Chapter 6 Resource Masters


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1 Chapter 6 Resource Masters
2 Consumable Workbooks Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook ANSWERS FR WRKBKS The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet. Glencoe/McGrawHill Copright b The McGrawHill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teacher, and families without charge; and be used solel in conjunction with Glencoe s Algebra. An other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGrawHill Companies 8787 rion Place Columbus, H ISBN: Algebra Chapter 6 Resource Masters
3 Contents Vocabular Builder vii Lesson 6 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 6 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 63 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 6 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 65 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 66 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 67 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Chapter 6 Assessment Chapter 6 Test, Form Chapter 6 Test, Form A Chapter 6 Test, Form B Chapter 6 Test, Form C Chapter 6 Test, Form D Chapter 6 Test, Form Chapter 6 penended Assessment Chapter 6 Vocabular Test/Review Chapter 6 Quizzes & Chapter 6 Quizzes 3 & Chapter 6 MidChapter Test Chapter 6 Cumulative Review Chapter 6 Standardized Test Practice Standardized Test Practice Student Recording Sheet A ANSWERS A A3 Glencoe/McGrawHill iii Glencoe Algebra
4 Teacher s Guide to Using the Chapter 6 Resource Masters The Fast File Chapter Resource sstem allows ou to convenientl file the resources ou use most often. The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, etensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra TeacherWorks CDRM. Vocabular Builder Pages vii viii include a student stud tool that presents up to twent of the ke vocabular terms from the chapter. Students are to record definitions and/or eamples for each term. You ma suggest that students highlight or star the terms with which the are not familiar. WHEN T USE Give these pages to students before beginning Lesson 6. Encourage them to add these pages to their Algebra Stud Notebook. Remind them to add definitions and eamples as the complete each lesson. Stud Guide and Intervention Each lesson in Algebra addresses two objectives. There is one Stud Guide and Intervention master for each objective. WHEN T USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN T USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closel follow the structure of the Practice and Appl section of the Student Edition eercises. These eercises are of average difficult. WHEN T USE These provide additional practice options or ma be used as homework for second da teaching of the lesson. Reading to Learn Mathematics ne master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the contet of and relationships among terms in the lesson. Finall, students are asked to summarize what the have learned using various representation techniques. WHEN T USE This master can be used as a stud tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one etension master for each lesson. These activities ma etend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. These are not written eclusivel for honors students, but are accessible for use with all levels of students. WHEN T USE These ma be used as etra credit, shortterm projects, or as activities for das when class periods are shortened. Glencoe/McGrawHill iv Glencoe Algebra
5 Assessment ptions The assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Chapter Assessment CHAPTER TESTS Form contains multiplechoice questions and is intended for use with basic level students. Forms A and B contain multiplechoice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms C and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with aes are provided for questions assessing graphing skills. Form 3 is an advanced level test with freeresponse questions. Grids without aes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. The penended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. A Vocabular Test, suitable for all students, includes a list of the vocabular words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate Assessment Four freeresponse quizzes are included to offer assessment at appropriate intervals in the chapter. A MidChapter Test provides an option to assess the first half of the chapter. It is composed of both multiplechoice and freeresponse questions. Continuing Assessment The Cumulative Review provides students an opportunit to reinforce and retain skills as the proceed through their stud of Algebra. It can also be used as a test. This master includes freeresponse questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which ma appear on the standardized tests that the ma encounter. This practice includes multiplechoice, gridin, and quantitativecomparison questions. Bubblein and gridin answer sections are provided on the master. Answers Page A is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages This improves students familiarit with the answer formats the ma encounter in test taking. The answers for the lessonblesson masters are provided as reduced pages with answers appearing in red. Fullsize answer kes are provided for the assessment masters in this booklet. Glencoe/McGrawHill v Glencoe Algebra
6 NAME DATE PERID 6 Reading to Learn Mathematics Vocabular Builder This is an alphabetical list of the ke vocabular terms ou will learn in Chapter 6. As ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. Add these pages to our Algebra Stud Notebook to review vocabular at the end of the chapter. Vocabular Term ais of smmetr Found on Page Definition/Description/Eample Vocabular Builder completing the square constant term discriminant dihs KRIH muh nuhnt linear term maimum value minimum value parabola puh RA buh luh quadratic equation kwah DRA tihk Quadratic Formula (continued on the net page) Glencoe/McGrawHill vii Glencoe Algebra
7 6 NAME DATE PERID Reading to Learn Mathematics Vocabular Builder (continued) Vocabular Term quadratic function Found on Page Definition/Description/Eample quadratic inequalit quadratic term roots Square Root Propert verte verte form Zero Product Propert zeros Glencoe/McGrawHill viii Glencoe Algebra
8 6 NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function Afunction defined b an equation of the form f () a b c, where a 0 b Graph of a Quadratic A parabola with these characteristics: intercept: c; ais of smmetr: ; a Function b coordinate of verte: a Eample Find the intercept, the equation of the ais of smmetr, and the coordinate of the verte for the graph of f() 3 5. Use this information to graph the function. a, b 3, and c 5, so the intercept is 5. The equation of the ais of smmetr is (3) 3 3 or. The coordinate of the verte is. () 3 Net make a table of values for near. Lesson f() (, f()) 0 0 3(0) 5 5 (0, 5) 3() 5 3 (, 3) , 3() 5 3 (, 3) 3 3 3(3) 5 5 (3, 5) f() Eercises For Eercises 3, complete parts a c for each quadratic function. a. Find the intercept, the equation of the ais of smmetr, and the coordinate of the verte. b. Make a table of values that includes the verte. c. Use this information to graph the function.. f() 6 8. f() 3. f() 3 8, 3, 3,, 3,, 3 f () f () f () () 8 f() 8 f() Glencoe/McGrawHill 33 Glencoe Algebra
9 6 NAME DATE PERID Stud Guide and Intervention (continued) Graphing Quadratic Functions Maimum and Minimum Values The coordinate of the verte of a quadratic function is the maimum or minimum value of the function. Maimum or Minimum Value The graph of f() a b c, where a 0, opens up and has a minimum of a Quadratic Function when a 0. The graph opens down and has a maimum when a 0. Eample Determine whether each function has a maimum or minimum value. Then find the maimum or minimum value of each function. a. f() For this function, a 3 and b 6. Since a 0, the graph opens up, and the function has a minimum value. The minimum value is the coordinate of the verte. The coordinate of the b 6 verte is. a (3) b. f() 00 For this function, a and b. Since a 0, the graph opens down, and the function has a maimum value. The maimum value is the coordinate of the verte. The coordinate of the verte b is. a () Evaluate the function at to find the minimum value. f() 3() 6() 7, so the minimum value of the function is. Evaluate the function at to find the maimum value. f() 00 () () 0, so the minimum value of the function is 0. Eercises Determine whether each function has a maimum or minimum value. Then find the maimum or minimum value of each function.. f() 0. f() 7 3. f() min., 9 min., min.,. f() 6 5. f() 7 6. f() 6 5 ma., 0 min., ma., 5 7. f() 5 8. f() f() min., ma., 9 min., 0. f() 0. f() 0 5. f() 6 ma., min., 0 ma., 7 3. f() f() f() 8 min., 50 min.,.5 ma., 7 3 Glencoe/McGrawHill 3 Glencoe Algebra
10 NAME DATE PERID 6 Skills Practice Graphing Quadratic Functions For each quadratic function, find the intercept, the equation of the ais of smmetr, and the coordinate of the verte.. f() 3. f() 3. f() 6 5 0; 0; 0 ; 0; 0 5; 3; 3. f() 5. f() f() 8 7 ; 0; 0 5; 5; 5 7; ; Complete parts a c for each quadratic function. a. Find the intercept, the equation of the ais of smmetr, and the coordinate of the verte. b. Make a table of values that includes the verte. c. Use this information to graph the function. Lesson 67. f() 8. f() 9. f() 6 8 0; 0; 0 ; ; 8; 3; 3 0 f () f () f () f() 6 f() f() 8 6 Determine whether each function has a maimum or a minimum value. Then find the maimum or minimum value of each function. 0. f() 6. f() 8. f() min.; 0 ma.; 0 min.; 3. f() 5. f() 5. f() 3 min.; ma.; 3 min.; 6. f() 3 7. f() f() ma.; min.; 9 min.; Glencoe/McGrawHill 35 Glencoe Algebra
11 NAME DATE PERID 6 Practice (Average) Graphing Quadratic Functions Complete parts a c for each quadratic function. a. Find the intercept, the equation of the ais of smmetr, and the coordinate of the verte. b. Make a table of values that includes the verte. c. Use this information to graph the function.. f() 8 5. f() 3. f() 5; ; ; ; ; 0.5; f () f () f () f() 6 8 f() Determine whether each function has a maimum or a minimum value. Then find the maimum or minimum value of each function.. f() 8 5. f() 6 6. v() 57 min.; 9 min.; 5 ma.; 8 7. f() 6 8. f() 9. f() 3 8 min.; 8 ma.; 3 ma.; 0 0. GRAVITATIN From feet above a swimming pool, Susan throws a ball upward with a velocit of 3 feet per second. The height h(t) of the ball t seconds after Susan throws it is given b h(t) 6t 3t. Find the maimum height reached b the ball and the time that this height is reached. 0 ft; s. HEALTH CLUBS Last ear, the SportsTime Athletic Club charged $0 to participate in an aerobics class. Sevent people attended the classes. The club wants to increase the class price this ear. The epect to lose one customer for each $ increase in the price. a. What price should the club charge to maimize the income from the aerobics classes? $5 b. What is the maimum income the SportsTime Athletic Club can epect to make? $05 Glencoe/McGrawHill 36 Glencoe Algebra
12 6 NAME DATE PERID Reading to Learn Mathematics Graphing Quadratic Functions PreActivit How can income from a rock concert be maimized? Reading the Lesson Read the introduction to Lesson 6 at the top of page 86 in our tetbook. Based on the graph in our tetbook, for what ticket price is the income the greatest? $0 Use the graph to estimate the maimum income. about $7,000. a. For the quadratic function f() 5 3, is the quadratic term, 5 is the linear term, and 3 is the constant term. b. For the quadratic function f() 3, a 3, b, and c. Lesson 6. Consider the quadratic function f() a b c, where a 0. a. The graph of this function is a parabola. b. The intercept is c. b c. The ais of smmetr is the line a. d. If a 0, then the graph opens upward and the function has a minimum value. e. If a 0, then the graph opens downward and the function has a maimum value. 3. Refer to the graph at the right as ou complete the following sentences. a. The curve is called a parabola. b. The line is called the ais of smmetr. c. The point (, ) is called the verte. (, ) f() (0, ) d. Because the graph contains the point (0, ), is the intercept. Helping You Remember. How can ou remember the wa to use the term of a quadratic function to tell whether the function has a maimum or a minimum value? Sample answer: Remember that the graph of f() (with a 0) is a Ushaped curve that opens up and has a minimum. The graph of g() (with a 0) is just the opposite. It opens down and has a maimum. Glencoe/McGrawHill 37 Glencoe Algebra
13 6 NAME DATE PERID Enrichment Finding the Ais of Smmetr of a Parabola As ou know, if f() a b c is a quadratic function, the values of b bac b b that make f() equal to zero are ac and. a a b The average of these two number values is a. The function f() has its maimum or minimum b value when. Since the ais of smmetr a of the graph of f () passes through the point where the maimum or minimum occurs, the ais of b smmetr has the equation a. f() b = a f() = a + b + c ( ( b, f b a a (( Eample Find the verte and ais of smmetr for f() b Use a. 0 The coordinate of the verte is. ( 5) Substitute in f() f() 5() 0() 7 The verte is (,). b The ais of smmetr is, or. a Find the verte and ais of smmetr for the graph of each function using b a.. f() 8. g() f() A() k() 6 Glencoe/McGrawHill 38 Glencoe Algebra
14 6 NAME DATE PERID Stud Guide and Intervention Solving Quadratic Equations b Graphing Solve Quadratic Equations Quadratic Equation A quadratic equation has the form a b c 0, where a 0. Roots of a Quadratic Equation solution(s) of the equation, or the zero(s) of the related quadratic function The zeros of a quadratic function are the intercepts of its graph. Therefore, finding the intercepts is one wa of solving the related quadratic equation. Eample Solve 6 0 b graphing. Graph the related function f() 6. b The coordinate of the verte is, and the equation of the a ais of smmetr is. Make a table of values using values around. f() 0 f() From the table and the graph, we can see that the zeros of the function are and 3. Eercises Lesson 6 Solve each equation b graphing.. 8 0, , , f() f() f() f() f() f() 3, 7 no real solutions Glencoe/McGrawHill 39 Glencoe Algebra
15 6 Estimate Solutions ften, ou ma not be able to find eact solutions to quadratic equations b graphing. But ou can use the graph to estimate solutions. Eample NAME DATE PERID Stud Guide and Intervention (continued) Solving Quadratic Equations b Graphing Solve 0 b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. The equation of the ais of smmetr of the related function is, so the verte has coordinate. Make a table of values. () f() 0 3 f () 3 The intercepts of the graph are between and 3 and between 0 and. So one solution is between and 3, and the other solution is between 0 and. Eercises Solve the equations b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located between 0 and ; between and ; between and 0; between 3 and between 5 and between and 3 f() f() f() between 3 and ; between and 3; between and ; between and between 3 and between 3 and f() f() f() Glencoe/McGrawHill 30 Glencoe Algebra
16 6 NAME DATE PERID Skills Practice Solving Quadratic Equations B Graphing Use the related graph of each equation to determine its solutions f() f() f() 6 9 f() f() 3 3 f() 3 3, 3 no real solutions Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located , 5 no real solutions between 0 and ; between 5 and 6 f() f() f() Lesson 6 Use a quadratic equation to find two real numbers that satisf each situation, or show that no such numbers eist. 7. Their sum is, and their product is Their sum is 0, and their product is 36. 0; 0, 36 0; 6, 6 f() 36 f() 6 6 Glencoe/McGrawHill 3 Glencoe Algebra
17 6 NAME DATE PERID Use the related graph of each equation to determine its solutions f() 3 3 Practice (Average) Solving Quadratic Equations B Graphing f() f() f(), no real solutions, Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located between 0 and ; 6, between and, between and 3 8 f() f() 3 3 f() f() 3 6 Use a quadratic equation to find two real numbers that satisf each situation, or show that no such numbers eist. 7. Their sum is, and their product is Their sum is 5, and their product is 8. f() 6 0; 3, 5 8 0; no such real numbers eist For Eercises 9 and 0, use the formula h(t) v 0 t 6t, where h(t) is the height of an object in feet, v 0 is the object s initial velocit in feet per second, and t is the time in seconds. 9. BASEBALL Marta throws a baseball with an initial upward velocit of 60 feet per second. Ignoring Marta s height, how long after she releases the ball will it hit the ground? 3.75 s 0. VLCANES A volcanic eruption blasts a boulder upward with an initial velocit of 0 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation from which it was ejected? 5 s Glencoe/McGrawHill 3 Glencoe Algebra
18 6 NAME DATE PERID Reading to Learn Mathematics Solving Quadratic Equations b Graphing PreActivit How does a quadratic function model a freefall ride? Read the introduction to Lesson 6 at the top of page 9 in our tetbook. Write a quadratic function that describes the height of a ball t seconds after it is dropped from a height of 5 feet. h(t) 6t 5 Reading the Lesson. The graph of the quadratic function f() 6 is shown at the right. Use the graph to find the solutions of the quadratic equation 6 0. and 3. Sketch a graph to illustrate each situation. a. A parabola that opens b. A parabola that opens c. A parabola that opens downward and represents a upward and represents a downward and quadratic function with two quadratic function with represents a real zeros, both of which are eactl one real zero. The quadratic function negative numbers. zero is a positive number. with no real zeros. Lesson 6 Helping You Remember 3. Think of a memor aid that can help ou recall what is meant b the zeros of a quadratic function. Sample answer: The basic facts about a subject are sometimes called the ABCs. In the case of zeros, the ABCs are the XYZs, because the zeros are the values that make the values equal to zero. Glencoe/McGrawHill 33 Glencoe Algebra
19 6 NAME DATE PERID Enrichment Graphing Absolute Value Equations You can solve absolute value equations in much the same wa ou solved quadratic equations. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZER feature in the CALC menu to find its real solutions, if an. Recall that solutions are points where the graph intersects the ais. For each equation, make a sketch of the related graph and find the solutions rounded to the nearest hundredth No solutions , 5 9, 3, ,, 3, Eplain how solving absolute value equations algebraicall and finding zeros of absolute value functions graphicall are related. Sample answer: values of when solving algebraicall are the intercepts (or zeros) of the function when graphed. Glencoe/McGrawHill 3 Glencoe Algebra
20 63 NAME DATE PERID Stud Guide and Intervention Solving Quadratic Equations b Factoring Solve Equations b Factoring When ou use factoring to solve a quadratic equation, ou use the following propert. Zero Product Propert For an real numbers a and b, if ab 0, then either a 0 or b 0, or both a and b 0. Eample a ( 5) 0 3 0or 5 0 Solve each equation b factoring. riginal equation Subtract 5 from both sides. Factor the binomial. Zero Product Propert 0or 5 Solve each equation. The solution set is {0, 5}. Eercises Solve each equation b factoring. b. 5 5 riginal equation 5 0 Subtract from both sides. ( 7)( 3) 0 Factor the trinomial. 7 0 or 3 0 Zero Product Propert 7 or 3 Solve each equation. The solution set is 7, , {0, 7} 0, , 0, 0, {5, 6} 3, {, 3} Lesson , 7 0,, ,, 6 {00, 5} ,,, ,,, ,,, 5 Glencoe/McGrawHill 35 Glencoe Algebra
21 63 NAME DATE PERID Stud Guide and Intervention (continued) Solving Quadratic Equations b Factoring Write Quadratic Equations To write a quadratic equation with roots p and q, let ( p)( q) 0. Then multipl using FIL. Eample Write a quadratic equation with the given roots. Write the equation in the form a b c 0. a. 3, 5 7 b., ( p)( q) 0 Write the pattern. 8 3 ( 3)[ (5)] 0 Replace p with 3, q with 5. ( p)( q) 0 ( 3)( 5) 0 Simplif Use FIL. The equation 5 0 has roots and 5. Eercises (8 7) 8 (3 ) 0 3 (8 7)(3 ) The equation has 7 roots and. 8 3 Write a quadratic equation with the given roots. Write the equation in the form a b c 0.. 3,. 8, 3., , 7 6., ,5 8., 9. 7, ,.,. 9, ,., 5., , 7., 8., Glencoe/McGrawHill 36 Glencoe Algebra
22 63 NAME DATE PERID Skills Practice Solving Quadratic Equations b Factoring Solve each equation b factoring.. 6 {8, 8} {0, 0} {, }. 3 0 {, 3} {, 3} {5, } {, 5} {0, 9} {0, 6} {, }. 5 {0, 5}. 9 0 {7} {, 3}. 8 8 {9} 5. {3, 7} , , , 5 Lesson 63 Write a quadratic equation with the given roots. Write the equation in the form a b c 0, where a, b, and c are integers. 9., , , , , , Find two consecutive integers whose product is 7. 6, 7 Glencoe/McGrawHill 37 Glencoe Algebra
23 63 NAME DATE PERID Practice (Average) Solving Quadratic Equations b Factoring Solve each equation b factoring.. 0 {6, } {8} {0} {, } {, } {, 7} 7. 0 {0, } , {5} ,. 99 {9, }. 36 {6} {3, } , {9, 5} , , {5, 3} , , , Write a quadratic equation with the given roots. Write the equation in the form a b c 0, where a, b, and c are integers.. 7, 3. 0, 3. 5, , , , , 9., 30. 0, , 3 3., 33., NUMBER THERY Find two consecutive even positive integers whose product is 6., NUMBER THERY Find two consecutive odd positive integers whose product is 33. 7, GEMETRY The length of a rectangle is feet more than its width. Find the dimensions of the rectangle if its area is 63 square feet. 7 ft b 9 ft 37. PHTGRAPHY The length and width of a 6inch b 8inch photograph are reduced b the same amount to make a new photograph whose area is half that of the original. B how man inches will the dimensions of the photograph have to be reduced? in. Glencoe/McGrawHill 38 Glencoe Algebra
24 NAME DATE PERID 63 Reading to Learn Mathematics Solving Quadratic Equations b Factoring PreActivit How is the Zero Product Propert used in geometr? Read the introduction to Lesson 63 at the top of page 30 in our tetbook. What does the epression ( 5) mean in this situation? It represents the area of the rectangle, since the area is the product of the width and length. Reading the Lesson. The solution of a quadratic equation b factoring is shown below. Give the reason for each step of the solution riginal equation Add to each side. ( 3)( 7) 0 Factor the trinomial. 3 0 or 7 0 Zero Product Propert 3 7 Solve each equation. The solution set is {3, 7}.. n an algebra quiz, students were asked to write a quadratic equation with 7 and 5 as its roots. The work that three students in the class wrote on their papers is shown below. Marla Rosa Larr ( 7)( 5) 0 ( 7)( 5) 0 ( 7)( 5) Who is correct? Rosa Lesson 63 Eplain the errors in the other two students work. Sample answer: Marla used the wrong factors. Larr used the correct factors but multiplied them incorrectl. Helping You Remember 3. A good wa to remember a concept is to represent it in more than one wa. Describe an algebraic wa and a graphical wa to recognize a quadratic equation that has a double root. Sample answer: Algebraic: Write the equation in the standard form a b c 0 and eamine the trinomial. If it is a perfect square trinomial, the quadratic function has a double root. Graphical: Graph the related quadratic function. If the parabola has eactl one intercept, then the equation has a double root. Glencoe/McGrawHill 39 Glencoe Algebra
25 63 NAME DATE PERID Enrichment Euler s Formula for Prime Numbers Man mathematicians have searched for a formula that would generate prime numbers. ne such formula was proposed b Euler and uses a quadratic polnomial,. Find the values of for the given values of. State whether each value of the polnomial is or is not a prime number Does the formula produce all prime numbers greater than 0? Give eamples in our answer.. Euler s formula produces primes for man values of, but it does not work for all of them. Find the first value of for which the formula fails. (Hint: Tr multiples of ten.) Glencoe/McGrawHill 330 Glencoe Algebra
26 6 NAME DATE PERID Stud Guide and Intervention Completing the Square Square Root Propert Use the following propert to solve a quadratic equation that is in the form perfect square trinomial constant. Square Root Propert For an real number if n, then n. Eample a ( ) or The solution set is {9, }. Solve each equation b using the Square Root Propert. or 5 5 b ( 5) or or 5 5 The solution set is 5. Eercises Solve each equation b using the Square Root Propert {, 6} {, 8}, , , 3, 3 {0.8,.} Lesson Glencoe/McGrawHill 33 Glencoe Algebra
27 6 NAME DATE PERID Complete the Square To complete the square for a quadratic epression of the form b, follow these steps. b Stud Guide and Intervention (continued) Completing the Square b. Find.. Square. 3. Add to b. b Eample Eample Find the value of c that makes c a perfect square trinomial. Then write the trinomial as the square of a binomial. b Step b ; Step Step 3 c The trinomial is, which can be written as ( ). Solve 8 0 b completing the square riginal equation Divide each side b. is not a perfect square. Add to each side. Since, add to each side. ( ) 6 Factor the square. 6 or The solution set is {6, }. Square Root Propert Solve each equation. Eercises Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.. 0 c. 60 c 3. 3 c 5; ( 5) 900; ( 30) 9 3 ;. 3. c 5. c 6..5 c.56; (.6) ;.565; (.5) Solve each equation b completing the square s 0s 0, 5 5, 3 3, ,, 7 9, t 3t Glencoe/McGrawHill 33 Glencoe Algebra
28 6 NAME DATE PERID Skills Practice Completing the Square Solve each equation b using the Square Root Propert , 5., ,. 9, Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square c 5; ( 5) 0. c 9; ( 7). c ; ( ) c ; c ;. c ; Solve each equation b completing the square , , , , , Lesson i. 0 i 3 Glencoe/McGrawHill 333 Glencoe Algebra
29 6 NAME DATE PERID Practice (Average) Completing the Square Solve each equation b using the Square Root Propert , 3, 9, , 0 5, Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 0. c. 0 c. c 36; ( 6) 00; ( 0) ; c.. c c 0.6; ( 0.).; (.) 0.03; ( 0.8) 6. 5 c 7. c 8. 5 c ; ; ; Solve each equation b completing the square , , , , i 5 5 i3 3 i GEMETRY When the dimensions of a cube are reduced b inches on each side, the surface area of the new cube is 86 square inches. What were the dimensions of the original cube? 6 in. b 6 in. b 6 in. 3. INVESTMENTS The amount of mone A in an account in which P dollars is invested for ears is given b the formula A P( r), where r is the interest rate compounded annuall. If an investment of $800 in the account grows to $88 in two ears, at what interest rate was it invested? 5% Glencoe/McGrawHill 33 Glencoe Algebra
30 6 NAME DATE PERID Reading to Learn Mathematics Completing the Square PreActivit How can ou find the time it takes an accelerating race car to reach the finish line? Reading the Lesson Read the introduction to Lesson 6 at the top of page 306 in our tetbook. Eplain what it means to sa that the driver accelerates at a constant rate of 8 feet per second square. If the driver is traveling at a certain speed at a particular moment, then one second later, the driver is traveling 8 feet per second faster.. Give the reason for each step in the following solution of an equation b using the Square Root Propert riginal equation ( 6) 8 Factor the perfect square trinomial or 6 9 Square Root Propert Rewrite as two equations. 5 3 Solve each equation.. Eplain how to find the constant that must be added to make a binomial into a perfect square trinomial. Sample answer: Find half of the coefficient of the linear term and square it. 3. a. What is the first step in solving the equation b completing the square? Divide the equation b 3. b. What is the first step in solving the equation 5 0 b completing the square? Add to each side. Helping You Remember. How can ou use the rules for squaring a binomial to help ou remember the procedure for changing a binomial into a perfect square trinomial? ne of the rules for squaring a binomial is ( ). In completing the square, ou are starting with b and need to find. b This shows ou that b, so. That is wh ou must take half of the coefficient and square it to get the constant that must be added to complete the square. Lesson 6 Glencoe/McGrawHill 335 Glencoe Algebra
31 6 NAME DATE PERID Enrichment The Golden Quadratic Equations A golden rectangle has the propert that its length can be written as a b, where a is the width of the rectangle and a b a. An golden rectangle can be a b divided into a square and a smaller golden rectangle, as shown. The proportion used to define golden rectangles can be used to derive two quadratic equations. These are sometimes called golden quadratic equations. a a a b b a Solve each problem.. In the proportion for the golden rectangle, let a equal. Write the resulting quadratic equation and solve for b.. In the proportion, let b equal. Write the resulting quadratic equation and solve for a. 3. Describe the difference between the two golden quadratic equations ou found in eercises and.. Show that the positive solutions of the two equations in eercises and are reciprocals. 5. Use the Pthagorean Theorem to find a radical epression for the diagonal of a golden rectangle when a. 6. Find a radical epression for the diagonal of a golden rectangle when b. Glencoe/McGrawHill 336 Glencoe Algebra
32 65 NAME DATE PERID Stud Guide and Intervention The Quadratic Formula and the Discriminant Quadratic Formula The Quadratic Formula can be used to solve an quadratic equation once it is written in the form a b c 0. Quadratic Formula The solutions of a b bac b c 0, with a 0, are given b. a Eample Solve 5 b using the Quadratic Formula. Rewrite the equation as 5 0. b bac a (5) (5) )( () () or The solutions are and 7. Eercises Quadratic Formula Replace a with, b with 5, and c with. Simplif. Solve each equation b using the Quadratic Formula , 7, 6 3, , 5, 5 3, ,, 3, r 3r ,, i3 3 Lesson 65 Glencoe/McGrawHill 337 Glencoe Algebra
33 65 Roots and the Discriminant Discriminant NAME DATE PERID Stud Guide and Intervention (continued) The Quadratic Formula and the Discriminant The epression under the radical sign, b ac, in the Quadratic Formula is called the discriminant. Roots of a Quadratic Equation Discriminant b ac 0 and a perfect square b ac 0, but not a perfect square b ac 0 b ac 0 Tpe and Number of Roots rational roots irrational roots rational root comple roots Eample Find the value of the discriminant for each equation. Then describe the number and tpes of roots for the equation. a. 5 3 The discriminant is b ac 5 ()(3) or. The discriminant is a perfect square, so the equation has rational roots. Eercises b. 3 5 The discriminant is b ac () (3)(5) or 56. The discriminant is negative, so the equation has comple roots. For Eercises, complete parts ac for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula.. p p 8; ; ; two irrational roots; one rational root; rational roots;, ; ; 6. 0 irrational roots; rational roots; 60; comple roots;, 7 i ; 8. m 8m 8; ; rational roots; irrational roots; rational root;, ; ;. 0 9; comple roots; rational roots; irrational roots; Glencoe/McGrawHill 338 Glencoe Algebra
34 65 NAME DATE PERID Skills Practice The Quadratic Formula and the Discriminant Complete parts ac for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula ; rational root; 5; rational roots;, ; rational roots; 0, 3 89; rational roots;, ; irrational roots; 30 5 ; irrational roots; ; irrational roots; 3 ; irrational roots; ; irrational roots; 3 96; comple roots; 7i ; comple roots; i 3 7; comple roots; 0 3 i 7 Solve each equation b using the method of our choice. Find eact solutions , , i i i i5 5. PARACHUTING Ignoring wind resistance, the distance d(t) in feet that a parachutist falls in t seconds can be estimated using the formula d(t) 6t.If a parachutist jumps from an airplane and falls for 00 feet before opening her parachute, how man seconds pass before she opens the parachute? about 8.3 s Lesson 65 Glencoe/McGrawHill 339 Glencoe Algebra
35 65 NAME DATE PERID Practice (Average) The Quadratic Formula and the Discriminant Complete parts ac for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula ; rational; 8 9; rational; 0, 3 0; rational; ; ; rational; 5, 8 irrational; ; rational; 0, ; ; ; 3 comple; i 9 rational;, comple; 3 i ; ; ; 3 i 5 rational;, irrational; 7 comple; ; ; irrational; ; rational; irrational; Solve each equation b using the method of our choice. Find eact solutions , , , ,. 5 8, i i i i GRAVITATIN The height h(t) in feet of an object t seconds after it is propelled straight up from the ground with an initial velocit of 60 feet per second is modeled b the equation h(t) 6t 60t. At what times will the object be at a height of 56 feet?.75 s, s 3. STPPING DISTANCE The formula d 0.05s.s estimates the minimum stopping distance d in feet for a car traveling s miles per hour. If a car stops in 00 feet, what is the fastest it could have been traveling when the driver applied the brakes? about 53. mi/h Glencoe/McGrawHill 30 Glencoe Algebra
36 65 NAME DATE PERID Reading to Learn Mathematics The Quadratic Formula and the Discriminant PreActivit How is blood pressure related to age? Read the introduction to Lesson 65 at the top of page 33 in our tetbook. Describe how ou would calculate our normal blood pressure using one of the formulas in our tetbook. Sample answer: Substitute our age for A in the appropriate formula (for females or males) and evaluate the epression. Reading the Lesson b bac. a. Write the Quadratic Formula. a b. Identif the values of a, b, and c that ou would use to solve 5 7, but do not actuall solve the equation. a b 5 c 7. Suppose that ou are solving four quadratic equations with rational coefficients and have found the value of the discriminant for each equation. In each case, give the number of roots and describe the tpe of roots that the equation will have. Value of Discriminant Number of Roots Tpe of Roots 6 real, rational 8 comple real, irrational 0 real, rational Helping You Remember 3. How can looking at the Quadratic Formula help ou remember the relationships between the value of the discriminant and the number of roots of a quadratic equation and whether the roots are real or comple? Sample answer: The discriminant is the epression under the radical in the Quadratic Formula. Look at the Quadratic Formula and consider what happens when ou take the principal square root of b ac and appl in front of the result. If b ac is positive, its principal square root will be a positive number and appling will give two different real solutions, which ma be rational or irrational. If b ac 0, its principal square root is 0, so appling in the Quadratic Formula will onl lead to one solution, which will be rational (assuming a, b, and c are integers). If b ac is negative, since the square roots of negative numbers are not real numbers, ou will get two comple roots, corresponding to the and in the smbol. Lesson 65 Glencoe/McGrawHill 3 Glencoe Algebra
37 65 NAME DATE PERID Enrichment Sum and Product of Roots Sometimes ou ma know the roots of a quadratic equation without knowing the equation itself. Using our knowledge of factoring to solve an equation, ou can work backward to find the quadratic equation. The rule for finding the sum and product of roots is as follows: Sum and Product of Roots If the roots of a b c 0, with a 0, are s and s, then s s b a and s s c. a Eample A road with an initial gradient, or slope, of 3% can be represented b the formula a c, where is the elevation and is the distance along the curve. Suppose the elevation of the road is 05 feet at points 00 feet and 000 feet along the curve. You can find the equation of the transition curve. Equations of transition curves are used b civil engineers to design smooth and safe roads. The roots are 3 and 8. 3 (8) 5 Add the roots. 3(8) Multipl the roots. Equation: ( 5, 30 ) 30 Write a quadratic equation that has the given roots.. 6, 9. 5, 3. 6, , Find k such that the number given is a root of the equation. 7. 7; k 0 8. ; 3 k 0 30 Glencoe/McGrawHill 3 Glencoe Algebra
38 66 NAME DATE PERID Stud Guide and Intervention Analzing Graphs of Quadratic Functions Analze Quadratic Functions The graph of a( h) k has the following characteristics: Verte: (h, k) Verte Form Ais of smmetr: h of a Quadratic pens up if a 0 Function pens down if a 0 Narrower than the graph of if a Wider than the graph of if a Lesson 66 Eample each graph. a. ( ) Identif the verte, ais of smmetr, and direction of opening of The verte is at (h, k) or (, ), and the ais of smmetr is. The graph opens up, and is narrower than the graph of. a. ( ) 0 The verte is at (h, k) or (, 0), and the ais of smmetr is. The graph opens down, and is wider than the graph of. Eercises Each quadratic function is given in verte form. Identif the verte, ais of smmetr, and direction of opening of the graph.. ( ) 6. ( 3) 7 3. ( 5) 3 (, 6); ; up (3, 7); 3; up (5, 3); 5; up. 7( ) 9 5. ( ) ( 6) 6 (, 9); ; down (, ); ; up (6, 6); 6; up 7. ( 9) ( 3) 9. 3( ) (9, ); 9; up (3, ); 3; up (, ); ; down 5 0. ( 5). ( 7). 6( ) 3 (5, ); 5; down (7, ); 7; up (, ); ; up 3. 3(.).7. 0.( 0.6) ( 0.8) 6.5 (.,.7);.; up (0.6, 0.); 0.6; (0.8, 6.5); 0.8; down up Glencoe/McGrawHill 33 Glencoe Algebra
39 66 Write Quadratic Functions in Verte Form A quadratic function is easier to graph when it is in verte form. You can write a quadratic function of the form a b c in verte from b completing the square. Eample NAME DATE PERID Stud Guide and Intervention (continued) Analzing Graphs of Quadratic Functions 5 ( 6) 5 ( 6 9) 5 8 ( 3) 7 Write 5 in verte form. Then graph the function. The verte form of the equation is ( 3) 7. Eercises Write each quadratic function in verte form. Then graph the function ( 5) 7 ( 3) 9 ( ) ( ) 5 3( ) 7 5( ) Glencoe/McGrawHill 3 Glencoe Algebra
40 66 NAME DATE PERID Skills Practice Analzing Graphs of Quadratic Functions Write each quadratic function in verte form, if not alread in that form. Then identif the verte, ais of smmetr, and direction of opening.. ( ) ( ) 0; ( 0) ; ( 0) 6; (, 0); ; up (0, ); 0; down (0, 6); 0; up Lesson ( 5) ( ) 8 3( 5) 0; 5( 0) 9; ( ) 8; (5, 0); 5; down (0, 9); 0; down (, 8); ; up ( ) 6; ( 3) 7; 3( ) 8; (, 6); ; up (3, 7); 3; up (, 8); ; down Graph each function. 0. ( 3). ( ). ( ) 3. ( ) Write an equation for the parabola with the given verte that passes through the given point. 6. verte: (, 36) 7. verte: (3, ) 8. verte: (, ) point: (0, 0) point: (, 0) point: (, 3) ( ) 36 ( 3) ( ) Glencoe/McGrawHill 35 Glencoe Algebra
41 66 NAME DATE PERID Practice (Average) Analzing Graphs of Quadratic Functions Write each quadratic function in verte form, if not alread in that form. Then identif the verte, ais of smmetr, and direction of opening.. 6( ) ( ) ; ( 0) ; ( ) ; (, ); ; down (0, ); 0; up (, ); ; down ( 5) 5; ( 3) ; (3, 0); 3( ) ; (5, 5); 5; up 3; up (, ); ; up ( ) ; 3( 3) 6; ( ) 3; (, 0); ; down (3, 6); 3; down (, 3); ; up Graph each function. 0. ( 3) Write an equation for the parabola with the given verte that passes through the given point. 3. verte: (, 3). verte: (3, 0) 5. verte: (0, ) point: (, 5) point: (3, 8) point: (5, 6) ( ) 3 ( 3) ( 0) 6. Write an equation for a parabola with verte at (, ) and intercept 6. ( ) 7. Write an equation for a parabola with verte at (3, ) and intercept. ( 3) 8. BASEBALL The height h of a baseball t seconds after being hit is given b h(t) 6t 80t 3. What is the maimum height that the baseball reaches, and when does this occur? 03 ft;.5 s 9. SCULPTURE A modern sculpture in a park contains a parabolic arc that starts at the ground and reaches a maimum height of 0 feet after a horizontal distance of feet. Write a quadratic function in verte form that describes the shape of the outside of the arc, where is the height of a point on the arc and is its horizontal distance from the lefthand starting point of the arc. 5 ( ) 0 Glencoe/McGrawHill 36 Glencoe Algebra ft 0 ft
42 66 NAME DATE PERID Reading to Learn Mathematics Analzing Graphs of Quadratic Equations PreActivit How can the graph of be used to graph an quadratic function? Read the introduction to Lesson 66 at the top of page 3 in our tetbook. What does adding a positive number to do to the graph of? It moves the graph up. What does subtracting a positive number to before squaring do to the graph of? It moves the graph to the right. Lesson 66 Reading the Lesson. Complete the following information about the graph of a( h) k. a. What are the coordinates of the verte? (h, k) b. What is the equation of the ais of smmetr? h c. In which direction does the graph open if a 0? If a 0? up; down d. What do ou know about the graph if a? It is wider than the graph of. If a? It is narrower than the graph of.. Match each graph with the description of the constants in the equation in verte form. a. a 0, h 0, k 0 iii b. a 0, h 0, k 0 iv c. a 0, h 0, k 0 ii d. a 0, h 0, k 0 i i. ii. iii. iv. Helping You Remember 3. When graphing quadratic functions such as ( ) and ( 5), man students have trouble remembering which represents a translation of the graph of to the left and which represents a translation to the right. What is an eas wa to remember this? Sample answer: In functions like ( ), the plus sign puts the graph ahead so that the verte comes sooner than the origin and the translation is to the left. In functions like ( 5), the minus puts the graph behind so that the verte comes later than the origin and the translation is to the right. Glencoe/McGrawHill 37 Glencoe Algebra
43 66 NAME DATE PERID Enrichment Patterns with Differences and Sums of Squares Some whole numbers can be written as the difference of two squares and some cannot. Formulas can be developed to describe the sets of numbers algebraicall. If possible, write each number as the difference of two squares. Look for patterns cannot cannot cannot cannot 6. 5 Even numbers can be written as n, where n is one of the numbers 0,,, 3, and so on. dd numbers can be written n. Use these epressions for these problems. 7. Show that an odd number can be written as the difference of two squares. n (n ) n 8. Show that the even numbers can be divided into two sets: those that can be written in the form n and those that can be written in the form n. Find n for n 0,,, and so on. You get {0,, 8,, }. For n, ou get {, 6, 0,, }. Together these sets include all even numbers. 9. Describe the even numbers that cannot be written as the difference of two squares. n, for n 0,,, 3, 0. Show that the other even numbers can be written as the difference of two squares. n (n ) (n ) Ever whole number can be written as the sum of squares. It is never necessar to use more than four squares. Show that this is true for the whole numbers from 0 through 5 b writing each one as the sum of the least number of squares Glencoe/McGrawHill 38 Glencoe Algebra
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