Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties.


 Brook Jennings
 2 years ago
 Views:
Transcription
1 Polynomial functions mctypolynomial Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties. In order to master the techniques eplained here it is vital that you undertake plenty of practice eercises so that they become second nature. Afterreadingthistet,and/orviewingthevideotutorialonthistopic,youshouldbeableto: recognisewhenaruledescribesapolynomialfunction,andwritedownthedegreeofthe polynomial, recognizethetypicalshapesofthegraphsofpolynomials,ofdegreeupto4, understandwhatismeantbythemultiplicityofarootofapolynomial, sketchthegraphofapolynomial,givenitsepressionasaproductoflinearfactors. Contents 1. Introduction 2 2. What is a polynomial? 2 3. Graphs of polynomial functions 3 4. Turning points of polynomial functions 6 5. Roots of polynomial functions 7 1 c mathcentre 2009
2 1. Introduction Apolynomialfunctionisafunctionsuchasaquadratic,acubic,aquartic,andsoon,involving onlynonnegativeintegerpowersof. Wecangiveageneraldefintionofapolynomial,and define its degree. 2. What is a polynomial? Apolynomialofdegree nisafunctionoftheform f() = a n n + a n 1 n a a 1 + a 0 where the a s are real numbers(sometimes called the coefficients of the polynomial). Although this general formula might look quite complicated, particular eamples are much simpler. For eample, f() = isapolynomialofdegree3,as3isthehighestpowerof intheformula.thisiscalledacubic polynomial, or just a cubic. And f() = isapolynomialofdegree7,as7isthehighestpowerof.noticeherethatwedon tneedevery powerof upto7: weneedtoknowonlythehighestpowerof tofindoutthedegree. An eampleofakindyoumaybefamiliarwithis f() = whichisapolynomialofdegree2,as2isthehighestpowerof.thisiscalledaquadratic. Functions containing other operations, such as square roots, are not polynomials. For eample, f() = isnotapolynomialasitcontainsasquareroot.and f() = / isnotapolynomialasitcontainsa divideby. Apolynomialisafunctionoftheform Key Point f() = a n n + a n 1 n a a 1 + a 0. The degree of a polynomial is the highest power of in its epression. Constant(nonzero) polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1, 2,3and4respectively.Thefunction f() = 0isalsoapolynomial,butwesaythatitsdegree is undefined. 2 c mathcentre 2009
3 3. Graphs of polynomial functions Wehavemetsomeofthebasicpolynomialsalready.Foreample, f() = 2isaconstantfunction and f() = 2 + 1isalinearfunction. f () f () = f () = 2 1 It is important to notice that the graphs of constant functions and linear functions are always straight lines. Wehavealreadysaidthataquadraticfunctionisapolynomialofdegree2. Herearesome eamples of quadratic functions: f() = 2, f() = 2 2, f() = 5 2. Whatistheimpactofchangingthecoefficientof 2 aswehavedoneintheseeamples? One waytofindoutistosketchthegraphsofthefunctions. f () f () = 5 2 f () = 2 2 f () = 2 Youcanseefromthegraphthat,asthecoefficientof 2 isincreased,thegraphisstretched vertically(that is, in the y direction). Whatwillhappenifthecoefficientisnegative?Thiswillmeanthatallofthepositive f()values willnowbecomenegative.sowhatwillthegraphsofthefunctionslooklike?thefunctionsare now f() = 2, f() = 2 2, f() = c mathcentre 2009
4 f () f () = 2 f () = 2 2 f () = 5 2 Noticeherethatallofthesegraphshaveactuallybeenreflectedinthe ais. Thiswillalways happenforfunctionsofanydegreeiftheyaremultipliedby 1. Nowletuslookatsomeotherquadratic functions toseewhathappens whenwevarythe coefficientof,ratherthanthecoefficientof 2.Weshalluseatableofvaluesinordertoplot thegraphs,butweshallfillinonlythosevaluesneartheturningpointsofthefunctions Youcanseethesymmetryineachrowofthetable,demonstratingthatwehaveconcentrated ontheregionaroundtheturningpointofeachfunction. Wecannowusethesevaluestoplot the graphs. f () f () = 2 + f () = f () = As you can see, increasing the positive coefficient of in this polynomial moves the graph down andtotheleft. 4 c mathcentre 2009
5 What happens if the coefficient of is negative? Againwecanusethesetablesofvaluestoplotthegraphsofthefunctions. f () f () = 2 f () = 2 4 f () = 2 6 Asyoucansee,increasingthenegativecoefficientof (inabsoluteterms)movesthegraph downandtotheright. Sonowweknowwhathappenswhenwevarythe 2 coefficient,andwhathappenswhenwe varythe coefficient. Butwhathappenswhenwevarytheconstanttermattheendofour polynomial? Wealreadyknowwhatthegraphofthefunction f() = 2 + lookslike,sohow doesthisdifferfromthegraphofthefunctions f() = ,or f() = ,or f() = 2 + 4?Asusual,atableofvaluesisagoodplacetostart Ourtableofvaluesisparticularlyeasytocompletesincewecanuseouranswersfromthe 2 + columntofindeverythingelse. Wecanusethesetablesofvaluestoplotthegraphsofthe functions. 5 c mathcentre 2009
6 f () f () = f () = f () = 2 + f () = Aswecanseestraightaway,varyingtheconstanttermtranslatesthe 2 + curvevertically. Furthermore,thevalueoftheconstantisthepointatwhichthegraphcrossesthe f()ais. 4. Turning points of polynomial functions Aturningpointofafunctionisapointwherethegraphofthefunctionchangesfromsloping downwards to sloping upwards, or vice versa. So the gradient changes from negative to positive, orfrompositivetonegative. Generallyspeaking,curvesofdegree ncanhaveupto (n 1) turning points. For instance, a quadratic has only one turning point. Acubiccouldhaveuptotwoturningpoints,and so would look something like this. However, some cubics have fewer turning points: for eample f() = 3. Butnocubichasmorethantwo turning points. 6 c mathcentre 2009
7 Inthesameway,aquarticcouldhaveuptothree turning turning points, and so would look something like this. Again, some quartics have fewer turning points, but none has more. Key Point Apolynomialofdegree ncanhaveupto (n 1)turningpoints. 5. Roots of polynomial functions Youmayrecallthatwhen ( a)( b) = 0,weknowthat aand barerootsofthefunction f() = ( a)( b). Nowwecanusetheconverseofthis,andsaythatif aand bareroots, thenthepolynomialfunctionwiththeserootsmustbe f() = ( a)( b),oramultipleof this. Foreample,ifaquadratichasroots = 3and = 2,thenthefunctionmustbe f() = ( 3)(+2),oraconstantmultipleofthis.Thiscanbeetendedtopolynomialsofanydegree. Foreample,iftherootsofapolynomialare = 1, = 2, = 3, = 4,thenthefunctionmust be f() = ( 1)( 2)( 3)( 4), or a constant multiple of this. Letusalsothinkaboutthefunction f() = ( 2) 2.Wecanseestraightawaythat 2 = 0, sothat = 2. Forthisfunctionwehaveonlyoneroot. Thisiswhatwecallarepeatedroot, andarootcanberepeatedanynumberoftimes. Foreample, f() = ( 2) 3 ( + 4) 4 has arepeatedroot = 2,andanotherrepeatedroot = 4. Wesaythattheroot = 2has multiplicity3,andthattheroot = 4hasmultiplicity4. Theusefulthingaboutknowingthemultiplicityofarootisthatithelpsuswithsketchingthe graphofthefunction.ifthemultiplicityofarootisoddthenthegraphcutsthroughthe ais atthepoint (, 0). Butifthemultiplicityiseventhenthegraphjusttouchesthe aisatthe point (, 0). For eample, take the function f() = ( 3) 2 ( + 1) 5 ( 2) 3 ( + 2) 4. Theroot = 3hasmultiplicity2,sothegraphtouchesthe aisat (3, 0). 7 c mathcentre 2009
8 Theroot = 1hasmultiplicity5,sothegraphcrossesthe aisat ( 1, 0). Theroot = 2hasmultiplicity3,sothegraphcrossesthe aisat (2, 0). Theroot = 2hasmultiplicity4,sothegraphtouchesthe aisat ( 2, 0). Totakeanothereample,supposewehavethefunction f() = ( 2) 2 ( + 1). Wecansee thatthelargestpowerof is3,andsothefunctionisacubic. Weknowthepossiblegeneral shapesofacubic,andasthecoefficientof 3 ispositivethecurvemustgenerallyincreaseto therightanddecreasetotheleft.wecanalsoseethattherootsofthefunctionare = 2and = 1. Theroot = 2hasevenmultiplicityandsothecurvejusttouchesthe aishere, whilst = 1hasoddmultiplicityandsoherethecurvecrossesthe ais.thismeanswecan sketch the graph as follows. f () 1 2 Key Point Thenumber aisarootofthepolynomialfunction f()if f(a) = 0,andthisoccurswhen ( a) isafactorof f(). If aisarootof f(),andif ( a) m isafactorof f()but ( a) m+1 isnotafactor,thenwe say that the root has multiplicity m. Atarootofoddmultiplicitythegraphofthefunctioncrossesthe ais,whereasatarootof even multiplicity the graph touches the ais. Eercises 1. What is a polynomial function? 2. Which of the following functions are polynomial functions? (a) f() = (b) f() = (c) f() = (d) f() = sin + 1 (e) f() = / (f) f() = c mathcentre 2009
9 3. Write down one eample of each of the following types of polynomial function: (a) cubic (b) linear (c) quartic (d) quadratic 4.Sketchthegraphsofthefollowingfunctionsonthesameaes: (a) f() = 2 (b) f() = 4 2 (c) f() = 2 (d) f() = Considerafunctionoftheform f() = 2 +a,where arepresentsarealnumber.thegraph of this function is represented by a parabola. (a)when a > 0,whathappenstotheparabolaas aincreases? (b)when a < 0,whathappenstotheparabolaas adecreases? 6.Writedownthemaimumnumberofturningpointsonthegraphofapolynomialfunctionof degree: (a) 2 (b) 3 (c) 12 (d) n 7. Write down a polynomial function with roots: (a) 1, 2, 3, 4 (b) 2, 4 (c) 12, 1, 6 8. Write down the roots and identify their multiplicity for each of the following functions: (a) f() = ( 2) 3 ( + 4) 4 (b) f() = ( 1)( + 2) 2 ( 4) 3 9. Sketch the following functions: (a) f() = ( 2) 2 ( + 1) (b) f() = ( 1) 2 ( + 3) Answers 1.Apolynomialfunctionisafunctionthatcanbewrittenintheform f() = a n n + a n 1 n 1 + a n 2 n a a 1 + a 0, whereeach a 0, a 1,etc.representsarealnumber,andwhere nisanaturalnumber(including0). 2. (a) f() = isapolynomial (b) f() = isnotapolynomial,becauseof (c) f() = isapolynomial (d) f() = sin + 1isnotapolynomial,becauseof sin (e) f() = /isnotapolynomial,becauseof 2/ (f) f() = isapolynomial 3. (a) Thehighestpowerof mustbe3,soeamplesmightbe f() = or f() = 3 2. (b) Thehighestpowerof mustbe1,soeamplesmightbe f() = or f() = 6 5. (c) Thehighestpowerof mustbe4,soeamplesmightbe f() = or f() = 4 5. (d) Thehighestpowerof mustbe2,soeamplesmightbe f() = 2 or f() = c mathcentre 2009
10 f () f () = 4 2 f () = 2 f () = 2 f () = (a) When a > 0,theparabolamovesdownandtotheleftas aincreases. (b) When a < 0,theparabolamovesdownandtotherightas adecreases. 6. (a) 1 turning point (d) (n 1) turning points. (b) 2 turning points (c) 11 turning points 7. (a) f() = ( 1)( 2)( 3)( 4)oramultiple (b) f() = ( 2)( + 4)oramultiple (c) f() = ( 12)( + 1)( + 6)oramultiple. 8. (a) = 2 = 4 (b) = 1 = 2 = 4 odd multiplicity even multiplicity odd multiplicity even multiplicity odd multiplicity 10 c mathcentre 2009
11 9) f () f () = ( 1) 2 ( + 3) f () = ( 2) 2 ( + 1) 11 c mathcentre 2009
Symmetry. A graph is symmetric with respect to the yaxis if, for every point (x, y) on the graph, the point (x, y) is also on the graph.
Symmetry When we graphed y =, y = 2, y =, y = 3 3, y =, and y =, we mentioned some of the features of these members of the Library of Functions, the building blocks for much of the study of algebraic functions.
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More informationRoots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)
Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationAfterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: explainwhycubicequationspossesseitheronerealrootorthreerealroots
Cubic equations Acubicequationhastheform mctycubicequations2009 ax 3 + bx 2 + cx + d = 0 where a 0 Allcubicequationshaveeitheronerealroot,orthreerealroots.Inthisunitweexplorewhythis isso. Thenwelookathowcubicequationscanbesolvedbyspottingfactorsandusingamethodcalled
More informationCompleting the square
Completing the square mctycompletingsquare0091 In this unit we consider how quadratic expressions can be written in an equivalent form using the technique known as completing the square. This technique
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationInvestigation. So far you have worked with quadratic equations in vertex form and general. Getting to the Root of the Matter. LESSON 9.
DA2SE_73_09.qd 0/8/0 :3 Page Factored Form LESSON 9.4 So far you have worked with quadratic equations in verte form and general form. This lesson will introduce you to another form of quadratic equation,
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationAlgebra II A Final Exam
Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More information2.4. Factoring Quadratic Expressions. Goal. Explore 2.4. Launch 2.4
2.4 Factoring Quadratic Epressions Goal Use the area model and Distributive Property to rewrite an epression that is in epanded form into an equivalent epression in factored form The area of a rectangle
More informationObjectives: To graph exponential functions and to analyze these graphs. None. The number A can be any real constant. ( A R)
CHAPTER 2 LESSON 2 Teacher s Guide Graphing the Eponential Function AW 2.6 MP 2.1 (p. 76) Objectives: To graph eponential functions and to analyze these graphs. Definition An eponential function is a function
More informationAlgebra II Semester Exam Review Sheet
Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph
More informationAn Insight into Division Algorithm, Remainder and Factor Theorem
An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the
More informationFunctions: Piecewise, Even and Odd.
Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 2122, 2015 Tutorials, Online Homework.
More informationSection 2.3. Learning Objectives. Graphing Quadratic Functions
Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic
More informationCalculus Card Matching
Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationObjective 1: Identify the characteristics of a quadratic function from its graph
Section 8.2 Quadratic Functions and Their Graphs Definition Quadratic Function A quadratic function is a seconddegree polynomial function of the form, where a, b, and c are real numbers and. Every quadratic
More informationNotes on Curve Sketching. B. Intercepts: Find the yintercept (f(0)) and any xintercepts. Skip finding xintercepts if f(x) is very complicated.
Notes on Curve Sketching The following checklist is a guide to sketching the curve y = f(). A. Domain: Find the domain of f. B. Intercepts: Find the yintercept (f(0)) and any intercepts. Skip finding
More informationSection 1.3: Transformations of Graphs
CHAPTER 1 A Review of Functions Section 1.3: Transformations of Graphs Vertical and Horizontal Shifts of Graphs Reflecting, Stretching, and Shrinking of Graphs Combining Transformations Vertical and Horizontal
More informationSTRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula
UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving
More informationPreCalculus III Linear Functions and Quadratic Functions
Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
More informationQuadratic Functions and Models
Quadratic Functions and Models MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: analyze the graphs of quadratic functions, write
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
More informationIn this lesson you will learn to find zeros of polynomial functions that are not factorable.
2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
More informationThe degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =
Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial
More informationSection V: Quadratic Equations and Functions. Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and CompletingtheSquare
Haberman / Kling MTH 95 Section V: Quadratic Equations and Functions Module : Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and CompletingtheSquare DEFINITION: A quadratic equation
More information8 Graphs of Quadratic Expressions: The Parabola
8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More informationFunctions and Graphs
PSf Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form. 2 2. hsn.uk.net Page 1 Questions
More informationAnswer Key Building Polynomial Functions
Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,
More informationCPM Educational Program
CPM Educational Program A California, NonProfit Corporation Chris Mikles, National Director (888) 8084276 email: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few
More informationPolynomial and Rational Functions
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
More informationIntegrating algebraic fractions 2
Integrating algebraic fractions 2 mctyalgfrac22009 Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fractionasthesumofitspartialfractionsinthisunitwelookatthecasewherethedenominator
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationIntegration of Rational Expressions by Partial Fractions
Integration of Rational Epressions by Partial Fractions INTRODUTION: We start with a few definitions rational epression is formed when a polynomial is divided by another polynomial In a proper rational
More informationSection 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
More informationFactoring Polynomials
Factoring Polynomials 412014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall
More information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
More informationLines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line
Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Hperbolic functions The hperbolic functions have similar names to the trigonmetric functions, but the are defined in terms of the eponential function In this unit we define the three main hperbolic functions,
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More informationA polynomial of degree n is a function of the form. f(x) = 3x 22x 4. Also, identify domain & range.
f(x) = 3x 22x 4. Also, identify domain & range. Identify polynomial functions. Recognize characteristics of graphs of polynomials. Determine end behavior. Use factoring to find zeros of polynomials.
More informationTaylor Polynomials. for each dollar that you invest, giving you an 11% profit.
Taylor Polynomials Question A broker offers you bonds at 90% of their face value. When you cash them in later at their full face value, what percentage profit will you make? Answer The answer is not 0%.
More informationGraphing Quadratic Functions
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationMidterm 1. Solutions
Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationPolynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI8 can be used
More informationFactorising quadratics
Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationExponential and Logarithmic Functions
Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationINTEGRATION FINDING AREAS
INTEGRTIN FINDING RES Created b T. Madas Question 1 (**) = 4 + 10 3 The figure above shows the curve with equation = 4 + 10, R. Find the area of the region, bounded b the curve the coordinate aes and the
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationUnit 3: Day 2: Factoring Polynomial Expressions
Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationCubic Functions: Global Analysis
Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavitysign, 232 Slopesign, 234 Extremum, 235 Heightsign, 236 0Concavity Location, 237 0Slope Location, 239 Extremum Location,
More information3.1 Quadratic Functions
Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain
More information2.1 Equations of Lines
Section 2.1 Equations of Lines 1 2.1 Equations of Lines The SlopeIntercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is
More informationMath 119 Pretest Review Answers
Math 9 Pretest Review Answers Linear Equations Find the slope of the line passing through the given points:. (, 5); (0, ) 7/. (, 4); (, 0) Undefined. (, ); (, 4) 4. Find the equation of the line with
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More information1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.
Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1.
More informationHomework Assignment and Teacher Comments 1 st 9weeks There are 41 teaching days. Learning Target Goal Students will be able to
Algebra 2 Pacing Guide 20152016 Teacher Name Period This Curriculum Guide is a suggestion for you to use as a guide. Each class should progress at its own pace. (written June 2015) Stards Benchmarks 1
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More information2.3 Quadratic Functions
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
More informationContents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...
Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima
More information