Keystone Exams: Algebra II - Pennsylvania Department Of Education
Keystone Exams: Algebra IIAssessment Anchors and Eligible Contentwith Sample Questions and GlossaryPennsylvania Department of Educationwww.education.state.pa.usApril 2014
PENNSYLVANIA DEPARTMENT OF EDUCATIONGeneral Introduction to the Keystone Exam Assessment AnchorsIntroductionSince the introduction of the Keystone Exams, the Pennsylvania Department of Education (PDE) has beenworking to create a set of tools designed to help educators improve instructional practices and betterunderstand the Keystone Exams. The Assessment Anchors, as defined by the Eligible Content, are one of themany tools the Department believes will better align curriculum, instruction, and assessment practicesthroughout the Commonwealth. Without this alignment, it will not be possible to significantly improve studentachievement across the Commonwealth.How were Keystone Exam Assessment Anchors developed?Prior to the development of the Assessment Anchors, multiple groups of PA educators convened to create aset of standards for each of the Keystone Exams. Enhanced Standards, derived from a review of existingstandards, focused on what students need to know and be able to do in order to be college and career ready.(Note: Since that time, PA Core Standards have replaced the Enhanced Standards and reflect the collegeand career-ready focus.) Additionally, the Assessment Anchors and Eligible Content statements were createdby other groups of educators charged with the task of clarifying the standards assessed on the KeystoneExams. The Assessment Anchors, as defined by the Eligible Content, have been designed to hold together, oranchor, the state assessment system and the curriculum/instructional practices in schools.Assessment Anchors, as defined by the Eligible Content, were created with the following design parameters: Clear: The Assessment Anchors are easy to read and are user friendly; they clearly detail whichstandards are assessed on the Keystone Exams. Focused: The Assessment Anchors identify a core set of standards that can be reasonablyassessed on a large-scale assessment; this will keep educators from having to guess whichstandards are critical. Rigorous: The Assessment Anchors support the rigor of the state standards by assessing higherorder and reasoning skills. Manageable: The Assessment Anchors define the standards in a way that can be easilyincorporated into a course to prepare students for success.How can teachers, administrators, schools, and districts use these Assessment Anchors?The Assessment Anchors, as defined by the Eligible Content, can help focus teaching and learning becausethey are clear, manageable, and closely aligned with the Keystone Exams. Teachers and administrators will bebetter informed about which standards will be assessed. The Assessment Anchors and Eligible Contentshould be used along with the Standards and the Curriculum Framework of the Standards Aligned System(SAS) to build curriculum, design lessons, and support student achievement.The Assessment Anchors and Eligible Content are designed to enable educators to determine when they feelstudents are prepared to be successful in the Keystone Exams. An evaluation of current course offerings,through the lens of what is assessed on those particular Keystone Exams, may provide an opportunity for analignment to ensure student preparedness.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 2
How are the Assessment Anchors organized?The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, eachstructured with a common labeling system that can be read like an outline. This framework is organized firstby module, then by Assessment Anchor, followed by Anchor Descriptor, and then finally, at the greatest levelof detail, by an Eligible Content statement. The common format of this outline is followed across the KeystoneExams.Here is a description of each level in the labeling system for the Keystone Exams: Module: The Assessment Anchors are organized into two thematic modules for each of theKeystone Exams. The module title appears at the top of each page. The module level is importantbecause the Keystone Exams are built using a module format, with each of the Keystone Examsdivided into two equal-size test modules. Each module is made up of two or more AssessmentAnchors. Assessment Anchor: The Assessment Anchor appears in the shaded bar across the top of eachAssessment Anchor table. The Assessment Anchors represent categories of subject matter thatanchor the content of the Keystone Exams. Each Assessment Anchor is part of a module and hasone or more Anchor Descriptors unified under it. Anchor Descriptor: Below each Assessment Anchor is a specific Anchor Descriptor. The AnchorDescriptor level provides further details that delineate the scope of content covered by theAssessment Anchor. Each Anchor Descriptor is part of an Assessment Anchor and has one or moreEligible Content statements unified under it. Eligible Content: The column to the right of the Anchor Descriptor contains the Eligible Contentstatements. The Eligible Content is the most specific description of the content that is assessed onthe Keystone Exams. This level is considered the assessment limit and helps educators identify therange of the content covered on the Keystone Exams. PA Core Standard: In the column to the right of each Eligible Content statement is a coderepresenting one or more PA Core Standards that correlate to the Eligible Content statement. SomeEligible Content statements include annotations that indicate certain clarifications about the scopeof an Eligible Content. “e.g.” (“for example”)—sample approach, but not a limit to the Eligible Content“Note”—content exclusions or definable range of the Eligible ContentHow do the K–12 Pennsylvania Core Standards affect this document?Assessment Anchor and Eligible Content statements are aligned to the PA Core Standards; thus, the formerenhanced standards are no longer necessary. Within this document, all standard references reflect the PACore Standards.Standards Aligned System—www.pdesas.orgPennsylvania Department of Education—www.education.state.pa.usPennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 3
Keystone Exams: Algebra IIFORMULA SHEETFormulas that you may need to work questions in this document are found below.You may use calculator π or the number 3.14.ShapesLogarithmic Propertiesloga x y x a yA lwwlog x y x 10 yIn x y x eyloga (x · y ) loga x loga ylloga x p p · loga xxloga y loga x loga yhV lwhwlQuadratic Functionsf (x) ax 2 bx cGeneral Formula:Data Analysisf (x) a(x h )2 kStandard (Vertex) Form:Permutation:nPrCombination: n!(n r)!n!nCr r!(n r)!f(x ) a(x x 1)(x x 2)Factored Form:Quadratic Formula:x ˉb b 2 4ac2awhen ax 2 bx c 0 and a Þ 0Exponential Propertiesam · an am n(a m )n a m · nam am nana 1 1aCompound Interest EquationsAnnual:A P (1 r ) tPeriodic:A P 1 i3 ii 2 1ntnP principal amountr annual rate of interestt time (years)Continuous: 1r( )Powers of the Imaginary Uniti A account total after t yearsA Pertn number of periods interestis compounded per yeari4 1Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 4
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.1Operations with Complex NumbersAnchor DescriptorA2.1.1.1Represent and/or useimaginary numbersin equivalent forms(e.g., square roots andexponents).Eligible ContentA2.1.1.1.1Simplify/writesquareroots in terms of i}}(e.g., Ï-24 2iÏ6 ).A2.1.1.1.2Simplify/evaluate expressions involvingpowers of i (e.g., i6 i3 –1 – i).PA CoreStandardsCC.2.1.HS.F.6Sample Exam QuestionsStandard A2.1.1.1.1}Standard A2.1.1.1.2}The expression Ïx is equivalent to 14i Ï 3 . What isthe value of x?An expression is shown below.x5 6x3 8xA. –588B. –588iWhich value of x makes the expression equal to 0?C. 588A. –2i2D. 588iB. –2iC. 4iD. 4i2Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 5
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.1Operations with Complex NumbersAnchor DescriptorA2.1.1.2Apply the orderof operations incomputation andin problem-solvingsituations.Eligible ContentA2.1.1.2.1Add and subtract complex numbers(e.g., (7 – 3i ) – (2 i ) 5 – 4i ).A2.1.1.2.2Multiply and divide complex numbers(e.g., (7 – 3i )(2 i ) 17 i ).PA CoreStandardsCC.2.1.HS.F.6Sample Exam QuestionsStandard A2.1.1.2.1An equation with real numbers a, b, c, and d isshown below.Standard A2.1.1.2.2An equation is shown below.(a bi )(4 – 2i ) 40(4i ab) – (6i cd) –2iWhat is the value of b?Which relationship must be true?A. 2A. ab –cdB. ab cdC. a cB. 4C. 10D. 20D. (a – c) (b – d)Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 6
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.1Operations with Complex NumbersSample Exam QuestionsStandard A2.1.1Lily is practicing multiplying complex numbers using the complex number (2 i ).To determine the value of (2 i )2, Lily performs the following operations:step 1:(2 i )2 4 i2step 2:4 i2 4 (–1)step 3:4 (–1) 3Lily made an error.A.Explain Lily’s error and correct the step which contains her error.Lily says that (2 i )n is a complex number for every positive integer value of n.B.Explain how you know that Lily is correct.Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 7
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsContinued. Please refer to the previous page for task explanation.Lily is continuing to explore different ways in which complex numbers can be multiplied so the answeris not a complex number. Lily multiplies (2 i ) and (a bi ), where a and b are real numbers, and findsthat her answer is not a complex number.C.Write an equation that expresses the relationship between a and b.equation:D.Explain why the expression (c di )2 is always a complex number for nonzero, real values ofc and d.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 8
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsStandard A2.1.1To find the roots of a quadratic equation, ax2 bx c, where a, b, and c are real numbers, Jan uses thequadratic formula.Jan finds that a quadratic equation has 2 distinct roots, but neither are real numbers.A.Write an inequality using the variables a, b, and c that must always be true for Jan’squadratic equation.inequality:}–The expression 3 Ï 4 is a solution of the quadratic equation x2 – 6x 13 0.B.}–What is 3 Ï 4 written as a complex number?}–3 Ï4 Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 9
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsContinued. Please refer to the previous page for task explanation.C.What is (5 2i )2 expressed as a complex number? Use the form a bi, where a and b are realnumbers.(5 2i )2 D.}}What is a possible solution to the equation 5 Ï (a – bi )(a bi ) when a and b are whole numbersgreater than zero?a b Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 10
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.2Non-Linear ExpressionsAnchor DescriptorA2.1.2.1Use exponents, roots,and/or absolute valuesto represent equivalentforms or to solveproblems.Eligible ContentA2.1.2.1.1Use exponential expressions to representrational numbers.A2.1.2.1.2Simplify/evaluate expressions involvingpositive and negative exponents and/orroots (may contain all types of realnumbers—exponents should not exceedpower of 10).Simplify/evaluate expressions involvingmultiplying with exponents(e.g., x6 x7 x13), powers of powers(e.g., (x6)7 x42), and powers of products(e.g., (2x2)3 8x6).Note: Limit to rational exponents.Simplify or evaluate expressions involvinglogarithms and exponents (e.g., log28 3or log42 1}2).A2.1.2.1.3A2.1.2.1.4PA CoreStandardsCC.2.1.HS.F.1CC.2.2.HS.D.2Sample Exam QuestionsStandard A2.1.2.1.11 ?Which expression is equivalent to25Standard A2.1.2.1.2An expression is shown below.8-8x x–2A. 4 10–B. 25 10 2–C. 4 10 1–D. 25 10 15 x 5xWhich value of x makes the expression equal to 1}?2A. 0B. 1C. 2D. 4Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 11
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsStandard A2.1.2.1.3Standard A2.1.2.1.4–4log–x10n(x5 (x 5 )n) x 10Which is the value of n?A. –3An expression is shown below.An equation is shown below.16x y4What is the value of the expression when log x 8and log y 1?A. 7B. 15B. –1C. 16D. 311C. – }2D. 2Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 12
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.2Non-Linear ExpressionsAnchor DescriptorA2.1.2.2Simplify expressionsinvolving polynomials.PA .3CC.2.2.HS.D.4CC.2.2.HS.D.5Eligible ContentA2.1.2.2.1Factor algebraic expressions, includingdifference of squares and trinomials.Note: Trinomials limited to the formax2 bx c where a is not equal to 0.A2.1.2.2.2Simplify rational algebraic expressions.Sample Exam QuestionsStandard A2.1.2.2.1An expression is shown below.Standard A2.1.2.2.2An expression is shown below.23x – 4x – 15 ; x Þ – 1, 3}26x2 – 19x 10Which is a factor of the expression?A. 2x 2B. 2x 5C. 3x – 22x – 5x – 32Which expression is equivalent to the one shown?3x 5A.2x 1–5B. 3x}2x – 1D. 3x – 5C. x2 x – 12D. 5x2 – 9x – 18Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 13
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.2Non-Linear ExpressionsSample Exam QuestionsStandard A2.1.2The expression (10d)3/2 is used to find how many times more energy is released by an earthquake ofgreater magnitude than by an earthquake of lesser magnitude, where d is the difference in magnitudes.A.How many times more energy is given off by an earthquake with magnitude 5.2 than by anearthquake with magnitude 3.2?times more energy:B.What is the difference (d) when 100 times more energy is released by an earthquake of greatermagnitude than by an earthquake of lesser magnitude?d Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 14
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsContinued. Please refer to the previous page for task explanation.C.What is an equivalent exponential expression to (10d)3/2 with a base of 1,000?equivalent exponential expression:D.}Explain why (Ï 10 )3d is equivalent to (10d)3/2.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 15
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsStandard A2.1.2Beatriz is simplifying exponential and radical expressions.A.3What rational number is the result of simplifying 16-3/2 ?rational number:The exponential expression 52x 3 can be simplified to the form a(bx) where a and b are integers.B.What are the values of a and b?a b Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 16
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsContinued. Please refer to the previous page for task explanation.The variable c represents a whole number between 1 and 100. The values of the expressions c1/2 andc2/3 are both whole numbers for only one value of c.C.What whole number does c represent?c Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 17
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.3Non-Linear EquationsAnchor DescriptorA2.1.3.1PA CoreStandardsCC.2.2.HS.C.2Write and/or solve quadratic equationsCC.2.2.HS.C.3(including factoring and using theCC.2.2.HS.C.4Quadratic Formula).CC.2.2.HS.C.5Solve equations involving rational and/orCC.2.2.HS.C.6radical expressionsCC.2.2.HS.D.5(e.g., 10/(x 3) 12/(x – 2) 1 orCC.2.2.HS.D.6}CC.2.2.HS.D.7Ïx2 21x 14).Write and/or solve a simple exponential or CC.2.2.HS.D.8CC.2.2.HS.D.9logarithmic equation (including commonCC.2.2.HS.D.10and natural logarithms).Eligible ContentWrite and/or solve non- A2.1.3.1.1linear equations usingvarious methods.A2.1.3.1.2A2.1.3.1.3A2.1.3.1.4Write, solve, and/or apply linear orexponential growth or decay (includingproblem situations).Sample Exam QuestionsStandard A2.1.3.1.1Standard A2.1.3.1.2The equation x2 bx c 0 has exactly 1 realsolution when b and c are real numbers. Whichequation describes b in terms of c?An equation is shown below.A. b c2What is the solution set of the equation?}48xB. b ÏcA. {–7, –1}C. b 2cB. {–7, 1}}D. b 2Ïc364x 8x x2 83C. {–1, 7}D. {1, 7}Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 18
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsStandard A2.1.3.1.3Standard A2.1.3.1.4An equation is shown below.35x 92x – 1A patient is given a 100-milligram dosage of a drugthat decays exponentially, with a half-life of 6 hours.Which equation could be used to find the milligramsof drug remaining (y) after x hours?Which equation has the same solution?A. y 100(6)0.5xA. 3x 10x – 5B. 5x 4x – 2C. 8x 11x – 1B. y 100(x)0.5/6C. y 100(0.5)x/6D. y 100(0.5x)1/6D. 15x 18x – 9Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 19
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.3Non-Linear EquationsAnchor DescriptorA2.1.3.2Describe and/ordetermine change.Eligible ContentA2.1.3.2.1Determine how a change in one variablerelates to a change in a second variable(e.g., y 4/x; if x doubles, what happensto y?).A2.1.3.2.2Use algebraic processes to solve aformula for a given variable(e.g., solve d rt for r).PA .4CC.2.2.HS.D.7CC.2.2.HS.D.8CC.2.2.HS.D.9Sample Exam QuestionsStandard A2.1.3.2.1Standard A2.1.3.2.2A moving object’s kinetic energy (Ek ) is dependenton the mass of the object (m) and the object’svelocity (v), as shown in the equation below.Physicists use the formula shown below todetermine total energy (E) of a body by usingmomentum (p), mass (m), and the speed of light (c).}}1 mv2Ek 2How does the value of Ek change when the valueof m is unchanged and the value of v is multipliedby 2?A. The value of Ek is squared.B. The value of Ek is multiplied by 2.C. The value of Ek is multiplied by 4.D. The value of Ek is multiplied by 8.E Ï(pc)2 (mc2)2A physicist knows the speed of light, the mass ofthe body, and the total energy used by the body.Which formula could be used to determine themomentum of the body?}E2 – m2c4A. p Ï}c}E – mc2B. p Ï}cE – mcC. p cD. p E – mcPennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 20
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsASSESSMENT ANCHORA2.1.3Non-Linear EquationsSample Exam QuestionsStandard A2.1.3Michaela is solving rational equations.A.B.x2 – 7x 12 3? Show or explain all your work.What is the solution set of the equationx2 x – 122 bx – 18 4 is x –17. What is the value of b?The only solution of the equation x}}–2x 4b Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 21
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsContinued. Please refer to the previous page for task explanation.Michaela solved for x in the rational equation as shown below.2x 2x – 15 3x–3x2 2x – 15 3 (x – 3)(x – 3) x–3x2 2x – 15 3x – 9x2 – x – 6 0(x – 3)(x 2) 0x – 3 0 or x 2 0x {3, –2}The solution set for x is incorrect.C.Explain Michaela’s error.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 22
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsStandard A2.1.3A fully-charged lithium-ion computer battery loses 20% of its permanent capacity each year of storage.A.Write an exponential equation showing the capacity (c) remaining in a fully-charged lithium-ioncomputer battery after y years.c B.What capacity is remaining in a fully-charged lithium-ion battery after 1.5 years?capacity:Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 23
Keystone Exams: Algebra IIMODULE 1—Number Systems and Non-Linear Expressions & EquationsContinued. Please refer to the previous page for task explanation.C.Solve the exponential equation from part A for y.y D.For how many years will a fully-charged lithium-ion computer battery have been stored when it haslost exactly half of its capacity?years:Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 24
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisASSESSMENT ANCHORA2.2.1Patterns, Relations, and FunctionsAnchor DescriptorA2.2.1.1Analyze and/or usepatterns or numbers.PA CoreStandardsCC.2.1.HS.F.7Analyze a set of data for the existence ofa pattern, and represent the pattern with a CC.2.2.HS.C.1CC.2.2.HS.C.2rule algebraically and/or graphically.CC.2.2.HS.C.3Identify and/or extend a pattern as eitherCC.2.2.HS.C.5an arithmetic or geometric sequenceCC.2.2.HS.C.6(e.g., given a geometric sequence, find the CC.2.3.HS.A.10CC.2.4.HS.B.220th term).Eligible ermine the domain, range, or inverse ofa relation.Identify and/or determine thecharacteristics of an exponential,quadratic, or polynomial function(e.g., intervals of increase/decrease,intercepts, zeros, and asymptotes).Sample Exam QuestionsStandard A2.2.1.1.1Terms 1 through 5 of a pattern are listed below.Standard A2.2.1.1.2Terms 1 through 5 of a sequence are shown below.8}814 3 4 7 12The pattern continues. Which expression could beused to determine the nth term in the pattern?A. 2n 2B. Zn – 2Z 3C. n2 – 4n 7D. n3 – 5n2 7n 14}272}91}31}2What is the 10th term in the sequence?81A. }64B. 81}32C. 243}64243D.32Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 25
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisStandard A2.2.1.1.3What is the inverse of y ln (x – 15) 3?A. y ex – 3 15B. y e 12xC. y ex 12 15D. y ex – 15 3Standard A2.2.1.1.4When is f(x) x2 – x – 12 increasing?1A. x }2B. x 1}2C. x –3D. x 4Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 26
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisASSESSMENT ANCHORA2.2.1Patterns, Relations, and FunctionsSample Exam QuestionsStandard A2.2.1The path of a roller coaster after it has reached the top of the first hill follows a polynomial function, asshown in the graph below.Path of a Roller Coasterf(x)Height (in feet)3002001000100200300400500xDistance (in feet)A.Over what interval is f(x) increasing? x Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 27
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisContinued. Please refer to the previous page for task explanation.B.At what value of x is there a minimum of f(x) over the interval 0 x 300?value of x:C.At what value of x is there a zero of f(x)?value of x:Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 28
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisContinued. Please refer to the previous page for task explanation.D.Explain why f(x) cannot be a quadratic function.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 29
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisStandard A2.2.1The function below describes the graph of a quadratic function where c is a positive real number.y x2 – cA.What are the x-intercept(s) of the graph of the quadratic function?x-intercept(s):B.What is a y-value, in terms of c, which cannot be in the range of the quadratic function?y-value:Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 30
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisContinued. Please refer to the previous page for task explanation.C.What is the domain of the inverse of the quadratic function?domain of the inverse of the quadratic function:Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 31
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisASSESSMENT ANCHORA2.2.2Applications of FunctionsAnchor DescriptorA2.2.2.1Eligible ContentCreate, interpret,and/or use polynomial,exponential, and/orlogarithmic functionsand their equations,graphs, or ate, interpret, and/or use the equation,graph, or table of a polynomial function(including quadratics).Create, interpret, and/or use the equation,graph, or table of an exponential orlogarithmic function (including commonand natural logarithms).Determine, use, and/or interpret minimumand maximum values over a specifiedinterval of a graph of a polynomial,exponential, or logarithmic function.Translate a polynomial, exponential,or logarithmic function from onerepresentation of a function to another(graph, table, and equation).PA .D.7CC.2.3.HS.A.10Sample Exam QuestionsStandard A2.2.2.1.1The table below represents a quadratic function.xf(x)53401308115Which describes a complete list where the zeros off(x) occur?A. x 8 and x 4B. x 4 and x 2C. x –8 and x –4D. x –4 and x –2Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 32
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisStandard A2.2.2.1.2Standard A2.2.2.1.4A logarithmic function is graphed below.A function of x is graphed below.yf(x)86424321– 4 – 3 – 2 –1–1–2–3–41 2 3 4xWhat is the value of f(8)?A. 3– 8 – 6 – 4 –2–2–4–6–82 4 6 8xWhich equation best describes the graph?A. y x2 5B. 4B. y (x – 2)2 1C. 16C. y (x 2)2 1D. 256D. y (x 2)(x – 1)Standard A2.2.2.1.3A function of x is shown below.y –3(x – 2)(x 4)What is the maximum value of the function over theinterval –3 x 2?A. 0B. 15C. 24D. 27Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 33
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisASSESSMENT ANCHORA2.2.2Applications of FunctionsAnchor DescriptorA2.2.2.2Describe and/ordetermine families offunctions.Eligible ContentA2.2.2.2.1Identify or describe the effect of changingparameters within a family of functions(e.g., y x2 and y x2 3, or y x2 andy 3x2).PA .6Sample Exam QuestionStandard A2.2.2.2.1The graph of the equation y 3x2 has its vertex atthe coordinate point (0, 0). What coordinate pointdescribes the vertex of the graph of the equationy 3x2 – 3?A. (0, –3)B. (0, 3)C. (–3, 0)D. (3, 0)Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 34
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisASSESSMENT ANCHORA2.2.2Applications of FunctionsSample Exam QuestionsStandard A2.2.2An exponential function of the form f(x) a bx c is represented by the pairs of values shown in thetable below.xf(x)21,215113501512A.53527Determine the exponential function that contains the 5 points shown in the table.f(x) Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 35
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisContinued. Please refer to the previous page for task explanation.B.What is the minimum value of f(x) over the interval –5 x 5?minimum:C.Describe the difference in the graph of the exponential function g(x) a bx 2 c and the graph ofthe exponential function f(x) a bx c when a, b, and c remain unchanged.Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 36
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisContinued. Please refer to the previous page for task explanation.D.What is the value of g(0)?g(0) Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 37
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisStandard A2.2.2The number of meals a restaurant serves is a function of the price of each meal. The restaurant found it willserve 72 meals when it charges a price of 7.00 per meal. It will serve 52 meals when it charges a price of 12.00 per meal. The relationship between the number of meals served and the price of each meal is linear.A.Write a linear function that represents the relationship between the number of meals served (f(x) )and the price of each meal (x).f(x) Continued on next page.Pennsylvania Department of Education—Assessment Anchors and Eligible ContentPage 38
Keystone Exams: Algebra IIMODULE 2—Functions and Data AnalysisContinued. Please refer to the previous pag
Keystone Exams. The module title appears at the top of each page. The module level is important because the Keystone Exams are built using a module format, with each of the Keystone Exams divided into two equal-size test modules. Each module is made up of two or more Assessment Anchors.
General Introduction to the Keystone Exam Assessment Anchors. Introduction . Since the introduction of the Keystone Exams, the Pennsylvania Department of Education (PDE) has been working to create a setof tools designed to help educators improve instructional practices and better understand the Keystone Exams.
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KE01053 Keystone Literature - English as a Second Language X KV01053 Keystone Literature - Visually Impaired X KH01053 Keystone Literature - Hearing Impaired X KAE01053 Keystone Literature - Alternative Education X K01005 Keystone – AP English Language and Composition X X X K01006 Keystone – AP Engl
Pennsylvania Keystone Algebra I Item and Scoring Sampler 2014 1 Keystone Algebra I Sampler Information About Algebra I INTRODUCTION The Pennsylvania Department of Education (PDE) provides districts and schools with tools to assist in delivering . Each Keystone item has been through a rigorous review process to ensure that it is as demanding .
Robert Gerver, Ph.D. North Shore High School 450 Glen Cove Avenue Glen Head, NY 11545 gerverr@northshoreschools.org Rob has been teaching at . Algebra 1 Financial Algebra Geometry Algebra 2 Algebra 1 Geometry Financial Algebra Algebra 2 Algebra 1 Geometry Algebra 2 Financial Algebra ! Concurrently with Geometry, Algebra 2, or Precalculus
Keystone STARS Program Manual I. About Keystone STARS Keystone STARS is a program of Pennsylvania's Office of Child Development and Early Learning (OCDEL). Keystone STARS has four primary goals: To improve the quality of early care and education; To support early care and education providers in meeting their quality improvement goals;
The Keystone Exams are end‑of‑course assessments currently designed to assess proficiencies in Algebra I, Biology, and Literature. For detailed information about how the Keystone Exams are being integrated into the Pennsylvania graduation requirements, please contact the Pennsylvania Department of Education or visit the
Nama Mata Kuliah : Akuntansi Keuangan Lanjutan Kode Mata Kuliah : AKM 145001 Semester : 5 (lima) Sks/jam perminggu : 3 SKS/ 6 jam Jurusan/ Program Studi : Jurusan Akuntansi/ DIV Akuntansi Manajemen Dosen Pengampu : 1. Novi Nugrahani, SE., M.Ak., Ak 2. Drs. Bambang Budi Prayitno, M.Si., Ak 3. Marlina Magdalena, S.Pd. MSA Capaian Pembelajaran Lulusan yang dibebankan pada mata kuliah :Setelah .
