Algebra, Functions, And Data Analysis Vocabulary Cards

Factoring: Sum andDifference of Cubes3322a b (a b)(a – ab b )3322a – b (a – b)(a ab b )Examples:33327y 1 (3y) (1)2 (3y 1)(9y – 3y 1)3332x – 64 x – 4 (x – 4)(x 4x 16)Virginia Department of Education, 2014AFDA Vocabulary CardsPage 43

Difference of Squares22a – b (a b)(a – b)2a2a –babb(a b)(a – b)a(a – b) b(a – b)a baa–ba–bba–bVirginia Department of Education, 2014AFDA Vocabulary CardsPage 44

Divide PolynomialsDivide each term of the dividendby the monomial divisorExample:(12x – 36x 16x) 4x323233 x2412 – 3 412 41 142 3x – 9x 4Virginia Department of Education, 2014AFDA Vocabulary CardsPage 45

Divide Polynomialsby BinomialsFactor and simplifyExample:(7w 3w – 4) (w 1)22 3 –41–411 7w – 4Virginia Department of Education, 2014AFDA Vocabulary CardsPage 46

Prime PolynomialCannot be factored into a product oflesser degree polynomial factorsExampler3t 92x 125y – 4y 3NonexampleFactors2x –4(x 2)(x – 2)23x – 3x 6 3(x 1)(x – 2)23xxxVirginia Department of Education, 2014AFDA Vocabulary CardsPage 47

Square Root2radical symbolradicand or argumentSimply square root expressions.Examples:2-2 3 x2 23x 3x2– 3 -(x – 3) -x 3Squaring a number and taking a squareroot are inverse operations.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 48

Cube Rootindex33radical symbolradicand or argumentSimplify cube root expressions.Examples:3334 -2 33343 43-3 -3 xCubing a number and taking a cuberoot are inverse operations.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 49

thn Rootindex radicand or argumentradical symbolExamples:34 2Virginia Department of Education, 201443 4 3AFDA Vocabulary Cards32Page 50

Product Property ofRadicalsThe square root of a product equalsthe product of the square rootsof the factors. a 0 and b 0Examples:4 4 33 3Virginia Department of Education, 2014 23 3AFDA Vocabulary Cards a33 2 2 2Page 51

Quotient Propertyof RadicalsThe square root of a quotient equals thequotient of the square roots of thenumerator and denominator. a 0 and b 0Example:2Virginia Department of Education, 2014 2AFDA Vocabulary Cards,y 0Page 52

Zero ProductPropertyIf ab 0,then a 0 or b 0.Example:(x 3)(x – 4) 0(x 3) 0 or (x – 4) 0x -3 or x 4The solutions are -3 and 4, alsocalled roots of the equation.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 53

Solutions or Roots2x 2x 3Solve using the zero product property.2x 2x – 3 0(x 3)(x – 1) 0x 3 0orx–1 0x -3 or x 1The solutions or roots of thepolynomial equation are -3 and 1.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 54

ZerosThe zeros of a function f(x) are thevalues of x where the function is equalto zero.f(x) x2 2x – 3Find f(x) 0.20 x 2x – 30 (x 3)(x – 1)x -3 or x 1The zeros are -3 and 1located at (-3,0) and (1,0).The zeros of a function are also thesolutions or roots of the related equation.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 55

x-InterceptsThe x-intercepts of a graph are locatedwhere the graph crosses the x-axis andwhere f(x) 0.2f(x) x 2x – 30 (x 3)(x – 1)0 x 3 or 0 x – 1x -3 or x 1The zeros are -3 and 1.The x-intercepts are: -3 or (-3,0) 1 or (1,0)Virginia Department of Education, 2014AFDA Vocabulary CardsPage 56

Coordinate PlaneVirginia Department of Education, 2014AFDA Vocabulary CardsPage 57

Linear EquationAx By C(A, B and C are integers; A and B cannot bothequal zero.)Example:6y543-2x y -321x0-3-2-1-1012345-2-3-4-5-6-7-8The graph of the linear equation is astraight line and represents all solutions(x, y) of the equation.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 58

Linear Equation:Standard FormAx By C(A, B, and C are integers;A and B cannot both equal zero.)Examples:4x 5y -24x – 6y 9Virginia Department of Education, 2014AFDA Vocabulary CardsPage 59

Literal EquationA formula or equation whichconsists primarily of variablesExamples:ax b cA 12V lwhF C 32A πrVirginia Department of Education, 20142AFDA Vocabulary CardsPage 60

Vertical Linex a(where a can be any real number)x -4Example:4y321x0-5-4-3-2-10123-1-2-3-4Vertical lines have an undefined slope.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 61

Horizontal Liney c(where c can be any real number)y 6Example:7y654321x0-4-3-2-101234-1-2Horizontal lines have a slope of 0.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 62

Quadratic Equation2ax bx c 0a 0Example: x2 – 6x 8 0Solve by factoringSolve by graphingx2 – 6x 8 0Graph the relatedfunction f(x) x2 – 6x 8.y(x – 2)(x – 4) 0(x – 2) 0 or (x – 4) 0x 2 or x 4xSolutions to the equation are 2 and 4;the x-coordinates where the curve crosses the x-axis.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 63

Quadratic Equation2ax bx c 0a 0Example solved by factoring:x2 – 6x 8 0Quadratic equation(x – 2)(x – 4) 0Factor(x – 2) 0 or (x – 4) 0 Set factors equal to 0x 2 or x 4Solve for xSolutions to the equation are 2 and 4.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 64

Quadratic Equation2ax bx c 0a 0Example solved by graphing:x2 – 6x 8 0Graph the relatedfunction2f(x) x – 6x 8.Solutions to the equation are thex-coordinates (2 and 4) of the pointswhere the curve crosses the x-axis.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 65

Quadratic Equation:Number of Real Solutions2ax bx c 0,ExamplesGraphs2x –x 3Number of RealSolutions/Roots21 distinctroot2x 16 8xwith a multiplicityof two2x2 – 2x 3 0Virginia Department of Education, 2014a 00AFDA Vocabulary CardsPage 66

Identity Property ofAdditiona 0 0 a aExamples:3.8 0 3.86x 0 6x0 (-7 r) -7 rZero is the additive identity.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 67

Inverse Property ofAdditiona (-a) (-a) a 0Examples:4 (-4) 00 (-9.5) 9.5x (-x) 00 3y (-3y)Virginia Department of Education, 2014AFDA Vocabulary CardsPage 68

CommutativeProperty ofAdditiona b b aExamples:2.76 3 3 2.76x 5 5 x(a 5) – 7 (5 a) – 711 (b – 4) (b – 4) 11Virginia Department of Education, 2014AFDA Vocabulary CardsPage 69

AssociativeProperty ofAddition(a b) c a (b c)Examples:31 1031103x (2x 6y) (3x 2x) 6yVirginia Department of Education, 2014AFDA Vocabulary CardsPage 70

Identity Property ofMultiplicationa 1 1 a aExamples:3.8 (1) 3.86x 1 6x1(-7) -7One is the multiplicative identity.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 71

Inverse Property ofMultiplication11a a 1a 0Examples:1 1 1, x 0-13 (-3p) 1p p1The multiplicative inverse of a is .Virginia Department of Education, 2014AFDA Vocabulary CardsPage 72

CommutativeProperty ofMultiplicationab baExamples:(-8)23 23(-8)y y4(2x 3 4 3 2x)8 5x 8 x Virginia Department of Education, 2014AFDA Vocabulary CardsPage 73

AssociativeProperty ofMultiplication(ab)c a(bc)Examples:(1 8 334 1 8 334)(3x)x 3(x x)Virginia Department of Education, 2014AFDA Vocabulary CardsPage 74

DistributivePropertya(b c) ab acExamples:1 3 1 32 x 2 5 2(x 5)3.1a (1)(a) (3.1 1)aVirginia Department of Education, 2014AFDA Vocabulary CardsPage 75

DistributiveProperty4(y 2) 4y 4(2)44(y 2)y 244y 4(2)yVirginia Department of Education, 2014AFDA Vocabulary Cards2Page 76

MultiplicativeProperty of Zeroa 0 0 or 0 a 0Examples:283 · 0 00 · (-13y – 4) 0Virginia Department of Education, 2014AFDA Vocabulary CardsPage 77

SubstitutionPropertyIf a b, then b can replace a in agiven equation or inequality.Examples:GivenGivenSubstitutionr 93r 273(9) 27b 5a24 b 824 5a 8y 2x 12y 3x – 22(2x 1) 3x – 2Virginia Department of Education, 2014AFDA Vocabulary CardsPage 78

Reflexive Propertyof Equalitya aa is any real numberExamples:-4 -43.4 3.49y 9yVirginia Department of Education, 2014AFDA Vocabulary CardsPage 79

Symmetric Propertyof EqualityIf a b, then b a.Examples:If 12 r, then r 12.If -14 z 9, then z 9 -14.If 2.7 y x, then x 2.7 y.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 80

Transitive Propertyof EqualityIf a b and b c,then a c.Examples:If 4x 2y and 2y 16,then 4x 16.If x y – 1 and y – 1 -3,then x -3.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 81

InequalityAn algebraic sentence comparing twoquantitiesSymbolMeaning less thanless than or equal togreater thangreater than or equal tonot equal toExamples:-10 -9.9 – 1.28 3t 2x – 5y -12r 3Virginia Department of Education, 2014AFDA Vocabulary CardsPage 82

Graph of anInequalitySymbol Examples or x 3 or -3 y t -2Virginia Department of Education, 2014AFDA Vocabulary CardsGraphPage 83

Transitive Propertyof InequalityIfThena b and b ca b and b ca ca cExamples:If 4x 2y and 2y 16,then 4x 16.If x y – 1 and y – 1 3,then x 3.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 84

Addition/SubtractionProperty ofInequalityIfa ba ba ba bThena c b ca c b ca c b ca c b cExample:d – 1.9 -8.7d – 1.9 1.9 -8.7 1.9d -6.8Virginia Department of Education, 2014AFDA Vocabulary CardsPage 85

MultiplicationProperty ofInequalityIfa ba ba ba bCasec 0, positivec 0, positivec 0, negativec 0, negativeThenac bcac bcac bcac bcExample: if c -25 -35(-2) -3(-2)-10 6Virginia Department of Education, 2014AFDA Vocabulary CardsPage 86

Division Property ofInequalityIfCaseThena bc 0, positive a bc 0, positive a bc 0, negative a bc 0, negative Example: if c -4-90 -4t- 0-4 -4-422.5 tVirginia Department of Education, 2014AFDA Vocabulary CardsPage 87

Linear Equation:Slope-Intercept Formy mx b(slope is m and y-intercept is b)Example:y -43x 5(0,5)m --43b -5Virginia Department of Education, 2014AFDA Vocabulary CardsPage 88

Linear Equation:Point-Slope Formy – y1 m(x – x1)where m is the slope and (x1,y1) is the pointExample:Write an equation for the line thatpasses through the point (-4,1) and hasa slope of 2.y – 1 2(x – -4)y – 1 2(x 4)y 2x 9Virginia Department of Education, 2014AFDA Vocabulary CardsPage 89

SlopeA number that represents the rate ofchange in y for a unit change in x3Slope 223The slope indicates thesteepness of a line.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 90

Slope FormulaThe ratio of vertical change tohorizontal changeyx2 – x1B(x2, y2)y2 – y1A(x1, y1)xslope m Virginia Department of Education, 2014AFDA Vocabulary CardsPage 91

Slopes of LinesLine phas a positiveslope.Line nhas a negativeslope.Vertical line s hasan undefinedslope.Horizontal line thas a zero slope.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 92

Perpendicular LinesLines that intersect to form a right anglePerpendicular lines (not parallel toeither of the axes) have slopes whoseproduct is -1.Example:1The slope of line n -2. The slope of line p .21-2 -1, therefore, n is perpendicular to p.2Virginia Department of Education, 2014AFDA Vocabulary CardsPage 93

Parallel LinesLines in the same plane that do notintersect are parallel.Parallel lines have the same slopes.ybaxExample:The slope of line a -2.The slope of line b -2.-2 -2, therefore, a is parallel to b.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 94

MathematicalNotationSet BuilderNotationRead{x 0 x 3}The set of all xsuch that x isgreater than orequal to 0 and xis less than 3.{y: y -5}Virginia Department of Education, 2014The set of all ysuch that y isgreater than orequal to -5.AFDA Vocabulary CardsOtherNotation0 x 3(0, 3]y -5[-5, )Page 95

System of LinearEquationsSolve by graphing:-x 2y 32x y 4The solution,(1, 2), is theonly orderedpair thatsatisfies bothequations(the point ofintersection).Virginia Department of Education, 2014AFDA Vocabulary CardsPage 96

System of LinearEquationsSolve by substitution:x 4y 17y x–2Substitute 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.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 97

System of LinearEquationsSolve by elimination:-5x – 6y 85x 2y 4Add or subtract the equations to eliminate onevariable.-5x – 6y 8 5x 2y 4-4y 12y -3Now substitute -3 for y in either original equationto find the value of x, the eliminated variable.-5x – 6(-3) 8x 2The solution to the linear system is (2,-3), theordered pair that satisfies both equations.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 98

System of LinearEquationsIdentifying the Number of SolutionsNumber ofSolutionsSlopes andy-interceptsGraphyOnesolutionDifferent slopesxySame slope andNo solutiondifferent yinterceptsInfinitelymanysolutionsSame slope andsame yinterceptsVirginia Department of Education, 2014AFDA Vocabulary CardsxyxPage 99

Graphing LinearInequalitiesExampleGraphyy x 2xyxy -x – 1Virginia Department of Education, 2014AFDA Vocabulary CardsPage 100

System of LinearInequalitiesSolve by graphing:y x–3y -2x 3yThe solution regioncontains all orderedpairs that are solutionsto both inequalities inthe system.x(-1,1) is one solution tothe system located inthe solution region.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 101

Linear ProgrammingAn optimization process consisting of a systemof constraints and an objective quantity thatcan be maximized or minimizedExample:Find the minimum and maximum value of theobjective function C 4x 5y, subject to thefollowing constraints.x 0(0,6)y 0x y 6x y 6feasible region(6,0)(0,0)The maximum or minimum value for C 4x 5ywill occur at a corner point of the feasible region.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 102

Dependent andIndependentVariablex, independent variable(input values or domain set)Example:y 2x 7y, dependent variable(output values or range set)Virginia Department of Education, 2014AFDA Vocabulary CardsPage 103

Dependent andIndependentVariableDetermine the distance a carwill travel going 55 mph.d 55hindependentVirginia Department of Education, 2014h0123d055110165AFDA Vocabulary CardsdependentPage 104

Graph of a QuadraticEquation2y ax bx ca 0Example:132y x 2x – 3y12111098765line of symmetry43210-6-5-4-3-2-1-101234x-2-3vertex-4-5The graph of the quadratic equation is a curve(parabola) with one line of symmetry and one vertex.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 105

Quadratic FormulaUsed to find the solutions toany quadratic equation of the2form, y ax bx cx Virginia Department of Education, 2014-2- 4a2aAFDA Vocabulary CardsPage 106

RelationsRepresentations ofrelationshipsx-30125y40-62-1Example 2Example 1{(0,4), (0,3), (0,2), (0,1)}Example 3Virginia Department of Education, 2014AFDA Vocabulary CardsPage 107

FunctionsRepresentations of functionsx320-1y2422Example 2yExample 1{(-3,4), (0,3), (1,2), (4,6)}Example 3xExample 4Virginia Department of Education, 2014AFDA Vocabulary CardsPage 108

FunctionA relationship between two quantities inwhich every input corresponds toexactly one outputX210476583Y10A relation is a function if and only if eachelement in the domain is paired with aunique element of the range.Virginia Department of Education, 2014AFDA Vocabulary CardsPage 109

DomainA set of input values of arelationExamples:input 1234-2The domain of g(x)is {-2, -1, 0, 1}.Virginia Department of Education, 2014The domain of f(x) isall real numbers.AFDA Vocabulary CardsPage 110

RangeA set of output values ofa 387654321x0-4-3-2-1-101234-2The range of g(x)is {0, 1, 2, 3}.Virginia Department of Education, 2014The range of f(x) is allreal numbers greaterthan or equal to zero.AFDA Vocabulary CardsPage 111

Function Notationf(x)f(x) is read“the value of f at x” or “f of x”Example:f(x) -3x 5, find f(2).f(2) -3(2) 5f(2) -6Letters other than f can be used toname functions, e.g., g(x) and h(x)Virginia Department of Education, 2014AFDA Vocabulary CardsPage 112

Parent Functionsy4Linear32f(x) x10-4-3-2-1012341234x-1-2-3-49Quadraticyy872f(x) x6543210-4-3-2-10x-1-2Virginia Department of Education, 2014AFDA Vocabulary CardsPage 113

Parent FunctionsAbsolute Valuef(x) x Square Rootf(x) Virginia Department of Education, 2014AFDA Vocabulary CardsPage 114

Parent FunctionsCubic3f(x) xCube Rootf(x) 3Virginia Department of Education, 2014AFDA Vocabulary CardsPage 115

Parent FunctionsExponentialf(x) bxb 1Logarithmicf(x) lob 1Virginia Department of Education, 2014AFDA Vocabulary CardsPage 116

Transformations ofParent FunctionsTranslationsPar

Factoring (greatest common factor) Factoring (perfect square trinomials) Factoring (difference of squares) Factoring (sum and difference of cubes) Difference of Squares (model) Divide Polynomials (monomial divisor) Divide Polynomials (binomial divisor) Prime Polynomial Square Root th Root Product Property of Radicals Quotient Property of Radicals