERLINK \l "literal_equation" Literal EquationVertical LineHorizontal Line HYPERLINK \l "quadratic_eq" Quadratic Equation (solve by factoring and graphing)Quadratic Equation (number of solutions)InequalityGraph of an InequalityTransitive Property for InequalityAddition/Subtraction Property of InequalityMultiplication Property of InequalityDivision Property of InequalityLinear Equation (standard form)Linear Equation (slope intercept form)Linear Equation (point-slope form)Equivalent Forms of a Linear EquationSlopeSlope FormulaSlopes of LinesPerpendicular LinesParallel LinesMathematical NotationSystem of Linear Equations (graphing)System of Linear Equations (substitution)System of Linear Equations (elimination)System of Linear Equations (number of solutions)Graphing Linear InequalitiesSystem of Linear InequalitiesLinear ProgrammingDependent and Independent VariableDependent and Independent Variable (application)Graph of a Quadratic EquationVertex of a Quadratic FunctionQuadratic FormulaRelations and Functions HYPERLINK \l "raltion_examples" Relations (definition and examples)Functions (definition) Function (example)DomainRangeFunction NotationHYPERLINK \l "end_behavior"End BehaviorIncreasing/DecreasingExtremaParent FunctionsLinear, QuadraticExponential, LogarithmicTransformations of Parent Functions HYPERLINK \l "trans_parent_func" TranslationReflectionDilationLinear Function (transformational graphing)HYPERLINK \l "linear_func_translation"Translation HYPERLINK \l "linear_func_dilation" Dilation (m>0) HYPERLINK \l "linear_func_dilation_reflection" Dilation/reflection (m<0)Quadratic Function (transformational graphing) HYPERLINK \l "quad_func_vert_trans" Vertical translationHorizontal translation HYPERLINK \l "quad_func_dilation" Dilation (a>0)Dilation/reflection (a<0)Arithmetic SequenceGeometric SequenceStatisticsProbabilityProbability of Independent EventsProbability of Dependent EventsProbability (mutually exclusive events)Fundamental Counting PrinciplePermutationPermutation (formula)CombinationCombination (formula)Statistics NotationMeanMedianModeSummationVarianceStandard Deviation (definition)Standard Deviation (graphic)z-Score (definition)z-Score (graphic)Empirical RuleElements within One Standard Deviation of the Mean (graphic)Scatterplot HYPERLINK \l "_Positive_Linear_Relationship" Positive Linear Relationship (Correlation)Negative Linear Relationship (Correlation)No CorrelationCurve of Best Fit (linear)Curve of Best Fit (quadratic) HYPERLINK \l "curve_best_fit_exp" Curve of Best Fit (exponential)Outlier Data (graphic)Real NumbersThe set of all rational and irrational numbersNatural Numbers{1, 2, 3, 4 …}Whole Numbers{0, 1, 2, 3, 4 …}Integers{… -3, -2, -1, 0, 1, 2, 3 …}Rational Numbersthe set of all numbers that can be written as the ratio of two integers with a non-zero denominator (e.g., 235, -5, 0.3, 16 , 137)Irrational Numbersthe set of all nonrepeating, nonterminating decimals (e.g, 7 , π, -.23223222322223…)center17911900Complex Numberscenter929583514366483539Imaginary Numbers00Imaginary Numbers1037314669843Real Numbers020000Real NumbersThe set of all real andimaginary numbersComplex Number(Examples)a ± bia and b are real numbers and i = 1A complex number consists of both real (a) and imaginary (bi) but either part can be 0CaseExamplesa = 0-i, 0.01i, b = 05, 4, -12.8a ≠ 0, b ≠ 039 – 6i, -2 + πiAbsolute Value|5| = 5 |-5| = 5 center97155 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 65 units5 units00 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 65 units5 unitsThe distance between a numberand zeroOrder of Operations Grouping Symbols294005130810( ) { } [ ] ( ) { } [ ] ExponentsanMultiplicationDivision10668035623500Left to RightAdditionSubtraction9525037846000Left to Right-26352565659000ExpressionA representation of a quantity that may contain numbers, variables or operation symbolsx 3(y + 3.9)4 – 89Variable222885013017500 9 + log x = 2.08d = 7c - 5A = r 2Coefficient24707858128000(-4) + 2 log x 23 ab – 12πr2Term1505447613355003 log x + 2y – 83 terms-5x2 – x 2 terms 1 termScientific Notationa x 10n1 ≤ |a| < 10 and n is an integercenter432501Examples:Standard NotationScientific Notation17,500,0001.75 x 107-84,623-8.4623 x 1040.00000262.6 x 10-6-0.080029-8.0029 x 10-2(4.3 x 105) (2 x 10-2)(4.3 x 2) (105 x 10-2) =8.6 x 105+(-2) = 8.6 x 1036.6 ×1062×1036.62×106103=3.3 ×106-3=3.3 ×10300Examples:Standard NotationScientific Notation17,500,0001.75 x 107-84,623-8.4623 x 1040.00000262.6 x 10-6-0.080029-8.0029 x 10-2(4.3 x 105) (2 x 10-2)(4.3 x 2) (105 x 10-2) =8.6 x 105+(-2) = 8.6 x 1036.6 ×1062×1036.62×106103=3.3 ×106-3=3.3 ×103center24701500Exponential Form1005840-1270exponent00exponent2222500937260n factors00n factors240665936625base00basean = a?a?a?a…, a0center7937500Examples:2 ? 2 ? 2 = 23 = 8n ? n ? n ? n = n4 3?3?3?x?x = 33x2 = 27x2Negative Exponenta-n = 1an , a 0center4597600Examples:4-2 = 142 = 116x4y-2 = x41y2 = x41? y21 = x4y2(2 – a)-2 = 1(2 – a)2 , a ≠2Zero Exponentcenter102485700a0 = 1, a 0Examples:(-5)0 = 1 (3x + 2)0 = 1(x2y-5z8)0 = 14m0 = 4 ? 1 = 4Product of Powers Propertyam ? an = am + n center41021000Examples: x4 ? x2 = x4+2 = x6a3 ? a = a3+1 = a4w7 ? w-4 = w7 + (-4) = w3Power of a Power Property(am)n = am · ncenter54184500Examples: center235016(y4)2 = y4?2 = y8center1891470 (g2)-3 = g2?(-3) = g-6 = 1g6Power of a Product Property(ab)m = am ? bm -450574240700Examples: Quotient of Powers Propertyaman = am – n, a 095802569469009582153377755095894120248900center1714500Examples:x6x5 = x6 – 5 = x1 = xy-3y-5 = y-3 – (-5) = y2a4a4 = a4-4 = a0 = 1Power of Quotient Propertyabm= ambm , b0-1200154749800020790484357Examples:y34= y434 = y812286031559505t-3= 5-3t-3 = 1531t3 = 153?t31 = t353 = t3125Polynomial ExampleNameTerms76xmonomial1 term3t – 112xy3 + 5x4ybinomial2 terms2x2 + 3x – 7trinomial3 termsNonexampleReason102616016510005mn – 8variable exponent833120571500n-3 + 9negative exponentDegree of a PolynomialPolynomialDegree of Each TermDegree of Polynomial-7m3n5 -7m3n5 → degree 882x + 32x → degree 13 → degree 016a3 + 3a2b3 – 216a3 → degree 33a2b3 → degree 5-21 → degree 0 5The largest exponent or the largest sum of exponents of a term within a polynomialLeading CoefficientThe coefficient of the first term of a polynomial written in descending order of exponents-9144058229400Examples:7a3 – 2a2 + 8a – 1-3n3 + 7n2 – 4n + 1016t – 1Add Polynomials(Group Like Terms – Horizontal Method)center1841500 Example: (2g2 + 6g – 4) + (g2 – g) = 2g2 + 6g – 4 + g2 – gcenter300355(Group like terms and add)020000(Group like terms and add)= (2g2 + g2) + (6g – g) – 4 3g2 + 5g – 4 Add Polynomials(Align Like Terms – Vertical Method)-91440889000 Example: (2g3 + 6g2 – 4) + (g3 – g – 3) 127063512065(Align like terms and add)020000(Align like terms and add) 2g3 + 6g2 – 4174752055308500 + g3 – g – 33g3 + 6g2 – g – 7Subtract Polynomials(Group Like Terms - Horizontal Method)-723904254500 Example: (4x2 + 5) – (-2x2 + 4x -7)(Add the inverse)= (4x2 + 5) + (2x2 – 4x +7)= 4x2 + 5 + 2x2 – 4x + 7(Group like terms and add.)= (4x2 + 2x2) – 4x + (5 + 7) 6x2 – 4x + 12Subtract Polynomials(Align Like Terms --1524075818900Vertical Method) Example: (4x2 + 5) – (-2x2 + 4x -7)(Align like terms then add the inverse and add the like terms.)393001531432500 4x2 + 5 4x2 + 539516053079750040944805480050070485058864500–(-2x2 + 4x – 7) + 2x2 – 4x + 76x2 – 4x + 12Multiply BinomialsApply the distributive property.(a + b)(c + d) = a(c + d) + b(c + d) =ac + ad + bc + bdcenter24892000392906346767600358140010985500Example: (x + 3)(x + 2)39528753302000= (x + 3)(x + 2)= x(x + 2) + 3(x + 2)= x2 + 2x + 3x + 6= x2 + 5x + 6Multiply PolynomialsApply the distributive property.(a + b)(d + e + f)158686522733000(a + b)( d + e + f )= a(d + e + f) + b(d + e + f)= ad + ae + af + bd + be + bfMultiply Binomials(Model)Apply the distributive property. -6223026479500Example: (x + 3)(x + 2)42303701625601 =x =Key:x2 =001 =x =Key:x2 =93980149225x + 3x + 200x + 3x + 2 x2 + 2x + 3x + 6 = x2 + 5x + 6Multiply Binomials(Graphic Organizer)Apply the distributive property.-9525025336500Example: (x + 8)(2x – 3) = (x + 8)(2x + -3)23844252374902x + -3002x + -31564640286385x + 800x + 82x2-3x16x-242x2 + 16x + -3x + -24 = 2x2 + 13x – 24Multiply Binomials(Squaring a Binomial)(a + b)2 = a2 + 2ab + b2center71818500(a – b)2 = a2 – 2ab + b2Examples:28956013335 (3m + n)2 = 9m2 + 2(3m)(n) + n2 = 9m2 + 6mn + n2 2895602266950(y – 5)2 = y2 – 2(5)(y) + 25 = y2 – 10y + 25Multiply Binomials(Sum and Difference)(a + b)(a – b) = a2 – b22978154851400016808454673600Examples:(2b + 5)(2b – 5) = 4b2 – 2511251953273140(7 – w)(7 + w) = 49 – w2 Factors of a MonomialThe number(s) and/or variable(s) that are multiplied together to form a monomialExamples:FactorsExpanded Form5b25?b25?b?b6x2y6?x2?y2?3?x?x?y-5p2q32-52 ?p2?q312 ·(-5)?p?p?q?q?qFactoring(Greatest Common Factor)Find the greatest common factor (GCF) of all terms of the polynomial and then apply the distributive property.2164283391000 Example: 20a4 + 8a89250515875002 ? 2 ? 5 ? a ? a ? a ? a + 2 ? 2 ? 2 ? a21602709525common factors00common factorsGCF = 2 ? 2 ? a = 4a20a4 + 8a = 4a(5a3 + 2)Factoring(Perfect Square Trinomials)a2 + 2ab + b2 = (a + b)2a2 – 2ab + b2 = (a – b)21157936900550023241144069000Examples: x2 + 6x +9 = x2 + 2?3?x +32= (x + 3)22324106858000 4x2 – 20x + 25 = (2x)2 – 2?2x?5 + 52 = (2x – 5)2Factoring(Difference of Squares)a2 – b2 = (a + b)(a – b)641268745400Examples:center1739900 x2 – 49 = x2 – 72 = (x + 7)(x – 7)3752852508250 4 – n2 = 22 – n2 = (2 – n) (2 + n)9213851600200 9x2 – 25y2 = (3x)2 – (5y)2 = (3x + 5y)(3x – 5y)Difference of Squares(Model)-180340673735002139315590550baab00baaba2 – b2 = (a + b)(a – b)106362572390a2 – b200a2 – b2-252095499745a(a – b) + b(a – b)00a(a – b) + b(a – b)2940050496570(a + b)(a – b)00(a + b)(a – b)-67945566420003084830200025a + ba – b 00a + ba – b -217170253365baa – b a – b 00baa – b a – b Factoring(Sum and Difference of Cubes)a3 + b3 = (a + b)(a2 – ab + b2)a3 – b3 = (a – b)(a2 + ab + b2)-1009651397000Examples:27y3 + 1 = (3y)3 + (1)3 = (3y + 1)(9y2 – 3y + 1)x3 – 64 = x3 – 43 = (x – 4)(x2 + 4x + 16) Factoring (By Grouping)For trinomials of the formax2+bx+ccenter4006850030340302292350Example: 3x2+8x+4center125730ac = 3 4 = 12Find factors of ac that add to equal b12 = 2 6 2 + 6 = 800ac = 3 4 = 12Find factors of ac that add to equal b12 = 2 6 2 + 6 = 8341503027432000center99695005221605466090Rewrite 8x as 2x + 6x00Rewrite 8x as 2x + 6x52025551675765Factor out a common binomial00Factor out a common binomial52184301094740Group factors00Group factors3x2+2x+6x+4(3x2+2x)+(6x+4)x(3x+2)+2(3x+2)(3x+2)(x+2)Divide Polynomials(Monomial Divisor)Divide each term of the dividend by the monomial divisor451251276600 Example: (12x3 – 36x2 + 16x) 4x= 12x3 – 36x2 + 16x4x= 12x34x – 36x24x + 16x4x 3x2 – 9x + 4Divide Polynomials (Binomial Divisor)-13300459526700Factor and simplifyExample: (7w2 + 3w – 4) (w + 1)= 7w2 + 3w – 4w + 1= 7w – 4(w + 1)w + 1 7w – 4 Prime PolynomialCannot be factored into a product of lesser degree polynomial factorsExampler3t + 9x2 + 15y2 – 4y + 3NonexampleFactorsx2 – 4(x + 2)(x – 2)3x2 – 3x + 63(x + 1)(x – 2)x3x?x2Square Rootright778453radical symbolradicand or argument 00radical symbolradicand or argument x2 -11049050291900Simplify square root expressions.Examples: 9x2 = 32?x2 = (3x)2 = 3x-(x – 3)2 = -(x – 3) = -x + 3Squaring a number and taking a square root are inverse operations.Cube Root125095095885index00index3787149876308radicand or argument00radicand or argument469900705485radical symbol00radical symbol3x3 Simplify cube root expressions.1647710454380center2032000Examples:364 = 343 = 4164738014732003-27 = 3(-3)3 = -3164674515240003x3 = xCubing a number and taking a cube root are inverse operations.nth Root613410139065index00index3023235845820radicand or argument00radicand or argument-133985782320radical symbol00radical symbolnxm= xmn-463552032000Examples:564 = 543 = 435 6729x9y6 = 3x32ySimplify Radical Expressions Simplify radicals and combine like terms where possible.246421-103100Examples:590806246669012+3-32-112-8=-102-234-22=-5-234-22590806165199018-2327=23-23 =23-6Add and Subtract Radical ExpressionsAdd or subtract the numerical factors of the like radicals.center5651500Examples:2a+5a=2+5a=7a63xy-43xy-3xy=6-4-13xy=3xy24c+72-24c=2-24c+72=72Product Property of RadicalsThe nth root of a product equals the product of the nth roots.nab= na ? nb a ≥ 0 and b ≥ 0center1397000Examples:4x = 4 ? x = 2x5a3 = 5 ? a3 = a5a316 = 38?2 = 38 ? 32 = 232Quotient Propertyof RadicalsThe nth root of a quotient equals the quotient of the nth roots of the numerator and denominator.nab=nanb a ≥ 0 and b ? 02419353492500Examples:5y2 = 5y2 = 5y, y ≠ 0253=53?33=533Zero Product PropertyIf ab = 0,then a = 0 or b = 0.center2222500 Example:(x + 3)(x – 4) = 0(x + 3) = 0 or (x – 4) = 0x = -3 or x = 4The solutions or roots of the polynomial equation are -3 and 4.Solutions or Rootsx2 + 2x = 3Solve using the zero product property.-2349533337500x2 + 2x – 3 = 0(x + 3)(x – 1) = 0x + 3 = 0 or x – 1 = 0x = -3 or x = 1The solutions or roots of the polynomial equation are -3 and 1.Zeros The zeros of a function f(x) are the values of x where the function is equal to zero. -15240128905f(x) = x2 + 2x – 3Find f(x) = 0.0 = x2 + 2x – 30 = (x + 3)(x – 1)x = -3 or x = 100f(x) = x2 + 2x – 3Find f(x) = 0.0 = x2 + 2x – 30 = (x + 3)(x – 1)x = -3 or x = 1369951112890500-914406413500-34290610235The zeros of the function f(x) = x2 + 2x – 3 are -3 and 1 and are located at the x-intercepts (-3,0) and (1,0).00The zeros of the function f(x) = x2 + 2x – 3 are -3 and 1 and are located at the x-intercepts (-3,0) and (1,0).-34290353060The zeros of a function are also the solutions or roots of the related equation00The zeros of a function are also the solutions or roots of the related equationx-InterceptsThe x-intercepts of a graph are located where the graph crosses the x-axis and where f(x) = 0. -25400299720f(x) = x2 + 2x – 30 = (x + 3)(x – 1)0 = x + 3 or 0 = x – 1 x = -3 or x = 1The zeros are -3 and 1.The x-intercepts are:-3 or (-3,0)1 or (1,0)00f(x) = x2 + 2x – 30 = (x + 3)(x – 1)0 = x + 3 or 0 = x – 1 x = -3 or x = 1The zeros are -3 and 1.The x-intercepts are:-3 or (-3,0)1 or (1,0)-38735205740003592467153398Coordinate Plane270510354330Literal EquationA formula or equation that consists primarily of variables-6350047752000Examples:Ax + By = CA = 12bhV = lwhF = 95 C + 32A = πr2Vertical Linex = a (where a can be any real number)-33020317500Example: x = -4359918084455y00y14753592317754906645307975x00x-5403851088390Vertical lines have undefined slope. 00Vertical lines have undefined slope. Horizontal Liney = c(where c can be any real number)-8763012700003098800531495y00yExample:y = 61451229816614712970529590x00x-257810286385Horizontal lines have a slope of 0.00Horizontal lines have a slope of 0.Quadratic Equationax2 + bx + c = 0a 0-19875518542000Example: x2 – 6x + 8 = 0Solve by factoringSolve by graphingx2 – 6x + 8 = 0(x – 2)(x – 4) = 0(x – 2) = 0 or (x – 4) = 0 x = 2 or x = 4159004031927800028898853081020x020000x1017905781050y020000y503555900430 Graph the related function f(x) = x2 – 6x + 8. -358140405765Solutions (roots) to the equation are 2 and 4; the x-coordinates where the function crosses the x-axis.020000Solutions (roots) to the equation are 2 and 4; the x-coordinates where the function crosses the x-axis.Quadratic Equation (Number/Type of Solutions)ax2 + bx + c = 0, a 0ExamplesGraph of the related functionNumber and Type of Solutions/Rootsx2 – x = 371380661242 Real rootsx2 + 16 = 8x82397418411 distinct Real rootwith a multiplicity of two2x2 – 2x + 3 = 093414946760 Real roots;2 Complex rootsInequalityAn algebraic sentence comparing two quantitiesSymbolMeaning<less thanless than or equal togreater thangreater than or equal tonot equal to-869953429000Examples:-10.5 ? -9.9 – 1.28 > 3t + 2x – 5y -12 r 3Graph of an InequalitySymbolExamplesGraph< ; x < 3168275524510 ; -3 y121920393700t -2106456489324 Transitive Property of InequalityIfThena b and b c a ca b and b c a ccenter2540000Examples: If 4x 2y and 2y 16, then 4x 16.If x y – 1 and y – 1 3, then x 3.Addition/Subtraction Property of InequalityIfThena > ba + c > b + ca ba + c b + ca < ba + c < b + ca ba + c b + ccenter44767500Example:d – 1.9 -8.7d – 1.9 + 1.9 -8.7 + 1.9d -6.8Multiplication Property of InequalityIfCaseThen a < bc > 0, positiveac < bca > bc > 0, positiveac > bca < bc < 0, negativeac > bca > bc < 0, negativeac < bccenter44132500Example: If c = -25 > -33297555146050005(-2) < -3(-2)-10 < 6Division Property of InequalityIfCaseThen a < bc > 0, positiveac < bca > bc > 0, positiveac > bca < bc < 0, negativeac > bca > bc < 0, negativeac < bccenter2730500Example: If c = -4337566068199000-90 -4t-90-4 -4t-422.5 tAbsolute Value InequalitiesAbsolute Value InequalityEquivalent Compound Inequality “and” statement“or” statement 28263317772600Example: Linear Equation (Standard Form) Ax + By = C(A, B and C are integers; A and B cannot both equal zero)118110255906003754755150657y00y292862030289500 Example: -2x + y = -3519828739193x020000xThe graph of the linear equation is a straight line and represents all solutions (x, y) of the equation.Linear Equation (Slope-Intercept Form)y = mx + b(slope is m and y-intercept is b)8064536576000Example: y = -43 x + 53550920436880(0,5)-4300(0,5)-432181202241389445135417195m = -43b = 500m = -43b = 5Linear Equation(Point-Slope Form)y – y1 = m(x – x1)where m is the slope and (x1,y1) is the point12065051689000Example: Write an equation for the line that passes through the point (-4,1) and has a slope of 2.y – 1 = 2(x – -4)y – 1 = 2(x + 4)y = 2x + 9Equivalent Forms of a Linear EquationForms of a Linear Equation3y = 2 – 4xSlope-Intercepty = -43x+2Point-Slopey--2=-43(x-3)Standard4x+3y=2SlopeA number that represents the rate of change in y for a unit change in x 50165447040002406652184404327525280035Slope = 2300Slope = 231049655130810320032The slope indicates the steepness of a line.Slope Formula The ratio of vertical change tohorizontal change655320128905AB(x1, y1)(x2, y2)x2 – x1y2 – y1 xy00AB(x1, y1)(x2, y2)x2 – x1y2 – y1 xy 2671998199044slope = m = Slopes of Lines-228592396586Line phas a positive slope.Line n has a negative slope.Vertical line s has an undefined slope.Horizontal line t has a zero slope.00Line phas a positive slope.Line n has a negative slope.Vertical line s has an undefined slope.Horizontal line t has a zero slope.30384751720853145155445135Perpendicular LinesLines that intersect to form a right angle99949067945Perpendicular lines (not parallel to either of the axes) have slopes whose product is -1.-17335553340Example: The slope of line n = -2. The slope of line p = 12.-2 ? 12 = -1, therefore, n is perpendicular to p.00Example: The slope of line n = -2. The slope of line p = 12.-2 ? 12 = -1, therefore, n is perpendicular to p.Parallel LinesLines in the same plane that do not intersect are parallel.1011555447040yxba00yxbaParallel lines have the same slopes.-19050189230Example: The slope of line a = -2. The slope of line b = -2.-2 = -2, therefore, a is parallel to b.00Example: The slope of line a = -2. The slope of line b = -2.-2 = -2, therefore, a is parallel to b.Mathematical NotationEquation or InequalitySet NotationInterval Notation0 < x 3{x|0<x≤3}(0, 3]y ≥ -5{y: y ≥ -5}[-5, +∞)z<-1 or z ≥ 3 {z| z<-1 or z ≥ 3}(-∞,-1) ∪[3, +∞)x < 5 or x > 5{x: x 5}(-∞,5)∪(5, +∞)System of Linear Equations(Graphing)20002501460500-x + 2y = 32x + y = 4center3175000296481557150122555237490The solution, (1, 2), is the only ordered pair that satisfies both equations (the point of intersection).00The solution, (1, 2), is the only ordered pair that satisfies both equations (the point of intersection).System of Linear Equations(Substitution)20008853175000x + 4y = 17y = x – 2center9525000Substitute x – 2 for y in the first equation.x + 4(x – 2) = 17x = 5Now substitute 5 for x in the second equation.y = 5 – 2y = 3The solution to the linear system is (5, 3),the ordered pair that satisfies both equations.System of Linear Equations(Elimination)20453353746500-5x – 6y = 85x + 2y = 4-23622022225000Add or subtract the equations to eliminate one variable. -5x – 6y = 8+ 5x + 2y = 420758153746500 -4y = 12 y = -3Now substitute -3 for y in either original equation to find the value of x, the eliminated variable.-5x – 6(-3) = 8 x = 2The solution to the linear system is (2,-3), the ordered pair that satisfies both equations.System of Linear Equations(Number of Solutions)Number of SolutionsSlopes and y-intercepts906145763270xy00xyGraphOne solutionDifferent slopes9867901513840xy00xy42418017780No solutionSame slope anddifferent -intercepts9632951577340xy00xy44577028575Infinitely many solutionsSame slope andsame y-intercepts8001413950530044373825527Graphing Linear Inequalitiescenter6101500The graph of the solution of a linear inequality is a half-plane bounded by the graph of its related linear equation. Points on the boundary are included unless the inequality contains only < or >.00The graph of the solution of a linear inequality is a half-plane bounded by the graph of its related linear equation. Points on the boundary are included unless the inequality contains only < or >.ExampleGraphy x + 228892595251669415-31750y00y31235652035810x00xy > -x – 11715770-56515y00y3166745727075x00x222885-53975System of Linear InequalitiesSolve by graphing:185166010477500y x – 34241165557530y00yy -2x + 3282395269843400-5524580645003714387276950174625327025The solution region contains all ordered pairs that are solutions to both inequalities in the system.(-1,1) is one solution to the system located in the solution region.00The solution region contains all ordered pairs that are solutions to both inequalities in the system.(-1,1) is one solution to the system located in the solution region.3990340389890005942965539750x00xLinear ProgrammingAn optimization process consisting of a system of constraints and an objective quantity that can be maximized or minimized-8382020256500 Example:Find the minimum and maximum value of the objective function C = 4x + 5y, subject to the following constraints.245027836344x 075819013970(6,0)(0,0)(0,6)feasible region00(6,0)(0,0)(0,6)feasible regiony 04227195372745x + y 600x + y 6x + y 6The maximum or minimum value for C = 4x + 5y will occur at a corner point of the feasible region.Dependent andIndependent Variablex, independent variable(input values or domain set)y, dependent variable2546351175385Example:y = 2x + 7 00Example:y = 2x + 7 (output values or range set)Dependent andIndependent Variable(Application)Determine the distance a car will travel going 55 mph.hd0015521103165d = 55h4415155186055dependent020000dependent-400685192405independent020000independentGraph of a Quadratic Equationy = ax2 + bx + ca 0-438154699000430339523495004446905118745y00y2708529264795Example: y = x2 + 2x – 3483235190500line of symmetry020000line of symmetry27774904178300056972201343025x00x1678305763905vertex020000vertex2793365100203000423545091694000The graph of the quadratic equation is a curve (parabola) with one line of symmetry and one vertex.Vertex of a Quadratic FunctionFor a given?quadratic?y = ax2+ bx + c, the?vertex?(h, k) is found by computing h = -b2a and then evaluating y at h to find k.-103723-5175003798258484457Example: y=x2+2x-8h=-b2a=-22(1)=-1k=-12+2-1-8 =-9The vertex is (-1,-9).Line of symmetry is x=h.x=-1Quadratic Formula Used to find the solutions to any quadratic equation of the form, f(x) = ax2 + bx + cx = -b ± b2- 4ac 2acenter41211500Example: g(x)=2x2-4x-3x=--4±-42-42-322x=2+102, 2-102RelationA set of ordered pairsExamples:x167202333278y-34001-6171894542037000222534285574675005261048516510Example 2020000Example 2-19922004668Example 1020000Example 1{(0,4), (0,3), (0,2), (0,1)}258064038100Example 3020000Example 3Function(Definition)A relationship between two quantities in which every input corresponds to exactly one output5814060499110yrange00yrange-300990441960xdomain00xdomain423608515875001426210527050023114004768850019227807048524681000246810469011022034510753001075324218905784850023945854616450024555453803650025870563407600A relation is a function if and only if each element in the domain is paired with a unique element of the range.Functions(Examples)3043984143407xy322402-126445252696845Example 1020000Example 1-2203453648075{(-3,4), (0,3), (1,2), (4,6)}020000{(-3,4), (0,3), (1,2), (4,6)}48482255750560Example 4020000Example 410096504236720Example 3020000Example 336969701754505Example 2020000Example 23723640165989000307086010572750064160404584700x020000x54165502032000y020000y41186102228850 Domain the set of all possible values of the independent variable center19685003481070580390Examples:inputoutputxg(x)-20113030-55372000-1102134557395153035f(x)020000f(x)3764915563880005803900478790x020000x132080351790The domain of g(x) is {-2, -1, 0, 1}.020000The domain of g(x) is {-2, -1, 0, 1}.3420110361950The domain of f(x) is all real numbers.020000The domain of f(x) is all real numbers.Rangethe set of all possible values of the dependent variable-44143222534004792980239395f(x)020000f(x)3731623165826 -5270510795004792980239395f(x)020000f(x)3731623165826Examples:inputoutputxg(x)-20-1254635-44577000102134984115168910006059805675005x020000x3515360485775The range of f(x) is all real numbers greater than or equal to zero.020000The range of f(x) is all real numbers greater than or equal to zero.353060539750The range of g(x) is {0, 1, 2, 3}.020000The range of g(x) is {0, 1, 2, 3}.Function Notation f(x)f(x) is read “the value of f at x” or “f of x”center38544500Example:f(x) = -3x + 5, find f(2).f(2) = -3(2) + 5f(2) = -6Letters other than f can be used to name functions, e.g., g(x) and h(x)End BehaviorThe value of a function as x approaches positive or negative infinity807513665520067161238542300 Examples:12705914029469f(x) approaches 0 as the values of x approach -∞. f(x) is approaches +∞ as the values of x approach +∞. 00f(x) approaches 0 as the values of x approach -∞. f(x) is approaches +∞ as the values of x approach +∞. 3221530488447f(x) approaches +∞ as the values of x approach -∞. f(x) is approaches +∞ as the values of x approach +∞.020000f(x) approaches +∞ as the values of x approach -∞. f(x) is approaches +∞ as the values of x approach +∞.3738055260168600Increasing/ DecreasingA function can be described as increasing, decreasing, or constant over a specified interval or the entire domain. 249174012001500755015305104y00y-100965571500 Examples: 1198880112395y00y-71252682590045942341596580056451505715y00y21811952227388f(x) is decreasing over {x|-∞<x<0} because the values of f(x) decrease as the values of x increase. f(x) is increasing over {x|0<x<+∞} because the values of f(x) increase as the values of x increase.00f(x) is decreasing over {x|-∞<x<0} because the values of f(x) decrease as the values of x increase. f(x) is increasing over {x|0<x<+∞} because the values of f(x) increase as the values of x increase.2208530224790x00x761421648823f(x) is decreasing over the entire domain because the values of f(x) decrease as the values of x increase.00f(x) is decreasing over the entire domain because the values of f(x) decrease as the values of x increase.47952731590477f(x) is constant over the entire domain because the values of f(x) remain constant as the values of x increase.00f(x) is constant over the entire domain because the values of f(x) remain constant as the values of x increase.6560267112947x00xAbsolute ExtremaThe largest (maximum) and smallest (minimum) value of a function on the entire domain of a function (the absolute or global extrema)5243162398681y00y-9036837314900 412493528893400 Examples: 609603940810A function, f, has an absolute maximum located at x = a if f(a) is the largest value of f over its domain.A function, f, has an absolute minimum located at x = a if f(a) is the smallest value of f over its domain.00A function, f, has an absolute maximum located at x = a if f(a) is the largest value of f over its domain.A function, f, has an absolute minimum located at x = a if f(a) is the smallest value of f over its domain.1888077387820absolute maximum00absolute maximum1809486574675003433517888234x00x149517481760y00y25995539092038672402824167absolute minimum00absolute minimum512979428651780061700721183929x00xParent Functions(Linear, Quadratic)3571512158684yxyxLinear f(x) = x35115175715yxyx Quadratic f(x) = x2Parent Functions(Exponential, Logarithmic)326657120219300Exponential f(x) = bxb > 1301388538862000Logarithmicf(x) = logbxb > 1Transformations of Parent Functions(Translation)Parent functions can be transformed to create other members in a family of graphs.Translationsg(x) = f(x) + kis the graph of f(x) translated vertically –k units up when k > 0.k units down when k < 0.g(x) = f(x ? h)is the graph of f(x) translated horizontally –h units right when h > 0.h units left when h < 0.Transformations of Parent Functions(Reflection)Parent functions can be transformed to create other members in a family of graphs.Reflectionsg(x) = -f(x)is the graph of f(x) –reflected over the x-axis.g(x) = f(-x)is the graph of f(x) –reflected over the y-axis.Transformations of Parent Functions(Dilations)Parent functions can be transformed to create other members in a family of graphs.Dilationsg(x) = a · f(x)is the graph of f(x) –vertical dilation (stretch) if a > 1.vertical dilation (compression) if 0 < a < 1.g(x) = f(ax)is the graph of f(x) –horizontal dilation (compression) if a > 1.horizontal dilation (stretch) if 0 < a < 1.Linear Function(Transformational Graphing)Translationg(x) = x + b2565402114550034377093126924693285136525y020000y311785622300Examples:f(x) = xt(x) = x + 4h(x) = x – 2 00Examples:f(x) = xt(x) = x + 4h(x) = x – 2 6029960167640x020000xVertical translation of the parent function, f(x) = xLinear Function(Transformational Graphing)Dilation (m>0)g(x) = mx4566285347345y020000y32086553625851993901524000173355472440Examples:f(x) = xt(x) = 2xh(x) = 12x00Examples:f(x) = xt(x) = 2xh(x) = 12x5814695718185x020000xVertical dilation (stretch or compression) of the parent function, f(x) = x Linear Function(Transformational Graphing)Dilation/Reflection (m<0)g(x) = mx90170405130004257040328930y020000y251460457200Examples:f(x) = xt(x) = -xh(x) = -3xd(x) = -13x00Examples:f(x) = xt(x) = -xh(x) = -3xd(x) = -13x31934861250945526405735965x020000xVertical dilation (stretch or compression) with a reflection of f(x) = x Quadratic Function(Transformational Graphing)Vertical Translationh(x) = x2 + c46621705715y020000y374659271000333527413208059759852699385x020000x147955291465Examples:f(x) = x2g(x) = x2 + 2t(x) = x2 – 300Examples:f(x) = x2g(x) = x2 + 2t(x) = x2 – 3Vertical translation of f(x) = x2Quadratic Function(Transformational Graphing)Horizontal Translation h(x) = (x + c)2-19685478790004283710511175y020000y25885148509019050458470Examples:f(x) = x2g(x) = (x + 2)2t(x) = (x – 3)200Examples:f(x) = x2g(x) = (x + 2)2t(x) = (x – 3)26137275189230x020000xHorizontal translation of f(x) = x2Quadratic Function(Transformational Graphing)Dilation (a>0)h(x) = ax24597400411480y020000y292633441338580645203200061372752802890x020000x31432591440Examples: f(x) = x2 g(x) = 2x2 t(x) = 13x200Examples: f(x) = x2 g(x) = 2x2 t(x) = 13x2Vertical dilation (stretch or compression) of f(x) = x2Quadratic Function(Transformational Graphing)Dilation/Reflection (a<0)h(x) = ax24951095326390y020000y8890380365003757549328930539750480695Examples: f(x) = x2 g(x) = -2x2 t(x) = -13x2020000Examples: f(x) = x2 g(x) = -2x2 t(x) = -13x25984875156845x020000xVertical dilation (stretch or compression) with a reflection of f(x) = x2Arithmetic SequenceA sequence of numbers that has a common difference between every two consecutive terms-6413552705001551940421640+5+5+5+500+5+5+5+5Example: -4, 1, 6, 11, 16 …2645410243840common difference00common difference39706555403854429125390525y020000yPositionxTermy1779780140970+5+5+5+500+5+5+5+5-4213641151649434751454151515001515 628967537465x020000x608330975995The common difference is the slope of the line of best fit.020000The common difference is the slope of the line of best fit.Geometric SequenceA sequence of numbers in which each term after the first term is obtained by multiplying the previous term by a constant ratiocenter29718000ProbabilityThe likelihood of an event occurringProbability of an event = number of favorable outcomesnumber of possible outcomes-2413053657500Example: What is the probability of drawing an A from the bag of letters shown?-77470127050800403860A C CA B A B020000A C CA B A B P(A) = 371377315336550370 12 100370 12 1Probability of Independent Events-68580425450003152775281940What is the probability of landing on green on the first spin and then landing on yellow on the second spin?00What is the probability of landing on green on the first spin and then landing on yellow on the second spin?4123055227965020000Example:1683385619125001185545553085Y020000Y1793240547370G020000G3007272565182283460274955B020000B776605261620Y020000Y2289175381635B020000B732155339725G020000G1222375242570B020000B1798955267335G020000GP(green and yellow) =P(green) ? P(yellow) = 38?14 = 332Probability of Dependent Events-66040389255004320540116840Candy JarRRGYPBRRGYPBCandy JarRRGYPBRRGYPBExample: 88265190500What is the probability of selecting a red jelly bean on the first pick and without replacing it, selecting a blue jelly bean on the second pick?00What is the probability of selecting a red jelly bean on the first pick and without replacing it, selecting a blue jelly bean on the second pick?P(red and blue) = 35852102487332329815469900001201420810895“blue after red”00“blue after red” P(red) ? P(blue|red) =Probability(Mutually Exclusive)-53340105791000Events that cannot occur at the same timeExamples:A. Tossing a coin and getting heads. B. Tossing a coin and getting tails. 453771014478000A. Turning left.B. Turning right.1197595904500PA and B= 018919632219If two events are mutually exclusive, then the probability of them both occurring at the same time is 0.020000If two events are mutually exclusive, then the probability of them both occurring at the same time is 0.Fundamental Counting PrincipleIf there are m ways for one event to occur and n ways for a second event to occur, then there are m?n ways for both events to occur.-889002349500Example: How many outfits can Joey make using 3 pairs of pants and 4 shirts?3 ? 4 = 12 outfits7915744492600390053154046100PermutationAn ordered arrangement of a group of objects47952562576621st 2nd 3rd 001st 2nd 3rd 1282542339111st 2nd3rd 001st 2nd3rd is different from Both arrangements are included in possible outcomes. -539753937000Example:5 people to fill 3 chairs (order matters). How many ways can the chairs be filled?1st chair – 5 people to choose from2nd chair – 4 people to choose from3rd chair – 3 people to choose from# possible arrangements are 5 ? 4 ? 3 = 60Permutation(Formula)To calculate the number of permutations1816100119380n and r are positive integers, n ≥ r, and n is the total number of elements in the set and r is the number to be ordered.-16862939480500Example: There are 30 cars in a car race. The first-, second-, and third-place finishers win a prize. How many different arrangements (order matters)of the first three positions are possible?30P3 = 30!(30-3)! = 30!27! = 24360CombinationThe number of possible ways to select or arrange objects when there is no repetition and order does not matter-66675508000Example: If Sam chooses 2 selections from triangle, square, circle and pentagon. How many different combinations are possible?48907703721101500997371475Order (position) does not matter so450278525771196682225245 is the same as40119726025119945353556000453898034925001513840311150049301403937000There are 6 possible bination(Formula)To calculate the number of possible combinations using a formula1780540254635n and r are positive integers, n ≥ r, and n is the total number of elements in the set and r is the number to be ordered.left173350013754101349375Example: In a class of 24 students, how many ways can a group of 4 students be arranged (order does not matter)?Statistics NotationSymbolRepresentationxiith element in a data set μmean of the data set σ2variance of the data setσstandard deviation of the data setnnumber of elements in the data setMeanA measure of central tendencycenter27114500Example: Find the mean of the given data set.Data set: 0, 2, 3, 7, 8Balance Point8445505016544231 0 1 2 3 4 5 6 7 80044231 0 1 2 3 4 5 6 7 8Numerical Average111823536766500541147073914000MedianA measure of central tendency-1714543561000Examples: Find the median of the given data sets.323913544259500Data set: 6, 7, 8, 9, 9The median is 8.311975543561000Data set: 5, 6, 8, 9, 11, 12 The median is 8.5.ModeData SetsMode3, 4, 6, 6, 6, 6, 10, 11, 1460, 3, 4, 5, 6, 7, 9, 10none5.2, 5.2, 5.2, 5.6, 5.8, 5.9, 6.05.21, 1, 2, 5, 6, 7, 7, 9, 11, 121, 7bimodalA measure of central tendency-342907747000Examples:5374640-1290955A1, A2, AFDA020000A1, A2, AFDASummation3403600271145004378325-45085stopping pointupper limit00stopping pointupper limit2966085269240849630252095summation sign00summation sign232092515367000395033523431500437896027940typical element00typical element232854517081500932180116840index of summation00index of summation438023084455starting pointlower limit00starting pointlower limit353758517780000This expression means sum the values of x, starting at x1 and ending at xn.1385272167266229362010160= x1 + x2 + x3 + … + xn00= x1 + x2 + x3 + … + xn10731535814000 Example: Given the data set {3, 4, 5, 5, 10, 17}i=16xi=3 + 4 + 5 + 5 + 10 + 17 = 44VarianceA measure of the spread of a data set 86550548895The mean of the squares of the differences between each element and the mean of the data setNote: The square root of the variance is equal to the standard deviation.Standard Deviation(Definition)A measure of the spread of a data set42291045720The square root of the mean of the squares of the differences between each element and the mean of the data set or the square root of the varianceStandard Deviation(Graphic)A measure of the spread of a data set6038851447805880101282704462145584835Smaller Larger00Smaller Larger33345175746300Comparison of two distributions with same mean () and different standard deviation () valuesz-Score(Definition)The number of standard deviations an element is away from the mean983524106227where x is an element of the data set, μ is the mean of the data set, and σ is the standard deviation of the data set.-558806350000Example: Data set A has a mean of 83 and a standard deviation of 9.74. What is the z‐score for the element 91 in data set A? z = 91-839.74 = 0.821z-Score(Graphic)The number of standard deviations an element is from the mean126873012827000-273129751200698269136913z = 1z = 2z = 3z = -1z = -2z = -3z = 000z = 1z = 2z = 3z = -1z = -2z = -3z = 0Empirical Rule-26329470940-406684174784Normal Distribution Empirical Rule (68-95-99.7 rule) – approximate percentage of element distribution020000Normal Distribution Empirical Rule (68-95-99.7 rule) – approximate percentage of element distribution927100273437608270875-4760595Z=100Z=16051550-5868670Given μ = 45 and σ = 24020000Given μ = 45 and σ = 24156210-6472555Elements within one standard deviation of the mean00Elements within one standard deviation of the meanElements within One Standard Deviation (σ) of the Mean (?)(Graphic)2022526985000ScatterplotGraphical representation of the relationship between two numerical sets of data1531620585470xy00xyPositive Linear Relationship (Correlation)In general, a relationship where the dependent (y) values increase as independent values (x) increase1308735509270xy00xyNegative Linear Relationship (Correlation)In general, a relationship where the dependent (y) values decrease as independent (x) values increase.1483360250825xy00xyNo Correlation No relationship between the dependent (y) values and independent (x) values. 14484359525 xy00 xyCurve of Best Fit(Linear)-426720386715400053194050Equation of Curve of Best Fity = 11.731x + 193.850Equation of Curve of Best Fity = 11.731x + 193.85Curve of Best Fit(Quadratic)354330691515-1009654095115Equation of Curve of Best Fity = -0.01x2 + 0.7x + 60Equation of Curve of Best Fity = -0.01x2 + 0.7x + 6Curve of Best Fit(Exponential)394735824628681068784215106Equation of Curve of Best Fit 0Equation of Curve of Best Fit center484357Bacteria Growth Over Time00Bacteria Growth Over Time637606839586002567305348107000Outlier Data(Graphic)811530939808210552609850 ................
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