Algebra Vocabulary Word Wall Cards - Brain Page 206



Algebra I Vocabulary CardsTable of ContentsExpressions and OperationsNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal NumbersAbsolute ValueOrder of OperationsExpressionVariableCoefficientTermScientific NotationExponential FormNegative ExponentZero ExponentProduct of Powers PropertyPower of a Power PropertyPower of a Product PropertyQuotient of Powers PropertyPower of a Quotient PropertyPolynomialDegree of PolynomialLeading CoefficientAdd Polynomials (group like terms)Add Polynomials (align like terms)Subtract Polynomials (group like terms)Subtract Polynomials (align like terms)Multiply PolynomialsMultiply BinomialsMultiply Binomials (model)Multiply Binomials (graphic organizer)Multiply Binomials (squaring a binomial)Multiply Binomials (sum and difference)Factors of a MonomialFactoring (greatest common factor)Factoring (perfect square trinomials)Factoring (difference of squares)Difference of Squares (model)Divide Polynomials (monomial divisor)Divide Polynomials (binomial divisor)Prime PolynomialSquare RootCube RootProduct Property of RadicalsQuotient Property of RadicalsZero Product PropertySolutions or RootsZeros x-InterceptsEquations and InequalitiesCoordinate PlaneLinear EquationLinear Equation (standard form)Literal EquationVertical LineHorizontal LineQuadratic EquationQuadratic Equation (solve by factoring)Quadratic Equation (solve by graphing)Quadratic Equation (number of solutions)Identity Property of AdditionInverse Property of AdditionCommutative Property of AdditionAssociative Property of AdditionIdentity Property of MultiplicationInverse Property of MultiplicationCommutative Property of MultiplicationAssociative Property of MultiplicationDistributive PropertyDistributive Property (model)Multiplicative Property of ZeroSubstitution PropertyReflexive Property of EqualitySymmetric Property of EqualityTransitive Property of EqualityInequalityGraph of an InequalityTransitive Property for InequalityAddition/Subtraction Property of InequalityMultiplication Property of InequalityDivision Property of InequalityLinear Equation (slope intercept form)Linear Equation (point-slope form)SlopeSlope 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 InequalitiesDependent and Independent VariableDependent and Independent Variable (application)Graph of a Quadratic EquationQuadratic FormulaRelations and FunctionsRelations (examples)Functions (examples)Function (definition)DomainRangeFunction NotationParent FunctionsLinear, QuadraticTransformations of Parent FunctionsTranslationReflectionDilationLinear Function (transformational graphing)TranslationDilation (m>0)Dilation/reflection (m<0)Quadratic Function (transformational graphing)Vertical translationDilation (a>0)Dilation/reflection (a<0)Horizontal translationDirect VariationInverse VariationStatisticsStatistics NotationMeanMedianModeBox-and-Whisker PlotSummationMean Absolute DeviationVarianceStandard Deviation (definition)z-Score (definition)z-Score (graphic)Elements within One Standard Deviation of the Mean (graphic)ScatterplotPositive CorrelationNegative CorrelationConstant CorrelationNo CorrelationCurve of Best Fit (linear/quadratic)Outlier Data (graphic)Natural NumbersThe set of numbers 1, 2, 3, 4…840740450850Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers00Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers1797050636905Natural Numbers00Natural NumbersWhole NumbersThe set of numbers 0, 1, 2, 3, 4…837565-5715Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers00Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal NumbersIntegersThe set of numbers…-3, -2, -1, 0, 1, 2, 3…886460458470Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers00Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal NumbersRational Numbers82931079375Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers00Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal NumbersThe set of all numbers that can be written as the ratio of two integers with a non-zero denominator235 , -5 , 0.3, 16 , 137 Irrational Numbers845185271145Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal Numbers00Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal NumbersThe set of all numbers that cannot be expressed as the ratio of integers7 , π , -0.23223222322223… Real Numbers824865156210Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational Numbers00Whole NumbersIntegersRational NumbersIrrational NumbersNatural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersThe set of all rational and irrational numbersAbsolute Value|5| = 5 |-5| = 5 -33718574295 -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 Symbols( ){ }[ ]|absolute value|fraction barExponentsanMultiplicationDivision10668035623500Left to RightAdditionSubtraction9525037846000Left to Right-26352565659000Expressionx-2634 + 2m 3(y + 3.9)2 – 89Variable1993900183515002(y + 3)9 + x = 2.08d = 7c - 5A = r 2Coefficient34607503175000(-4) + 2x 25355551841500-7y 221761454038600023 ab – 1226670006223000πr2Term4152900605155003096895494665001929765494665003x + 2y – 83 terms375158052768500247078537020500-5x2 – x 2 terms29864056210300023ab1 termScientific Notationa x 10n1 ≤ |a| < 10 and n is an integer355603403600090170375285Examples:Standard NotationScientific Notation17,500,0001.75 x 107-84,623-8.4623 x 1040.00000262.6 x 10-6-0.080029-8.0029 x 10-200Examples:Standard NotationScientific Notation17,500,0001.75 x 107-84,623-8.4623 x 1040.00000262.6 x 10-6-0.080029-8.0029 x 10-2Exponential Form1005840-1270exponent00exponent2041525899160factors00factors240665936625base00basean = a?a?a?a…, a0-863607937500Examples:2 ? 2 ? 2 = 23 = 8n ? n ? n ? n = n4 3?3?3?x?x = 33x2 = 27x2Negative Exponenta-n = 1an , a 0-1168402222500Examples:4-2 = 142 = 116x4y-2 = x41y2 = x41y2? y2y2 = x4y2(2 – a)-2 = 1(2 – a)2 , a ≠2Zero Exponent-76835128905000a0 = 1, a 0Examples:(-5)0 = 1 (3x + 2)0 = 1(x2y-5z8)0 = 14m0 = 4 ? 1 = 4Product of Powers Propertyam ? an = am + n -9842541021000Examples: x4 ? x2 = x4+2 = x6a3 ? a = a3+1 = a4w7 ? w-4 = w7 + (-4) = w3Power of a Power Property(am)n = am · n-89535-444500Examples: (y4)2 = y4?2 = y8(g2)-3 = g2?(-3) = g-6 = 1g6Power of a Product Property(ab)m = am ? bm-13779512446000 Examples: (-3ab)2 = (-3)2?a2?b2 = 9a2b2-1(2x)3 = -123? x3 = -18x3Quotient of Powers Propertyaman = am – n, a 0-869951714500Examples:x6x5 = x6 – 5 = x1 = xy-3y-5 = y-3 – (-5) = y2a4a4 = a4-4 = a0 = 1Power of Quotient Propertyabm= ambm , b0-6858047752000Examples:y34= y434 5t-3= 5-3t-3 = 1531t3 = t353 = t3125Polynomial ExampleNameTerms76xmonomial1 term3t – 112xy3 + 5x4ybinomial2 terms2x2 + 3x – 7trinomial3 termsNonexampleReason102616016510005mn – 8variable exponent833120571500n-3 + 9negative exponentDegree of a Polynomial-83185198437500The largest exponent or the largest sum of exponents of a term within a polynomialExample:TermDegree6a3 + 3a2b3 – 216a333a2b35-210Degree of polynomial:5Leading CoefficientThe coefficient of the first term of a polynomial written in descending order of exponents-895352095500Examples:7a3 – 2a2 + 8a – 1-3n3 + 7n2 – 4n + 1016t – 1Add PolynomialsCombine like terms.-9017046037500Example: (2g2 + 6g – 4) + (g2 – g) = 2g2 + 6g – 4 + g2 – g193675328930(Group like terms and add.)020000(Group like terms and add.)= (2g2 + g2) + (6g – g) – 4 = 3g2 + 5g2 – 4 Add PolynomialsCombine like terms.-17272053467000Example:(2g3 + 6g2 – 4) + (g3 – g – 3) 1156335361315(Align like terms and add.)020000(Align like terms and add.)2g3 + 6g2 – 4124269558166000 + g3 – g – 33g3 + 6g2 – g – 7Subtract PolynomialsAdd the inverse.-14478034417000Example: (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 PolynomialsAdd the inverse.-21082034417000Example:(4x2 + 5) – (-2x2 + 4x -7)(Align like terms then add the inverse and add the like terms.) 4x2 + 5 4x2 + 537134805765800012382558864500289052031750000–(2x2 + 4x – 7) + 2x2 – 4x + 7 6x2 – 4x + 12Multiply PolynomialsApply the distributive property.(a + b)(d + e + f)2763520-43815002847975647700002398395-58420000(a + b)( d + e + f )= a(d + e + f) + b(d + e + f)= ad + ae + af + bd + be + bfMultiply BinomialsApply the distributive property.(a + b)(c + d) = a(c + d) + b(c + d) =ac + ad + bc + bd-9271028003500Example: (x + 3)(x + 2)= x(x + 2) + 3(x + 2)= x2 + 2x + 3x + 6= x2 + 5x + 6Multiply BinomialsApply the distributive property. -4762529400500Example: (x + 3)(x + 2)42303701625601 =x =Key:x2 =001 =x =Key:x2 =93980149225x + 3x + 200x + 3x + 2 31337251524006006x2 + 2x + 3x + = x2 + 5x + 6Multiply BinomialsApply the distributive property.-9525025336500Example: (x + 8)(2x – 3) = (x + 8)(2x + -3)22955252374902x + -3002x + -31355090178435x + 800x + 82x2-3x8x-242x2 + 8x + -3x + -24 = 2x2 + 5x – 24Multiply Binomials:Squaring a Binomial(a + b)2 = a2 + 2ab + b2(a – b)2 = a2 – 2ab + b2-990603619500Examples:(3m + n)2 = 9m2 + 2(3m)(n) + n2 = 9m2 + 6mn + n2 (y – 5)2 = y2 – 2(5)(y) + 25 = y2 – 10y + 25Multiply Binomials: Sum and Difference(a + b)(a – b) = a2 – b2-130810508000Examples:(2b + 5)(2b – 5) = 4b2 – 25(7 – w)(7 + w) = 49 + 7w – 7w – w2 = 49 – w2Factors 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 FactorFind the greatest common factor (GCF) of all terms of the polynomial and then apply the distributive property.-685801016000Example: 20a4 + 8a48069519685002 ? 2 ? 5 ? a ? a ? a ? a + 2 ? 2 ? 2 ? a3119755393700018535659525common factors020000common factorsGCF = 2 ? 2 ? a = 4a20a4 + 8a = 4a(5a3 + 2)Factoring: Perfect Square Trinomialsa2 + 2ab + b2 = (a + b)2a2 – 2ab + b2 = (a – b)2-9017073279000Examples: x2 + 6x +9 = x2 + 2?3?x +32= (x + 3)2 4x2 – 20x + 25 = (2x)2 – 2?2x?5 + 52 = (2x – 5)2Factoring: Difference of Two Squaresa2 – b2 = (a + b)(a – b)-12509551371500Examples: x2 – 49 = x2 – 72 = (x + 7)(x – 7)4 – n2 = 22 – n2 = (2 – n) (2 + n)9x2 – 25y2 = (3x)2 – (5y)2 = (3x + 5y)(3x – 5y)Difference of Squares-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 Divide PolynomialsDivide each term of the dividend by the monomial divisor-787401206500Example:(12x3 – 36x2 + 16x) 4x= 12x3 – 36x2 + 16x4x= 12x34x – 36x24x + 16x4x= 3x2 – 9x + 4Divide Polynomials by BinomialsFactor and simplify-1270004508500Example:(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 Root407670695325radical symbolradicand or argument 00radical symbolradicand or argument x2 Simply square root expressions.-1085854000500Examples:9x2 = 32?x2 = (3x)2 = 3x-(x – 3)2 = -(x – 3) = -x + 3Squaring a number and taking a square root are inverse operations.Cube Root111950595885index00index309245705485radical symbol00radical symbol3595370757555radicand or argument00radicand or argument3x3 Simplify cube root expressions.-463552032000Examples:364 = 343 = 43-27 = 3(-3)3 = -33x3 = xCubing a number and taking a cube root are inverse operations.Product Property of RadicalsThe square root of a product equals the product of the square roots of the factors.ab = a ? ba ≥ 0 and b ≥ 0-914401397000Examples:4x = 4 ? x = 2x5a3 = 5 ? a3 = a5a316 = 38?2 = 38 ? 32 = 232Quotient Propertyof RadicalsThe square root of a quotient equals the quotient of the square roots of the numerator and denominator.ab = ab a ≥ 0 and b ? 0-1320803937000Example:5y2 = 5y2 = 5y, y ≠ 0Zero Product PropertyIf ab = 0,then a = 0 or b = 0.-654052222500Example:(x + 3)(x – 4) = 0(x + 3) = 0 or (x – 4) = 0x = -3 or x = 4The solutions are -3 and 4, also called roots of the equation.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. -1054108001000-235585111760f(x) = x2 + 2x – 3Find f(x) = 0.0 = x2 + 2x – 30 = (x + 3)(x – 1)x = -3 or x = 1The zeros are -3 and 1 located at (-3,0) and (1,0).00f(x) = x2 + 2x – 3Find f(x) = 0.0 = x2 + 2x – 30 = (x + 3)(x – 1)x = -3 or x = 1The zeros are -3 and 1 located at (-3,0) and (1,0).349377066040The zeros of a function are also the solutions or roots of the related equation.x-InterceptsThe x-intercepts of a graph are located where the graph crosses the x-axis and where f(x) = 0. -3873520574000-127635299720f(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)3489960149225Coordinate PlaneLinear EquationAx + By = C(A, B and C are integers; A and B cannot both equal zero.)3404235199390y020000y-69850280670002426335289179Example: -2x + y = -35031740255905x020000xThe graph of the linear equation is a straight line and represents all solutions (x, y) of the equation.Linear Equation: Standard Form Ax + By = C (A, B, and C are integers; A and B cannot both equal zero.)-8953554927500Examples:4x + 5y = -24x – 6y = 9Literal EquationA formula or equation which consists primarily of variables-6350047752000Examples:ax + b = cA = 12bhV = lwhF = 95 C + 32A = πr2Vertical Linex = a (where a can be any real number)-33020317500Example: x = -4359918084455y00y14753592317754906645307975x00x-5403851088390Vertical lines have an undefined slope. 00Vertical lines have an 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 0-2863851206500Example: 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. -467995138430Solutions to the equation are 2 and 4; the x-coordinates where the curve crosses the x-axis.020000Solutions to the equation are 2 and 4; the x-coordinates where the curve crosses the x-axis.Quadratic Equationax2 + bx + c = 0 a 0-23558553213000Example solved by factoring:x2 – 6x + 8 = 0Quadratic equation(x – 2)(x – 4) = 0Factor(x – 2) = 0 or (x – 4) = 0Set factors equal to 0x = 2 or x = 4Solve for x Solutions to the equation are 2 and 4.Quadratic Equationax2 + bx + c = 0a 0-450851079500Example solved by graphing: 2884805390525x2 – 6x + 8 = 0-425453056890Solutions to the equation are the x-coordinates (2 and 4) of the points where the curve crosses the x-axis.020000Solutions to the equation are the x-coordinates (2 and 4) of the points where the curve crosses the x-axis.90805220980Graph the related function f(x) = x2 – 6x + 8.00Graph the related function f(x) = x2 – 6x + 8.Quadratic Equation: Number of Real Solutionsax2 + bx + c = 0, a 0ExamplesGraphsNumber of Real Solutions/Rootsx2 – x = 371380661242x2 + 16 = 8x82397418411 distinct rootwith a multiplicity of two2x2 – 2x + 3 = 093414946760Identity Property of Additiona + 0 = 0 + a = a-10795031686500Examples: 3.8 + 0 = 3.86x + 0 = 6x0 + (-7 + r) = -7 + rZero is the additive identity.Inverse Property of Additiona + (-a) = (-a) + a = 0-115570000Examples: 4 + (-4) = 00 = (-9.5) + 9.5x + (-x) = 00 = 3y + (-3y) Commutative Property of Additiona + b = b + a-7556545656500Examples:2.76 + 3 = 3 + 2.76x + 5 = 5 + x(a + 5) – 7 = (5 + a) – 711 + (b – 4) = (b – 4) + 11Associative Property of Addition (a + b) + c = a + (b + c)-673101841500Examples:5 + 35+ 110= 5 +35 + 1103x + (2x + 6y) = (3x + 2x) + 6y Identity Property of Multiplicationa ? 1 = 1 ? a = a-8128035877500Examples:3.8 (1) = 3.86x ? 1 = 6x1(-7) = -7One is the multiplicative identity.Inverse Property of Multiplicationa ? 1a = 1a ? a = 1a 0-222252032000Examples:7 ? 17 = 15x ? x5 = 1, x 0-13 ? (-3p) = 1p = pThe multiplicative inverse of a is mutative Property of Multiplicationab = ba-9906044259500Examples:(-8)23 = 23(-8)y ? 9 = 9 ? y 4(2x ? 3) = 4(3 ? 2x)8 + 5x = 8 + x ? 5Associative Property of Multiplication (ab)c = a(bc)-381006731000Examples:(1 ? 8) ? 334 = 1 ? (8 ? 334) (3x)x = 3(x ? x)Distributive Propertya(b + c) = ab + ac -323851143000Examples:5y – 13 = 5 ? y – 5 ?13 2 ? x + 2 ? 5 = 2(x + 5)3.1a + (1)(a) = (3.1 + 1)aDistributive Property4(y + 2) = 4y + 4(2)129540035369544(y + 2)0044(y + 2)26130251458595y + 2020000y + 2127317520148554y24y + 4(2)004y24y + 4(2)Multiplicative Property of Zero a ? 0 = 0 or 0 ? a = 0-5080073152000Examples:823 · 0 = 00 · (-13y – 4) = 0Substitution PropertyIf a = b, then b can replace a in a given equation or inequality. -1905002540000Examples:GivenGivenSubstitutionr = 93r = 273(9) = 27b = 5a24 < b + 824 < 5a + 8y = 2x + 12y = 3x – 22(2x + 1) = 3x – 2Reflexive Property of Equalitya = aa is any real number-628652794000Examples:-4 = -43.4 = 3.49y = 9ySymmetric Property of EqualityIf a = b, then b = a.-762002032000Examples:If 12 = r, then r = 12.If -14 = z + 9, then z + 9 = -14.If 2.7 + y = x, then x = 2.7 + y.Transitive Property of EqualityIf a = b and b = c, then a = c.-6286534480500Examples: If 4x = 2y and 2y = 16, then 4x = 16.If x = y – 1 and y – 1 = -3, then x = -3.InequalityAn 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< or x < 3168275524510 or -3 y121920393700t -2106456489324 Transitive Property of InequalityIfThena b and b c a ca b and b c a c-895352540000Examples: 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 + c-21272545720000Example: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 < bc-11239544132500Example: if c = -25 > -35(-2) < -3(-2) -10 < 6Division Property of InequalityIfCaseThen a < bc > 0, positiveac < bca > bc > 0, positiveac > bca < bc < 0, negativeac > bca > bc < 0, negativeac < bc-666752730500Example: if c = -4-90 -4t-90-4 -4t-422.5 tLinear Equation: Slope-Intercept Formy = mx + b(slope is m and y-intercept is b)-889008445500Example: y = -43 x + 53550920436880(0,5)-4300(0,5)-432181202241389445135417195m = -43b = -500m = -43b = -5Linear Equation: Point-Slope Formy – y1 = m(x – x1)where m is the slope and (x1,y1) is the point-8382051689000Example: 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 + 9SlopeA number that represents the rate of change in y for a unit change in x -110490447040002406652184404327525280035Slope = 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-531495306705Line 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.26955751530352745105435610Perpendicular 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 NotationSet BuilderNotationReadOther Notation{x|0 < x 3}The set of all x such that x is greater than or equal to 0 and x is less than 3.0 < x 3(0, 3]{y: y ≥ -5}The set of all y such that y is greater than or equal to -5.y ≥ -5[-5, ∞)System of Linear EquationsSolve by graphing:20002501460500-x + 2y = 32x + y = 4285051519875510096515240000122555237490The 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 EquationsSolve by substitution:20008853175000x + 4y = 17y = x – 2-6286521780500Substitute 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 EquationsSolve by elimination:20453353746500-5x – 6y = 85x + 2y = 4-8445522352000Add 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 EquationsIdentifying the Number of SolutionsNumber of SolutionsSlopes and y-intercepts906145763270xy00xyGraphOne solutionDifferent slopes9867901513840xy00xy42418017780No solutionSame slope anddifferent y-intercepts9632951577340xy00xy44577028575Infinitely many solutionsSame slope andsame y-intercepts44373825527Graphing Linear InequalitiesExampleGraphy x + 228892595251669415-31750y00y31235652035810x00xy > -x – 11715770-56515y00y3166745727075x00x222885-53975System of Linear InequalitiesSolve by graphing:185166010477500y x – 34241165557530y00yy -2x + 33714387276950281368570040500-1143008064500174625327025The 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.3990340389890005942965539750x00xDependent andIndependent Variablex, independent variable(input values or domain set)-88265952500Example:y = 2x + 7 y, dependent variable(output values or range set)Dependent andIndependent VariableDetermine the distance a car will travel going 55 mph.hd0015521103165d = 55h4415155186055dependent020000dependent-400685192405independent020000independentGraph of a Quadratic Equationy = ax2 + bx + ca 0-742956223000430339523495004446905118745y00y2708529264795Example: 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.Quadratic Formula Used to find the solutions to any quadratic equation of the form, y = ax2 + bx + cx = -b ± b2- 4ac 2aRelationsRepresentations of relationshipsx167202333278y-34001-6171894542037000222534285574675005261048516510Example 2020000Example 2-18972555080Example 1020000Example 1{(0,4), (0,3), (0,2), (0,1)}258064038100Example 3020000Example 3FunctionsRepresentations of functions3043984143407xy322402-1248482255750560Example 4020000Example 410096504236720Example 3020000Example 336969701754505Example 2020000Example 24851402708275Example 1020000Example 13723640165989000307086010572750064160404584700x020000x54165502032000y020000y41186102228850-4235453721100{(-3,4), (0,3), (1,2), (4,6)}020000{(-3,4), (0,3), (1,2), (4,6)}FunctionA relationship between two quantities in which every input corresponds to exactly one output5618480476885Y00Y1067435567690X00X423608515875001426210527050023114004768850019227807048524681000246810469011022034510753001075324218905784850023945854616450024555453803650025514304064000A relation is a function if and only if each element in the domain is paired with a unique element of the range.DomainA set of input values of a relation-4191037465003481070580390Examples:-717556350000inputoutputxg(x)-20-1102134557395153035f(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.RangeA set of output values of a relation-5270510795004792980239395f(x)020000f(x)3731623165826Examples:inputoutputxg(x)-20-1102134984115168910001601470119380006059805675005x020000x3515360485775The 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”-8001038544500Example: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)Parent Functions4439920300990y020000y3091180446405Linear5706745219710x020000x f(x) = x4411345571500y020000y3027045464820 Quadratic f(x) = x25714365979805x020000xTransformations of Parent FunctionsParent 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 FunctionsParent 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 FunctionsParent 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.Transformational GraphingLinear functionsg(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) = xTransformational GraphingLinear functionsg(x) = mx4566285347345y020000y3208655362585m>01993901524000173355472440Examples: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 Transformational GraphingLinear functionsg(x) = mx90170405130004257040328930y020000ym < 0251460457200Examples: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 Transformational GraphingQuadratic functionsh(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) = x2Transformational GraphingQuadratic functionsh(x) = ax24597400411480y020000y2926334413385a > 080645203200061372752802890x020000x31432591440Examples: 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) = x2Transformational GraphingQuadratic functionsh(x) = ax24951095326390y020000y8890380365003757549328930a < 0539750480695Examples: 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) = x2Transformational GraphingQuadratic functions 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) = x2Direct Variationy = kx or k = yxconstant of variation, k 0-48895328930004122420320675y00y266167563623Example: y = 3x or 3 = yx572389027305x00xThe graph of all points describing a direct variation is a line passing through the origin.Inverse Variationy = kx or k = xyconstant of variation, k 0-18415313690004363720262890y00y2659634422910 Example: y = 3x or xy = 36064250155575x00xThe graph of all points describing an inverse variation relationship are 2 curves that are reflections of each other. Statistics Notationxiith 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 tendency-10414027114500Example: 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 8508508080772000Numerical AverageMedianA measure of central tendency-4635543561000Examples: 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-234957302500Examples:Box-and-Whisker PlotA graphical representation of the five-number summary-303530262890LowerQuartile (Q1)LowerExtremeUpperQuartile (Q3)UpperExtremeMedianInterquartile Range (IQR)510152000LowerQuartile (Q1)LowerExtremeUpperQuartile (Q3)UpperExtremeMedianInterquartile Range (IQR)51015205374640-1290955A1, 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 = 44Mean Absolute DeviationA measure of the spread of a data set Mean Absolute Deviation=The mean of the sum of the absolute value of the differences between each element and the mean of the data setVarianceA 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 setStandard DeviationA 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 variancez-ScoreThe 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-ScoreThe number of standard deviations an element is from the mean12687301282700024159121462267627530480z = 1z = 2z = 3z = -1z = -2z = -300z = 1z = 2z = 3z = -1z = -2z = -332245301545590Mean020000Mean31057851847850z=0020000z=067729105029835μ=12σ=3.4900μ=12σ=3.4951066703287395X020000X24866602870200X020000X399859522218650047879005727065μ+σz=1020000μ+σz=116478255725795μ-σz=-1020000μ-σz=-124422104662170Elements within one standard deviation of the mean020000Elements within one standard deviation of the mean28784553295650X020000X33616903288030X020000X47250353276600X020000X42741853281680X020000X20135853272790X020000X15373353275330X020000X69462653255010X020000X38030153275330X020000X6184903265805X020000X-2686053935095510152000510152023958552524125005535295251650500238252024955500055486302536825X020000X55492652905760X020000X60528203260725X020000X55492653262630X020000X47244002540000X020000X28797252524125X020000X47244002893060X020000X42748202896870X020000X38049202896870X020000X33616902877820X020000X15386052865120X020000X28784552879725X020000X24853903287395X020000XElements within One Standard Deviation (σ)of the Mean (?)ScatterplotGraphical representation of the relationship between two numerical sets of data1344930585470xy00xyPositive CorrelationIn general, a relationship where the dependent (y) values increase as independent values (x) increase1308735509270xy00xyNegative CorrelationIn general, a relationship where the dependent (y) values decrease as independent (x) values increase.1496060394335xy00xyConstant Correlation The dependent (y) values remain about the same as the independent (x) values increase. 1369060117475xy00xyNo Correlation No relationship between the dependent (y) values and independent (x) values. 1463675598805xy00xyCurve of Best Fit1996440140970Calories and Fat Content020000Calories and Fat Content3689353761740421640385445Outlier Data3853180730885003008630495935Outlier020000Outlier786765513715Miles per GallonFrequencyGas Mileage for Gasoline-fueled CarsOutlier00Miles per GallonFrequencyGas Mileage for Gasoline-fueled CarsOutlier1049020475615 ................
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