# Adding and Subtracting Rational Expressions

Save this PDF as:
Size: px
Start display at page:

## Transcription

1 6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy tht you dd nd subtrct frctions Sum of rtionl epressions + Difference of rtionl epressions Adding nd Subtrcting Rtionl Epressions Work with prtner. Find the sum or difference of the two rtionl epressions. Then mtch the sum or difference with its domin. Eplin your resoning. Sum or Difference Domin. b. c. d. e. + A. ll rel numbers ecept + B. ll rel numbers ecept nd + C. ll rel numbers ecept + D. ll rel numbers ecept E. ll rel numbers ecept nd + + f. + F. ll rel numbers ecept 0 nd CONSTRUCTING VIABLE ARGUMENTS To be proficient in mth, you need to justify your conclusions nd communicte them to others. g. + G. ll rel numbers ecept h. + + H. ll rel numbers ecept 0 nd Writing Sum or Difference Work with prtner. Write sum or difference of rtionl epressions tht hs the given domin. Justify your nswer.. ll rel numbers ecept b. ll rel numbers ecept nd c. ll rel numbers ecept, 0, nd Communicte Your Answer. How cn you determine the domin of the sum or difference of two rtionl epressions? 4. Your friend found sum s follows. Describe nd correct the error(s) Section 6.4 Adding nd Subtrcting Rtionl Epressions

2 6.4 Lesson Wht You Will Lern Core Vocbulry comple frction, p. Previous rtionl numbers reciprocl Add or subtrct rtionl epressions. Rewrite rtionl functions. Simplify comple frctions. Adding or Subtrcting Rtionl Epressions As with numericl frctions, the procedure used to dd (or subtrct) two rtionl epressions depends upon whether the epressions hve like or unlike denomintors. To dd (or subtrct) rtionl epressions with like denomintors, simply dd (or subtrct) their numertors. Then plce the result over the common denomintor. Core Concept Adding or Subtrcting with Like Denomintors Let, b, nd c be epressions with c 0. Addition Subtrction c + b c + b c c b c b c Adding or Subtrcting with Like Denomintors b Add numertors nd simplify. Subtrct numertors. Monitoring Progress Find the sum or difference Help in English nd Spnish t BigIdesMth.com To dd (or subtrct) two rtionl epressions with unlike denomintors, find common denomintor. Rewrite ech rtionl epression using the common denomintor. Then dd (or subtrct). Core Concept Adding or Subtrcting with Unlike Denomintors Let, b, c, nd d be epressions with c 0 nd d 0. Addition Subtrction c + b d d + bc d + bc c b d d bc d bc Chpter 6 Rtionl Functions You cn lwys find common denomintor of two rtionl epressions by multiplying the denomintors, s shown bove. However, when you use the lest common denomintor (LCD), which is the lest common multiple (LCM) of the denomintors, simplifying your nswer my tke fewer steps.

3 To find the LCM of two (or more) epressions, fctor the epressions completely. The LCM is the product of the highest power of ech fctor tht ppers in ny of the epressions. Finding Lest Common Multiple (LCM) Find the lest common multiple of 4 6 nd Step Fctor ech polynomil. Write numericl fctors s products of primes ( 4) ( )( + )( ) ( 4 + 4) ()()( ) Step The LCM is the product of the highest power of ech fctor tht ppers in either polynomil. LCM ( )()( + )( ) ( + )( ) Find the sum Adding with Unlike Denomintors Method Use the definition for dding rtionl epressions with unlike denomintors ( + ) + (9 ) 9 ( + ) ( + ) ( ) 9 ( + )() ( + ) c + b d d + bc Distributive Property Fctor. Divide out common fctors. Simplify. Method Find the LCD nd then dd. To find the LCD, fctor ech denomintor nd write ech fctor to the highest power tht ppers in either denomintor. Note tht 9 nd + ( + ), so the LCD is 9 ( + ) ( + ) ( + ) ( + ) + 9 ( + ) ( + ) Fctor second denomintor. LCD is 9 ( + ). Multiply. Add numertors. Note in Emples nd tht when dding or subtrcting rtionl epressions, the result is rtionl epression. In generl, similr to rtionl numbers, rtionl epressions re closed under ddition nd subtrction. Section 6.4 Adding nd Subtrcting Rtionl Epressions

4 COMMON ERROR When subtrcting rtionl epressions, remember to distribute the negtive sign to ll the terms in the quntity tht is being subtrcted. Find the difference Subtrcting with Unlike Denomintors ( ) ( )( ) + ( ) ( )( ) 6 ( )( ) 4 ( )( ) 6 ( 4 ) ( )( ) + 4 ( )( ) ( )( + 4) ( )( ) Fctor ech denomintor. LCD is ( )( ). Multiply. + 4, Simplify. ( ) Subtrct numertors. Simplify numertor. Fctor numertor. Divide out common fctor. Monitoring Progress Help in English nd Spnish t BigIdesMth.com. Find the lest common multiple of nd 0. Find the sum or difference Rewriting Rtionl Functions Rewriting rtionl function my revel properties of the function nd its grph. In Emple 4 of Section 6., you used long division to rewrite rtionl function. In the net emple, you will use inspection. Rewriting nd Grphing Rtionl Function Rewrite g() + in the form g() + k. Grph the function. Describe the + h grph of g s trnsformtion of the grph of f(). 4 g 4 y Rewrite by inspection: ( + ) + ( + ) The rewritten function is g() +. The grph of g is trnsltion unit + left nd units up of the grph of f(). Monitoring Progress Help in English nd Spnish t BigIdesMth.com 9. Rewrite g() 4 in the form g() + k. Grph the function. h Describe the grph of g s trnsformtion of the grph of f(). 4 Chpter 6 Rtionl Functions

5 Comple Frctions A comple frction is frction tht contins frction in its numertor or denomintor. A comple frction cn be simplified using either of the methods below. Core Concept Simplifying Comple Frctions Method If necessry, simplify the numertor nd denomintor by writing ech s single frction. Then divide by multiplying the numertor by the reciprocl of the denomintor. Method Multiply the numertor nd the denomintor by the LCD of every frction in the numertor nd denomintor. Then simplify. + 4 Simplify Method Simplifying Comple Frction ( + 4) ( + 4) ( + 4) ( + 4)( + 8) + 8, 4, 0 Add frctions in denomintor. Multiply by reciprocl. Divide out common fctors. Simplify. Method The LCD of ll the frctions in the numertor nd denomintor is ( + 4) ( + 4) ( + 4) + 4 ( + 4) + 4 ( + 4) + Divide ( + 4) Monitoring Progress + ( + 4) + 8, 4, 0 Multiply numertor nd denomintor by the LCD. out common fctors. Simplify. Simplify. Help in English nd Spnish t BigIdesMth.com Simplify the comple frction Section 6.4 Adding nd Subtrcting Rtionl Epressions

6 6.4 Eercises Dynmic Solutions vilble t BigIdesMth.com Vocbulry nd Core Concept Check. COMPLETE THE SENTENCE A frction tht contins frction in its numertor or denomintor is clled (n).. WRITING Eplin how dding nd subtrcting rtionl epressions is similr to dding nd subtrcting numericl frctions. Monitoring Progress nd Modeling with Mthemtics In Eercises 8, find the sum or difference. (See Emple.) In Eercises 9 6, find the lest common multiple of the epressions. (See Emple.) 9., ( ) 0., 4 +., ( ). 4, 8 6., , , , + 7 ERROR ANALYSIS In Eercises 7 nd 8, describe nd correct the error in finding the sum ( + ) ( + )( ) In Eercises 9 6, find the sum or difference. (See Emples nd 4.) REASONING In Eercises 7 nd 8, tell whether the sttement is lwys, sometimes, or never true. Eplin. 7. The LCD of two rtionl epressions is the product of the denomintors. 8. The LCD of two rtionl epressions will hve degree greter thn or equl to tht of the denomintor with the higher degree. 9. ANALYZING EQUATIONS How would you begin to rewrite the function g() 4 + to obtin the form + g() h + k? 4( + ) 7 A g() + 4( + ) + B g() + ( + ) + ( ) C g() + D g() ANALYZING EQUATIONS How would you begin to rewrite the function g() to obtin the form g() h + k? ( + )( ) A g() B g() + C g() + D g() 6 Chpter 6 Rtionl Functions

7 In Eercises 8, rewrite the function in the form g() + k. Grph the function. Describe the h grph of g s trnsformtion of the grph of f(). (See Emple.) 7. g() g() + 6. g(). g() two resistors in prllel circuit with resistnces R nd R (in ohms) is given by the eqution shown. Simplify the comple frction. Then find the totl resistnce when R 000 ohms nd R 600 ohms. Rt + R R 8 +. g() R 4 6. g() + Rt g() 8. g() R In Eercises 9 44, simplify the comple frction. (See Emple 6.) REWRITING A FORMULA The totl resistnce Rt of PROBLEM SOLVING You pln trip tht involves 40-mile bus ride nd trin ride. The entire trip is 40 miles. The time (in hours) the bus trvels is 40 y, where is the verge speed (in miles per hour) of the bus. The time (in hours) the trin trvels 00 is y. Write nd simplify model tht shows + 0 the totl time y of the trip. 48. PROBLEM SOLVING You prticipte in sprint trithlon tht involves swimming, bicycling, nd running. The tble shows the distnces (in miles) nd your verge speed for ech portion of the rce PROBLEM SOLVING The totl time T (in hours) needed to fly from New York to Los Angeles nd bck cn be modeled by the eqution below, where d is the distnce (in miles) ech wy, is the verge irplne speed (in miles per hour), nd j is the verge speed (in miles per hour) of the jet strem. Simplify the eqution. Then find the totl time it tkes to fly 468 miles when 0 miles per hour nd j miles per hour. d d T+ j +j Distnce (miles) Speed (miles per hour) Swimming 0. r Bicycling r Running 6 r+. Write model in simplified form for the totl time (in hours) it tkes to complete the rce. b. How long does it tke to complete the rce if you cn swim t n verge speed of miles per hour? Justify your nswer. 49. MAKING AN ARGUMENT Your friend clims tht NY LA NY LA A j j j +j Section 6.4 Int_Mth_PE_0604.indd 7 the lest common multiple of two numbers is lwys greter thn ech of the numbers. Is your friend correct? Justify your nswer. Adding nd Subtrcting Rtionl Epressions 7 /0/ 4: PM

8 0. HOW DO YOU SEE IT? Use the grph of the function f() h + k to determine the vlues of h nd k.. REWRITING A FORMULA You borrow P dollrs to buy cr nd gree to repy the lon over t yers t monthly interest rte of i (epressed s deciml). Your monthly pyment M is given by either formul below. Pi M t ( + i) Pi( + i) or M t ( + i) t. Show tht the formuls re equivlent by simplifying the first formul. b. Find your monthly pyment when you borrow \$,00 t monthly interest rte of 0.% nd repy the lon over 4 yers.. THOUGHT PROVOKING Is it possible to write two rtionl functions whose sum is qudrtic function? Justify your nswer y f. PROBLEM SOLVING You re hired to wsh the new crs t cr delership with two other employees. You tke n verge of 40 minutes to wsh cr (R /40 cr per minute). The second employee wshes cr in minutes. The third employee wshes cr in + 0 minutes.. Write single epression R for the combined rte of crs wshed per minute by the group. b. Evlute your epression in prt () when the second employee wshes cr in minutes. How mny crs per hour does this represent? Eplin your resoning. 6. USING TOOLS The epression w + 40 models the w perimeter of the corrl in Section. Emple. Find the sum of the terms. Then use grph to justify the vlue of w found in the emple. How is the grph different from previous grphs of rtionl functions? 7. MODELING WITH MATHEMATICS The mount A (in milligrms) of spirin in person s bloodstrem cn be modeled by 9t A t t + where t is the time (in hours) fter one dose is tken. A first dose second dose A combined effect. USING TOOLS Use technology to rewrite the (97.6)(0.04) + (0.00) function g() in the. + form g() + k. Describe the grph of g s h trnsformtion of the grph of f(). 4. MATHEMATICAL CONNECTIONS Find n epression for the surfce re of the bo Mintining Mthemticl Proficiency Solve f() g() by grphing nd lgebric methods. (Section.). A second dose is tken hour fter the first dose. Write n eqution to model the mount of the second dose in the bloodstrem. b. Write model for the totl mount of spirin in the bloodstrem fter the second dose is tken. 8. FINDING A PATTERN Find the net two epressions in the pttern shown. Then simplify ll five epressions. Wht vlue do the epressions pproch? + +, +, +, Reviewing wht you lerned in previous grdes nd lessons 9. f() f() + 6. f() 4 + g() g() g() + 8 Chpter 6 Rtionl Functions

### Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s

### fractions Let s Learn to

5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

### SEE the Big Idea. Cost of Fuel (p. 397) Galapagos Penguin (p. 382) Lightning Strike (p. 371) 3-D Printer (p. 369) Volunteer Project (p.

7 Rtionl Functions 7. Inverse Vrition 7. Grphing Rtionl Functions 7.3 Multipling nd Dividing Rtionl Epressions 7. Adding nd Subtrcting Rtionl Epressions 7. Solving Rtionl Equtions Cost of Fuel (p. 397)

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### Identify graphs of linear inequalities on a number line.

COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

### 12.1 Introduction to Rational Expressions

. Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

### Review Factoring Polynomials:

Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4

### Consolidation Worksheet

Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

### MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

### A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

### 5.2 Exponent Properties Involving Quotients

5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

### Lesson 25: Adding and Subtracting Rational Expressions

Lesson 2: Adding nd Subtrcting Rtionl Expressions Student Outcomes Students perform ddition nd subtrction of rtionl expressions. Lesson Notes This lesson reviews ddition nd subtrction of frctions using

### Equations and Inequalities

Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

### Chapter 1: Fundamentals

Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

### STRAND B: NUMBER THEORY

Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,

### THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

### SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

### 7h1 Simplifying Rational Expressions. Goals:

h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & -, no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd

### Chapter 1: Logarithmic functions and indices

Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

### Sample pages. 9:04 Equations with grouping symbols

Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity

Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

### Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1

Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS - Algebr Comprehensive Pre-Post Assessment CRS - Algebr Comprehensive Midterm Assessment Algebr Bsics CRS - Algebr Quik-Piks

### Summary Information and Formulae MTH109 College Algebra

Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

### Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q

### Name Date. In Exercises 1 6, tell whether x and y show direct variation, inverse variation, or neither.

1 Prctice A In Eercises 1 6, tell whether nd show direct vrition, inverse vrition, or neither.. 7. 6. 10. 8 6. In Eercises 7 10, tell whether nd show direct vrition, inverse vrition, or neither. 8 10 8.

### Exponents and Logarithms Exam Questions

Eponents nd Logrithms Em Questions Nme: ANSWERS Multiple Choice 1. If 4, then is equl to:. 5 b. 8 c. 16 d.. Identify the vlue of the -intercept of the function ln y.. -1 b. 0 c. d.. Which eqution is represented

### Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

### Precalculus Spring 2017

Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

### The graphs of Rational Functions

Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### Lesson 2.4 Exercises, pages

Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0 - ( ) () 0 c) -7 + d) (7) ( ) 7 - + 8 () ( 8). Expnd nd simplify. ) b) - 7 - + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing

### I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics

AAT-A Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #17-45 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped

### Mathematics Extension 1

04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

### TO: Next Year s AP Calculus Students

TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

### AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

### Advanced Algebra & Trigonometry Midterm Review Packet

Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

### Linear Inequalities. Work Sheet 1

Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges \$ 15 per week plus \$ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

### Each term is formed by adding a constant to the previous term. Geometric progression

Chpter 4 Mthemticl Progressions PROGRESSION AND SEQUENCE Sequence A sequence is succession of numbers ech of which is formed ccording to definite lw tht is the sme throughout the sequence. Arithmetic Progression

### 1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

### AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

### Unit 1 Exponentials and Logarithms

HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

### List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

### 7-1: Zero and Negative Exponents

7-: Zero nd Negtive Exponents Objective: To siplify expressions involving zero nd negtive exponents Wr Up:.. ( ).. 7.. Investigting Zero nd Negtive Exponents: Coplete the tble. Write non-integers s frctions

### MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### x ) dx dx x sec x over the interval (, ).

Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

### x means to use x as a factor five times, or x x x x x (2 c ) means to use 2c as a factor four times, or

14 DAY 1 CHAPTER FIVE Wht fscinting mthemtics is now on our gend? We will review the pst four chpters little bit ech dy becuse mthemtics builds. Ech concept is foundtion for nother ide. We will hve grph

### Matrices and Determinants

Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

### Section 3.2: Negative Exponents

Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.

### SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive)

Mth 0-1 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify

### Bridging the gap: GCSE AS Level

Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

### AP Calculus AB Summer Packet

AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

### The use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.

ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### Scientific notation is a way of expressing really big numbers or really small numbers.

Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific

### MAC 1105 Final Exam Review

1. Find the distnce between the pir of points. Give n ect, simplest form nswer nd deciml pproimtion to three plces., nd, MAC 110 Finl Em Review, nd,0. The points (, -) nd (, ) re endpoints of the dimeter

### SECTION 9-4 Translation of Axes

9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.

HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property

### Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

### Lesson 5.3 Graph General Rational Functions

Copright Houghton Mifflin Hrcourt Publishing Compn. All rights reserved. Averge cost (\$) C 8 6 4 O 4 6 8 Number of people number of hits.. number of times t bt.5 n n 4 b. 4.5 4.5.5; No, btting verge of.5

### Exponents and Polynomials

C H A P T E R 5 Eponents nd Polynomils ne sttistic tht cn be used to mesure the generl helth of ntion or group within ntion is life epectncy. This dt is considered more ccurte thn mny other sttistics becuse

### Chapters Five Notes SN AA U1C5

Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

### 3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.

SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd

### Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

### MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

### PHYS Summer Professor Caillault Homework Solutions. Chapter 2

PHYS 1111 - Summer 2007 - Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement

### Calculus 2: Integration. Differentiation. Integration

Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

### Chapter 6 Continuous Random Variables and Distributions

Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

### SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

### 3.1 EXPONENTIAL FUNCTIONS & THEIR GRAPHS

. EXPONENTIAL FUNCTIONS & THEIR GRAPHS EXPONENTIAL FUNCTIONS EXPONENTIAL nd LOGARITHMIC FUNCTIONS re non-lgebric. These functions re clled TRANSCENDENTAL FUNCTIONS. DEFINITION OF EXPONENTIAL FUNCTION The

### 1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### REVIEW Chapter 1 The Real Number System

Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

### Exponentials - Grade 10 [CAPS] *

OpenStx-CNX module: m859 Exponentils - Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work

### MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

### Prerequisites CHAPTER P

CHAPTER P Prerequisites P. Rel Numers P.2 Crtesin Coordinte System P.3 Liner Equtions nd Inequlities P.4 Lines in the Plne P.5 Solving Equtions Grphiclly, Numericlly, nd Algericlly P.6 Comple Numers P.7

### 11.1 Exponential Functions

. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

### CHAPTER 9. Rational Numbers, Real Numbers, and Algebra

CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers

### Obj: SWBAT Recall the many important types and properties of functions

Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions One-to-One nd Inverse Functions

### 1.1 Reviewing the Exponent Laws

. Reviewing the Exponent Lws INVESTIGATE & INQUIRE An order of gnitude is n pproxite size of quntity, expressed s power of 0. The tble shows soe speeds in etres per second, expressed to the nerest order

### Read section 3.3, 3.4 Announcements:

Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

### Math 130 Midterm Review

Mth 130 Midterm Review April 6, 2013 1 Topic Outline: The following outline contins ll of the mjor topics tht you will need to know for the exm. Any topic tht we ve discussed in clss so fr my pper on the

### sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to

### and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

### Mathematics Extension 2

00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd

### Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.

Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the

### 4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

### CH 9 INTRO TO EQUATIONS

CH 9 INTRO TO EQUATIONS INTRODUCTION I m thinking of number. If I dd 10 to the number, the result is 5. Wht number ws I thinking of? R emember this question from Chpter 1? Now we re redy to formlize the

### The Fundamental Theorem of Calculus, Particle Motion, and Average Value

The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### 3.1 Exponential Functions and Their Graphs

. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

### The practical version

Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

### Equations, expressions and formulae

Get strted 2 Equtions, epressions nd formule This unit will help you to work with equtions, epressions nd formule. AO1 Fluency check 1 Work out 2 b 2 c 7 2 d 7 2 2 Simplify by collecting like terms. b

### along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne