Algebra I Vocabulary Word Wall Cards
Algebra I Vocabulary 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. The cards are designed for print use only.Table of ContentsExpressions and OperationsReal 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 Binomials HYPERLINK \l "multiply_polynomials" Multiply PolynomialsHYPERLINK \l "mult_binomials_model"Multiply Binomials (model)Multiply Binomials (graphic organizer)Multiply Binomials (squaring a binomial)Multiply Binomials (sum and difference)Factors of a MonomialFactoring (greatest common factor)Factoring (by grouping)Factoring (perfect square trinomials)Factoring (difference of squares)Difference of Squares (model)Divide Polynomials (monomial divisor)Divide Polynomials (binomial divisor)Square RootCube RootSimplify Numerical Expressions Containing Square or Cube RootsAdd and Subtract Monomial Radical ExpressionsProduct Property of RadicalsQuotient Property of RadicalsEquations and InequalitiesZero Product PropertySolutions or RootsZerosx-InterceptsCoordinate PlaneLiteral EquationVertical LineHorizontal LineQuadratic Equation (solve by factoring)Quadratic Equation (solve by graphing)Quadratic Equation (number of real solutions)InequalityGraph of an InequalityTransitive Property for InequalityAddition/Subtraction Property of InequalityMultiplication Property of InequalityDivision Property of InequalityLinear Equation (standard form) HYPERLINK \l "linear_equation_slope_intercept" Linear Equation (slope intercept form) HYPERLINK \l "point_slope" Linear Equation (point-slope form)Equivalent Forms of a Linear EquationHYPERLINK \l "slope"SlopeSlope FormulaSlopes of LinesPerpendicular LinesParallel LinesMathematical Notation HYPERLINK \l "system_lin_eq_graphing" System of Linear Equations (graphing) HYPERLINK \l "system_lin_eq_substitution" 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 EquationVertex of a Quadratic FunctionQuadratic FormulaFunctionsRelations (definition and examples)Function (definition) HYPERLINK \l "function_examples" Functions (examples)DomainRangeFunction NotationParent Functions - Linear, QuadraticTransformations of Parent Functions HYPERLINK \l "trans_parent_func" TranslationReflectionDilationLinear Functions (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)Vertical translation HYPERLINK \l "quad_func_dilation" Dilation (a>0) HYPERLINK \l "quad_func_dilation_reflection" Dilation/reflection (a<0) HYPERLINK \l "quad_func_horizontal_translation" Horizontal translationMultiple Representations of FunctionsStatisticsDirect VariationInverse VariationScatterplotPositive Linear RelationshipNegative Linear Relationship HYPERLINK \l "_No_Linear_Relationship" No Linear RelationshipCurve of Best Fit (linear)Curve of Best Fit (quadratic) HYPERLINK \l "outlier_data" 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…)left17911900Absolute Value|5| = 5 |-5| = 5 left74295 -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-2634 + 2m ax2 + bx + c3(y + 3.9)2 – 89Variable2350160140731002(y + 3)9 + x = 2.08d = 7c - 5A = r 2Coefficient38288853175000(-4) + 2x 27482802794000-7y525324054751120023 ab – 1230470113847900πr2Term2330865467842003463013490468004476772536819003x + 2y – 83 terms407193848664400297436733044800-5x2 – x 2 termscenter6210310023ab1 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 Form1005840-1270exponent00exponent2041525899160n 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 center47117001297334804000Examples: (-3a4b)2 = (-3)2?(a4)2?b2 = 9a8b25211455010150-1(2x)3 = -123? x3 = -18x3Quotient 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 , b0-1217144728280020790484357Examples:y34= y434 = y812330533210505t-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 exponentscenter4191000Examples:104206831952913863958242307a3 – 2a2 + 8a – 11045845764730-3n3 + 7n2 – 4n + 1016t – 1Add Polynomials(Group Like Terms – Horizontal Method)center13017500Example: (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)center56324500Example:(2g3 + 6g2 – 4) + (g3 – g – 3) 1156335361315(Align like terms and add)020000(Align like terms and add)2g3 + 6g2 – 4155596256472700 + g3 – g – 33g3 + 6g2 – g – 7Subtract Polynomials(Group Like Terms - Horizontal Method)center34417000Example: (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 -Vertical Method)center24066500Example:(4x2 + 5) – (-2x2 + 4x -7)(Align like terms then add the inverse and add the like terms.) 4x2 + 5 4x2 + 537325303175000040944805480050070485058864500–(-2x2 + 4x – 7) + 2x2 – 4x + 7 6x2 – 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 Polynomials258603953625600Apply the distributive property.31502363301900(x + 2)(3x2 + 5x + 1)262890031495900(x + 2)( 3x2 + 5x + 1)= x(3x2 + 5x + 1) + 2(3x2 + 5x + 1)= x·3x2 + x·5x + x·1 + 2·3x2 + 2·5x + 2·1= 3x3 + 5x2 + x + 6x2 + 10x + 2= 3x3 + 11x2 + 11x + 2Multiply Binomials(Model)Apply the distributive property.center17018000Example: (x + 3)(x + 2)180975147955x + 3x + 200x + 3x + 242303701625601 =x =Key:x2 =001 =x =Key:x2 = 37814241257306006x2 + 2x + 3x + = x2 + 5x + 6Multiply Binomials(Graphic Organizer)Apply the distributive property.left27495500Example: (x + 8)(2x – 3) = (x + 8)(2x + -3)center2425702x + -3002x + -31821815262890x + 800x + 82x2-3x16x-242x2 + 16x + -3x + -24 = 2x2 + 13x – 24Multiply Binomials(Squaring a Binomial)(a + b)2 = a2 + 2ab + b234290071818500(a – b)2 = a2 – 2ab + b2Examples:5789307142(3m + n)2 = 9m2 + 2(3m)(n) + n2 = 9m2 + 6mn + n2 5778502271400(y – 5)2 = y2 – 2(5)(y) + 25 = y2 – 10y + 25Multiply Binomials(Sum and Difference)(a + b)(a – b) = a2 – b22978154851400014714854673600Examples:(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.3124205778500Example: 20a4 + 8a89090517780002 ? 2 ? 5 ? a ? a ? a ? a + 2 ? 2 ? 2 ? a198628013335 common factors020000 common factors34750383302000GCF = 2 ? 2 ? a = 4a20a4 + 8a = 4a(5a3 + 2)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 = 8341503027432000center996950053549551694815Factor out a common binomial00Factor out a common binomial53454301094740Group factors00Group factors5259705456565Rewrite 8x as 2x + 6x00Rewrite 8x as 2x + 6x3x2+2x+6x+4(3x2+2x)+(6x+4)x(3x+2)+2(3x+2)(3x+2)(x+2)Factoring(Perfect Square Trinomials)a2 + 2ab + b2 = (a + b)2a2 – 2ab + b2 = (a – b)21157936900550035330044459900Examples: x2 + 6x +9 = x2 + 2?3?x +32= (x + 3)23585857112000 4x2 – 20x + 25 = (2x)2 – 2?2x?5 + 52 = (2x – 5)2Factoring(Difference of Squares)a2 – b2 = (a + b)(a – b)641268745400Examples:6264322314040 x2 – 49 = x2 – 72 = (x + 7)(x – 7)62643231218804 – n2 = 22 – n2 = (2 – n) (2 + n)96920024130009x2 – 25y2 = (3x)2 – (5y)2 = (3x + 5y)(3x – 5y)Difference of Squares (Model)right60325000center581025baab00baaba2 – b2 = (a + b)(a – b)106362572390a2 – b200a2 – b2309880575945a(a – b) + b(a – b)00a(a – b) + b(a – b)4073525553720(a + b)(a – b)00(a + b)(a – b)center5029200087630253365baa – b a – b 00baa – b a – b 3380105171450a + ba – b 00a + ba – b Divide Polynomials(Monomial Divisor)Divide each term of the dividend by the monomial divisorcenter36149600Example:(12x3 – 36x2 + 16x) 4x= 12x3 – 36x2 + 16x4x= 12x34x – 36x24x + 16x4x= 3x2 – 9x + 4Divide Polynomials (Binomial Divisor)Factor and simplifycenter4508500Example:(7w2 + 3w – 4) (w + 1)= 7w2 + 3w – 4w + 1= 7w – 4(w + 1)w + 1= 7w – 4 Square Root779145752475radical symbolradicand or argument 00radical symbolradicand or argument x2 Simplify square root expressions.center4213200768540433705Examples:9x2 = 32?x2 = (3x)2 = 3x7689361585200-(x – 3)2 = -(x – 3) = -x + 3Squaring a number and taking a square root are inverse operations.Cube Root138620538735index00index3900170852805radicand or argument00radicand or argument661670743585radical 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.Simplify Numerical Expressions ContainingSquare or Cube RootsSimplify radicals and combine like terms where possible.center7556500Examples:586740248920012-32-112+8=-102-42+22=-5-22590550172720018-2327=32-23 =32-6Add and Subtract Monomial Radical ExpressionsAdd or subtract the numerical factors of the like radicals.center5651500Examples:635-435-35=6-4-135=352x3+5x3=2+5x3=7x323+72-23=2-23+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 ? 02464213462000Example:5y2 = 5y2 = 5y, y ≠ 0Zero Product PropertyIf ab = 0,then a = 0 or b = 0.center2222500Example:(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.center33337500x2 + 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. center4127500-144666114404f(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 = 135958436604062230332105The 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).The 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. 319405175895002628904445f(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) and 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) and 1 or (1,0)39566856350Coordinate PlaneLiteral EquationA formula or equation that consists primarily of variablescenter47752000Examples:Ax + By = CA = 12bhV = lwhF = 95 C + 32A = πr2Vertical Linex = a (where a can be any real number)center317500Example: x = -4359918084455y00y14753592317754906645307975x00x-1879601145540Vertical lines have undefined slope. 00Vertical lines have undefined slope. Horizontal Liney = c(where c can be any real number)center12700003098800531495y00yExample:y = 61689100622304893945339090x00x-635314960Horizontal lines have a slope of 0.00Horizontal lines have a slope of 0.Quadratic Equation(Solve by Factoring)ax2 + bx + c = 0 a 0center25987900Example 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. Solutions are {2, 4}Quadratic Equation(Solve by Graphing)ax2 + bx + c = 0a 0center1079500Example solved by graphing: 2884805390525x2 – 6x + 8 = 063503140710Solutions to the equation are the x-coordinates{2, 4} of the points where the function crosses the x-axis.020000Solutions to the equation are the x-coordinates{2, 4} of the points where the function 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/Type of Real Solutions)ax2 + bx + c = 0, a 0ExamplesGraph of the related functionNumber and Type of Solutions/Rootsx2 – x = 32098151231902 distinct Real roots(crosses x-axis twice)x2 + 16 = 8x2343151358901 distinct Real root with a multiplicity of two (double root)(touches x-axis but does not cross)12x2 – 2x + 3 = 03600451327150 Real rootsInequalityAn algebraic sentence comparing two quantitiesSymbolMeaning<less thanless than or equal togreater thangreater than or equal tonot equal to35750515494000Examples:-10.5 ? -9.9 – 1.28 < 3t + 2x – 5y ≥ -12x ≤ -11 r 3Graph of an InequalitySymbolExampleGraph< ; 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 > -33305175152400005(-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 = -4336804065151000-90 -4t-90-4 -4t-422.5 tLinear Equation(Standard Form)Ax + By = C(A, B and C are integers; A and B cannot both equal zero)4411980179070y00ycenter2578100035953701651000Example: -2x + y = -3585533586360x020000xThe 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)center47815500Example: 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 pointcenter51689000Example: 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 EquationExample3y = 6 – 4xSlope-Intercepty = mx + b y=-43x+2Point-Slopey – y1 = m(x – x1)y--2=-43(x-3)StandardAx + By= C4x+3y=6SlopeA number that represents the rate of change in y for a unit change in x 2432052216150053594088904327525280035Slope = 2300Slope = 231049655130810320032The slope indicates the steepness of a line.Slope Formula The ratio of vertical change tohorizontal changecenter128905AB(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.center53340Example: 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.center189230Example: 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/InequalitySet Notationx=-5-5x=5 or x=-3.45, -3.4y>83y∶y>83x≤2.34x| x≤2.34Empty (null) set ?{ }All Real Numbers x∶x All Real NumbersSystem of Linear Equations(Graphing)24003002413000-x + 2y = 32x + y = 42850515198755center15240000122555237490The 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)21628103175000x + 4y = 17y = x – 2center21780500Substitute 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 = 4-15684622733000Add or subtract the equations to eliminate one variable. -5x – 6y = 8+ 5x + 2y = 423806153746500 -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 InequalitiesExampleGraphy x + 2290195146050030365701703705x00x1669415-31750y00yy > -x – 13075940637540x00x1715770-56515y00y222885-53975590555847080The 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 >.System of Linear InequalitiesSolve by graphing:185166010477500y x – 34241165557530y00yy -2x + 3271970562801400167005351155The solution region contains all ordered pairs that are solutions to both inequalities in the system.(-1,1) is one of the solutions 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 of the solutions to the system located in the solution region.center850900037143872769503990340389890005942965539750x00xDependent 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 = 55h178130211455independent020000independent4996180243205dependent020000dependentGraph of a Quadratic Equationy = ax2 + bx + ca 0center6223000430339523495004446905118745y00y2708529264795Example: 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.109855195580004112260550545Example: y=x2+2x-8h=-b2a=-22(1)=-1k=-12+2-1-8k=-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, 2-102 RelationA set of ordered pairsExamples:x167202333278y-34001-6171894542037000222534285574675005261048516510Example 2020000Example 2-194239711207Example 1020000Example 12761615581025Example 3020000Example 3{(0,4), (0,3), (0,2), (0,1)} Function(Definition)A relationship between two quantities in which every input corresponds to exactly one output5662930337819y00y1081405401320x00x423608515875001426210527050023114004768850019227807048524681000246810469011022034510753001075324218905784850023945854616450024555453803650025514304064000A relation is a function if and only if each element in the domain is paired with a unique element of the range.Functions(Examples)3043984143407xy322402-1253351135714934Example 4020000Example 445461222181348-1282703787775{(-3,4), (0,3), (1,2), (4,6)}020000{(-3,4), (0,3), (1,2), (4,6)}10096504236720Example 3020000Example 336969701754505Example 2020000Example 24851402708275Example 1020000Example 13723640165989000307086010572750064160404584700x020000x54165502032000y020000yDomainA set of input values of a relationcenter162873004069781327165f(x)xf(x)xExamples:inputoutputxg(x)-52705-381000-20-110213630844395959The domain of g(x) is {-2, -1, 0, 1}.020000The domain of g(x) is {-2, -1, 0, 1}.3764495394244The domain of f(x) is all real numbers.020000The domain of f(x) is all real numbers.RangeA set of output values of a relationcenter10795004792980239395f(x)020000f(x)3731623165826Examples:inputoutputxg(x)-20-1113665-49911000102134984115168910006059805675005x020000x3515360485775The 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”center38163500Example:f(x) = -3x + 5, find f(2).f(2) = -3(2) + 5f(2) = -6 + 5f(2) = -1Letters other than f can be used to name functions, e.g., g(x) and h(x)Parent Functions(Linear, Quadratic)3571512158684yxyxLinear f(x) = x35115175715yxyx Quadratic f(x) = x2Transformations 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(Vertical 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.Stretches away from the x-axisvertical dilation (compression) if 0 < a < presses toward the x-axisLinear Function(Transformational Graphing)Translationg(x) = x + b822424836056Examples:f(x) = xt(x) = x + 4h(x) = x – 2 00Examples:f(x) = xt(x) = x + 4h(x) = x – 2 3701687117855yxyxcenter21145500Vertical translation of the parent function, f(x) = xLinear Function(Transformational Graphing)Vertical Dilation (m > 0)g(x) = mx765159389312Examples:f(x) = xt(x) = 2xh(x) = 12x00Examples:f(x) = xt(x) = 2xh(x) = 12x359482617277yxyxcenter1524000Vertical dilation (stretch or compression) of the parent function, f(x) = x Linear Function(Transformational Graphing)Vertical Dilation/Reflection (m < 0)g(x) = mx5028936269553y020000ycenter40513000773974457200Examples:f(x) = xt(x) = -xh(x) = -3xd(x) = -13x00Examples:f(x) = xt(x) = -xh(x) = -3xd(x) = -13x39999481087346332855676085x020000xVertical dilation (stretch or compression) with a reflection of f(x) = x Quadratic Function(Transformational Graphing)Vertical Translationh(x) = x2 + c503030517590y020000y3643779146619center927100046858942059Examples:f(x) = x2g(x) = x2 + 2t(x) = x2 – 300Examples:f(x) = x2g(x) = x2 + 2t(x) = x2 – 36403497431099x020000xVertical translation of f(x) = x2Quadratic Function(Transformational Graphing)Vertical Dilation (a>0)h(x) = ax24597400411480y020000y2926334413385center203200061372752802890x020000x31432591440Examples: 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)Vertical Dilation/Reflection (a<0)h(x) = ax24951095326390y020000ycenter380365003757549328930539750480695Examples: 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) = x2Quadratic Function(Transformational Graphing)Horizontal Translation h(x) = (x ± c)2center478790004283710511175y020000y29801458509041093613722Examples: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) = x2Multiple Representations of Functionscenter6985Equationy=12x-200Equationy=12x-2242998457711Tablexy-2-30-22-14000Tablexy-2-30-22-1403056956401056Graph00Graphcenter1063625Wordsy equals one-half x minus 2020000Wordsy equals one-half x minus 2Direct Variationy = kx or k = yxconstant of variation, k 04903470320675y00ycenter328930003347085149225Example: y = 3x or 3 = yxxy-2-6-1-300132664725559525x00x3 = -6-2=-3-1=31=62 The graph of all points describing a direct variation is a line passing through the origin.Inverse Variationy = kx or k = xy2835242365150xyxyconstant of variation, k 0center1714500 Example: y = 3x or xy = 3center588010The graph of all points describing an inverse variation relationship are two curves that are reflections of each other.020000The graph of all points describing an inverse variation relationship are two curves that are reflections of each other.ScatterplotGraphical representation of the relationship between two numerical sets of data1344930585470xy00xyPositive 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.1496060394335xy00xyNo Linear Relationship (Correlation)No relationship between the dependent (y) values and independent (x) values.1287285804809xy00xyCurve 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 + 6Outlier Data(Graphic)6872017370400703055248622200 ................
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