# Quadratic Functions. Analyze and describe the characteristics of quadratic functions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Section.3 - Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze sctter plots, nd determine nd pply the equtions for the curves of best fit, using pproprite technology C8 Describe nd trnslte between grphicl, tbulr, written, nd symbolic representtions of qudrtic reltionships B Demonstrte n understnding of the reltionships tht exist between rithmetic opertions nd the opertions used when solving equtions. C9 Trnslte between different forms of qudrtic equtions A7 Describe nd interpret domins nd rnges using set nottion C3 Demonstrte n understnding of how prmeter chnges ffect the grphs of qudrtic functions Assumed Prior Knowledge enerl form of the qudrtic function Effects of the coefficient on the shpe of grph (prbol) of qudrtic function Vertex of prbol Mximum nd minimum y-vlues Fctoring nd expnding qudrtic expressions Using lgebr tiles to fctor nd represent qudrtic expressions Using technology to grph qudrtic functions nd to find the eqution of the curve of best fit Pge --

2 Dignostic Worksheet for Selected Assumed Prior Knowledge (Section.3). Which is qudrtic function in generl form? ) y = (x-3) + 4 b) y-4 = (x-3) c) y = x + 3x -5. Which of these sttements is true bout the grph of -c(y + ) = (x - ) when compred to the grph of y = x? ) Verticl stretch -8, Trnsltion left, up b) Reflection in y-xis, Verticl stretch of c, Trnsltion right, down c) Reflection in x-xis, Verticl stretch of 8, Trnsltion right, down 3. Wht is the mpping rule tht describes the trnsformtion of y = x into ¼(y+) = (x - 9)? 4. Expnd: ) (x-)(x+3) b) (x-4)(x+) c) (3x - )(-x-) 5. Fctor ech of the following completely. Check your work by multiplying your fctors. ) x - 9x - 36 b) 5x + 5x c) x - 6x + 5 d) x + x + e) y + 3y + f) y + 8y + 6 g) x - 4x + 40 h) x - 6x -7 i) 5 - b j) 9n - 6 k) y For which vlues of k cn ech of the following be fctored? ) 5x + kx + 4 b) 3x + kx - c) -8x + kx - 6 Pge --

3 7. Remove common fctor first, then fctor further if possible. Check your work by multiplying your fctors. ) 3y + 5y + b) 5x - 0x + 5 c) 5x 5-80xy 4 d) 49x 3 + 5xy 8. Fctor completely. Check your work by multiplying your fctors. ) 3x + 3x + 4 b) x + 7x - 4 c) 8x - x - 0 d) -c +9c - 4 Answers #. c #. c #3. (x, y) ö (x + 9, 4y - ) #4. ) x + x -6 b) x - 3x - 4 c) -6x - x + #5. ) (x - )(x + 3) b) 5x(x + 5) c) (x - 5)(x - ) d) (x + ) e) (y + )(y + ) f) (y + 4) g) (x - 0)(x - 4) i) (5 - b)(5 + b) j) (3n - 4)(3n + 4) k) (y - 9)(y + 9) #6. ) ±, ±9, ± b) ±5, ± c) ±49, ±6, ±9, ±6, ±4 #7. ) 3(y + 4)(y + ) b) 5(x - 3)(x - ) c) 5x(5x - 6y )(5x + 6y ) d) x(49x + 5y ) #8. ) (3x + )(x + 4) b) (x - )(x + 4) c) (8x + 5)(x - ) d) - (4c - )(3c - 4) Pge -3-

4 Focus C - Forms of Qudrtic Functions LOSSARY TERMS (see text p. 4-30) Prbol, Trnsformtionl Form of Qudrtic Function, Vertex of Prbol, Axis of Symmetry, Verticl Stretch, Stndrd Form of Qudrtic Function, enerl Form of Qudrtic Function, This Focus is included minly to provide you with n opportunity to review some things you lerned in Mth 04 bout prbols, mpping rules, nd trnsformtions. Some of the bove terminology my be new to you. In prticulr, the Stndrd Form of Qudrtic is new but not difficult to understnd if you re comfortble with trnsformtionl form. Wht you should hve lerned from this Focus: Qudrtic Function A. enerl form y = x +bx+c, 0 L is the verticl stretch, (0,c) is the y-intercept B. Stndrd Form y = (x-h) +k, 0 L is the verticl stretch, vertex is (h,k) C. Trnsformtionl Form (y k) = (x h), 0 L is the verticl stretch, vertex is (h,k). The mpping rule when compred to y = x is (x,y) º(x + h, y + k). How to write qudrtic equtions in both stndrd nd generl forms How to chnge qudrtic function from Stndrd Form to Trnsformtionl Form nd vice vers. How to find the verticl stretch of grphed prbol by compring with the grph of y = x Pge -4-

5 Focus D - Creting the Trnsformtionl Form of Qudrtic As you hve lredy lerned, there re three forms of the Qudrtic function: enerl form y = x +bx+c, 0 Stndrd Form y = (x-h) +k, 0 Trnsformtionl Form (y k) = (x h), 0 Ech of these forms re useful becuse we cn esily red certin informtion tht is different from tht using nother form. It will therefore be helpful to be ble to convert from one form to nother, nd in Focus C you lredy lerned how to chnge from Stndrd Form to Trnsformtionl Form nd vice vers. In this Focus you will lern how to convert from enerl Form to either of the other forms. First, however, we need to look t specil types of trinomils which cn be fctored s perfect squres Wht is Perfect Squre Trinomil? Recll tht perfect squre number is one tht hs two fctors which re EXACTLY the sme e.g. 4 = =, 5 = 5 5 = 5. In similr mnner, some trinomils cn be fctored s the product of two fctors which re EXACTLY the sme e.g. x + 0x + 5 = (x+5)(x+5) = (x+5) Prctice - Fctor ech of the following perfect squre trinomils: ) x + x + 36 b) x - 0x + 5 c) x - 8x + 8 In ech of the bove perfect squre trinomils, there is specil reltionship between the middle nd lst coefficients (tht is, the vlues of b nd c): Tke hlf of the middle coefficient, then squre it nd you should obtin the lst coefficient. For exmple, in question b), -0 is the middle coefficient; hlf of tht is -5; (-5) = 5, which is the lst coefficient. In generl, for perfect squre trinomil of the form x + bx + c, the reltionship is (b/) = c. Pge -5-

6 Prctice - Find the vlue tht should be plced in ech blnk in order to crete perfect squre trinomil. ) x + 4x + b) x - 8x + c) x - 4x + d) x - x + e) x - 3x + f) d + 9d + g) h) q - q + i) p p + 49 Chnging from enerl Form to Trnsformtionl Form If we look t the Trnsformtionl or Stndrd Forms of Qudrtic Function, we cn see the result of fctoring perfect squre trinomil e.g. EXAMPLE: Chnge to Trnsformtionl Form: y = x - 8x + 5 Step y - 5 = x - 8x Bring the constnt term (the c vlue) to the other side of the eqution Step y - 5 = (x - 8x) The coefficient of the qudrtic term (the vlue) must be removed s common fctor Step 3 y ( 6) = (x - 8x + 6) Chnge the expression in the brckets to perfect squre trinomil. Blnce the eqution by dding equivlent mounts to both sides; in this cse, 6 hs been dded to both sides. Step 4 y + 7 = (x - 4) Fctor the perfect squre trinomil Step 5 ½(y + 7) = (x - 4) OR y = (x - 4) - 7 Move the cross the = sign using its reciprocl to crete Trnsformtionl Form Move the 7 cross the = sign to crete the Stndrd Form. Pge -6-

7 Wht you should hve lerned from this Focus: How to identify nd crete perfect squre trinomil written in the form x +bx+ c How to convert from enerl Form to either Trnsformtionl or Stndrd Form How to use the Trnsformtionl or Stndrd Form to solve problems, including those involving mxim nd minim (See text p. 33, #8, 9,, ) Extr Prctice - Revisit the Extr Prctice problems from Investigtion 4 nd solve them by creting the Trnsformtionl or Stndrd Form of the qudrtic function insted of using the grphing clcultor s you did previously. Pge -7-

8 Focus E - Determining Qudrtic Functions from Prbols Red p Another exmple using both methods is provided below. Fill in the spces to complete the explntions nd/or the steps Method (y k) = (x h) y = x, Vertex is (0,0) From vertex, Over up Over up 4 Over 3 up is the trnsformtionl form. Method For the grph, Vertex is (3,4). From vertex, Over down Verticl stretch is therefore Since the grph is opening down nd the grph of y = x opens up, then there is reflection cross the. The qudrtic function is -½(y - ) = (x - ) Vertex (h,k) = (3,4). Therefore, we know (y ) = (x _) The grph psses through the point (4,). Substitute these coordintes for x nd y, then solve for : ( _) = ( _ ) ( ) = () = = Therefore, the eqution is Pge -8-

9 Extr Prctice Find the Trnsformtionl nd the Stndrd Form of the function for ech grph shown: Pge -9-

10 Answers Trnsformtionl Form. (y-) = x. (y+) = x 3. 3y = (x+) 4. -8(y-) = (x+5) 5. -/4. (y-5.8) = (x-.5) or -5/(y-5.8) = (x-.5) 6. /. (y + 0.3) = (x-.4) or 0.8 (y + 0.3) = (x-.4). y = x +. y = 3x - 3. y = (x+) 4. y = -/8(x+5) + 5. y = -4.(x-.5) y=.(x-.4) -0.3 Stndrd Form Wht you should hve lerned from this Focus: How to find the verticl stretch of Qudrtic Function from its grph (review from Focus C) How to find Qudrtic Function in either Trnsformtionl or Stndrd Form using the grph How to find Qudrtic Function in either Trnsformtionl or Stndrd Form given the vertex (or informtion to help you find it) nd nother point (e.g. see text p.38 #39, 4) How to find Qudrtic Function in either Trnsformtionl or Stndrd Form from problem sitution (e.g. see text, p.38, #38, 45) Pge -0-

### Section 7-4 Translation of Axes

62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

### Math 135 Circles and Completing the Square Examples

Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

### Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

### Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

### Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

### SPECIAL PRODUCTS AND FACTORIZATION

MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

### PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

### Factoring Polynomials

Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

### Curve Sketching. 96 Chapter 5 Curve Sketching

96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

### Vectors 2. 1. Recap of vectors

Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

### 4.11 Inner Product Spaces

314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

### LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

### Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

### Basic Analysis of Autarky and Free Trade Models

Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

### Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

### Warm-up for Differential Calculus

Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

### Binary Representation of Numbers Autar Kaw

Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

### Algebra Review. How well do you remember your algebra?

Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

### P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

### 6.2 Volumes of Revolution: The Disk Method

mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

### Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

### Integration by Substitution

Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

### 1 Numerical Solution to Quadratic Equations

cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

### 9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

### LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

### Section 5-4 Trigonometric Functions

5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

### 5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

### Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

### FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

### MATLAB Workshop 13 - Linear Systems of Equations

MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB

### Exponential and Logarithmic Functions

Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

### Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

### Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

### Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

### Experiment 6: Friction

Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

### Volumes of solids of revolution

Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

### Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

### 9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

### Double Integrals over General Regions

Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

### A new algorithm for generating Pythagorean triples

A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

### Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

### Plotting and Graphing

Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

### Review guide for the final exam in Math 233

Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

### The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

### Applications to Physics and Engineering

Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

### and thus, they are similar. If k = 3 then the Jordan form of both matrices is

Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

### Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

### Lecture 2: Matrix Algebra. General

Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots

### EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

### Integration. 148 Chapter 7 Integration

48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

### Square Roots Teacher Notes

Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

### Section A-4 Rational Expressions: Basic Operations

A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

### MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

### Lecture 5. Inner Product

Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

### AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

### Solving BAMO Problems

Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only

### THE RATIONAL NUMBERS CHAPTER

CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions

### Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

### Numeracy across the Curriculum in Key Stages 3 and 4. Helpful advice and suggested resources from the Leicestershire Secondary Mathematics Team

Numercy cross the Curriculum in Key Stges 3 nd 4 Helpful dvice nd suggested resources from the Leicestershire Secondry Mthemtics Tem 1 Contents pge The development of whole school policy 3 A definition

### Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

### Mathematics Higher Level

Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

### Second-Degree Equations as Object of Learning

Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct Second-Degree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,

### Introduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans.

Introduction Introduction The Key Stge 3 Mthemtics series covers the new Ntionl Curriculum for Mthemtics (SCAA: The Ntionl Curriculum Orders, DFE, Jnury 1995, 0 11 270894 3). Detiled curriculum references

### 5.6 POSITIVE INTEGRAL EXPONENTS

54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

### Physics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.

Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from

### Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

### Project Recovery. . It Can Be Done

Project Recovery. It Cn Be Done IPM Conference Wshington, D.C. Nov 4-7, 200 Wlt Lipke Oklhom City Air Logistics Center Tinker AFB, OK Overview Mngement Reserve Project Sttus Indictors Performnce Correction

### CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

### The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

### RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

### Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems

Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch

### 1.2 The Integers and Rational Numbers

.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

### Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

### The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx

The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner

### MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

### Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

### 1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

### Dispersion in Coaxial Cables

Dispersion in Coxil Cbles Steve Ellingson June 1, 2008 Contents 1 Summry 2 2 Theory 2 3 Comprison to Welch s Result 4 4 Findings for RG58 t LWA Frequencies 5 Brdley Dept. of Electricl & Computer Engineering,

### Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.

Text questions, Chpter 5, problems 1-5: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed

### Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

### www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

### Section 1: Crystal Structure

Phsics 927 Section 1: Crstl Structure A solid is sid to be crstl if toms re rrnged in such w tht their positions re ectl periodic. This concept is illustrted in Fig.1 using two-dimensionl (2D) structure.

### LECTURE #05. Learning Objectives. How does atomic packing factor change with different atom types? How do you calculate the density of a material?

LECTURE #05 Chpter : Pcking Densities nd Coordintion Lerning Objectives es How does tomic pcking fctor chnge with different tom types? How do you clculte the density of mteril? 2 Relevnt Reding for this

### 4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

### Techniques for Requirements Gathering and Definition. Kristian Persson Principal Product Specialist

Techniques for Requirements Gthering nd Definition Kristin Persson Principl Product Specilist Requirements Lifecycle Mngement Elicit nd define business/user requirements Vlidte requirements Anlyze requirements

### A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

### Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

### Straight pipe model. Orifice model. * Please refer to Page 3~7 Reference Material for the theoretical formulas used here. Qa/Qw=(ηw/ηa) (P1+P2)/(2 P2)

Air lek test equivlent to IX7nd IX8 In order to perform quntittive tests Fig. shows the reltionship between ir lek mount nd wter lek mount. Wter lek mount cn be converted into ir lek mount. By performing

### Linear Equations in Two Variables

Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then

### Unit 6: Exponents and Radicals

Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

### Answer, Key Homework 4 David McIntyre Mar 25,

Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his print-out should hve 18 questions. Multiple-choice questions my continue on the next column or pe find ll choices before mkin your selection.

### Hillsborough Township Public Schools Mathematics Department Computer Programming 1

Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,

### Increasing Q of Waveguide Pulse-Compression Cavities

Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide Pulse-Compression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque

### Volumes as integrals of cross-sections (Sect. 6.1) Volumes as integrals of cross-sections (Sect. 6.1)

Volumes s integrls of cross-sections (ect. 6.1) Te volume of simple regions in spce Volumes integrting cross-sections: Te generl cse. Certin regions wit oles. Volumes s integrls of cross-sections (ect.