THE POWER RULES. Raising an Exponential Expression to a Power

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "THE POWER RULES. Raising an Exponential Expression to a Power"

Transcription

1 8 (5-) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar of the Rules Applications In Section 5. ou learned some of the basic rules for working with eponents. All of the rules of eponents are designed to make it easier to work with eponential epressions. In this section we will etend our list of rules to include three new ones. Raising an Eponential Epression to a Power An epression such as ( ) consists of the eponential epression raised to the power. We can use known rules to simplif this epression. ( ) Eponent indicates two factors of. Product rule: Note that the eponent is the product of the eponents and. This eample illustrates the power of a power rule. Power of a Power Rule If m and n are an integers and a 0, then (a m ) n a mn. E X A M P L E A graphing cannot prove that the power of a power rule is correct, but it can provide numerical support for it. Using the power of a power rule Use the rules of eponents to simplif each epression. Write the answer with positive eponents onl. Assume all variables represent nonzero real numbers. a) ( ) 5 b) ( ) c) ( ) 5 d) a) ( ) 5 5 Power of a power rule b) ( ) Power of a power rule Definition of a negative eponent ( ) ( ) c) ( ) 5 5 Power of a power rule Product rule d) ( ) 9 ( ) Power of a power rule 7 Quotient rule

2 5. The Power Rules (5-) 9 You can use a graphing to illustrate the power of a product rule. Raising a Product to a Power Consider how we would simplif a product raised to a positive power and a product raised to a negative power using known rules. factors of () 8 (a) a (a) (a)(a)(a) a In each of these cases the original eponent is applied to each factor of the product. These eamples illustrate the power of a product rule. Power of a Product Rule If a and b are nonzero real numbers and n is an integer, then (ab) n a n b n. E X A M P L E You can use a graphing to illustrate the power of a quotient rule. Using the power of a product rule Simplif. Assume the variables represent nonzero real numbers. Write the answers with positive eponents onl. a) () 4 b) ( ) c) ( ) a) () 4 () 4 4 Power of a product rule 8 4 b) ( ) () ( ) Power of a product rule 8 Power of a power rule c) ( ) () ( ) ( ) Raising a Quotient to a Power Now consider an eample of appling known rules to a power of a quotient: We get a similar result with a negative power: In each of these cases the original eponent applies to both the numerator and denominator. These eamples illustrate the power of a quotient rule. 5 Power of a Quotient Rule If a and b are nonzero real numbers and n is an integer, then a n a n. b n b

3 70 (5-4) Chapter 5 Eponents and Polnomials E X A M P L E Using the power of a quotient rule Use the rules of eponents to simplif each epression. Write our answers with positive eponents onl. Assume the variables are nonzero real numbers. a) b) c) d) 4 helpful hint The eponent rules in this section appl to epressions that involve onl multiplication and division. This is not too surprising since eponents, multiplication, and division are closel related. Recall that a a a a and a b a b. a) Power of a quotient rule 8 () b) 9 Because ( ) 9 and ( ) c) 8 d) 4 ) 4 (4 ( ) 9 A fraction to a negative power can be simplified b using the power of a quotient rule as in Eample. Another method is to find the reciprocal of the fraction first, then use the power of a quotient rule as shown in the net eample. E X A M P L E 4 Negative powers of fractions Simplif. Assume the variables are nonzero real numbers and write the answers with positive eponents onl. a) 4 b) 5 c) a) 4 4 The reciprocal of 4 is 4. b) 5 4 Power of a quotient rule ( 5 ) 4 c) 9 4 Variable Eponents So far, we have used the rules of eponents onl on epressions with integral eponents. However, we can use the rules to simplif epressions having variable eponents that represent integers. E X A M P L E 5 Epressions with variables as eponents Simplif. Assume the variables represent integers. a) 4 5 b) (5 ) 5n c) n m

4 5. The Power Rules (5-5) 7 Did we forget to include the rule (a b) n a n b n?you can easil check with a that this rule is not correct. a) Product rule: b) (5 ) 5 Power of a power rule: n c) 5n ( m n ) 5n Power of a quotient rule ( m ) 5n 5n Power of a power rule 5mn Summar of the Rules The definitions and rules that were introduced in the last two sections are summarized in the following bo. For these rules m and n are integers and a and b are nonzero real numbers.. a n Definition of negative eponent a n. a n a n, a a, and an Negative eponent rules. a 0 Definition of zero eponent 4. a m a n a mn Product rule m 5. a an a mn Quotient rule. (a m ) n a mn Power of a power rule 7. (ab) n a n b n Power of a product rule 8. a b n a b n n Rules for Integral Eponents a n Power of a quotient rule helpful hint In this section we use the amount formula for interest compounded annuall onl. But ou probabl have mone in a bank where interest is compounded dail. In this case r represents the dail rate (APR5) and n is the number of das that the mone is on deposit. E X A M P L E Applications Both positive and negative eponents occur in formulas used in investment situations. The amount of mone invested is the principal, and the value of the principal after a certain time period is the amount. Interest rates are annual percentage rates. Amount Formula The amount A of an investment of P dollars with interest rate r compounded annuall for n ears is given b the formula A P( r) n. Finding the amount According to Fidelit Investments of Boston, U.S. common stocks have returned an average of 0% annuall since 9. If our great-grandfather had invested $00 in the stock market in 9 and obtained the average increase each ear, then how much would the investment be worth in the ear 00 after 80 ears of growth?

5 7 (5-) Chapter 5 Eponents and Polnomials With a graphing ou can enter 00( 0.0) 80 almost as it appears in print. Use n 80, P $00, and r 0.0 in the amount formula: A P( r) n A 00( 0.0) 80 00(.) 80 04,840.0 So $00 invested in 9 would have amounted to $04,840.0 in 00. When we are interested in the principal that must be invested toda to grow to a certain amount, the principal is called the present value of the investment. We can find a formula for present value b solving the amount formula for P: A P( r) n A P Divide each side b ( r) n. ( r) n P A( r) n Present Value Formula Definition of a negative eponent The present value P that will amount to A dollars after n ears with interest compounded annuall at annual interest rate r is given b P A( r) n. E X A M P L E 7 Finding the present value If our great-grandfather wanted ou to have $,000,000 in 00, then how much could he have invested in the stock market in 9 to achieve this goal? Assume he could get the average annual return of 0% (from Eample ) for 80 ears. Use r 0.0, n 80, and A,000,000 in the present value formula: P A( r) n P,000,000( 0.0) 80 P,000,000(.) 80 Use a with an eponent ke. P A deposit of $488.9 in 9 would have grown to $,000,000 in 80 ears at a rate of 0% compounded annuall. WARM-UPS True or false? Eplain our answer. Assume all variables represent nonzero real numbers.. ( ) 5 False. ( ) 8 True. ( ) 9 True 4. ( ) 7 False 5. () False. ( ) 9 9 False 7. True True 7 True 9. 8 True

6 5. The Power Rules (5-7) 7 5. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the power of a power rule? The power of a power rule sas that (a m ) n a mn.. What is the power of a product rule? The power of a product rule sas that (ab) m a m b m.. What is the power of a quotient rule? The power of a quotient rule sas that (ab) m a m b m. 4. What is principal? Principal is the amount of mone invested initiall. 5. What formula is used for computing the amount of an investment for which interest is compounded annuall? To compute the amount A when interest is compounded annuall, use A P( i) n, where P is the principal, i is the annual interest rate, and n is the number of ears.. What formula is used for computing the present value of an amount in the future with interest compounded annuall? To compute the present value P for the amount A in n ears at annual interest rate i, use P A( i) n. For all eercises in this section, assume the variables represent nonzero real numbers and use positive eponents onl in our answers. Use the rules of eponents to simplif each epression. See Eample. 7. ( ) 8. ( ) 9. ( ) ( ). ( ) 4. ( ) (m ) 4. (a ) 5. ( ) ( ) m 8 a 9. (m ) (m ) 4 7. ( 4 ) 8. ( a ) 5 ( ) ( a ) 4 5 m a Simplif. See Eample. 9. (9) 8 0. (a) 8a. (5w ) 5w. (w 5 ) w ( ) 4. (a b ) a b 5. (ab ) b. ( ) 9 a 8 7. ab 8. 5a b ( ) (5 ab ) 9. ( ab) 0. ( ) ab 8a b Simplif. See Eample.. w w. 8 m 5 m 5. a 4 7a b 8b a b 7 8a b Simplif. See Eample a b c 4. a b a b c a b a b 8 Simplif each epression. Assume that the variables represent integers. See Eample t 5 4t 5 t 48. n 4n 49. ( w ) w w ( ) m p 5. 7 m 7 7 m p 4p 5. 8 a (8 a4 ) 8 5a 54. (5 4 ) (5 ) 5 87 Use the rules of eponents to simplif each epression. If possible, write down onl the answer ( 4 ) 57. ( ) z z ( ). ( ) 7 Use the rules of eponents to simplif each epression ( 4 5a b ) b 70. (m n ) 4 4 ( 5ab ) a mn 7 m n 7

7 74 (5-8) Chapter 5 Eponents and Polnomials 7. ( ) ( ) (7) 7. ( ) ( (9 9 5 ) ) b 7. a 4 c (a b ) 74. (7z ) 4 7 z ac 8 7 z Write each epression as raised to a power. Assume that the variables represent integers n 80. n5 5n n n5 8. m n m 8. n 8 4m n Use a to evaluate each epression. Round approimate answers to three decimal places (. 5) (. 5) (0.0) (4.9) 88. (4.7) 5(0.47) (5.7) ( 4. 9) 90. [5.9 (0.74) ] (. 7) Solve each problem. See Eamples and Deeper in debt. Melissa borrowed $40,000 at % compounded annuall and made no paments for ears. How much did she owe the bank at the end of the ears? (Use the compound interest formula.) $5, Comparing stocks and bonds. According to Fidelit Investments of Boston, throughout the 990s annual returns on common stocks averaged 9%, whereas annual returns on bonds averaged 9%. a) If ou had invested $0,000 in bonds in 990 and achieved the average return, then what would our investment be worth after 0 ears in 000? $,7.4 Value of $0,000 investment (in thousands of dollars) Stocks Bonds Number of ears after 990 FIGURE FOR EXERCISE 9 b) How much more would our $0,000 investment be worth in 000 if ou had invested in stocks? $, Saving for college. Mr. Watkins wants to have $0,000 in a savings account when his little Wanda is read for college. How much must he deposit toda in an account paing 7% compounded annuall to have $0,000 in 8 ears? $, Saving for retirement. In the 990s returns on Treasur Bills fell to an average of 4.5% per ear (Fidelit Investments). Wilma wants to have $,000,000 when she retires in 45 ears. If she assumes an average annual return of 4.5%, then how much must she invest now in Treasur Bills to achieve her goal? $75, Life epectanc of white males. Strange as it ma seem, our life epectanc increases as ou get older. The function L 7.(.00) a can be used to model life epectanc L for U.S. white males with present age a (National Center for Health Statistics, a) To what age can a 0-ear-old white male epect to live? 75. ears b) To what age can a 0-ear-old white male epect to live? (See also Chapter Review Eercises 5 and 54.) 8.4 ears 9. Life epectanc of white females. Life epectanc improved more for females than for males during the 940s and 950s due to a dramatic decrease in maternal mortalit rates. The function L 78.5(.00) a can be used to model life epectanc L for U.S. white females with present age a. a) To what age can a 0-ear-old white female epect to live? 80. ears b) Bob, 0, and Ashle,, are an average white couple. How man ears can Ashle epect to live as a widow? 7.9 ears c) Wh do the life epectanc curves intersect in the accompaning figure? At 80 both males and females can epect about 5 more ears. Life epectanc (ears) White females White males Present age FIGURE FOR EXERCISES 95 AND 9

8 5. Addition, Subtraction, and Multiplication of Polnomials (5-9) 75 GETTING MORE INVOLVED 97. Discussion. For which values of a and b is it true that (ab) a b? Find a pair of nonzero values for a and b for which (a b) a b. 98. Writing. Eplain how to evaluate in three different was. 99. Discussion. Which of the following epressions has a value different from the others? Eplain. a) b) 0 c) d) () e) () d 00. True or False? Eplain our answer. a) The square of a product is the product of the squares. b) The square of a sum is the sum of the squares. a) True b) False GRAPHING CALCULATOR EXERCISES 0. At % compounded annuall the value of an investment of $0,000 after ears is given b 0,000(.). a) Graph 0,000(.) and the function 0,000 on a graphing. Use a viewing window that shows the intersection of the two graphs. b) Use the intersect feature of our to find the point of intersection. c) The -coordinate of the point of intersection is the number of ears that it will take for the $0,000 investment to double. What is that number of ears? b) (., 0,000) c). ears 0. The function 7.(.00) gives the life epectanc of a U.S. white male with present age. (See Eercise 95.) a) Graph 7.(.00) and 8 on a graphing. Use a viewing window that shows the intersection of the two graphs. b) Use the intersect feature of our to find the point of intersection. c) What does the -coordinate of the point of intersection tell ou? b) (87.54, 8) c) At ears of age ou can epect to live until 8. The model fails here. In this section Polnomials Evaluating Polnomials Addition and Subtraction of Polnomials Multiplication of Polnomials 5. ADDITION, SUBTRACTION, AND MULTIPLICATION OF POLYNOMIALS A polnomial is a particular tpe of algebraic epression that serves as a fundamental building block in algebra. We used polnomials in Chapters and, but we did not identif them as polnomials. In this section ou will learn to recognize polnomials and to add, subtract, and multipl them. Polnomials The epression 5 7 is an eample of a polnomial in one variable. Because this epression could be written as (5 ) 7 (), we sa that this polnomial is a sum of four terms:, 5, 7, and. A term of a polnomial is a single number or the product of a number and one or more variables raised to whole number powers. The number preceding the variable in each term is called the coefficient of that variable. In 5 7 the coefficient of is, the coefficient of is 5, and the coefficient of is 7. In algebra a number is frequentl referred to as a constant, and so the last term is called the constant term. A polnomial is defined as a single term or a sum of a finite number of terms.

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m 0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

More information

SECTION 5-1 Exponential Functions

SECTION 5-1 Exponential Functions 354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational

More information

Simplification of Rational Expressions and Functions

Simplification of Rational Expressions and Functions 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2 4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

More information

9.3 OPERATIONS WITH RADICALS

9.3 OPERATIONS WITH RADICALS 9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in

More information

a > 0 parabola opens a < 0 parabola opens

a > 0 parabola opens a < 0 parabola opens Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(

More information

Are You Ready? Simplify Radical Expressions

Are You Ready? Simplify Radical Expressions SKILL Are You Read? Simplif Radical Epressions Teaching Skill Objective Simplif radical epressions. Review with students the definition of simplest form. Ask: Is written in simplest form? (No) Wh or wh

More information

Exponential and Logarithmic Functions

Exponential 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 information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

A.1 Radicals and Rational Exponents

A.1 Radicals and Rational Exponents APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational

More information

The Natural Logarithmic Function: Integration. Log Rule for Integration

The Natural Logarithmic Function: Integration. Log Rule for Integration 6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate

More information

Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

More information

SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT

SLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT . Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail

More information

DIVISION OF POLYNOMIALS

DIVISION OF POLYNOMIALS 5.5 Division of Polynomials (5-33) 89 5.5 DIVISION OF POLYNOMIALS In this section Dividing a Polynomial by a Monomial Dividing a Polynomial by a Binomial Synthetic Division Division and Factoring We began

More information

Chapter 3 Section 6 Lesson Polynomials

Chapter 3 Section 6 Lesson Polynomials Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.

More information

SECTION P.5 Factoring Polynomials

SECTION P.5 Factoring Polynomials BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

More information

5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED

5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given

More information

Exponents. Learning Objectives 4-1

Exponents. Learning Objectives 4-1 Eponents -1 to - Learning Objectives -1 The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient We can simplify by combining the like

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 0 C)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 0 C) Sample/Practice Final Eam MAT 09 Beginning Algebra Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the epression. 1) 2 - [ - ( 3-8)]

More information

The majority of college students hold credit cards. According to the Nellie May

The majority of college students hold credit cards. According to the Nellie May CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

More information

Polynomial 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.

Polynomial 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 information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

Notes Algebra 2 Chapter 7 Exponential and Logarithmic Functions. Name. Period. Date 7.1 Graph Exponential Growth Functions

Notes Algebra 2 Chapter 7 Exponential and Logarithmic Functions. Name. Period. Date 7.1 Graph Exponential Growth Functions Notes Algera 2 Chapter 7 Eponential and Logarithmic Functions Date 7. Graph Eponential Growth Functions An eponential function has the form = a. Name Period If a > 0 and >, then the function = a is an

More information

Rational Functions, Equations, and Inequalities

Rational Functions, Equations, and Inequalities Chapter 5 Rational Functions, Equations, and Inequalities GOALS You will be able to Graph the reciprocal functions of linear and quadratic functions Identif the ke characteristics of rational functions

More information

FACTORING ax 2 bx c WITH a 1

FACTORING ax 2 bx c WITH a 1 296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In

More information

Exponential equations will be written as, where a =. Example 1: Determine a formula for the exponential function whose graph is shown below.

Exponential equations will be written as, where a =. Example 1: Determine a formula for the exponential function whose graph is shown below. .1 Eponential and Logistic Functions PreCalculus.1 EXPONENTIAL AND LOGISTIC FUNCTIONS 1. Recognize eponential growth and deca functions 2. Write an eponential function given the -intercept and another

More information

North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Community College System Diagnostic and Placement Test Sample Questions North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

More information

THE POINT-SLOPE FORM

THE POINT-SLOPE FORM . The Point-Slope Form (-) 67. THE POINT-SLOPE FORM In this section In Section. we wrote the equation of a line given its slope and -intercept. In this section ou will learn to write the equation of a

More information

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM . Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

More information

Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form

Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form 8 (-) Chapter Linear Equations in Two Variables and Their Graphs In this section Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications

More information

Graphing Quadratic Functions

Graphing 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 information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

GRAPHS OF RATIONAL FUNCTIONS

GRAPHS OF RATIONAL FUNCTIONS 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

More information

6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3

6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3 0 (6-18) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) One-half of 1 b) One-third of c) One-half of d) One-half of 1 a) b) c) d) 8 7. Eploration. Let R

More information

Exponents and Polynomials

Exponents and Polynomials Because of permissions issues, some material (e.g., photographs) has been removed from this chapter, though reference to it may occur in the tet. The omitted content was intentionally deleted and is not

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

SECTION 2-5 Combining Functions

SECTION 2-5 Combining Functions 2- Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps

More information

SAMPLE. Polynomial functions

SAMPLE. 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 information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: 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 information

Unit 7 Polynomials. 7 1 Naming Polynomials. 7 2 Adding/Subtracting Polynomials. 7 3 Multiplying Monomials. 7 4 Dividing Monomials

Unit 7 Polynomials. 7 1 Naming Polynomials. 7 2 Adding/Subtracting Polynomials. 7 3 Multiplying Monomials. 7 4 Dividing Monomials Unit 7 Polnomials 7 1 Naming Polnomials 7 Adding/Subtracting Polnomials 7 Multipling Monomials 7 Dividing Monomials 7 Multipling (Monomial b Pol) 7 6 Multipling Polnomials 7 7 Special Products 0 Section

More information

Quadratic Equations and Functions

Quadratic Equations and Functions Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

More information

Review of Intermediate Algebra Content

Review 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 information

4.1 Exponential Functions and Their Graphs

4.1 Exponential Functions and Their Graphs . Eponential Functions and Their Graphs In this section ou will learn to: evaluate eponential functions graph eponential functions use transformations to graph eponential functions use compound interest

More information

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

More information

Simplifying Exponential Expressions

Simplifying Exponential Expressions Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write

More information

Multiplying and Dividing Algebraic Fractions

Multiplying and Dividing Algebraic Fractions . Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple

More information

Partial Fractions. and Logistic Growth. Section 6.2. Partial Fractions

Partial Fractions. and Logistic Growth. Section 6.2. Partial Fractions SECTION 6. Partial Fractions and Logistic Growth 9 Section 6. Partial Fractions and Logistic Growth Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system. _.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

More information

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square

More information

Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

More information

Modifying Functions - Families of Graphs

Modifying Functions - Families of Graphs Worksheet 47 Modifing Functions - Families of Graphs Section Domain, range and functions We first met functions in Sections and We will now look at functions in more depth and discuss their domain and

More information

MULTIPLE REPRESENTATIONS through 4.1.7

MULTIPLE REPRESENTATIONS through 4.1.7 MULTIPLE REPRESENTATIONS 4.1.1 through 4.1.7 The first part of Chapter 4 ties together several was to represent the same relationship. The basis for an relationship is a consistent pattern that connects

More information

CHAPTER 7: FACTORING POLYNOMIALS

CHAPTER 7: FACTORING POLYNOMIALS CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

More information

Chapter 4: Exponential and Logarithmic Functions

Chapter 4: Exponential and Logarithmic Functions Chapter 4: Eponential and Logarithmic Functions Section 4.1 Eponential Functions... 15 Section 4. Graphs of Eponential Functions... 3 Section 4.3 Logarithmic Functions... 4 Section 4.4 Logarithmic Properties...

More information

If (a)(b) 5 0, then a 5 0 or b 5 0.

If (a)(b) 5 0, then a 5 0 or b 5 0. chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic

More information

HI-RES STILL TO BE SUPPLIED

HI-RES STILL TO BE SUPPLIED 1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems

11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems. 11.7 MATHEMATICAL

More information

MATH 102 College Algebra

MATH 102 College Algebra FACTORING Factoring polnomials ls is simpl the reverse process of the special product formulas. Thus, the reverse process of special product formulas will be used to factor polnomials. To factor polnomials

More information

3.1 Graphically Solving Systems of Two Equations

3.1 Graphically Solving Systems of Two Equations 3.1 Graphicall Solving Sstems of Two Equations (Page 1 of 24) 3.1 Graphicall Solving Sstems of Two Equations Definitions The plot of all points that satisf an equation forms the graph of the equation.

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions 0 0. Algebra and Composition of Functions 0. Inverse Functions 0. Eponential Functions 0. Logarithmic Functions 0. Properties of Logarithms 0. The Irrational Number

More information

Polynomial and Rational Functions

Polynomial 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 information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. 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 information

Polynomials and Factoring

Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

More information

Algebra Concept-Readiness Test, Form A

Algebra Concept-Readiness Test, Form A Algebra Concept-Readiness Test, Form A Concept : The Distributive Property Study the concept, and then answer the test questions on the net page. You can use the distributive property to simplify an epression

More information

Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Polynomials. 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 information

6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH

6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH 6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model real-life situations. Partial Fractions

More information

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers. 1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

More information

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button. Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.

More information

Graphing Nonlinear Systems

Graphing 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 information

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 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 information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

CSE 1400 Applied Discrete Mathematics Conversions Between Number Systems

CSE 1400 Applied Discrete Mathematics Conversions Between Number Systems CSE 400 Applied Discrete Mathematics Conversions Between Number Systems Department of Computer Sciences College of Engineering Florida Tech Fall 20 Conversion Algorithms: Decimal to Another Base Conversion

More information

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia

More information

Florida Algebra I EOC Online Practice Test

Florida Algebra I EOC Online Practice Test Florida Algebra I EOC Online Practice Test Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end

More information

Chapter 4 Fractions and Mixed Numbers

Chapter 4 Fractions and Mixed Numbers Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

Summer Review For Students Entering Algebra 2

Summer Review For Students Entering Algebra 2 Summer Review For Students Entering Algebra Board of Education of Howard Count Frank Aquino Chairman Ellen Flnn Giles Vice Chairman Larr Cohen Allen Der Sandra H. French Patricia S. Gordon Janet Siddiqui

More information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets

More information

Introduction to the Practice Exams

Introduction to the Practice Exams Introduction to the Practice Eams The math placement eam determines what math course you will start with at North Hennepin Community College. The placement eam starts with a 1 question elementary algebra

More information

The Quadratic Function

The 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 information

More Equations and Inequalities

More Equations and Inequalities Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities

More information

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

In 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 information

Contents. How You May Use This Resource Guide

Contents. How You May Use This Resource Guide Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................

More information

A5.1 Exponential growth and decay

A5.1 Exponential growth and decay Applications 5.1 Eponential growth and deca A5.1 Eponential growth and deca Before ou start You need to be able to: work out scale factors for percentage increase and decrease. draw a graph given in terms

More information

SOLVING SYSTEMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS SOLVING SYSTEMS OF EQUATIONS 4.. 4..4 Students have been solving equations even before Algebra. Now the focus on what a solution means, both algebraicall and graphicall. B understanding the nature of solutions,

More information

Five 5. Rational Expressions and Equations C H A P T E R

Five 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 information