Quadratic Functions and Inequalities Chapter Overview and Pacing

Transcription

1 Quadratic Functions and Inequalities Chapter verview and Pacing PACING (das) Regular Block Basic/ Basic/ Average Advanced Average Advanced Graphing Quadratic Functions (pp. 86 9) Graph quadratic functions. Find and interpret the maimum and minimum values of a quadratic function. LESSN BJECTIVES Solving Quadratic Equations b Graphing (pp. 9 00) 0.5 Solve quadratic equations b graphing. (with 6- Estimate solutions of quadratic equations b graphing. Follow-Up) Follow-Up: Modeling Real-World Data Solving Quadratic Equations b Factoring (pp. 0 05) Solve quadratic equations b factoring. Write a quadratic equation with given roots. Completing the Square (pp. 06 ) 0.5 Solve quadratic equations b using the Square Root Propert. Solve quadratic equations b completing the square. The Quadratic Formula and the Discriminant (pp. 9) Solve quadratic equations b using the Quadratic Formula. Use the discriminant to determine the number and tpe of roots of a quadratic equation. Analzing Graphs of Quadratic Functions (pp. 0 8) Preview: Families of Parabolas (with 6-6 (with 6-6 Analze quadratic functions of the form a( h) k. Preview) Preview) Write a quadratic function in the form a( h) k. Graphing and Solving Quadratic Inequalities (pp. 9 5) Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable. Stud Guide and Practice Test (pp. 6 ) Standardized Test Practice (pp. ) Chapter Assessment TTAL Pacing suggestions for the entire ear can be found on pages T0 T. 8A Chapter 6 Quadratic Functions and Inequalities

2 Timesaving Tools Chapter Resource Manager All-In-ne Planner and Resource Center See pages T T. CHAPTER 6 RESURCE MASTERS Stud Guide and Intervention Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Applications* 5-Minute Check Transparencies Interactive Chalkboard AlgePASS: Tutorial Plus (lessons) Materials SC 6-6- (Follow-Up: graphing calculator) grid paper , algebra tiles GCS , posterboard (Preview: graphing calculator) graphing calculator, inde cards GCS 7, SC 55 68, 7 7 *Ke to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual Chapter 6 Quadratic Functions and Inequalities 8B

3 Mathematical Connections and Background Continuit of Instruction Prior Knowledge In the previous chapter students eplored how to factor quadratic epressions, how to work with radicals, and how to perform operations on comple numbers. The have also solved linear equations and inequalities, and the are familiar with using formulas. This Chapter Students solve quadratic equations b graphing, b factoring, b completing the square, and b using the Quadratic Formula. The eplore how values of a quadratic equation are reflected in the parabola that represents it, and use equations and graphs to eplore quadratic equations that have 0,, or roots. The relate the value of the discriminant to the number of roots and to whether the roots are rational, irrational, or comple. Future Connections Students will continue to eplore roots and zeros of equations, eamining higher-order polnomial equations in Chapter 7. Students will learn to recognize other tpes of equations that can be solved using the Quadratic Formula. The will eplore sstems of quadratic inequalities in Chapter 8. Graphing Quadratic Functions When graphing a quadratic function, use all the available information to produce the most accurate possible graph. This includes a table of values containing the verte and the -intercept. Be sure the points on the graph are connected with a smooth curve and that the graph has a U-shape and not a V-shape at the verte of the graph. Unlike the letter U, however, the graph should become progressivel wider and have arrows indicating that the graph continues to infinit. Solving Quadratic Equations b Graphing It is important to distinguish between finding solutions or roots of an equation and finding zeros of its related function. An equation of the form a b c 0 has a related function, f() a b c. The zeros of f() are the -coordinates of the points where the graph crosses the -ais. These values are the solutions of the related quadratic equation. Without the use of a graphing calculator, this method of solving quadratic equations will usuall provide onl an estimate of solutions. Solutions that appear to be integers should be verified b substituting them into the original equation. Solving Quadratic Equations b Factoring When solving a quadratic equation b factoring, it is important to review factoring techniques. These include the techniques for factoring general trinomials, perfect square trinomials, and a difference of squares. Also remember to look for a greatest common factor (GCF) that might be factored out or the possibilit of factoring b grouping. Before factoring, the equation should be rewritten so that one side of the equation is 0. If the GCF of the terms of the polnomial being factored is a variable or the product of a number and a variable, such as, one solution to the equation is 0. Complete the Square Completing the square is a technique most often used to solve quadratic equations that are not factorable. To use this technique it is desirable to rewrite the equation so that it is equal to a constant. Then divide the coefficient of the linear term b and square the result. Add this value to both sides of the equation. ne side of the equation will now be a perfect square trinomial that can be rewritten as the square of a binomial. To help isolate the variable on one side of the equation, take the square root 8C Chapter 6 Quadratic Functions and Inequalities

4 of both sides, remembering that the square root of the constant on one side of the equation will result in two values, one positive and one negative, represented b a sign. Finall, isolate the variable using the Addition, Subtraction, Multiplication, and/or Division Properties of Equalit. Simplif an solution that involves the smbol b simplifing two separate epressions, one using the plus sign and the other using the minus sign. When using the technique of completing the square, be sure that the coefficient of the quadratic term is. If it is not, divide both sides of the equation b that coefficient. You can then solve the equation b completing the square. The Quadratic Formula and the Discriminant While the technique of completing the square can be used to solve an quadratic equation, implementing this technique can lead to operations involving unwield fractions. The Quadratic Formula can also be used to solve an quadratic equation. Using this formula, the variable is isolated in the ver first step and the other steps involve simplifing the solutions b simplifing radicals and fractions. The discriminant is simpl the portion of the Quadratic Formula that appears underneath the radical, b ac. This value alone will determine the number and tpe of roots (solutions) of the equation. This is because the square root of this value could result in a rational number, such as 6, an irrational number, such as, the value 0, or an imaginar number, such as i. B finding and eamining just the value of the discriminant, ou can tell ver quickl what tpe of solutions a quadratic equation will have. This can serve as a check when solving the equation. Analzing Graphs of Quadratic Functions To write a quadratic equation in the form a( h) k, called verte form, it is important to remember that an equation is a statement of equalit. When an equation is rewritten in a different form, this equalit must be maintained. In previous lessons, if a value was added to one side of an equation, it was also added to the other side of the equation in order to maintain equalit. Another wa to maintain equalit is to add a value to one side and then subtract that same value from that side. For eample, if an addition of 5 is shown on the right side of an equation, a subtraction of 5 would also be shown on the right side. So in essence this would be an overall addition of 5 5 or 0, which does not affect the equalit of the statement. For eample, is equivalent to the statement 5 5. Be especiall careful when adding a value inside a set of parentheses, since the use of parentheses often involves multiplication. For eample () is not equivalent to ( ), but instead to ( ) (). Graphing and Solving Quadratic Inequalities ne wa of solving a linear inequalit not discussed in Chapter is to first solve its related linear equation and then test values on either side of this value in the original inequalit. For eample, to solve, ou would solve the equation and find that. Testing a value less than and a value greater than in the inequalit reveals that the solution to the inequalit is the set of values greater than. The approach to solving a quadratic inequalit algebraicall is similar. The difference lies in the fact that man quadratic inequalities have not one but two solutions. This means that our number line is divided in three possible solution sets. Testing a value from each interval on the number line reveals which solution set or sets are correct. The solution set of a quadratic inequalit will often be a compound inequalit, so ou will want to review this topic from Chapter. Additional mathematical information and teaching notes are available in Glencoe s Algebra Ke Concepts: Mathematical Background and Teaching Notes, which is available at The lessons appropriate for this chapter are as follows. Solving Quadratic Equations b Graphing (Lesson ) Solving Equations b Factoring (Lesson 7) Solving Quadratic Equations b Completing the Square (Lesson 9) Solving Quadratic Equations b Using the Quadratic Formula (Lesson ) Graphing Technolog: Parent and Famil Graphs (Lesson 9) Graphing Quadratic Functions (Lesson 8) More on Ais of Smmetr and Vertices (Lesson 0) Chapter 6 Quadratic Functions and Inequalities 8D

5 and Assessment Tpe Student Edition Teacher Resources Technolog/Internet ASSESSMENT INTERVENTIN ngoing Prerequisite Skills, pp. 85, 9, 99, 05,, 9, 8 Practice Quiz, p. 05 Practice Quiz, p. 8 Mied Review Error Analsis Standardized Test Practice pen-ended Assessment Chapter Assessment pp. 9, 99, 05,, 9, 8, 5 Find the Error, pp. 0, 0, 5 Common Misconceptions, pp. 89, 08 pp. 9, 9, 99, 0, 0, 05,, 9, 7, 5,, Writing in Math, pp. 9, 99, 05,, 9, 7, pen Ended, pp. 90, 97, 0, 7, 5, Stud Guide, pp. 6 0 Practice Test, p. 5-Minute Check Transparencies Quizzes, CRM pp Mid-Chapter Test, CRM p. 7 Stud Guide and Intervention, CRM pp., 9 0, 5 6,, 7 8,, 9 50 Cumulative Review, CRM p. 7 Find the Error, TWE pp. 0, 0, 5 Unlocking Misconceptions, TWE pp. 88, 95 Tips for New Teachers, TWE pp. 88, 05,, TWE p. 0 Standardized Test Practice, CRM pp. 7 7 Modeling: TWE pp. 99, 9 Speaking: TWE pp. 9, 05, 5 Writing: TWE pp., 8 pen-ended Assessment, CRM p. 67 Multiple-Choice Tests (Forms, A, B), CRM pp Free-Response Tests (Forms C, D, ), CRM pp Vocabular Test/Review, CRM p. 68 AlgePASS: Tutorial Plus Standardized Test Practice CD-RM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabular_review Ke to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review s Cracking the SAT & PSAT The Princeton Review s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment fleibilit: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management Sstem to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. 8E Chapter 6 Quadratic Functions and Inequalities

6 Intervention Technolog AlgePASS: Tutorial Plus CD-RM offers a complete, self-paced algebra curriculum. Algebra Lesson AlgePASS Lesson 6-0 Graphing Quadratic Equations 6-5 Solving Quadratic Equations Using the Quadratic Formula 6-5 Solving Word Problems Using Quadratic Equations ALEKS is an online mathematics learning sstem that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Log on for student stud help. For each lesson in the Student Edition, there are Etra Eamples and Self-Check Quizzes. For chapter review, there is vocabular review, test practice, and standardized test practice For more information on Intervention and Assessment, see pp. T8 T. Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Stud rganizer, p. 85 Concept Check questions require students to verbalize and write about what the have learned in the lesson. (pp. 90, 97, 0, 0, 7, 5,, 6) Writing in Math questions in ever lesson, pp. 9, 99, 05,, 9, 7, Reading Stud Tip, pp. 06,, 6 WebQuest, p. 8 Teacher Wraparound Edition Foldables Stud rganizer, pp. 85, 6 Stud Notebook suggestions, pp. 90, 97, 0, 0, 7, 5, Modeling activities, pp. 99, 9 Speaking activities, pp. 9, 05, 5 Writing activities, pp., 8 Differentiated Instruction, (Verbal/Linguistic), p. 96 ELL Resources, pp. 8, 9, 96, 98, 0,, 8, 7,, 6 Additional Resources Vocabular Builder worksheets require students to define and give eamples for ke vocabular terms as the progress through the chapter. (Chapter 6 Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter 6 Resource Masters, pp. 7,, 9, 5,, 7, 5) Vocabular PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabular lists that ou can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Chapter 6 Quadratic Functions and Inequalities 8F

7 Notes Quadratic Functions and Inequalities Have students read over the list of objectives and make a list of an words with which the are not familiar. Point out to students that this is onl one of man reasons wh each objective is important. thers are provided in the introduction to each lesson. Lesson 6- Graph quadratic functions. Lessons 6- through 6-5 Solve quadratic equations. Lesson 6- Write quadratic equations and functions. Lesson 6-6 Analze graphs of quadratic functions. Lesson 6-7 Graph and solve quadratic inequalities. Quadratic functions can be used to model real-world phenomena like the motion of a falling object. The can also be used to model the shape of architectural structures such as the supporting cables of a suspension bridge. You will learn to calculate the value of the discriminant of a quadratic equation in order to describe the position of the supporting cables of the Golden Gate Bridge in Lesson 6-5. Ke Vocabular root (p. 9) zero (p. 9) completing the square (p. 07) Quadratic Formula (p. ) discriminant (p. 6) NCTM Local Lesson Standards bjectives 6-,, 6, 8, 9, 0 6-,, 6, 8, 9, 0 6-, 5, 6, 8, 0 Follow-Up 6-,,, 6, 7, 8, 9 6-,,, 6, 7, 8, 9, 0 6-5,, 6, 8, 9 6-6, 8, 0 Preview 6-6, 6, 7, 8, 9, 0 6-7,, 6, 8, 9, 0 Ke to NCTM Standards: =Number & perations, =Algebra, =Geometr, =Measurement, 5=Data Analsis & Probabilit, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 0=Representation 8 Chapter 6 Quadratic Functions and Inequalities Vocabular Builder ELL The Ke Vocabular list introduces students to some of the main vocabular terms included in this chapter. For a more thorough vocabular list with pronunciations of new words, give students the Vocabular Builder worksheets found on pages vii and viii of the Chapter 6 Resource Masters. Encourage them to complete the definition of each term as the progress through the chapter. You ma suggest that the add these sheets to their stud notebooks for future reference when studing for the Chapter 6 test. 8 Chapter 6 Quadratic Functions and Inequalities

8 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter 6. For Lessons 6- and 6- Graph Functions Graph each equation b making a table of values. (For review, see Lesson -.) See pp. A-F. For Lessons 6-, 6-, and 6-5 Multipl Polnomials Find each product. (For review, see Lesson 5-.) 5. ( )(7 ) 6. ( 5) 7. ( ) 8. ( )( 9) For Lessons 6- and 6- Factor Polnomials Factor completel. If the polnomial is not factorable, write prime. (For review, see Lesson 5-.) prime ( 5) ( ) ( )( ) For Lessons 6- and 6-5 Simplif Radical Epressions Simplif. (For review, see Lessons 5-6 and 5-9.) i. i. 70 i 0. 5 i 5 9. ( 6)( 5) 0. ( )( 9). ( 8)( 7). ( )( 7) This section provides a review of the basic concepts needed before beginning Chapter 6. Page references are included for additional student help. Prerequisite Skills in the Getting Read for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Prerequisite Lesson Skill 6- Evaluating Functions (p. 9) 6- Factoring Trinomials (p. 99) 6- Simplifing Radicals (p. 05) 6-5 Evaluating Epressions (p. ) 6-6 Perfect Square Trinomials (p. 9) 6-7 Inequalities (p. 8) Fold and Cut Make this Foldable to record information about quadratic functions and inequalities. Begin with one sheet of " 7" paper. Refold and Label Fold in half lengthwise. Then fold in fourths crosswise. Cut along the middle fold from the edge to the last crease as shown. Refold along lengthwise fold and staple uncut section at top. Label the section with a lesson number and close to form a booklet Vocab. Reading and Writing As ou read and stud the chapter, fill the journal with notes, diagrams, and eamples for each lesson. Chapter 6 Quadratic Functions and Inequalities 85 For more information about Foldables, see Teaching Mathematics with Foldables. TM Sequencing Information and Progression of Knowledge After students make their Foldable, have them label a section for each lesson in Chapter 6 and a section for vocabular. As students progress through the lessons, have them summarize ke concepts and note the order in which the are presented. Ask students to write about wh the concepts were presented in this sequence. If the cannot see the logic in the sequence, have them reorder the ke concepts and justif their reasoning. Chapter 6 Quadratic Functions and Inequalities 85

9 Lesson Notes Graphing Quadratic Functions Focus 5-Minute Check Transparenc 6- Use as a quiz or a review of Chapter 5. Mathematical Background notes are available for this lesson on p. 8C. Building on Prior Knowledge In Chapter 5, students wrote and solved various equations and inequalities. In this lesson, the will relate quadratic equations to their graphs. can income from a rock concert be maimized? Ask students: How is the income represented in the given function? b P() What is significant about the value of P() when 0 (the ticket price of $0)? The value of P() is greatest when 0, or the income is at its maimum value when 0. Vocabular quadratic function quadratic term linear term constant term parabola ais of smmetr verte maimum value minimum value Graph quadratic functions. Find and interpret the maimum and minimum values of a quadratic function. GRAPH QUADRATIC FUNCTINS A quadratic function is described b an equation of the following form. f() a b c, where a 0 The graph of an quadratic function is called a parabola. ne wa to graph a quadratic function is to graph ordered pairs that satisf the function. Eample can income from a rock concert be maimized? Rock music managers handle publicit and other business issues for the artists the manage. ne group s manager has found that based on past concerts, the predicted income for a performance is P() , where is the price per ticket in dollars. The graph of this quadratic function is shown at the right. Notice that at first the income increases as the price per ticket increases, but as the price continues to increase, the income declines. quadratic term linear term constant term Graph a Quadratic Function C06-5p Graph f() 8 9 b making a table of values. First, choose integer values for. Then, evaluate the function for each value. Graph the resulting coordinate pairs and connect the points with a smooth curve. Income (thousands of dollars) Rock Concert Income P() Ticket Price (dollars) 8 9 f() (, f ()) 0 (0) 8(0) 9 9 (0, 9) () 8() 9 (, ) () 8() 9 (, ) () 8() 9 (, ) () 8() 9 9 (, 9) f () f () Chapter 6 Quadratic Functions and Inequalities Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. 5 Practice, p. 6 Reading to Learn Mathematics, p. 7 Enrichment, p. 8 Teaching Algebra With Manipulatives Masters, p. Transparencies 5-Minute Check Transparenc 6- Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson 0 Interactive Chalkboard

10 TEACHING TIP Tell students the will derive the equation for the ais of smmetr in Lesson 6-6, Eercise 5, after the have learned about a technique called completing the square. All parabolas have an ais of smmetr. If ou were to fold a parabola along its ais of smmetr, the portions of the parabola on either side of this line would match. The point at which the ais of smmetr intersects a parabola is called the verte. The -intercept of a quadratic function, the equation of the ais of smmetr, and the -coordinate of the verte are related to the equation of the function as shown below. Words Consider the graph of a b c, where a 0. The -intercept is a(0) b(0) c or c. b The equation of the ais of smmetr is a. b The -coordinate of the verte is a. Model Graph of a Quadratic Function Teach GRAPH QUADRATIC FUNCTINS In-Class Eamples Power Point Graph f() b making a table of values. 0 f() f() intercept: c verte ais of smmetr: b a Knowing the location of the ais of smmetr, -intercept, and verte can help ou graph a quadratic function. Eample Ais of Smmetr, -Intercept, and Verte Consider the quadratic function f() 9 8. a. Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte. Begin b rearranging the terms of the function so that the quadratic term is first, the linear term is second, and the constant term is last. Then identif a, b, and c. f() a b c f() 9 8 f() 8 9 So, a, b 8, and c 9. The -intercept is 9. You can find the equation of the ais of smmetr using a and b. b a Equation of the ais of smmetr 8 ( ) a, b 8 Simplif. The equation of the ais of smmetr is. Therefore, the -coordinate of the verte is. Lesson 6- Graphing Quadratic Functions 87 f() Teaching Tip Make sure students understand that the graph shows values for all the points that satisf the function, even when the value is not an integer. For eample, the verte is (.5,.5). Consider the quadratic function f(). 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. 0 f() c. Use this information to graph the function. f() (, ) (, ) (, ) Lesson 6- Graphing Quadratic Functions 87

11 Intervention When discussing Eample, make sure students realize that f() and can be used interchangeabl, and also that the maimum or minimum value of the function is given b the -coordinate of the verte of the parabola. New MAXIMUM AND MINIMUM VALUES In-Class Eample Power Point Consider the function f(). a. Determine whether the function has a maimum or a minimum value. maimum b. State the maimum or minimum value of the function. Teaching Tip Remind students to use the equation for the ais of smmetr to determine the -coordinate of the verte and then to find the -coordinate of the verte. Stud Tip Smmetr Sometimes it is convenient to use smmetr to help find other points on the graph of a parabola. Each point on a parabola has a mirror image located the same distance from the ais of smmetr on the other side of the parabola. f () b. Make a table of values that includes the verte. Choose some values for that are less than and some that are greater than. This ensures that points on each side of the ais of smmetr are graphed. 8 9 f() (, f ()) 6 ( 6) 8( 6) 9 ( 6, ) 5 ( 5) 8( 5) 9 6 ( 5, 6) ( ) 8( ) 9 7 (, 7) ( ) 8( ) 9 6 (, 6) ( ) 8( ) 9 (, ) c. Use this information to graph the function. Graph the verte and -intercept. Then graph the points from our table connecting them and the -intercept with a smooth curve. As a check, draw the ais of smmetr,, as a dashed line. The graph of the function should be smmetrical about this line. MAXIMUM AND MINIMUM VALUES The -coordinate of the verte of a quadratic function is the maimum value or minimum value obtained b the function. Maimum and Minimum Value Words The graph of f() a b c, where a 0, opens up and has a minimum value when a 0, and opens down and has a maimum value when a 0. Models a is positive. a is negative. Verte f () (0, 9) 8 8 (, 7) 8 Interactive Chalkboard PowerPoint Presentations This CD-RM is a customizable Microsoft PowerPoint presentation that includes: Step-b-step, dnamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The 5-Minute Check Transparencies Hot links to Glencoe nline Stud Tools Eample 88 Chapter 6 Quadratic Functions and Inequalities Maimum or Minimum Value Consider the function f() 9. a. Determine whether the function has a maimum or a minimum value. For this function, a, b, and c 9. Since a 0, the graph opens up and the function has a minimum value. Unlocking Misconceptions Minimum and Maimum Values Make sure students understand that a parabola which opens upward is the graph of a function with a minimum value and that a parabola which opens downward is the graph of a function with a maimum value. Compare these parabolas to valles (where the altitude of the valle floor is a minimum) and hills (where the peak of the hill is the maimum altitude). 88 Chapter 6 Quadratic Functions and Inequalities

12 Stud Tip Common Misconception The terms minimum point and minimum value are not interchangeable. The minimum point on the graph of a quadratic function is the set of coordinates that describe the location of the verte. The minimum value of a function is the -coordinate of the minimum point. It is the smallest value obtained when f() is evaluated for all values of. b. State the maimum or minimum value of the function. The minimum value of the function is the -coordinate of the verte. The -coordinate of the verte is or. () Find the -coordinate of the verte b evaluating the function for. f() 9 riginal function f() () () 9 or 5 Therefore, the minimum value of the function is 5. When quadratic functions are used to model real-world situations, their maimum or minimum values can have real-world meaning. Eample Find a Maimum Value FUND-RAISING Four hundred people came to last ear s winter pla at Sunnbrook High School. The ticket price was $5. This ear, the Drama Club is hoping to earn enough mone to take a trip to a Broadwa pla. The estimate that for each $0.50 increase in the price, 0 fewer people will attend their pla. a. How much should the tickets cost in order to maimize the income from this ear s pla? 8 f () f () 9 8 In-Class Eample Power Point ECNMICS A souvenir shop sells about 00 coffee mugs each month for $6 each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 0 fewer coffee mugs per month. a. How much should the owner charge for each mug in order to maimize the monthl income from their sales? $8 b. What is the maimum monthl income the owner can epect to make from these items? $80 Words Variables The income is the number of tickets multiplied b the price per ticket. Let the number of $0.50 price increases. Then the price per ticket and 00 0 the number of tickets sold. Let I() = income as a function of. Fund-Raising The London Marathon, which has been run through the streets of London, England, annuall since 98, has historicall raised more mone than an other charit sports event. In 000, this event raised an estimated 0 million ($.6 million U.S. dollars). Source: Guinness World Records The the number multiplied the price income is of tickets b per ticket. Equation I() (00 0) (5 0.50) 00(5) 00(0.50) 0(5) 0(0.50) Multipl Simplif. Rewrite in a b c form. I() is a quadratic function with a 5, b 50, and c 000. Since a 0, the function has a maimum value at the verte of the graph. Use the formula to find the -coordinate of the verte. b -coordinate of the verte Formula for the -coordinate of the verte a 50 a 5, b 50 ( 5) 5 Simplif. This means the Drama Club should make 5 price increases of $0.50 to maimize their income. Thus, the ticket price should be (5) or $.50. (continued on the net page) Lesson 6- Graphing Quadratic Functions 89 Differentiated Instruction Auditor/Musical Ask students to suggest the kinds of musical sounds the might associate with a parabola that has a maimum value, and have them contrast this to a sound that might be associated with a parabola that has a minimum value. For eample, an orchestra piece that rises to a crescendo and then graduall returns to the previous volume might be seen as being related to a parabola with a maimum value. Lesson 6- Graphing Quadratic Functions 89

13 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. add the information in the Ke Concept bo on p. 87 about the graph of a quadratic function to their notebook. make labeled sketches similar to those on p. 88 illustrating the maimum and minimum values of the graphs of quadratic functions. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Graph Quadratic Functions:, Maimum and Minimum Values:, 5 5 dd/even Assignments Eercises and are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercises 58 6 require a graphing calculator. Assignment Guide Basic: 5 7 odd, odd, 7, 5 57, 6 78 Average: 5 odd, 6 50, 5 57, 6 78 (optional: 58 6) Advanced: even, 6 7 (optional: 75 78) Concept Check. Sample answer: f() 5 6;, 5, 6 Guided Practice GUIDED PRACTICE KEY Eercises Eamples 9, 0 90 Chapter 6 Quadratic Functions and Inequalities b. What is the maimum income the Drama Club can epect to make? To determine maimum income, find the maimum value of the function b evaluating I() for 5. I() I(5) 5(5) 50(5) Income function Use a calculator. Thus, the maimum income the Drama Club can epect is $5. CHECK Graph this function on a graphing calculator, and use the CALC menu to confirm this solution. KEYSTRKES: nd [CALC] 0 ENTER 5 ENTER ENTER At the bottom of the displa are the coordinates of the maimum point on the graph of The value of these coordinates is the maimum value of the function, or 5. [ 5, 50] scl: 5 b [ 00, 000] scl: 500. PEN ENDED Give an eample of a quadratic function. Identif its quadratic term, linear term, and constant term.. Identif the verte and the equation of the ais of smmetr for each function graphed below. a. (, ); b. (, ); a. f () b.. State whether the graph of each quadratic function opens up or down. Then state whether the function has a maimum or minimum value. a. f() 5 up; min. b. f() 9 down; ma. c. f() 5 8 down; ma. d. f() 6 5 up; min. 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. 9. See margin.. f() 5. f() 6. f() 7. f() 8 8. f() 9. f() 0 f () Answers c. f() 5a. 0; ; 5c. f() a. 0; 0; 0 b. f() 0 0 ( 0, 0) f() 5b. f() f() (, ) 90 Chapter 6 Quadratic Functions and Inequalities

14 Application Determine whether each function has a maimum or a minimum value. Then find the maimum or minimum value of each function. 0. f() 7. f() 6. f() 9 5 min.; 0 ma.; 7 min.;. NEWSPAPERS Due to increased production costs, the Dail News must increase its subscription rate. According to a recent surve, the number of subscriptions will decrease b about 50 for each 5 increase in the subscription rate. What weekl subscription rate will maimize the newspaper's income from subscriptions? $8.75 Subscription Rate $7.50/wk Current Circulation 50,000 Answers 8a. ; ; 8b. f() indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples 9 0, 5 5 Etra Practice See page 89. Architecture The Echange House in London, England, is supported b two interior and two eterior steel arches. V-shaped braces add stabilit to the structure. Source: Council on Tall Buildings and Urban Habitat 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.. See pp. A F.. f() 5. f() 5 6. f() 7. f() 9 8. f() 9. f() 0. f(). f() 9 9. f() 5. f() 6. f() 6 5. f() 8 6. f() 7. f() 5 8. f() f() f() 9. f() 8 9 Determine whether each function has a maimum or a minimum value. Then find the maimum or minimum value of each function.. f() min.; 0. f() 9 ma.; 9. f() 8 min.; 5. f() 6 min.; 6. f() ma.; 5 7. f() 6 ma.; 8. f() 5 min.; 9 9. f() ma.; f() 7 ma.; 5. f() min.;. f() ma.; 5. f() 5 min.; 0 ARCHITECTURE For Eercises and 5, use the following information. The shape of each arch supporting the Echange House can be modeled b h() 0.05, where h() represents the height of the arch and represents the horizontal distance from one end of the base in meters.. Write the equation of the ais of smmetr, and find the coordinates of the verte of the graph of h(). 0; (0, 0) 5. According to this model, what is the maimum height of the arch? 0 m Lesson 6- Graphing Quadratic Functions 9 8c a. 0; ; 9b. f() 9c. f() f() (, ) f() f() 0 (, ) 6a. ; ; 6b. f() 0 6c. f() (, ) f() 7a. ; ; 7b. f() c. f() f() 8 (, ) Lesson 6- Graphing Quadratic Functions 9

15 6- NAME DATE PERID Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function 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() 5. Use this information to graph the function. a, b, and c 5, so the -intercept is 5. The equation of the ais of smmetr is ( ) or. The -coordinate of the verte is. () Net make a table of values for near. 5 f() (, f()) 0 0 (0) 5 5 (0, 5) () 5 (, ) Stud Stud Guide and and Intervention Intervention, p. (shown) and p. 5, () 5 (, ) () 5 5 (, 5) Eercises For Eercises, 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(). f() 8,,,,,, f() Lesson 6- PHYSICS For Eercises 6 and 7, use the following information. An object is fired straight up from the top of a 00-foot tower at a velocit of 80 feet per second. The height h(t) of the object t seconds after firing is given b h(t) 6t 80t Find the maimum height reached b the object and the time that the height is reached. 00 ft,.5 s 7. Interpret the meaning of the -intercept in the contet of this problem. The -intercept is the initial height of the object. CNSTRUCTIN For Eercises 8 50, use the following information ft b 0 ft Steve has 0 feet of fence to make a rectangular kennel for his dogs. He will use his house as one side. 8. Write an algebraic epression for the kennel s length What dimensions produce a kennel with the greatest area? 50. Find the maimum area of the kennel. 800 ft ft ft 0 0 f () (, ) 8 f() f () (, ) 8 8 f() 8 (, ) 8 NAME DATE PERID Practice (Average) Skills Practice, p. 5 and Practice, p. 6 (shown) Graphing Quadratic Functions Complete parts a c for each quadratic function. f () 9 8 f() TURISM For Eercises 5 and 5, use the following information. A tour bus in the historic district of Savannah, Georgia, serves 00 customers a da. The charge is $8 per person. The owner estimates that the compan would lose 0 passengers a da for each $ fare increase. 5. What charge would give the most income for the compan? $ If the compan raised their fare to this price, how much dail income should the epect to bring in? $65 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(). f() 5; ; ; ; ; 0.5; () 5 5 f 6 8 f() 6 8 (, ) 6 0 f () (, 6) 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() 8 min.; 8 ma.; ma.; f() () f f() (0.5, 0.5) Tourism Known as the Hostess Cit of the South, Savannah, Georgia, is a popular tourist destination. ne of the first planned cities in the Americas, Savannah s Historic District is based on a grid-like pattern of streets and alles surrounding open spaces called squares. Source: savannah-online.com 5. GEMETRY A rectangle is inscribed in an isosceles triangle as shown. Find the dimensions of the inscribed rectangle with maimum area. (Hint: Use similar triangles.) 5 in. b in. 5. CRITICAL THINKING Write an epression for the minimum value of a function of the form a c, where a 0. Eplain our reasoning. Then use this function to find the minimum value of c; See margin for eplanation;.5. 0 in. 8 in. 0. GRAVITATIN From feet above a swimming pool, Susan throws a ball upward with a velocit of feet per second. The height h(t) of the ball t seconds after Susan throws it is given b h(t) 6t t. 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 Reading to Learn Mathematics, p NAME 6 DATE PERID Reading to Learn Mathematics Graphing Quadratic Functions ELL Pre-Activit How can income from a rock concert be maimized? 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 Reading the Lesson. a. For the quadratic function f() 5, is the quadratic term, 5 is the linear term, and is the constant term. b. For the quadratic function f(), a, b,and c.. 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.. Refer to the graph at the right as ou complete the f() following sentences. (, ) a. The curve is called a parabola. b. The line is called the ais of smmetr. (0, ) c. The point (, ) is called the verte. 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 U-shaped 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. 6- Enrichment, p. 8 Finding the Ais of Smmetr of a Parabola As ou know, if f() a b c is a quadratic function, the values of b b ac 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. Eample Use b. Standardized Test Practice 9 Chapter 6 Quadratic Functions and Inequalities NAME DATE PERID Find the verte and ais of smmetr for f() WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can income from a rock concert be maimized? Include the following in our answer: an eplanation of wh income increases and then declines as the ticket price increases, and an eplanation of how to algebraicall and graphicall determine what ticket price should be charged to achieve maimum income. 56. The graph of which of the following equations is smmetrical about the -ais? C f() A C B 6 9 D b = a f() = a + b + c ( b, f ( b a a (( Answer 5. The -coordinate of the verte of a 0 c is or 0, so the a -coordinate of the verte, the minimum of the function, is a(0) c or c. 9 Chapter 6 Quadratic Functions and Inequalities

16 Graphing Calculator 57. Which of the following tables represents a quadratic relationship between the two variables and? C A C MAXIMA AND MINIMA You can use the MINIMUM or MAXIMUM feature on a graphing calculator to find the minimum or maimum value of a quadratic function. This involves defining an interval that includes the verte of the parabola. A lower bound is an value left of the verte, and an upper bound is an value right of the verte. Step Graph the function so that the verte of the parabola is visible. Step Select :minimum or :maimum from the CALC menu. Step Using the arrow kes, locate a left bound and press ENTER. Step Locate a right bound and press ENTER twice. The cursor appears on the maimum or minimum value of the function, and the coordinates are displaed. Find the coordinates of the maimum or minimum value of each quadratic function to the nearest hundredth. 58. f() f() f() f() f() f() 5.56 B D Assess pen-ended Assessment Speaking Ask students to eplain how to tell b eamining a quadratic function whether its graph will have a maimum or minimum value. Then ask them to give an eample of what such a value might mean in a real-world problem. Getting Read for Lesson 6- PREREQUISITE SKILL Lesson 6- presents solving quadratic equations b graphing. Finding points on the graph of the function involves evaluating quadratic functions. Eercises should be used to determine our students familiarit with evaluating functions. Maintain Your Skills Mied Review Simplif. (Lesson 5-9) 6. i 65. ( i) (5 6i) 66. (7 i)( i) 9 5i Solve each equation. i (Lesson 5-8) b n n Answer 7. (, ) Perform the indicated operations. (Lesson -) 70. [ ] [6 5 8] [0 5] 7. [ 5 7] [ 8 ] [5 8] Getting Read for the Net Lesson 7. Graph the sstem of equations and. State the solution. Is the sstem of equations consistent and independent, consistent and dependent, or inconsistent? (Lesson -) See margin for graph; (, ); consistent and independent. PREREQUISITE SKILL Evaluate each function for the given value. (To review evaluating functions, see Lesson -.) 75. f(), f() 5, f() 7, 78. f(), Lesson 6- Graphing Quadratic Functions 9 Answer 55. If a quadratic function can be used to model ticket price versus profit, then b finding the -coordinate of the verte of the parabola ou can determine the price per ticket that should be charged to achieve maimum profit. Answers should include the following. If the price of a ticket is too low, then ou won t make enough mone to cover our costs, but if the ticket price is too high fewer people will bu them. You can locate the verte of the parabola on the graph of the function. It occurs when 0. Algebraicall, this is found b calculating b 000 which, for this case, is or 0. Thus a ( 50) the ticket price should be set at $0 each to achieve maimum profit. Lesson 6- Graphing Quadratic Functions 9

17 Lesson Notes Solving Quadratic Equations b Graphing Focus 5-Minute Check Transparenc 6- Use as a quiz or a review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 8C. does a quadratic function model a free-fall ride? Ask students: The acceleration of a free-falling object due to Earth s gravit is ft/sec. It is given as a negative value because the acceleration is downward, toward Earth s surface. How is this fact represented in the height function? The coefficient 6 is the one half of the acceleration due to gravit in a downward direction. How far has a person fallen second after beginning a free fall? after seconds? after seconds? 6 ft; 6 ft; ft Vocabular quadratic equation root zero Stud Tip Reading Math In general, equations have roots, functions have zeros, and graphs of functions have -intercepts. Solve quadratic equations b graphing. Estimate solutions of quadratic equations b graphing. does a quadratic function model a free-fall ride? As ou speed to the top of a free-fall ride, ou are pressed against our seat so that ou feel like ou re being pushed downward. Then as ou free-fall, ou fall at the same rate as our seat. Without the force of our seat pressing on ou, ou feel weightless. The height above the ground (in feet) of an object in free-fall can be determined b the quadratic function h(t) 6t h 0, where t is the time in seconds and the initial height is h 0 feet. SLVE QUADRATIC EQUATINS When a quadratic function is set equal to a value, the result is a quadratic equation. A quadratic equation can be written in the form a b c 0, where a 0. The solutions of a quadratic equation are called the roots of the equation. ne method for finding the roots of a quadratic equation is to find the zeros of the related quadratic function. The zeros of the function are the -intercepts of its graph. These are the solutions of the related equation because f() 0 at those points. The zeros of the function graphed at the right are and. Eample Two Real Solutions Solve b graphing. Graph the related quadratic function f() 6 8. The equation of the ais 6 of smmetr is or. Make a table using values around. Then, ( ) graph each point. f() 5 f() 0 0 From the table and the graph, we can see that the zeros of the function are and. Therefore, the solutions of the equation are and. CHECK Check the solutions b substituting each solution into the equation to see if it is satisfied ( ) 6( ) 8? 0 ( ) 6( ) 8? f() (, 0) (, 0) f() 6 8 The graph of the related function in Eample had two zeros; therefore, the quadratic equation had two real solutions. This is one of the three possible outcomes when solving a quadratic equation. 9 Chapter 6 Quadratic Functions and Inequalities Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 9 0 Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. Enrichment, p. Assessment, p. 69 Transparencies School-to-Career Masters, p. 5-Minute Check Transparenc 6- Answer Ke Transparencies Technolog Interactive Chalkboard

18 Stud Tip ne Real Solution When a quadratic equation has one real solution, it reall has two solutions that are the same number. Words Models Eample Solutions of a Quadratic Equation A quadratic equation can have one real solution, two real solutions, or no real solution. ne Real Solution Solve 8 6 b graphing. Write the equation in a b c 0 form Subtract 6 from each side. Graph the related quadratic function f() 8 6. f() ne Real Solution f() Notice that the graph has onl one -intercept,. Thus, the equation s onl solution is. Two Real Solutions f() No Real Solution f() f() f() 8 6 Teach SLVE QUADRATIC EQUATINS In-Class Eamples Solve 0 b graphing. and f() f() Solve b graphing. f() Power Point Eample No Real Solution NUMBER THERY Find two real numbers whose sum is 6 and whose product is 0 or show that no such numbers eist. Eplore Plan Solve Eamine Let one of the numbers. Then 6 the other number. Since the product of the two numbers is 0, ou know that (6 ) 0. (6 ) 0 riginal equation Distributive Propert Subtract 0 from each side. You can solve b graphing the related function f() f() Notice that the graph has no -intercepts. This means that the original equation has no real solution. Thus, it is not possible for two numbers to have a sum of 6 and a product of 0. Tr finding the product of several pairs of numbers whose sum is 6. Is the product of each pair less than 0 as the graph suggests? f() f() 6 0 f() Teaching Tip In Eample, inform students that while this equation does not have an real solutions, it does have a solution in the set of comple numbers, the topic of Lesson 5-9. Such equations will be discussed again in Lessons 6- and 6-5. NUMBER THERY Find two real numbers whose sum is and whose product is 5 or show that no such numbers eist. f() Lesson 6- Solving Quadratic Equations b Graphing 95 Unlocking Misconceptions Equations and Functions Some students ma notice that the equation derived in Eample, 8 6 0, is equivalent to the equation Either equation can be produced from the other b multipling each side b. These two equations have the same solution,. However, stress that the related functions, f() 8 6 and f() 8 6 are not equivalent. This can be seen b looking at their graphs, which open in opposite directions. f() 5 The graph of the related function does not intersect the -ais. Therefore, no such numbers eist. Lesson 6- Solving Quadratic Equations b Graphing 95

19 ESTIMATE SLUTINS In-Class Eamples Power Point Solve 6 0 b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. ne solution is between 0 and, and the other is between 5 and 6. f() Stud Tip Location of Roots Notice in the table of values that the value of the function changes from negative to positive between the values of 0 and, and and. ESTIMATE SLUTINS ften eact roots cannot be found b graphing. In this case, ou can estimate solutions b stating the consecutive integers between which the roots are located. Eample Estimate Roots 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 or. ( ) f() f() 0 f() The -intercepts of the graph are between 0 and and between and. So, one solution is between 0 and, and the other is between and. f() 6 For man applications, an eact answer is not required, and approimate solutions are adequate. Another wa to estimate the solutions of a quadratic equation is b using a graphing calculator. 5 RYAL GRGE BRIDGE The highest bridge in the United States is the Roal Gorge Bridge in Colorado. The deck of the bridge is 05 feet above the river below. Suppose a marble is dropped over the railing from a height of feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h(t) 6t h 0, where t is the time in seconds and h 0 is the initial height above the water in feet. about 8 s Eample 5 Write and Solve an Equation EXTREME SPRTS n March, 999, Adrian Nicholas broke the world record for the longest human flight. He flew 0 miles from his drop point in minutes 55 seconds using a speciall designed, aerodnamic suit. Using the information at the right and ignoring air resistance, how long would Mr. Nicholas have been in free-fall had he not used this special suit? Use the formula h(t) 6t h 0, where the time t is in seconds and the initial height h 0 is in feet. We need to find t when h 0 5,000 and h(t) 500. Solve 500 6t 5, t 5,000 riginal equation 0 6t,500 Subtract 500 from each side. Graph the related function 6t,500 using a graphing calculator. Adjust our window so that the -intercepts of the graph are visible. Jumps from plane at 5,000 ft Free-fall pens parachute at 500 ft Use the ZER feature, nd [CALC], to find the positive zero of the function, since time cannot be negative. Use the arrow kes to locate a left bound for the zero and press ENTER. Then, locate a right bound and press ENTER twice. The positive zero of the function is approimatel 6.. Mr. Nicholas would have been in free-fall for about 6 seconds. [ 60, 60] scl: 5 b [ 0000, 0000] scl: Chapter 6 Quadratic Functions and Inequalities Differentiated Instruction ELL Verbal/Linguistic Have students discuss with a partner or in a small group the methods for multipling and dividing monomial epressions with eponents, and also numbers written in scientific notation. Ask them to work together to develop a list of common errors for such problems, and to suggest was to correct and avoid these errors. 96 Chapter 6 Quadratic Functions and Inequalities

20 Concept Check Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6 7. between and 0; between and Application indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples Etra Practice See page Define each term and eplain how the are related. See margin. a. solution b. root c. zero of a function d. -intercept. PEN ENDED Give an eample of a quadratic function and state its related quadratic equation. Sample answer: f() ; 0. Eplain how ou can estimate the solutions of a quadratic equation b eamining the graph of its related function. See margin. Use the related graph of each equation to determine its solutions.. 0, 5. 0, f() f() f() f() 8 6 Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located , , , between and ;. NUMBER THERY Use a quadratic equation to find two real numbers whose sum is 5 and whose product is, or show that no such numbers eist., 7 f() Use the related graph of each equation to determine its solutions , , f() 8 f () , 9. 0 no real solutions f() f() f() f () f() f () 6 9 f () 5 f () 6 f() f() Lesson 6- Solving Quadratic Equations b Graphing 97 f () Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. add the graphic representations of the three possible tpes of real solutions of a quadratic equation shown in the Ke Concept bo on p. 95. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Solve Quadratic Equations:,, 6 Estimate Solutions:,, 5, 6 dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercises 5 56 require a graphing calculator. Assignment Guide Basic: 5 5 odd, 7 50, 57 7 Average: 5 5 odd, 7 50, 57 7 (optional: 5 56) Advanced: 6 even, 7 66 (optional: 67 7) Answers a. The solution is the value that satisfies an equation. b. A root is a solution of an equation. c. A zero is the value of a function that makes the function equal to 0. d. An -intercept is the point at which a graph crosses the -ais. The solutions, or roots, of a quadratic equation are the zeros of the related quadratic function. You can find the zeros of a quadratic function b finding the -intercepts of its graph.. The -intercepts of the related function are the solutions to the equation. You can estimate the solutions b stating the consecutive integers between which the -intercepts are located. Lesson 6- Solving Quadratic Equations b Graphing 97

21 Stud Stud Guide and and Intervention Intervention, p. 9 (shown) and p NAME DATE PERID 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. f() 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. 0 f() From the table and the graph, we can see that the zeros of the function are and. Eercises Solve each equation b graphing.. 8 0, ,. 5 0, f() f() f(), 7 no real solutions f() f() f() Lesson 6-. between 5 and ; between 0 and. between and 0; between and. between and ; between 0 and. between 0 and ; between and. between and 0; between and 5. between and ; between and Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located ,. 0 0, , , , , 0. 5,. 5, no real solutions no real solutions NUMBER THERY Use a quadratic equation to find two real numbers that satisf each situation, or show that no such numbers eist. 8. Their sum is 7, and their product is 7. 8, 9 9. Their sum is 7, and their product is. See pp. A F. 0. Their sum is 9, and their product is. See pp. A F.. Their sum is, and their product is 8., 6- NAME 9 DATE PERID Practice (Average) Skills Practice, p. and Practice, p. (shown) Use the related graph of each equation to determine its solutions 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 f() Solving Quadratic Equations B Graphing f() f() f() f() f() 0 f() f() f() f() f() 6 For Eercises, 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.. ARCHERY An arrow is shot upward with a velocit of 6 feet per second. Ignoring the height of the archer, how long after the arrow is released does it hit the ground? s. TENNIS A tennis ball is hit upward with a velocit of 8 feet per second. Ignoring the height of the tennis plaer, how long does it take for the ball to fall to the ground? s. BATING A boat in distress launches a flare straight up with a velocit of 90 feet per second. Ignoring the height of the boat, how man seconds will it take for the flare to hit the water? about s 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; f() 5 8 0; f() 6, f() no such real 5 8 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?.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 Reading to Learn Mathematics, p. 6- NAME DATE PERID Reading to Learn Mathematics Solving Quadratic Equations b Graphing Pre-Activit How does a quadratic function model a free-fall 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 ELL Empire State Building Located on the 86th floor, 050 feet (0 meters) above the streets of New York Cit, the bservator offers panoramic views from within a glassenclosed pavilion and from the surrounding open-air promenade. Source: 5. LAW ENFRCEMENT Police officers can use the length of skid marks to help determine the speed of a vehicle before the brakes were applied. If the skid s marks are on dr concrete, the formula d can be used. In the formula, s represents the speed in miles per hour, and d represents the length of the skid marks in feet. If the length of the skid marks on dr concrete are 50 feet, how fast was the car traveling? about 5 mph 6. EMPIRE STATE BUILDING Suppose ou could conduct an eperiment b dropping a small object from the bservator of the Empire State Building. How long would it take for the object to reach the ground, assuming there is no air resistance? Use the information at the left and the formula h(t) 6t h 0, where t is the time in seconds and the initial height h 0 is in feet. about 8 s 7. CRITICAL THINKING A quadratic function has values f( ), f( ) 9, and f(0) 5. Between which two values must f() have a zero? Eplain our reasoning. and ; See margin for eplanation. 98 Chapter 6 Quadratic Functions and Inequalities. 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. Helping You Remember. 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. NAME DATE PERID 6- Enrichment, p. 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 7 Answer 7. The value of the function changes from negative to positive, therefore the value of the function is zero between these two numbers. 98 Chapter 6 Quadratic Functions and Inequalities

22 Standardized Test Practice Etending the Lesson Maintain Your Skills Mied Review 8. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. A F. How does a quadratic function model a free-fall ride? Include the following in our answer: a graph showing the height at an given time of a free-fall ride that lifts riders to a height of 85 feet, and an eplanation of how to use this graph to estimate how long the riders would be in free-fall if the ride were allowed to hit the ground before stopping. 9. If one of the roots of the equation k 0 is, what is the value of k? A A B 0 C D 50. For what value of does f() 5 6 reach its minimum value? B A B 5 C D 5 SLVE ABSLUTE VALUE EQUATINS BY GRAPHING Similar to quadratic equations, ou can solve absolute value equations b graphing. Graph the related absolute value function for each equation using a graphing calculator. Then use the ZER feature, nd [CALC], to find its real solutions, if an, rounded to the nearest hundredth , , no real solutions Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte for each quadratic function. Then graph the function b making a table of values. (Lesson 6-) See margin for graphs. 57. f() f() f() ; ; ; ; ; 6; 6 Simplif. (Lesson 5-9) i 60. i 5 5 i 6. 0 i i 6. 5 i i 5 i Evaluate the determinant of each matri. (Lesson -) Assess pen-ended Assessment Modeling Have students draw parabolas in various positions and label them to show how man real roots the have and approimatel where those roots occur. Getting Read for Lesson 6- PREREQUISITE SKILL Lesson 6- presents solving quadratic equations b factoring. Frequentl this involves factoring a trinomial epression on one side of an equation. Eercises 67 7 should be used to determine our students familiarit with factoring trinomials. Assessment ptions Quiz (Lessons 6- and 6-) is available on p. 69 of the Chapter 6 Resource Masters. Answers 57. f() 68. ( 0)( 0) 69. ( 7)( ) Getting Read for the Net Lesson 7. ( )( ) 7. ( )( ) 66. CMMUNITY SERVICE A drug awareness program is being presented at a theater that seats 00 people. Proceeds will be donated to a local drug information center. If ever two adults must bring at least one student, what is the maimum amount of mone that can be raised? (Lesson -) $500 PREREQUISITE SKILL Factor completel. (To review factoring trinomials, see Lesson 5-.) ( 5) ( 9) Lesson 6- Solving Quadratic Equations b Graphing f() (, 5) f() 6 (, ) f() f() f() 8 8 ( 6, 5) Lesson 6- Solving Quadratic Equations b Graphing 99

23 Graphing Calculator Investigation A Follow-Up of Lesson 6- Getting Started Know Your Calculator When students use the procedure in Step to cop the regression equation from Step to the Y= list, the coefficients will have several more digits than the coefficients displaed on the home screen. The coefficients on the home screen are rounded versions of those in the Y= list. Scientific Notation In Step, the value of the coefficient a is displaed as.055e. Point out that this is how the calculator displas the scientific notation Teach Make sure students have cleared the L and L lists before entering new data. Also have them enter the WINDW dimensions shown. For Step, point out that ou can use the same kestrokes shown in Step, substituting for the first 5, to select LinReg. If an error message appears in Step, have students clear the Y= list before tring Step again. If students need to review entering data or selecting statistical plots, refer them to p. 87. Have students complete Eercises. Assess Ask students: What does it mean when the points on a scatter plot appear to lie along a curved path? The equation that best models the situation ma be quadratic, and is probabl not linear. Modeling Real-World Data You can use a TI-8 Plus to model data points whose curve of best fit is quadratic. 00 Chapter 6 Quadratic Functions and Inequalities Answers A Follow-Up of Lesson 6- FALLING WATER Water is allowed to drain from a hole made in a -liter bottle. The table shows the level of the water measured in centimeters from the bottom of the bottle after seconds. Find and graph a linear regression equation and a quadratic regression equation. Determine which equation is a better fit for the data. Time (s) Water level (cm) Find a linear regression equation. Enter the times in L and the water levels in L. Then find a linear regression equation. KEYSTRKES: Review lists and finding a linear regression equation on page 87. Graph a scatter plot and the regression equation. KEYSTRKES: Review graphing a regression equation on page 87. [0, 60] scl: b [5, 5] scl: 5 Eercises. See margin. For Eercises, use the graph ofthe braking distances for dr pavement.. Find and graph a linear regression equation and a quadratic regression equation for the data. Determine which equation is a better fit for the data.. Use the CALC menu with each regression equation to estimate the braking distance at speeds of 00 and 50 miles per hour.. How do the estimates found in Eercise compare?. How might choosing a regression equation that does not fit the data well affect predictions made b using the equation? See pp. A F.. linear: (00, 5), (50, 56); quadratic: (00, 0), (50, 990). The quadratic estimates are much greater.. Sample answer: Choosing a model that does not fit the data well ma cause inaccurate predictions when the data are ver large or small. Find a quadratic regression equation. Find the quadratic regression equation. Then cop the equation to the Y= list and graph. KEYSTRKES: STAT 5 ENTER VARS 5 ENTER GRAPH [0, 60] scl: b [5, 5] scl: 5 The graph of the linear regression equation appears to pass through just two data points. However, the graph of the quadratic regression equation fits the data ver well. Distance (ft) Average Braking Distance on Dr Pavement Source: Missouri Department of Revenue Speed (mph) 00 Chapter 6 Quadratic Functions and Inequalities

24 Solving Quadratic Equations b Factoring Lesson Notes Solve quadratic equations b factoring. Write a quadratic equation with given roots. is the Zero Product Propert used in geometr? The length of a rectangle is 5 inches more than its width, and the area of the rectangle is square inches. To find the dimensions of the rectangle ou need to solve the equation ( 5) or 5. SLVE EQUATINS BY FACTRING In the last lesson, ou learned to solve a quadratic equation like the one above b graphing. Another wa to solve this equation is b factoring. Consider the following products. Words 7(0) 0 0( ) 0 (6 6)(0) 0 ( 5 5) 0 Notice that in each case, at least one of the factors is zero. These eamples illustrate the Zero Product Propert. Zero Product Propert For an real numbers a and b, if ab 0, then either a 0, b 0, or both a and b equal zero. Eample If ( 5)( 7) 0, then 5 0 and/or 7 0. Eample Two Roots Solve each equation b factoring. a. 6 6 riginal equation 6 0 ( 6) 0 Subtract 6 from each side. Factor the binomial. 5 Focus 5-Minute Check Transparenc 6- Use as a quiz or review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 8C. Building on Prior Knowledge In Lesson 6-, students solved quadratic equations b graphing. In this lesson, the use factoring as a method for finding the roots of a quadratic equation. is the Zero Product Propert used in geometr? Ask students: What is the product of the length and the width of the rectangle? in What is the difference between the length and the width of the rectangle? 5 in. 0 or 6 0 Zero Product Propert 6 Solve the second equation. The solution set is {0, 6}. CHECK Substitute 0 and 6 for in the original equation. 6 6 (0) 6(0) (6) 6(6) Lesson 6- Solving Quadratic Equations b Factoring 0 Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 5 6 Skills Practice, p. 7 Practice, p. 8 Reading to Learn Mathematics, p. 9 Enrichment, p. 0 Transparencies 5-Minute Check Transparenc 6- Real-World Transparenc 6 Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 0

25 Teach SLVE EQUATINS BY FACTRING In-Class Eamples In-Class Eample Power Point Teaching Tip In Eample a, some students ma suggest solving the equation b dividing both sides b. Point out that this cannot be done because the value of could be zero, and division b zero is undefined. Solve each equation b factoring. a. {0, } b. 5, Solve 6 9 b factoring. {} Teaching Tip Point out that the term repeated root is sometimes used as a substitute for the term double root. What is the positive solution of the equation 8 0? D A B 5 C 6 D 7 Teaching Tip Ask students wh dividing each side of the equation in this eample results in an equivalent equation, without the possibilit of losing a root. (The right side of the equation is 0, not f() or, and dividing b means that ou can be sure that ou are not dividing b zero.) WRITE QUADRATIC EQUATINS Power Point Write a quadratic equation with and 6 as its roots. Write the equation in the form a b c 0, where a, b, and c are integers. Sample answer: 6 0 Stud Tip Double Roots The application of the Zero Product Propert produced two identical equations, 8 0, both of which have a root of 8. For this reason, 8 is called the double root of the equation. Standardized Test Practice Test-Taking Tip Because the problem asked for a positive solution, choice A could have been eliminated even before the epression was factored. Eample 0 Chapter 6 Quadratic Functions and Inequalities Standardized Test Practice b Double Root Solve b factoring riginal equation ( 8)( 8) 0 Factor. 8 0 or 8 0 Zero Product Propert 8 8 Solve each equation. The solution set is {8}. CHECK Eample The graph of the related function, f() 6 6, intersects the -ais onl once. Since the zero of the function is 8, the solution of the related equation is 8. Multiple-Choice Test Item riginal equation Subtract 5 from each side. ( )( 5) 0 Factor the trinomial. 0 or 5 0 Zero Product Propert 5 Solve each equation. The solution set is 5,. Check each solution. Greatest Common Factor What is the positive solution of the equation 60 0? A B C 5 D 0 Read the Test Item You are asked to find the positive solution of the given quadratic equation. This implies that the equation also has a solution that is not positive. Since a quadratic equation can either have one, two, or no solutions, we should epect to find two solutions to this equation. Solve the Test Item Solve this equation b factoring. But before tring to factor 60 into two binomials, look for a greatest common factor. Notice that each term is divisible b riginal equation ( 0) 0 Factor. 0 0 Divide each side b. ( )( 5) 0 Factor. 0 or 5 0 Zero Product Propert 5 Solve each equation. Both solutions, and 5, are listed among the answer choices. Since the question asked for the positive solution, the answer is C. Eample Point out to students that b reading the question carefull and noting eactl what is asked for (the positive solution), the can quickl eliminate answer choice A because it is negative. Choice A is an attractive (though incorrect) choice because it is indeed a solution of the equation, just not the positive one. 0 Chapter 6 Quadratic Functions and Inequalities

26 Stud Tip Writing an Equation The pattern ( p)( q) 0 produces one equation with roots p and q. In fact, there are an infinite number of equations that have these same roots. WRITE QUADRATIC EQUATINS You have seen that a quadratic equation of the form ( p)( q) 0 has roots p and q. You can use this pattern to find a quadratic equation for a given pair of roots. Eample Write an Equation Given Roots Write a quadratic equation with and 5 as its roots. Write the equation in the form a b c 0, where a, b, and c are integers. ( p)( q) 0 Write the pattern. [ ( 5)] 0 Replace p with and q with 5. ( 5) Simplif. Use FIL. Multipl each side b so that b and c are integers. A quadratic equation with roots and 5 and integral coefficients is You can check this result b graphing the related function. Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. include an other item(s) that the find helpful in mastering the skills in this lesson. Concept Check. Sample answer: If the product of two factors is zero, then at least one of the factors must be zero.. Sample answer: roots 6 and 5; 0 0. Kristin; the Zero Product Propert applies onl when one side of the equation is 0. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 9, 0 Standardized Test Practice. Write the meaning of the Zero Product Propert.. PEN ENDED Choose two integers. Then, write an equation with those roots in the form a b c 0, where a, b, and c are integers.. FIND THE ERRR Lina and Kristin are solving 8. Who is correct? Eplain our reasoning. Solve each equation b factoring.. 0 {0, } { 8, } 6. 9 { 7, 7} {} 8., {, } 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. 0., 7.,. 5, Which of the following is the sum of the solutions of 8 0? D A Lina +=8 (+)=8 = 8 or + = 8 = 6 Kristin +=8 + 8=0 (+)( )=0 +=0 or =0 = = 6 B C D Lesson 6- Solving Quadratic Equations b Factoring 0 Differentiated Instruction Visual/Spatial Provide each student with a sheet of grid paper. Have students begin b first drawing a coordinate grid with two points on the -ais plotted as the roots of a quadratic equation. Then ask students to draw several different parabolas that might be the graphs of different equations having those two points as their solutions. Point out that this demonstrates that the steps demonstrated in Eample ield just one of the possible equations having the given roots. About the Eercises rganization b bjective Solve Equations b Factoring:, 6 Write Quadratic Equations: dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 5 odd, 5 odd, 7 68 Average: 5 odd, 7 68 Advanced: even, 5 6 (optional: 6 68) All: Practice Quiz ( 5) FIND THE ERRR To show that neither factor on the left side of Lina s second equation needs to be 8, ask students to name several pairs of numbers whose product is 8. Possible pairs are and, 0.5 and 6, and and 8. Lesson 6- Solving Quadratic Equations b Factoring 0

27 Stud Stud Guide and and Intervention Intervention, p. 5 (shown) and p NAME DATE PERID 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 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 b. 5 5 riginal equation 5 0 Subtract from both sides. ( 7)( ) 0 Factor the trinomial. 7 0 or 0 Zero Product Propert 7 or Solve each equation. The solution set is 7,. Solve each equation b factoring , 5 {0, 7} 0, , 7 0, 9 0, {5, 6}, {, } , 7 0,, ,, 6 {00, 5} ,,, ,,, ,, 7,5 6- NAME 5 DATE PERID Practice (Average) Skills Practice, p. 7 and Practice, p. 8 (shown) Solving Quadratic Equations b Factoring Solve each equation b factoring.. 0 {6, } {8} {0} {, } 5. 0 {, } {, 7} 7. 0 {0, } , {5} ,. 99 { 9, }. 6 { 6} {, } , {9, 5} 6. 0, , { 5, } , , , 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,. 0,. 5, , , 7., , 9., 0. 0, ,.,., NUMBER THERY Find two consecutive even positive integers whose product is 6., 6 5. NUMBER THERY Find two consecutive odd positive integers whose product is. 7, 9 6. GEMETRY The length of a rectangle is feet more than its width. Find the dimensions of the rectangle if its area is 6 square feet. 7 ft b 9 ft 7. PHTGRAPHY The length and width of a 6-inch b 8-inch 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. Reading to Learn Mathematics, p NAME 8 DATE PERID Reading to Learn Mathematics Solving Quadratic Equations b Factoring Pre-Activit How is the Zero Product Propert used in geometr? Read the introduction to Lesson 6- at the top of page 0 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. 0 riginal equation 0 0 Add to each side. ( )( 7) 0 Factor the trinomial. 0 or 7 0 Zero Product Propert 7 Solve each equation. The solution set is {, 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 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. 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. ELL Lesson 6- indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples,, Etra Practice See page , 7., Forestr A board foot is a measure of lumber volume. ne piece of lumber foot long b foot wide b inch thick measures one board foot. Source: 0 Chapter 6 Quadratic Functions and Inequalities NAME DATE PERID 6- Enrichment, p. 0 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.. 0.., prime, prime 7, prime , prime 6, prime 7, prime Solve each equation b factoring { 8,} {,7} 6. 5 { 5, 5} 7. 8 { 9, 9} 8. 8 { 6, } 9. {, 7} , 5. 0,. 6 {6}. 6 6 {8}. 7, 5. 7, , , {, } {, }. Find the roots of ( 6)( 5) 0. 0, 6, 5. Solve 9 b factoring. 0,, 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.., 5 5., 7 6., , 8 8., 9., 5 0.,., DIVING To avoid hitting an rocks below, a cliff diver jumps up and out. The equation h 6t t 6 describes her height h in feet t seconds after jumping. Find the time at which she returns to a height of 6 feet. s 6 ft h 6 ft. NUMBER THERY Find two consecutive even integers whose product is., 6. PHTGRAPHY A rectangular photograph is 8 centimeters wide and centimeters long. The photograph is enlarged b increasing the length and width b an equal amount in order to double its area. What are the dimensions of the new photograph? cm b 6 cm FRESTRY For Eercises 5 and 6, use the following information. Lumber companies need to be able to estimate the number of board feet that a given log will ield. ne of the most commonl used formulas for estimating board feet is L the Dole Log Rule, B (D 6 8D 6), where B is the number of board feet, D is the diameter in inches, and L is the length of the log in feet. 5. Rewrite Dole's formula for logs that are 6 feet long. B D 8D 6 6. Find the root(s) of the quadratic equation ou wrote in Eercise 5. What do the root(s) tell ou about the kinds of logs for which Dole s rule makes sense? See margin. 7. CRITICAL THINKING For a quadratic equation of the form ( p)( q) 0, show that the ais of smmetr of the related quadratic function is located halfwa between the -intercepts p and q. See margin. CRITICAL THINKING Find a value of k that makes each statement true. 8. is a root of k is a root of k. 6 Answer 6. ; The logs must have a diameter greater than in. for the rule to produce positive board feet values. 0 Chapter 6 Quadratic Functions and Inequalities

28 Standardized Test Practice Maintain Your Skills Mied Review 5. 5, between and 0; between and Getting Read for the Net Lesson P ractice Quiz 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. A F. How is the Zero Product Propert used in geometr? Include the following in our answer: an eplanation of how to find the dimensions of the rectangle using the Zero Product Propert, and wh the equation ( 5) is not solved b using and Which quadratic equation has roots and? D A B C D If the roots of a quadratic equation are 6 and, what is the equation of the ais of smmetr? B A B C D Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. (Lesson 6-) 5. f() 5 5. f() 55. f() Determine whether f() 7 has a maimum or a minimum value. Then find the maimum or minimum value. (Lesson 6-) min.; 9 Simplif. (Lesson 5-6) Solve each sstem of equations. (Lesson -) a b 6. r s 6. a b (, ) r s 8 (, 5), PREREQUISITE SKILL Simplif. (To review simplifing radicals, see Lesson 5-5.) i 67. i i. Find the -intercept, the equation of the ais of smmetr, and the -coordinate of the verte for f(). Then graph the function b making a table of values. (Lesson 6-), ; ; See margin for graph.. Determine whether f() 5 has a maimum or minimum value. Then find this maimum or minimum value. (Lesson 6-) ma.; 7 or 9. Solve 0 b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. (Lesson 6-),. Solve b factoring. (Lesson 6-) 5, 5. Write a quadratic equation with roots and. Write the equation in the form a b c 0, where a, b, and c are integers. (Lesson 6-) ( p)( q) p q pq (p q) pq a, b (p q), c pq b ais of smmetr: a (p q) () p q Lessons 6- through 6- Lesson 6- Solving Quadratic Equations b Factoring 05 The ais of smmetr is the average of the -intercepts. Therefore the ais of smmetr is located halfwa between the -intercepts. Assess pen-ended Assessment Speaking Ask students to give a verbal eplanation of the Zero Product Propert. The should discuss wh it is true and how it is used in finding the roots of a quadratic equation, demonstrating the technique using an eample. Intervention Suggest that students who have difficult understanding the Zero Product Propert tr to find two nonzero numbers whose product is zero. Students should quickl determine that at least one of the numbers must be zero in order for their product to be zero. New Getting Read for Lesson 6- PREREQUISITE SKILL Lesson 6- presents solving quadratic equations b completing the square. The process involves evaluating radicals. Eercises 6 68 should be used to determine our students familiarit with simplifing radicals. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 6- through 6-. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not et mastered. Answer (Practice Quiz ). 8 f() (, 8) f () 8 Lesson 6- Solving Quadratic Equations b Factoring 05

29 Lesson Notes Completing the Square Focus 5-Minute Check Transparenc 6- Use as a quiz or review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 8C. Building on Prior Knowledge In Lesson 6-, students solved quadratic equations b factoring. In this lesson the use two other methods to solve equations: the Square Root Propert and completing the square. can ou find the time it takes an accelerating race car to reach the finish line? Ask students: The number is a perfect square. What is the square root of? How does the number relate to the coefficient of the term? The coefficient of is twice. Vocabular completing the square Stud Tip Reading Math n is read plus or minus the square root of n. TEACHING TIP Have students solve Eample b factoring and compare the results. Point out that Eample cannot be solved b factoring. Solve quadratic equations b using the Square Root Propert. Solve quadratic equations b completing the square. can ou find the time it takes an accelerating race car to reach the finish line? Under a ellow caution flag, race car drivers slow to a speed of 60 miles per hour. When the green flag is waved, the drivers can increase their speed. Suppose the driver of one car is 500 feet from the finish line. If the driver accelerates at a constant rate of 8 feet per second squared, the equation t t 6 represents the time t it takes the driver to reach this line. To solve this equation, ou can use the Square Root Propert. SQUARE RT PRPERTY You have solved equations like 5 0 b factoring. You can also use the Square Root Propert to solve such an equation. This method is useful with equations like the one above that describes the race car s speed. In this case, the quadratic equation contains a perfect square trinomial set equal to a constant. For an real number n, if n, then n. Eample Equation with Rational Roots Solve b using the Square Root Propert riginal equation ( 5) 9 Factor the perfect square trinomial. 5 9 Square Root Propert Add 5 to each side. 5 7 or 5 7 Write as two equations. Solve each equation. Square Root Propert The solution set is {, }. You can check this result b using factoring to solve the original equation. Roots that are irrational numbers ma be written as eact answers in radical form or as approimate answers in decimal form when a calculator is used. 06 Chapter 6 Quadratic Functions and Inequalities Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. 5 Enrichment, p. 6 Assessment, pp. 69, 7 Teaching Algebra With Manipulatives Masters, pp., 5 6 Transparencies 5-Minute Check Transparenc 6- Answer Ke Transparencies Technolog Interactive Chalkboard

30 Eample Equation with Irrational Roots Solve 6 9 b using the Square Root Propert. 6 9 riginal equation ( ) Factor the perfect square trinomial. Square Root Propert Add to each side; or Write as two equations Use a calculator. The eact solutions of this equation are and. The approimate solutions are.7 and 8.7. Check these results b finding and graphing the related quadratic function. 6 9 riginal equation CHECK Subtract from each side. Related quadratic function Use the ZER function of a graphing calculator. The approimate zeros of the related function are.7 and 8.7. CMPLETE THE SQUARE The Square Root Propert can onl be used to solve quadratic equations when the side containing the quadratic epression is a perfect square. However, few quadratic epressions are perfect squares. To make a quadratic epression a perfect square, a method called completing the square ma be used. In a perfect square trinomial, there is a relationship between the coefficient of the linear term and the constant term. Consider the pattern for squaring a sum. ( 7) (7) 7 Square of a sum pattern 9 Simplif. 7 Notice that 9 is 7 and 7 is one-half of. You can use this pattern of coefficients to complete the square of a quadratic epression. Teach SQUARE RT PRPERTY In-Class Eamples Power Point Teaching Tip In Eample, point out that both constants in the equation, 5 and 9, are perfect squares. Solve 9 6 b using the Square Root Propert. { 5, } Teaching Tip Point out that the constant on the right side of the equation given in Eample, is not a perfect square. Stress that this occurrence means the roots will be irrational numbers involving radicals. Also emphasize the use of the smbol in the step where the Square Root Propert is utilized. Solve 0 5 b using the Square Root Propert. {5 } CMPLETE THE SQUARE Teaching Tip When discussing the steps for completing the square, emphasize that the coefficient of the quadratic term must be. Words Completing the Square To complete the square for an quadratic epression of the form b, follow the steps below. Step Find one half of b, the coefficient of. Step Square the result in Step. Step Add the result of Step to b. Smbols b b b Lesson 6- Completing the Square 07 Lesson 6- Completing the Square 07

31 In-Class Eamples Power Point Find the value of c that makes 6 c a perfect square. Then write the trinomial as a perfect square. 6; ( 8) Solve 0 b completing the square. { 6, } Teaching Tip Remind students to think carefull about the difference between multipling a quantit b a factor of and squaring a quantit. Eample Complete the Square Find the value of c that makes c a perfect square. Then write the trinomial as a perfect square. Step Find one half of. 6 Step Square the result of Step. 6 6 Step Add the result of Step to. 6 The trinomial 6 can be written as ( 6). You can solve an quadratic equation b completing the square. Because ou are solving an equation, add the value ou use to complete the square to each side. Completing the Square Use algebra tiles to complete the square for the equation 0. Represent 0 Add to each side of the on an equation mat. mat. Remove the zero pairs. 0 Begin to arrange the and tiles into a square. 0 To complete the square, add ellow tile to each side. The completed equation is or ( ). Model Use algebra tiles to complete the square for each equation.. 0 ( ) 5. 0 ( ). 6 5 ( ). ( ) 0 Stud Tip Common Misconception When solving equations b completing the square, don t forget to add b to each side of the equation. Eample Solve an Equation b Completing the Square Solve b completing the square Notice that 8 0 is not a perfect square Rewrite so the left side is of the form b. Since 8 6, add 6 to each side. ( ) 6 Write the left side as a perfect square b factoring. 08 Chapter 6 Quadratic Functions and Inequalities Algebra Activit Materials: algebra tiles, equation mat Ask students wh the choice was made to add unit tiles to each side of the equation mat in Step. Sample answer: In order to simplif the work arranging the tiles into a square in Step. Remind students that an tile is units long and unit wide. Stress that the width is the same as the length of each side of a unit tile. 08 Chapter 6 Quadratic Functions and Inequalities

32 When the coefficient of the quadratic term is not, ou must first divide the equation b that coefficient before completing the square. Eample Equation with a Square Root Propert Add to each side. 6 or 6 Write as two equations. 0 The solution set is { 0, }. You can check this result b using factoring to solve the original equation. Solve 5 0 b completing the square. 5 0 Notice that 5 is not a perfect square. 5 0 Divide b the coefficient of quadratic term,. 5 Subtract from each side Since , add 5 to each side. 6 5 Write the left side as a perfect square b factoring. 6 Simplif the right side. 5 Square Root Propert 5 Add 5 to each side. 5 or 5 Write as two equations. The solution set is,. Not all solutions of quadratic equations are real numbers. In some cases, the solutions are comple numbers of the form a bi, where b 0. Eample 6 Equation with Comple Solutions Solve 0 b completing the square. 0 Notice that is not a perfect square. Rewrite so the left side is of the form b. Since, add to each side. ( ) 7 Write the left side as a perfect square b factoring. 7 Square Root Propert i 7 i i 7 Subtract from each side. The solution set is { i 7, i 7 }. Notice that these are imaginar solutions. In-Class Eamples 5 6 Power Point Teaching Tip Some students ma notice that the left side of the equation in Eample 5 can be factored into the product of two binomials: ( )( ). Then the Zero Product Propert can be used to obtain the same solutions. This is a good time to point out that more than one method of solution is often possible when solving a quadratic equation. Solve 0 b completing the square., Solve 0 b completing the square. { i } Teaching Tip Ask students to name three possible was to check the solutions to an equation that the have solved b completing the square. graphing the related function, factoring the equation, substituting the solutions into the original equation CHECK A graph of the related function shows that the equation has no real solutions since the graph has no -intercepts. Imaginar solutions must be checked algebraicall b substituting them in the original equation. Lesson 6- Completing the Square 09 Differentiated Instruction Kinesthetic Have students work with algebra tiles to help them write five equations that can be solved b completing the square. Provide each student with one tile, several tiles, and several unit tiles. Have students begin b creating a square arrangement of their tiles and then work backwards through the steps shown in the Algebra Activit on p. 08 to find a quadratic equation. After students have written their five equations, ask them to trade their equations with another student and then use their algebra tiles to find the solutions of the equations the receive. Lesson 6- Completing the Square 09

33 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. include an other item(s) that the find helpful in mastering the skills in this lesson. Concept Check. Completing the square allows ou to rewrite one side of a quadratic equation in the form of a perfect square. nce in this form, the equation is solved b using the Square Root Propert.. Never; see margin for eplanation.. Eplain what it means to complete the square.. Determine whether the value of c that makes a b c a perfect square trinomial is sometimes, alwas, or never negative. Eplain our reasoning.. FIND THE ERRR Rashid and Tia are solving b completing the square. Rashid 8 +0=0 8 = = ( ) =6 =+ 6 =+ 6 Tia 8 +0=0 =0 5 + = 5 + ( ) = = + i = + i Who is correct? Eplain our reasoning. Tia; see margin for eplanation. About the Eercises rganization b bjective Square Root Propert: Complete the Square: 5, 5 dd/even Assignments Eercises 7 are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercise 5 involves research on the Internet or other reference materials. Assignment Guide Basic: 5 9 odd, 7 odd, 5, 5 7 Average: 5 7 odd, 5 7 Advanced: 8 even, 9 5, 5 68 (optional: 69 7) FIND THE ERRR Point out that, while it is possible to complete the square when the coefficient of the term is something other than, it is much easier to first divide each side b the coefficient and students will also be less likel to make an error like the one made b Rashid shown here. Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 5,,, 6, Application Practice and Appl 0 Chapter 6 Quadratic Functions and Inequalities Solve each equation b using the Square Root Propert { 0, } Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 6. c 6; ( 6) 7. c 9 ; Solve each equation b completing the square { 6, } { 5 } { i 5 }. 0 ASTRNMY For Eercises and, use the following information. The height h of an object t seconds after it is dropped is given b h gt h 0, where h 0 is the initial height and g is the acceleration due to gravit. The acceleration due to gravit near Earth s surface is 9.8 m/s, while on Jupiter it is. m/s. Suppose 7 8. an object is dropped from an initial height of 00 meters from the surface of each planet. 5. n which planet should the object reach the ground first? Jupiter Find the time it takes for the object to reach the ground on each planet to the nearest tenth of a second. Earth:.5 s, Jupiter:.9 s indicates increased difficult Homework Help For Eercises See Eamples, 8, 7, , 5 Etra Practice See page 80. Solve each equation b using the Square Root Propert.. 5 {, 7} {, } { 7 } { } , {.6, 0.}. MVIE SCREENS The area A in square feet of a projected picture on a movie screen is given b A 0.6d, where d is the distance from the projector to the screen in feet. At what distance will the projected picture have an area of 00 square feet? 5 ft Answers. The value of c that makes a b b c a perfect square trinomial is the square of and the square of a number can never be negative.. Before completing the square, ou must first check to see that the coefficient of the quadratic term is. If it is not, ou must first divide the equation b that coefficient. 0 Chapter 6 Quadratic Functions and Inequalities

34 Engineering Reverse ballistic testing accelerating a target on a sled to impact a stationar test item at the end of the track was pioneered at the Sandia National Laboratories Rocket Sled Track Facilit in Albuquerque, New Meico. This facilit provides a 0,000-foot track for testing items at ver high speeds. Source: i 7. 5 i 9., ENGINEERING In an engineering test, a rocket sled is propelled into a target. The sled s distance d in meters from the target is given b the formula d.5t 0, where t is the number of seconds after rocket ignition. How man seconds have passed since rocket ignition when the sled is 0 meters from the target? about 8.56 s Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square.. 6 c 6; ( 8) 5. 8 c 8; ( 9) 6. 5 c 5 ; c 9 ; c 0.09; ( 0.) 9.. c.; (.) 0. 8 c 6 ; 9. 5 c 5 ; 6 5 Solve each equation b completing the square {, 5}. 0 0 {, 0}. 6 0 { 7 } 5. 0 { } { i } { i} , 9. 0, {, 0.6} {0.7, } FRAMING A picture has a square frame that is inches wide. The area of the picture is one-third of the total area of the picture and frame. What are the dimensions of the picture to the nearest quarter of an inch? 5 in. b 5 in. 6- Enrichment, p. 6 The Golden Quadratic Equations A golden rectangle has the propert that its length a 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, a as shown. The proportion used to define golden rectangles can be used to derive two quadratic equations. These are a sometimes called golden quadratic equations. Solve each problem.. In the proportion for the golden rectangle, let a equal. Write the resulting quadratic equation and solve for b. b b 0 5 b let b equ GLDEN RECTANGLE For Eercises 9 5, use the A E B following information. A golden rectangle is one that can be divided into a square and a second rectangle that is geometricall similar to the original rectangle. The ratio of the length of the longer side to the shorter side of a golden rectangle is called the golden ratio. 9. Find the ratio of the length of the longer side to the length of the shorter side for rectangle ABCD and for rectangle EBCF. D F C 50. Find the eact value of the golden ratio b setting the two ratios in Eercise 9 equal and solving for. (Hint: The golden ratio is a positive value.) 5. RESEARCH Use the Internet or other reference to find eamples of the golden rectangle in architecture. What applications does the reciprocal of the golden ratio have in music? See margin. 5. CRITICAL THINKING Find all values of n such that b b n has a. one real root. n 0 b. two real roots. n 0 c. two imaginar roots. n Sample answers: The golden rectangle is found in much of ancient Greek architecture, such as the Parthenon, as well as in modern architecture, such as in the windows of the United Nations building. Man songs have their clima at a point occurring 6.8% of the wa through the piece, with 0.68 being about the reciprocal of the golden ratio. The reciprocal of the golden ratio is also used in the design of some violins. in. b b in. Lesson 6- Completing the Square NAME DATE PERID s a Stud Stud Guide and and Intervention Intervention, p. (shown) and p. 6- NAME DATE PERID 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 Eample a ( ) 5 5 For an real number if n, then n. Solve each equation b using the Square Root Propert. or or 5 The solution set is {9, }. Eercises b ( 5) 5 or 5 5 or 5 5 The solution set is 5. Solve each equation b using the Square Root Propert {, 6} {, 8}, , ,, { 0.8,.} NAME DATE PERID Practice (Average) Skills Practice, p. and Practice, p. (shown) Solve each equation b using the Square Root Propert ,, 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 6; ( 6) 00; ( 0) ;. 0.8 c.. c c 0.6; ( 0.).; (.) 0.0; ( 0.8) c 7. c 8. c ; ; ; Solve each equation b completing the square , 0. 0,. 5 0, , i 5 Completing the Square 5 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.. 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% Reading to Learn Mathematics, p NAME DATE PERID Reading to Learn Mathematics Completing the Square i 5 7 Pre-Activit How can ou find the time it takes an accelerating race car to reach the finish line? Read the introduction to Lesson 6- at the top of page 06 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. Reading the Lesson. Give the reason for each step in the following solution of an equation b using the Square Root Propert. 6 8 riginal equation ( 6) 8 Factor the perfect square trinomial or 6 9 Square Root Propert Rewrite as two equations. 5 Solve each equation. ELL. 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.. a. What is the first step in solving the equation 6 5 b completing the square? Divide the equation b. 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- Completing the Square Lesson 6- Lesson 6-

35 Assess pen-ended Assessment Writing Have students write a summar of the various techniques for solving quadratic equations using the Square Root Propert. Students should provide written eamples of each technique. Intervention Suggest students write a summar of the various methods that can be used to solve quadratic equations. Ask them which method the prefer to use, and wh the like that method the best. New Getting Read for Lesson 6-5 PREREQUISITE SKILL Lesson 6-5 presents the Quadratic Formula. The first step in evaluating the formula is to evaluate the epression under the radical sign. Use Eercises 69 7 to determine our students familiarit with evaluating epressions. Assessment ptions Quiz (Lessons 6- and 6-) is available on p. 69 of the Chapter 6 Resource Masters. Mid-Chapter Test (Lessons 6- through 6-) is available on p. 7 of the Chapter 6 Resource Masters. Standardized Test Practice Maintain Your Skills Mied Review 6. between and ; between 0 and Getting Read for the Net Lesson 5. KENNEL A kennel owner has 6 feet of fencing with which to enclose a rectangular region. He wants to subdivide this region into three smaller rectangles of equal length, as shown. If the total area to be enclosed is 576 square feet, find the dimensions of the entire enclosed region. (Hint: Write an epression for in terms of w.) 8 ft b ft or 6 ft b 9 ft 5. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can ou find the time it takes an accelerating race car to reach the finish line? Include the following in our answer: an eplanation of wh t t 6 cannot be solved b factoring, and a description of the steps ou would take to solve the equation t t What is the absolute value of the product of the two solutions for in 0? D A B 0 C D 56. For which value of c will the roots of c 0 be real and equal? D A B C D E 5 Write a quadratic equation with the given root(s). Write the equation in the form a b c 0, where a, b, and c are integers. (Lesson 6-) 57., 58., , 60., Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. (Lesson 6-) , 8 6., 6. Write the seventh root of 5 cubed using eponents. (Lesson 5-7) 5 Solve each sstem of equations b using inverse matrices. (Lesson -8) (, 5) 7, 6 7 CHEMISTRY For Eercises 67 and 68, use the following information. For hdrogen to be a liquid, its temperature must be within C of 57 C. (Lesson -) 67. Write an equation to determine the greatest and least temperatures for this substance. ( 57) 68. Solve the equation. greatest: 55 C; least: 59 C PREREQUISITE SKILL Evaluate b ac for the given values of a, b, and c. (To review evaluating epressions, see Lesson -.) 69. a, b 7, c a, b, c a, b 9, c 5 7. a, b, c w Chapter 6 Quadratic Functions and Inequalities Answer 5. To find the distance traveled b the accelerating race car in the given situation, ou must solve the equation t t 6 or t t 5 0. Answers should include the following. Since the epression t t 5 is prime, the solutions of t t 6 cannot be obtained b factoring. Rewrite t t as (t ). Solve (t ) 6 b appling the Square Root Propert. Then, subtract from each side. Using a calculator, the two solutions are about.7 or 6.7. Since time cannot be negative, the driver takes about.7 seconds to reach the finish line. Chapter 6 Quadratic Functions and Inequalities

36 The Quadratic Formula and the Discriminant Lesson Notes Vocabular Quadratic Formula discriminant Stud Tip Reading Math The Quadratic Formula is read equals the opposite of b, plus or minus the square root of b squared minus ac, all divided b a. Solve quadratic equations b using the Quadratic Formula. Use the discriminant to determine the number and tpe of roots of a quadratic equation. is blood pressure related to age? As people age, their arteries lose their elasticit, which causes blood pressure to increase. For health women, average sstolic blood pressure is estimated b P 0.0A 0.05A 07, where P is the average blood pressure in millimeters of mercur (mm Hg) and A is the person s age. For health men, average sstolic blood pressure is estimated b P 0.006A 0.0A 0. QUADRATIC FRMULA You have seen that eact solutions to some quadratic equations can be found b graphing, b factoring, or b using the Square Root Propert. While completing the square can be used to solve an quadratic equation, the process can be tedious if the equation contains fractions or decimals. Fortunatel, a formula eists that can be used to solve an quadratic equation of the form a b c 0. This formula can be derived b solving the general form of a quadratic equation. a b c 0 General quadratic equation c 0 Divide each side b a. b a a b a c a c Subtract a from each side. b a b c a a ba Complete the square. ba b ac a Factor the left side. Simplif the right side. b b ac a a b b ac a a b b ac a Square Root Propert b Subtract from each side. a Simplif. This equation is known as the Quadratic Formula. Quadratic Formula The solutions of a quadratic equation of the form a b c 0, where a 0, are given b the following formula. b b ac a Lesson 6-5 The Quadratic Formula and the Discriminant Focus 5-Minute Check Transparenc 6-5 Use as a quiz or review of Lesson 6-. Mathematical Background notes are available for this lesson on p. 8D. Building on Prior Knowledge In Lesson 6-, students solved quadratic equations b completing the square. In this lesson, students generalize this procedure as the complete the square for the general quadratic equation to derive the Quadratic Formula. is blood pressure related to age? Ask students: As the value of A increases in these equations, what happens to the value of P? It increases. Which wa do the parabolas open that are the graphs of these equations? upward Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. 7 8 Skills Practice, p. 9 Practice, p. 0 Reading to Learn Mathematics, p. Enrichment, p. Graphing Calculator and Spreadsheet Masters, p. 8 Transparencies 5-Minute Check Transparenc 6-5 Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lessons, Interactive Chalkboard Lesson - Lesson Title

37 Teach QUADRATIC FRMULA In-Class Eamples Power Point Solve 8 b using the Quadratic Formula., Teaching Tip Encourage students to write down the values of a, b, and c from the standard form of the quadratic equation before the begin substituting into the formula. Solve 89 0 b using the Quadratic Formula. 7 Stud Tip Quadratic Formula Although factoring ma be an easier method to solve the equations in Eamples and, the Quadratic Formula can be used to solve an quadratic equation. Eample Two Rational Roots Solve 8 b using the Quadratic Formula. First, write the equation in the form a b c 0 and identif a, b, and c. a b c Then, substitute these values into the Quadratic Formula. b b ac a Quadratic Formula ( ) ( ) ( 8) () () or 6 Replace a with, b with, and c with 8. Simplif. Simplif Simplif. Write as two equations. The solutions are and. Check b substituting each of these values into the original equation. When the value of the radicand in the Quadratic Formula is 0, the quadratic equation has eactl one rational root. Eample ne Rational Root Solve 0 b using the Quadratic Formula. Identif a, b, and c. Then, substitute these values into the Quadratic Formula. b b ac a Quadratic Formula () () ) ()( () Replace a with, b with, and c with. 0 Simplif. or 0 0 The solution is. CHECK A graph of the related function shows that there is one solution at. [ 5, 5] scl: b [ 5, 5] scl: Chapter 6 Quadratic Functions and Inequalities Teacher to Teacher Lori Haldorson & Cath Hokkanen Blaine H.S., Blaine, MN "To help students memorize the Quadratic Formula, we sing it to the tune of Pop Goes the Weasel." Chapter 6 Quadratic Functions and Inequalities

38 Stud Tip Using the Quadratic Formula Remember that to correctl identif a, b, and c for use in the Quadratic Formula, the equation must be written in the form a b c 0. You can epress irrational roots eactl b writing them in radical form. Eample Irrational Roots Solve 5 0 b using the Quadratic Formula. b b ac Quadratic Formula a () () 5) ()( () Replace a with, b with, and c with 5. Simplif. or 56 or The eact solutions are and. The approimate solutions are.9 and 0.9. CHECK 56 Check these results b graphing the related quadratic function, 5. Using the ZER function of a graphing calculator, the approimate zeros of the related function are.9 and 0.9. Comple Roots 5 [ 0, 0] scl: b [ 0, 0] scl: When using the Quadratic Formula, if the radical contains a negative value, the solutions will be comple. Comple solutions alwas appear in conjugate pairs. Eample Solve b using the Quadratic Formula. b b ac a Quadratic Formula ( ) ( ) () () () Replace a with, b with, and c with. 6 Simplif. 6i i 6 6( ) or 6i Simplif. The solutions are the comple numbers i and i. A graph of the related function shows that the solutions are comple, but it cannot help ou find them. In-Class Eamples Power Point Teaching Tip Some students ma have commented that the equations in Eamples and could have been solved b factoring. Before beginning Eample, stress that man quadratic equations cannot easil be solved b factoring. State that the quadratic equation presented in Eample is such an equation. Emphasize that the Quadratic Formula provides a wa to find the roots for an quadratic equation. Solve 6 0 b using the Quadratic Formula. 7, or approimatel 0. and 5.6 Teaching Tip Remind students that conjugate pairs are two comple numbers of the form a bi and a bi. Solve 6 b using the Quadratic Formula. i Concept Check Real Roots or Imaginar Roots Ask students to look back at the first four eamples in this lesson and see how the might predict whether the roots will be real or imaginar. If the values of a and c have the same sign and times their product is greater than the square of the value of b, then the roots will be imaginar. [ 5, 5] scl: b [, 8] scl: Lesson 6-5 The Quadratic Formula and the Discriminant 5 Lesson 6-5 The Quadratic Formula and the Discriminant 5

39 RTS AND THE DISCRIMINANT In-Class Eample Power Point 5 Find the value of the discriminant for each quadratic equation. Then describe the number and tpe of roots for the equation. a ; one rational root b. 5 0 ; two comple roots c ; two irrational roots d ; two rational roots Stud Tip Reading Math Remember that the solutions of an equation are called roots. CHECK To check comple solutions, ou must substitute them into the original equation. The check for i is shown below. riginal equation ( i) ( i) i i 9i 8 i Sum of a square; Distributive Propert 9i 9 Simplif. i RTS AND THE DISCRIMINANT In Eamples,,, and, observe the relationship between the value of the epression under the radical and the roots of the quadratic equation. The epression b ac is called the discriminant. b b ac discriminant a The value of the discriminant can be used to determine the number and tpe of roots of a quadratic equation. Value of Discriminant Consider a b c 0. Tpe and Number of Roots Discriminant Eample of Graph of Related Function b ac 0; b ac is a perfect square. real, rational roots b ac 0; b ac is not a perfect square. real, irrational roots b ac 0 real, rational root b ac 0 comple roots Stud Tip Using the Discriminant The discriminant can help ou check the solutions of a quadratic equation. Your solutions must match in number and in tpe to those determined b the discriminant. Eample 5 Describe Roots Find the value of the discriminant for each quadratic equation. Then describe the number and tpe of roots for the equation. a. 9 0 b. 6 0 a 9, b, c a, b 6, c b ac ( ) (9)() b ac (6) ()() The discriminant is 0, so The discriminant is negative, so there is one rational root. there are two comple roots. 6 Chapter 6 Quadratic Functions and Inequalities Differentiated Instruction Logical Have students use their classification skills to create a classroom poster listing the four different tpes of roots that can result when solving a quadratic equation. Each listing should include a sample equation that results in that tpe of roots and an eplanation of how the value of the discriminant is indicative of the root tpe. Graphs like those shown on p. 6 can be added to the poster. 6 Chapter 6 Quadratic Functions and Inequalities

40 Concept Check. The square root of a negative number is a comple number. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 7 5 8, c d a 5, b 8, c a 5, b 7, c b ac (8) ( 5)( ) b ac ( 7) (5)( ) or 7 The discriminant is, The discriminant is 89, which is not a perfect which is a perfect square. square. Therefore, there Therefore, there are are two irrational roots. two rational roots. You have studied a variet of methods for solving quadratic equations. The table below summarizes these methods. Graphing Factoring Method Square Root Propert Completing the Square Quadratic Formula Can be Used sometimes sometimes sometimes alwas alwas. PEN ENDED Sketch the graph of a quadratic equation whose discriminant is a. positive. b. negative. c. zero. a c. See margin.. Eplain wh the roots of a quadratic equation are comple if the value of the discriminant is less than 0.. Describe the relationship that must eist between a, b, and c in the equation a b c 0 in order for the equation to have eactl one solution. b ac must equal 0. Complete parts a c for each quadratic equation. a. Find the value of the discriminant. 7. See margin. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula Solving Quadratic Equations When to Use Use onl if an eact answer is not required. Best used to check the reasonableness of solutions found algebraicall. Use if the constant term is 0 or if the factors are easil determined. Eample 0 Use for equations in which a perfect square is equal to a constant. Eample ( ) 9 Useful for equations of the form b c 0, where b is even. Eample 9 0 Useful when other methods fail or are too tedious. Eample Lesson 6-5 The Quadratic Formula and the Discriminant 7 a. Sample answer: b. Sample answer: c. Sample answer: Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. cop the information provided in the Concept Summar on p. 7 into their notebook. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Quadratic Formula: 9, Roots and the Discriminant: 7, 0, dd/even Assignments Eercises 9 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 5 5 odd, 9 9 odd, 0,, 5 66 Average: 5 9 odd, 0, 5 66 Advanced: 8 even, 60 (optional: 6 66) Answers a. 8 b. rational 5 c., 5a. 8 5b. irrational 5c. 6a. 0 6b. one rational 6c. 7a. 7b. comple i 7c. Lesson 6-5 The Quadratic Formula and the Discriminant 7

41 Stud Stud Guide and and Intervention Intervention, p. 7 (shown) and p NAME DATE PERID 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 b ac 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 b ac Quadratic Formula a ( 5) ( 5) )( ( ) () or The solutions are and 7. Eercises Replace a with, b with 5, and c with. Simplif. Solve each equation b using the Quadratic Formula , 7, 6, , 5, 7 5, , 5,, r r ,, i NAME 7 DATE PERID Practice (Average) Skills Practice, p. 9 and Practice, p. 0 (shown) The Quadratic Formula and the Discriminant Complete parts a c 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, 0; rational; ; ; rational; 5, 8 irrational; ; rational; 0, ; ; ; comple; i 9 rational;, comple; i ; ; ; rational;, irrational; 7 6 i 5 comple; ; ; irrational; 05 0; rational; irrational; Solve each equation b using the method of our choice. Find eact solutions , , 9., ,. 5 8, i. 5 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. 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 Reading to Learn Mathematics, p. 6-5 NAME 0 DATE PERID Reading to Learn Mathematics The Quadratic Formula and the Discriminant Pre-Activit How is blood pressure related to age? Read the introduction to Lesson 6-5 at the top of page 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 b ac. a. Write the Quadratic Formula. a ELL 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. Lesson 6-5 Application. at about 0.7 s and again at about.6 s. Will the object ever reach a height of 0 feet? Eplain our reasoning. No; see margin for eplanation. indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples , 0 5 Etra Practice See page 8. Bridges The Golden Gate, located in San Francisco, California, is the tallest bridge in the world, with its towers etending 76 feet above the water and the floor of the bridge etending 0 feet above water. Source: 8 Chapter 6 Quadratic Functions and Inequalities Solve each equation using the method of our choice. Find eact solutions , , i 0 5 PHYSICS For Eercises and, use the following information. 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 85 feet per second is modeled b h(t) 6t 85t.. When will the object be at a height of 50 feet? Complete parts a c for each quadratic equation. a. Find the value of the discriminant. 7. See margin. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula See pp. A F. Solve each equation b using the method of our choice. Find eact solutions ( ) ( ) 5 BRIDGES For Eercises 0 and, use the following information. The supporting cables of the Golden Gate Bridge approimate the shape of a parabola. The parabola can be modeled b the quadratic function , where represents the distance from the ais of smmetr and represents the height of the cables. The related quadratic equation is Calculate the value of the discriminant What does the discriminant tell ou about the supporting cables of the Golden Gate Bridge? See pp. A F. FTBALL For Eercises and, use the following information. The average NFL salar A(t) (in thousands of dollars) from 975 to 000 can be estimated using the function A(t).t.t 7.7, where t is the number of ears since D: 0 t 5, R: 7.7 A(t) 0.. Determine a domain and range for which this function makes sense.. According to this model, in what ear did the average salar first eceed million dollars? 998 nline Research Data Update What is the current average NFL salar? How does this average compare with the average given b the function used in Eercises and? Visit to learn more. Value of Discriminant Number of Roots Tpe of Roots 6 real, rational 8 comple real, irrational 0 real, rational Helping You Remember. 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. NAME DATE PERID 6-5 Enrichment, p. 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 % can be represented b the formula a 0. 0 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 and 8. ( 8) 5 Add the roots. 0 ( 8) Multipl the roots. Equation: Answer. The discriminant of 6t 85t 0 is 55, indicating that the equation has no real solutions. 8 Chapter 6 Quadratic Functions and Inequalities

42 Standardized Test Practice Maintain Your Skills Mied Review Getting Read for the Net Lesson. HIGHWAY SAFETY Highwa safet engineers can use the formula d 0.05s.s to estimate the minimum stopping distance d in feet for a vehicle traveling s miles per hour. If a car is able to stop after 5 feet, what is the fastest it could have been traveling when the driver first applied the brakes? about 0. mph 5. CRITICAL THINKING Find all values of k such that k 9 0 has a. one real root. k 6 b. two real roots. c. no real roots. k 6 or k 6 6 k 6 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. A F. How is blood pressure related to age? Include the following in our answer: an epression giving the average sstolic blood pressure for a person of our age, and an eample showing how ou could determine A in either formula given a specific value of P. 7. If 5 9 0, then could equal which of the following? D A. B.5 C.6 D.7 8. Which best describes the nature of the roots of the equation 0? C A C real and equal comple B D real and unequal real and comple Solve each equation b using the Square Root Propert. (Lesson 6-) , 7 Solve each equation b factoring. (Lesson 6-) , , , 5 Simplif. (Lesson 5-5) 55. a 8 b 0 a b p q 0p 6 q 57. 6b 6 c 6 b c 58. ANIMALS The fastest-recorded phsical action of an living thing is the wing beat of the common midge. This tin insect normall beats its wings at a rate of,000 times per minute. At this rate, how man times would the midge beat its wings in an hour? Write our answer in scientific notation. (Lesson 5-) Solve each sstem of inequalities. (Lesson -) See pp. A F PREREQUISITE SKILL State whether each trinomial is a perfect square. If it is, factor it. (To review perfect square trinomials, see Lesson 5-.) no 6. 9 es; ( 7) Answers a. b. irrational c. 5a. 0 5b. irrational 5c es; ( ) es; (5 ) no es; (6 5) 6a. 6 6b. comple 6c. i 7a. 7b. comple 7c. i Lesson 6-5 The Quadratic Formula and the Discriminant 9 8a. 8b. rational 8c., 9a. 9 9b. rational 9c., 0a. 0 0b. irrational 0c. 5 a. b. irrational c. 6 Assess pen-ended Assessment Modeling Ask students to sketch graphs of parabolas that illustrate each of the four tpes of roots for quadratic equations. Have them label each sketch with the tpe of value the discriminant of the corresponding quadratic equation would have. Getting Read for Lesson 6-6 PREREQUISITE SKILL Lesson 6-6 presents the analsis of the graphs of quadratic functions. To graph a quadratic function, it is helpful if the function is written in verte form, which often requires students to complete the square. Recognition of perfect square trinomials is an important part of completing the square. Eercises 6 66 should be used to determine our students familiarit with perfect square trinomials. Answers a. 0 b. one rational c. a. 0 b. one rational 5 c. a. b. comple 9 i c. 8 5a. 5 5b. comple 5c. i 5 8 6a. 9 6b. irrational 6c a..8 7b. irrational 0.7 7c. 0.8 Lesson 6-5 The Quadratic Formula and the Discriminant 9

43 Graphing Calculator Investigation A Preview of Lesson 6-6 Getting Started Know Your Calculator Students can use the calculator to confirm the location of the verte of each parabola. A good wa to do this is to change the window settings for the -ais to [ 9., 9.]. Then use the Trace feature and smmetr properties of parabolas to check that the graph is smmetric with respect to the vertical line through the point that appears to be the verte. Families of Parabolas The general form of a quadratic equation is a( h) k. Changing the values of a, h, and k results in a different parabola in the famil of quadratic functions. You can use a TI-8 Plus graphing calculator to analze the effects that result from changing each of these parameters. Eample Graph each set ofequations on the same screen in the standard viewing window. Describe an similarities and differences among the graphs.,, 5 The graphs have the same shape, and all open up. The verte of each graph is on the -ais. However, the graphs have different vertical positions. A Preview of Lesson Teach Ask students to describe the three constants (a, h, and k) in the general form of a quadratic equation a( h) k. Sample answer: a: coefficient of the squared quantit involving the variable ; h: value subtracted from in the quantit being squared and then multiplied b a; k: value added at the end Before discussing the eamples, have students make a conjecture about the effect of the value of each of the constants a, h, and k on the graph of the parabola. After completing the discussion of Eample, have students compare the conjectures the made at the beginning of the investigation to the knowledge the gained during the discussions. Have students complete Eercises 5. Eample shows how changing the value of k in the equation a( h) k translates the parabola along the -ais. If k 0, the parabola is translated k units up, and if k 0, it is translated k units down. How do ou think changing the value of h will affect the graph of? Eample Graph each set ofequations on the same screen in the standard viewing window. Describe an similarities and differences among the graphs., ( ), ( 5) These three graphs all open up and have the same shape. The verte of each graph is on the -ais. However, the graphs have different horizontal positions. 0 Chapter 6 Quadratic Functions and Inequalities ( ) Eample shows how changing the value of h in the equation a( h) k translates the graph horizontall. If h 0, the graph translates to the right h units. If h 0, the graph translates to the left h units. ( 5) 0 Chapter 6 Quadratic Functions and Inequalities

44 How does the value a affect the graph of? Eample Graph each set ofequations on the same screen in the standard viewing window. Describe an similarities and differences among the graphs. a., The graphs have the same verte and the same shape. However, the graph of opens up and the graph of opens down. b.,, The graphs have the same verte, (0, 0), but each has a different shape. The graph of is narrower than the graph of. The graph of is wider than the graph of. Changing the value of a in the equation a( h) k can affect the direction of the opening and the shape of the graph. If a 0, the graph opens up, and if a 0, the graph opens down or is reflected over the -ais. If a, the graph is narrower than the graph of. If a, the graph is wider than the graph of. Thus, a change in the absolute value of a results in a dilation of the graph of. Eercises. See margin. Consider a( h) k.. How does changing the value of h affect the graph? Give an eample.. How does changing the value of k affect the graph? Give an eample.. How does using a instead of a affect the graph? Give an eample. Eamine each pair ofequations and predict the similarities and differences in their graphs. Use a graphing calculator to confirm our predictions. Write a sentence or two comparing the two graphs. 5. See pp. A F..,.5 5., 9 6., 7., 6 8., ( ) 9., 0., ( 7)., ( ) 7.,. ( ), ( ) 5. ( ), 5. ( ), 6( ) ( ) [ 0, 0] scl: b [ 5, 5] scl: Assess Ask students: In the general form of a quadratic equation, which constant would ou change to move the graph left or right? h Which constant would ou change to move the graph up or down? k Which constant would ou change to make the graph wider or narrower? a Answers. Changing the value of h moves the graph to the left and the right. If h 0, the graph translates to the right, and if h 0, it translates to the left. In, the verte is at (0, 0) and in ( ), the verte is at (, 0). The graph has been translated to the right.. Changing the value of k moves the graph up and down. If k 0, the graph translates upward, and if k 0, it translates downward. In, the verte is at (0, 0) and in, the verte is at (0, ). The graph has been translated downward.. Using a instead of a reflects the graph over the -ais. The graph of opens upward, while the graph of opens downward. Graphing Calculator Investigation Families of Parabolas Graphing Calculator Investigation Families of Parabolas

45 Lesson Notes Analzing Graphs of Quadratic Functions Focus 5-Minute Check Transparenc 6-6 Use as a quiz or review of Lesson 6-5. Mathematical Background notes are available for this lesson on p. 8D. can the graph of be used to graph an quadratic function? Ask students: For the function, what value of makes equal 0? 0 What value of makes equal 0 if the function is ( )? Compare the graph of with the graph of ( ). What difference does adding the within the parentheses make? Sample answer: Adding the inside the parentheses moves the graph units to the left rather than units up when compared to the graph of. Vocabular verte form Analze quadratic functions of the form a( h) k. Write a quadratic function in the form a( h) k. A famil of graphs is a group of graphs that displas one or more similar characteristics. The graph of is called the parent graph of the famil of quadratic functions. Stud the graphs of,, and ( ). Notice that adding a constant to moves the graph up. Subtracting a constant from before squaring it moves the graph to the right. ANALYZE QUADRATIC FUNCTINS Notice that each function above can be written in the form ( h) k, where (h, k) is the verte of the parabola and h is its ais of smmetr. This is often referred to as the verte form of a quadratic function. In Chapter, ou learned that a translation slides a figure on the coordinate plane without changing its shape or size. As the values of h and k change, the graph of a( h) k is the graph of translated h units left if h is negative or h units right if h is positive, and k units up if k is positive or k units down if k is negative. Eample can the graph of be used to graph an quadratic function? Equation Graph a Quadratic Function in Verte Form Analze ( ). Then draw its graph. This function can be rewritten as [ ( )]. Then h and k. The verte is at (h, k) or (, ), and the ais of smmetr is. The graph has the same shape as the graph of, but is translated units left and unit up. Now use this information to draw the graph. Step Plot the verte, (, ). (, 5) (0, 5) Step Draw the ais of smmetr,. ( ) Step Find and plot two points on one side of the (, ) (, ) ais of smmetr, such as (, ) and (0, 5). (, ) Step Use smmetr to complete the graph. or ( 0) 0 or ( 0) ( ) or ( ) 0 Verte ( ) Ais of Smmetr (0, 0) 0 (0, ) 0 (, 0) Chapter 6 Quadratic Functions and Inequalities Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. 5 Practice, p. 6 Reading to Learn Mathematics, p. 7 Enrichment, p. 8 Assessment, p. 70 Teaching Algebra With Manipulatives Masters, pp. 7 8 Transparencies 5-Minute Check Transparenc 6-6 Answer Ke Transparencies Technolog Interactive Chalkboard

46 Stud Tip Reading Math a means that a is a rational number between 0 and, such as, or a rational number 5 between and 0, such as 0.. How does the value of a in the general form a( h) k affect a parabola? Compare the graphs of the following functions to the parent function,. a. b. c. d. All of the graphs have the verte (0, 0) and ais of smmetr 0. Notice that the graphs of and are dilations of the graph of. The graph of is narrower than the graph of, while the graph of is wider. The graphs of and are reflections of each other over the -ais, as are the graphs of and. Changing the value of a in the equation a( h) k can affect the direction of the opening and the shape of the graph. If a 0, the graph opens up. If a 0, the graph opens down. If a, the graph is narrower than the graph of. If a, the graph is wider than the graph of. Verte and Ais of Smmetr h (h, k) a c b Quadratic Functions in Verte Form The verte form of a quadratic function is a( h) k. h and k Vertical Translation k 0 k k 0, k 0 d Teach ANALYZE QUADRATIC FUNCTINS In-Class Eample Power Point Analze ( ). Then draw its graph. The verte of the graph is at (, ) and the ais of smmetr is. The graph has the same shape as the graph of, but is translated units right and units up. f() (, 6) (5, 6) (, ) (, ) (, ) f() ( ) Teaching Tip To help students remember that as a increases, the graph gets narrower and not wider, discuss the fact that a greater multiplier for the quantit ( h) will make the corresponding value greater as well. Point out that greater values of result in a steeper (and thus narrower) graph. h Horizontal Translation, h 0 h 0 h 0 a 0 Direction of pening and Shape of Parabola a 0 a, a a a Lesson 6-6 Analzing Graphs of Quadratic Functions Intervention Encourage students to ask questions about an aspects the ma find confusing that are covered in the Concept Summar chart on this page. Ask them to write and use their own summar on an inde card. Eplain to students that a thorough understanding of these concepts will save them time since this knowledge will enable them to sketch approimate graphs quickl. New Lesson 6-6 Analzing Graphs of Quadratic Functions

47 WRITE QUADRATIC FUNCTINS IN VERTEX FRM In-Class Eamples Power Point Write in verte form. Then analze the function. ( ) ; verte: (, ); ais of smmetr: ; opens up; The graph has the same shape as the graph of, but it is translated unit left and units up. Write in verte form. Then analze and graph the function. ( ) ; verte: (, ); ais of smmetr: ; opens down; The graph is narrower than the graph of, and it is translated unit left and units up. f() f() ( ) Stud Tip Check As an additional check, graph the function in Eample to verif the location of its verte and ais of smmetr. WRITE QUADRATIC FUNCTINS IN VERTEX FRM Given a function of the form a b c, ou can complete the square to write the function in verte form. Eample Write b c in Verte Form Write 8 5 in verte form. Then analze the function. 8 5 Notice that 8 5 is not a perfect square. Complete the square b adding 8 or 6. ( 8 6) 5 6 Balance this addition b subtracting 6. ( ) Write 8 6 as a perfect square. This function can be rewritten as [ ( )] ( ). Written in this wa, ou can see that h and k. The verte is at (, ), and the ais of smmetr is. Since a, the graph opens up and has the same shape as the graph of, but it is translated units left and units down. CHECK You can check the verte and ais of smmetr using the formula b. In the original equation, a and b 8, so the ais of a 8 smmetr is or. Thus, the -coordinate of the verte is ( ), and the -coordinate of the verte is ( ) 8( ) 5 or. When writing a quadratic function in which the coefficient of the quadratic term is not in verte form, the first step is to factor out that coefficient from the quadratic and linear terms. Then ou can complete the square and write in verte form. Eample Write a b c in Verte Form, a Write 6 in verte form. Then analze and graph the function. 6 riginal equation ( ) Group a b and factor, dividing b a. Complete the square b adding inside the parentheses. ( ) ( )() Notice that this is an overall addition of (). Balance this addition b subtracting (). ( ) Write as a perfect square. The verte form of this function is ( ). So, h and k. The verte is at (, ), and the ais of smmetr is. Since a, the graph opens downward and is narrower than the graph of. It is also translated unit right and units up. Now graph the function. Two points on the graph to the right of are (.5,.5) and (, ). Use smmetr to complete the graph. ( ) (.5,.5) (, ) Chapter 6 Quadratic Functions and Inequalities Differentiated Instruction Naturalist Have students observe or research some natural events that can be modeled b parabolas, such as the fountain s water stream discussed in Eercise on p. 6. Students should report their observations and findings to the class. If students are able to determine a quadratic function that models the event, the should present the function and eplain how the characteristics of the equation can be used to analze its graph. Chapter 6 Quadratic Functions and Inequalities

48 Concept Check d. ( ) e. Sample answer: ( ) f. Sample answer: ( ). Sample answer: ( ). Jenn; when completing the square is used to write a quadratic function in verte form, the quantit added is then subtracted from the same side of the equation to maintain equalit. Guided Practice If the verte and one other point on the graph of a parabola are known, ou can write the equation of the parabola in verte form. Eample Write an Equation Given Points Write an equation for the parabola whose verte is at (, ) and passes through (, ). The verte of the parabola is at (, ), so h and k. Since (, ) is a point on the graph of the parabola, let and. Substitute these values into the verte form of the equation and solve for a. a( h) k Verte form a[ ( )] Substitute for, for, for h, and for k. a(9) Simplif. 9a Subtract from each side. a Divide each side b 9. The equation of the parabola in verte form is ( ). CHECK A graph of ( ) verifies that the parabola passes through the point at (, ). ( ) (, ). Write a quadratic equation that transforms the graph of ( ) so that it is: a. units up. ( ) 5 b. units down. ( ) c. units to the left. ( ) d. units to the right. e. narrower. f. wider. g. opening in the opposite direction. ( ). Eplain how ou can find an equation of a parabola using its verte and one other point on its graph. See margin.. PEN ENDED Write the equation of a parabola with a verte of (, ).. FIND THE ERRR Jenn and Ruben are writing 5 in verte form. Jenn = +5 = ( +)+5 = ( ) + Who is correct? Eplain our reasoning. Lesson 6-6 Analzing Graphs of Quadratic Functions 5 Ruben (, ) = +5 =( +)+5+ =( ) + 6 Write each quadratic function in verte form, if not alread in that form. Then identif the verte, ais of smmetr, and direction of opening See margin. 5. 5( ) In-Class Eample Practice/Appl Stud Notebook Power Point Write an equation for the parabola whose verte is at (, ) and passes through (, ). ( ) Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 6. write a summar in their own words of everthing ou can tell about the graph of a parabola when the function is written in verte form. include an other item(s) that the find helpful in mastering the skills in this lesson. FIND THE ERRR Students ma find that it will help them avoid errors of this tpe if the specificall write the addition and subtraction in a separate step before placing parentheses around the perfect square trinomial, as in 5. Answers. Substitute the -coordinate of the verte for h and the -coordinate of the verte for k in the equation a( h) k. Then substitute the -coordinate of the other point for and the -coordinate for into this equation and solve for a. Replace a with this value in the equation ou wrote with h and k. 5. (, ); ; up 6. ( ) 9, (, 9); ; up 7. ( ) 8; (, 8); ; down Lesson 6-6 Analzing Graphs of Quadratic Functions 5

49 About the Eercises rganization b bjective Analze Quadratic Functions: 5, 6, 7 0, 7, 8, 7, 5, 5 Write Quadratic Functions in Verte Form: 9 6, 6, 9 6, 8 50 dd/even Assignments Eercises 5 6 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 5 odd, 9 5 odd, 5 7 Average: 5 7 odd, 8 50, 5 7 Advanced: 6 6 even, 8 67 (optional: 68 7) All: Practice Quiz ( 0) Answers 8. GUIDED PRACTICE KEY Eercises Eamples , Application indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples , 7,, 8, , 9 Etra Practice See page Sample answer: the graphs have the same shape, but the graph of ( ) is unit to the left and 5 units below the graph of ( 5). Graph each function See margin. 8. ( ) 9. ( ) 0. 6 Write an equation for the parabola with the given verte that passes through the given point.. verte: (, 0). verte: (, 6). verte: (, ) point: (, ) point: ( 5, ) point: (, 5) ( ) ( ) 6 ( ). FUNTAINS The height of a fountain s water ft stream can be modeled b a quadratic function. Suppose the water from a jet reaches a maimum height of 8 feet at a distance foot awa from the jet. If the 8 ft water lands feet awa from the jet, find a quadratic function that models the height h(d) of the water at an given distance d feet from the jet. h(d) d d 6 Write each quadratic function in verte form, if not alread in that form. Then identif the verte, ais of smmetr, and direction of opening See margin. 5. ( ) (, 0); ; down 6. ( ) (, ); ; up (0, 6); 0; up 8. 8 (0, ); 0; down Graph each function See pp. A F. 7. ( ) 8. ( 5) 9. ( ) 0. ( ) ft 9. ( ) ( ) You can use a quadratic function to model the world population. Visit webquest to continue work on our WebQuest project. 6 Chapter 6 Quadratic Functions and Inequalities 7. Write one sentence that compares the graphs of 0.( ) and 0.( ). Sample answer: the graph of 0.( ) is narrower than the graph of 0.( ). 8. Compare the graphs of ( 5) and ( ). Write an equation for the parabola with the given verte that passes through the given point. 9. verte: (6, ) 9( 6) 0. verte: (, ) ( ) point: (5, 0) point: (, 6). verte: (, 0) ( ). verte: (5, ) ( 5) point: (6, 6) point: (, 8). verte: (0, 5) 5. verte: (, ) 5 ( ) point: (, 8) point: (, 8) ( ) ; (, ); ; down 0. ( ) 8; (, 8); ; up. ( ) ; (, ); ; down. ( ) 6; (, 6); ; up. ( ) 7; (, 7); ; up. ( 5) 5; (5, 5); 5; down ;, ; ; up 6. 0;, 0 ; ; up 6 Chapter 6 Quadratic Functions and Inequalities

50 Aerospace The KC5A has the nickname Vomit Comet. It starts its ascent at,000 feet. As it approaches maimum height, the engines are stopped, and the aircraft is allowed to free-fall at a determined angle. Zero gravit is achieved for 5 seconds as the plane reaches the top of its flight and begins its descent. Source: NASA 5. Angle A; the graph of the equation for angle A is higher than the other two since.7 is greater than.9 or Angle B; the verte of the equation for angle B is farther to the right than the other two since.57 is greater than.09 or.. Standardized Test Practice 5. Write an equation for a parabola whose verte is at the origin and passes through (, 8). 6. Write an equation for a parabola with verte at (, ) and -intercept 8. ( ) 7. AERSPACE NASA s KC5A aircraft flies in parabolic arcs to simulate the weightlessness eperienced b astronauts in space. The height h of the aircraft (in feet) t seconds after it begins its parabolic flight can be modeled b the equation h(t) 9.09(t.5),000. What is the maimum height of the aircraft during this maneuver and when does it occur?,000 feet;.5 s after the aircraft begins its parabolic flight DIVING For Eercises 8 50, use the following information. The distance of a diver above the water d(t) (in feet) t seconds after diving off a platform is modeled b the equation d(t) 6t 8t Find the time it will take for the diver to hit the water. about.6 s 9. Write an equation that models the diver s distance above the water if the platform were 0 feet higher. d(t) 6t 8t Find the time it would take for the diver to hit the water from this new height. about.0 s LAWN CARE For Eercises 5 and 5, use the following information. The path of water from a sprinkler can be modeled b a quadratic function. The three functions below model paths for three different angles of the water. Angle A: 0.8(.09).7 Angle B: 0.(.57).9 Angle C: 0.09(.).5 5. Which sprinkler angle will send water the highest? Eplain our reasoning. 5. Which sprinkler angle will send water the farthest? Eplain our reasoning CRITICAL THINKING Given a b c with a 0, derive the equation for the ais of smmetr b completing the square and rewriting the equation in the form a( h) k. See pp. A F. 5. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. A F. How can the graph be used to graph an quadratic function? Include the following in our answer: a description of the effects produced b changing a, h, and k in the equation a( h) k, and a comparison of the graph of and the graph of a( h) k using values of our own choosing for a, h, and k. 55. If f() 5 and f(n), then which of the following could be n? D A 5 B C D 56. The verte of the graph of ( 6) is located at which of the following points? B A (, ) B (6, ) C (6, ) D (, ) Lesson 6-6 Analzing Graphs of Quadratic Functions 7 NAME DATE PERID 6-6 Enrichment, p. 8 Stud Stud Guide and and Intervention Intervention, p. (shown) and p. 6-6 NAME DATE PERID 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 Eample Identif the verte, ais of smmetr, and direction of opening of each graph. a. ( ) 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. ( ) 7. ( 5) (, 6); ; up (, 7); ; up (5, ); 5; up. 7( ) 9 5. ( ) 6. 6( 6) 6 5 (, 9); ; down (, ); ; up ( 6, 6); 6; up 7. ( 9) 8. 8( ) 9. ( ) 5 (9, ); 9; up (, ); ; up (, ); ; down 5 0. ( 5). ( 7). 6( ) ( 5, ); 5; down (7, ); 7; up (, ); ; up. (.).7. 0.( 0.6) ( 0.8) 6.5 (.,.7);.; up (0.6, 0.); 0.6; ( 0.8, 6.5); 0.8; down up 6-6 NAME DATE PERID Practice (Average) Skills Practice, p. 5 and Practice, p. 6 (shown) 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( ).. 8 6( ) ; ( 0) ; ( ) ; (, ); ; down (0, ); 0; up (, ); ; down ( 5) 5; ( ) ;(, 0); ( ) ; ( 5, 5); 5; up ; up (, ); ; up ( ) ; ( ) 6; ( ) ; (, 0); ; down (, 6); ; down (, ); ; up Graph each function. 0. ( ) Write an equation for the parabola with the given verte that passes through the given point.. verte: (, ). verte: (, 0) 5. verte: (0, ) point: (, 5) point: (, 8) point: (5, 6) ( ) ( ) ( 0) 5 6. Write an equation for a parabola with verte at (, ) and -intercept 6. ( ) 7. Write an equation for a parabola with verte at (, ) and -intercept. ( ) 8. BASEBALL The height h of a baseball t seconds after being hit is given b h(t) 6t 80t. What is the maimum height that the baseball reaches, and when does this occur? 0 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 0 ft of a point on the arc and is its horizontal distance from the left-hand starting point of the arc. 5 ( ) 0 8 ft Reading to Learn Mathematics, p NAME 6 DATE PERID Reading to Learn Mathematics Analzing Graphs of Quadratic Equations Pre-Activit How can the graph of be used to graph an quadratic function? Read the introduction to Lesson 6-6 at the top of page in our tetbook. What does adding a positive number to do to the graph of? It moves the graph up. Reading the Lesson What does subtracting a positive number to before squaring do to the graph of? It moves the graph to the right.. 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 ELL Lesson 6-6 Lesson 6-6 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. i. ii. iii. iv. If possible, write each number as the difference of two squares. Look for patterns cannot cannot cannot cannot 6. 5 Helping You Remember. 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. Even numbers can be written as n, where n is one of the numbers 0,,,, and so on. dd numbers can be written n. Use these sio these problems Lesson 6-6 Analzing Graphs of Quadratic Functions 7

51 Assess pen-ended Assessment Writing Give students the equation of a specific parabola and ask them to put it in verte form, analze it, and sketch the graph. Instruct them to show all the steps in their procedures, with notes and eplanations for each step, similar to the notes in the right column of the Eamples. Getting Read for Lesson 6-7 PREREQUISITE SKILL Lesson 6-7 presents quadratic inequalities. Use Eercises 68 7 to determine our students familiarit with inequalities. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 6- through 6-6. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons 6-5 and 6-6) is available on p. 70 of the Chapter 6 Resource Masters. Answers 6. t t t 6. t t 65. n n 5n 66. Answers (Practice Quiz ) 8. ( ) ; (, ), ; up 9. ( 6) ; (6, 0), 6; down 0. ( ) 5; (, 5), ; up Maintain Your Skills Mied Review 67a. Sample answer using (99, 76,0) and (997, 99,8): 775 5,07,08 Getting Read for the Net Lesson P ractice Quiz 8 Chapter 6 Quadratic Functions and Inequalities Find the value of the discriminant for each quadratic equation. Then describe the number and tpe of roots for the equation. (Lesson 6-5) ; irrational 5; rational ; comple Solve each equation b completing the square. (Lesson 6-) { 5 } { i } Find each quotient. (Lesson 5-) See margin. 6. (t t ) (t ) 6. (t t ) (t ) 65. (n 8n 5n 05) (n 5) 66. ( ) ( ) 67. EDUCATIN The graph shows the number of U.S. students in stud-abroad programs. (Lesson -5) a. Write a prediction equation from the data given. b. Use our equation to predict the number of students in these programs in 005. Sample answer: 6,67 PREREQUISITE SKILL Determine whether the given value satisfies the inequalit. (To review inequalities, see Lesson -6.) 68. 0; 5 es 69. 0; no 70. 0; es ; 0 no Solve each equation b completing the square. (Lesson 6-). 7 0 { 7 }. 5 0 i Find the value of the discriminant for each quadratic equation. Then describe the number and tpe of roots for the equation. (Lesson 6-5). 5 0 ; comple ; rational Solve each equation b using the Quadratic Formula. (Lesson 6-5) i 7. Write an equation for a parabola with verte at (, 5) that passes through (, ). (Lesson 6-6) ( ) 5 Write each equation in verte form. Then identif the verte, ais of smmetr, and direction of opening. (Lesson 6-6) 8 0. See margin. More Americans stud abroad The number of U.S. college students in stud-abroad programs rose.% in the ear ending June 997 (latest available) to about % of students. Annual numbers: Source: Institute of International Education 76,0 8,0 89, 99,8 Lessons 6- through nline Lesson Plans USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to USA TDAY Snapshots Note: Includes an student getting credit at a U.S. school for stud abroad B Anne R. Care and Marc E. Mullins, USA TDAY 8 Chapter 6 Quadratic Functions and Inequalities

52 Graphing and Solving Quadratic Inequalities Lesson Notes Vocabular quadratic inequalit Stud Tip Look Back For review of graphing linear inequalities, see Lesson -7. TEACHING TIP Remind students that (0, 0) is a good point to use as a test point. Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable. Trampolining was first featured as an lmpic sport at the 000 lmpics in Sdne, Australia. The competitors performed two routines consisting of 0 different skills. Suppose the height h(t) in feet of a trampolinist above the ground during one bounce is modeled b the quadratic function h(t) 6t t.75. We can solve a quadratic inequalit to determine how long this performer is more than a certain distance above the ground. GRAPH QUADRATIC INEQUALITIES You can graph quadratic inequalities in two variables using the same techniques ou used to graph linear inequalities in two variables. Step Graph the related quadratic equation, a b c. Decide if the parabola should be solid or dashed. Step Step Eample Test a point (, ) inside the parabola. Check to see if this point is a solution of the inequalit. If (, ) is a solution, shade the region inside the parabola. If (, ) is not a solution, shade the region outside the parabola. Graph 6 7. Step can ou find the time a trampolinist spends above a certain height? Graph a Quadratic Inequalit Graph the related quadratic equation, 6 7. Since the inequalit smbol is, the parabola should be dashed. or Lesson 6-7 Graphing and Solving Quadratic Inequalities 9 (, ) is a solution. (, ) 6 7 or? a( ) b( ) c (, ) is not a solution. (continued on the net page) Focus 5-Minute Check Transparenc 6-7 Use as a quiz or review of Lesson 6-6. Mathematical Background notes are available for this lesson on p. 8D. Building on Prior Knowledge In Lesson 6-6, students analzed and graphed equations. In this lesson, students use the same techniques to graph and solve inequalities. can ou find the time a trampolinist spends above a certain height? Ask students: What is a trampoline and how does a trampolinist use it? Ask a student who is familiar with this sport to eplain it to those who ma not have seen it. Which wa does the parabola for the given quadratic equation open? downward Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 5 Practice, p. 5 Reading to Learn Mathematics, p. 5 Enrichment, p. 5 Assessment, p. 70 Graphing Calculator and Spreadsheet Masters, p. 7 School-to-Career Masters, p. Transparencies 5-Minute Check Transparenc 6-7 Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 9

53 Teach GRAPH QUADRATIC INEQUALITIES In-Class Eample Graph. Power Point Step Test a point inside the parabola, such as (, 0) ? ( ) 6( ) 7 0? ? Step So, (, 0) is not a solution of the inequalit. Shade the region outside the parabola. 6 7 (, 0) SLVE QUADRATIC INEQUALITIES To solve a quadratic inequalit in one variable, ou can use the graph of the related quadratic function. To solve a b c 0, graph a b c. Identif the values for which the graph lies below the -ais. a 0 a 0 Teaching Tip For In-Class Eample, encourage students to write the calculations as in Step of Eample when testing a point inside the parabola. Emphasize that the must write a question mark over each inequalit sign after substituting the coordinates of the point for the variables in the inequalit. For, include the -intercepts in the solution. To solve a b c 0, graph a b c. Identif the values for which the graph lies above the -ais. { } a 0 a 0 SLVE QUADRATIC INEQUALITIES In-Class Eample Solve 0 b graphing. Power Point Stud Tip Solving Quadratic Inequalities b Graphing A precise graph of the related quadratic function is not necessar since the zeros of the function were found algebraicall. { or } For, include the -intercepts in the solution. Eample Solve a b c 0 Solve 0 b graphing. The solution consists of the values for which the graph of the related quadratic function lies above the -ais. Begin b finding the roots of the related equation. 0 Related equation ( )( ) 0 Factor. 0 or 0 Zero Product Propert Solve each equation. Sketch the graph of a parabola that has -intercepts at and. The graph should open up since a 0. { or } Teaching Tip Suggest that students tr solving first b factoring, but if that does not quickl ield a solution, then use the Quadratic Formula. 0 Chapter 6 Quadratic Functions and Inequalities The graph lies above the -ais to the left of and to the right of. Therefore, the solution set is { or }. 0 Chapter 6 Quadratic Functions and Inequalities

54 Eample Solve a b c 0 Solve 0 7 b graphing. This inequalit can be rewritten as 7 0. The solution consists of the values for which the graph of the related quadratic function lies on and below the -ais. Begin b finding the roots of the related equation. 7 0 Related equation b b ac a Use the Quadratic Formula. ( 7) ( 7) ( ) () () Replace a with, b with 7, and c with. 7 6 or 7 6 Simplif and write as two equations Simplif. In-Class Eamples Power Point Solve 0 6 b graphing. 6 Sketch the graph of a parabola that has -intercepts of.7 and 0.. The graph should open up since a 0. The graph lies on and below the -ais at 0. and.7 and between these two values. Therefore, the solution set of the inequalit is approimatel { 0..7}. CHECK Test one value of less than 0., one between 0. and.7, and one greater than.7 in the original inequalit. Test. Test 0. Test ? ( ) 7( ) 0? (0) 7(0) 0? () 7() {.6 0.6} Teaching Tip When discussing Eample, be aware that some students ma not be familiar with all of the aspects of the game of football. Ask students who are familiar with the terms in this eample and the margin note to eplain them to the class. Football A long hang time allows the kicking team time to provide good coverage on a punt return. The suggested hang time for high school and college punters is.5.6 seconds. Source: Real-world problems that involve vertical motion can often be solved b using a quadratic inequalit. Eample Write an Inequalit FTBALL The height of a punted football can be modeled b the function H().9 0, where the height H() is given in meters and the time is in seconds. At what time in its flight is the ball within 5 meters of the ground? The function H() describes the height of the football. Therefore, ou want to find the values of for which H() 5. H() 5 riginal inequalit SPRTS The height of a ball above the ground after it is thrown upwards at 0 feet per second can be modeled b the function h() 0 6, where the height h() is given in feet and the time is in seconds. At what time in its flight is the ball within 5 feet of the ground? The ball is within 5 feet of the ground for the first 0.6 second of its flight and again after.0 seconds until the ball hits the ground at.5 seconds H() Subtract 5 from each side. Graph the related function.9 0 using a graphing calculator. The zeros of the function are about 0. and.87, and the graph lies below the -ais when 0. or.87. Thus, the ball is within 5 meters of the ground for the first 0. second of its flight and again after.87 seconds until the ball hits the ground at. seconds. [.5, 5] scl: b [ 5, 0] scl: 5 Lesson 6-7 Graphing and Solving Quadratic Inequalities Differentiated Instruction Intrapersonal Have students think about how the graph of a quadratic inequalit helps them understand what the inequalit means. Ask them to eplore which is more meaningful to them (and therefore easier for them to grasp), the quadratic inequalit itself or the graph of the inequalit. Ask them to give an eplanation of their choice. Lesson 6-7 Graphing and Solving Quadratic Inequalities

55 In-Class Eample 5 Practice/Appl Stud Notebook Power Point Teaching Tip An alternative wa to solve the inequalit in Eample 5 is to first subtract 6 from both sides of the inequalit, obtaining 6 0. After factoring the left side as ( )( ), point out that for the product ( )( ) to be greater than 0, either both binomials must be positive or both must be negative. This fact can be used to test the three intervals of the number line. Solve algebraicall. { } Have students complete the definitions/eamples for the remaining terms on their Vocabular Builder worksheets for Chapter 6. write the basic steps shown in Eample for solving a quadratic inequalit b graphing. include an other item(s) that the find helpful in mastering the skills in this lesson. Stud Tip Solving Quadratic Inequalities Algebraicall As with linear inequalities, the solution set of a quadratic inequalit can be all real numbers or the empt set,. The solution is all real numbers when all three test points satisf the inequalit. It is the empt set when none of the tests points satisf the inequalit. Concept Check You can also solve quadratic inequalities algebraicall. Eample 5 Solve a Quadratic Inequalit Solve 6 algebraicall. First solve the related quadratic equation 6. 6 Related quadratic equation 6 0 ( )( ) 0 Factor. Subtract 6 from each side. 0 or 0 Zero Product Propert Solve each equation. Plot and on a number line. Use circles since these values are not solutions of the original inequalit. Notice that the number line is now separated into three intervals Test a value in each interval to see if it satisfies the original inequalit. Test. Test 0. Test ( ) ( )? 6 0 0? 6? The solution set is { or }. This is shown on the number line below Determine which inequalit, ( ) or ( ), describes the graph at the right. ( ). PEN ENDED List three points ou might test to find the solution of ( )( 5) 0. Sample answer: one number less than, one number between and 5, and one number greater than 5. Eamine the graph of 5 at the right. a., 5 b. or 5 a. What are the solutions of 0 5? b. What are the solutions of 5 0? c. What are the solutions of 5 0? 5 8 ( ) 6 5 Chapter 6 Quadratic Functions and Inequalities Chapter 6 Quadratic Functions and Inequalities

56 Guided Practice GUIDED PRACTICE KEY Eercises Eamples 7 8, 9,, 5 Application indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples 5 6 9, 0,, 5 8 Etra Practice See page 8. Graph each inequalit. 7. See margin Use the graph of the related function of 6 5 0, which is shown at the right, to write the solutions of the inequalit. or 5 Solve each inequalit algebraicall { 7} 0. 0 { or } { }. BASEBALL A baseball plaer hits a high pop-up with an initial upward velocit of 0 meters per second,. meters above the ground. The height h(t) of the ball in meters t seconds after being hit is modeled b h(t).9t 0t.. How long does a plaer on the opposing team have to catch the ball if he catches it.7 meters above the ground? about 6. s Graph each inequalit. 5. See pp. A F. 0 m/s Use the graph of its related function to write the solutions of each inequalit m About the Eercises rganization b bjective Graph Quadratic Inequalities: 5 Solve Quadratic Inequalities: 6 8 dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercises 5 58 require a graphing calculator. Assignment Guide Basic: 5 odd, 5, 9 5, 59 7 Average: 5 5 odd, 9 5, 59 7 (optional: 5 58) Advanced: even, or or Lesson 6-7 Graphing and Solving Quadratic Inequalities Answers Lesson 6-7 Graphing and Solving Quadratic Inequalities

57 Stud Stud Guide and and Intervention Intervention, p. 9 (shown) and p NAME DATE PERID Graphing and Solving Quadratic Inequalities Graph Quadratic Inequalities To graph a quadratic inequalit in two variables, use the following steps:. Graph the related quadratic equation, a b c. Use a dashed line for or ; use a solid line for or.. Test a point inside the parabola. If it satisfies the inequalit, shade the region inside the parabola; otherwise, shade the region outside the parabola. Eample Graph the inequalit 6 7. First graph the equation 6 7. B completing the square, ou get the verte form of the equation ( ), so the verte is (, ). Make a table of values around, and graph. Since the inequalit includes, use a dashed line. Test the point (, 0), which is inside the parabola. Since ( ) 6( ) 7, and 0, (, 0) satisfies the inequalit. Therefore, shade the region inside the parabola. Eercises Graph each inequalit NAME 9 DATE PERID Practice (Average) Skills Practice, p. 5 and Practice, p. 5 (shown) Graph each inequalit Use the graph of its related function to write the solutions of each inequalit Graphing and Solving Quadratic Inequalities or 8 7 Solve each inequalit algebraicall { or 5} { 8} all reals { or 7} { 5 } all reals { } 0 or or 5 9. FENCING Vanessa has 80 feet of fencing that she intends to use to build a rectangular pla area for her dog. She wants the pla area to enclose at least 800 square feet. What are the possible widths of the pla area? 0 ft to 60 ft 0. BUSINESS A biccle maker sold 00 biccles last ear at a profit of $00 each. The maker wants to increase the profit margin this ear, but predicts that each $0 increase in profit will reduce the number of biccles sold b 0. How man $0 increases in profit can the maker add in and epect to make a total profit of at least $00,000? from 5 to 0 Reading to Learn Mathematics, p NAME 5 DATE PERID Reading to Learn Mathematics Graphing and Solving Quadratic Inequalities Pre-Activit How can ou find the time a trampolinist spends above a certain height? Read the introduction to Lesson 6-7 at the top of page 9 in our tetbook. How far above the ground is the trampoline surface?.75 feet Using the quadratic function given in the introduction, write a quadratic inequalit that describes the times at which the trampolinist is more than 0 feet above the ground. 6t t.75 0 Reading the Lesson. Answer the following questions about how ou would graph the inequalit 6. a. What is the related quadratic equation? 6 b. Should the parabola be solid or dashed? How do ou know? solid; The inequalit smbol is. c. The point (0, ) is inside the parabola. To use this as a test point, substitute 0 for and for in the quadratic inequalit. d. Is the statement true or false? true e. Should the region inside or outside the parabola be shaded? inside. The graph of is shown at the right. Match each of the following related inequalities with its solution set. a. 0 ii i. { 0 or } b. 0 iii ii. { 0 } c. 0 iv iii. { 0 or } d. 0 i iv. { 0 } ELL (, ) (0, 0) (, 0) Lesson 6-7 More About... Landscape Architect Landscape architects design outdoor spaces so that the are not onl functional, but beautiful and compatible with the natural environment. nline Research For information about a career as a landscape architect, visit: careers 6. P(n) n[5.5(60 n)] 55 or.5n 05n 55 Chapter 6 Quadratic Functions and Inequalities NAME DATE PERID 6-7 Enrichment, p. 5 Graphing Absolute Value Inequalities You can solve absolute value inequalities b graphing in much the same manner ou graphed quadratic inequalities. Graph the related absolute function for each inequalit b using a graphing calculator. For and, identif the -values, if an, for which the graph lies below the -ais. For and, identif the values, if an, for which the graph lies above the -ais. Solve each inequalit algebraicall { or 6}. 8 0 { 7 }. 5 { 5}. { 6 or }. 0 { } { 7 or } all reals { 7} all reals Solve ( )( )( ) 0. { or }. LANDSCAPING Kinu wants to plant a garden and surround it with decorative stones. She has enough stones to enclose a rectangular garden with a perimeter of 68 feet, but she wants the garden to cover no more than 0 square feet. What could the width of her garden be? 0 to 0 ft or to ft. BUSINESS A mall owner has determined that the relationship between monthl rent charged for store space r (in dollars per square foot) and monthl profit P(r) (in thousands of dollars) can be approimated b the function P(r) 8.r 6.9r 8.. Solve each quadratic equation or inequalit. Eplain what each answer tells about the relationship between monthl rent and profit for this mall. a d. See margin. a. 8.r 6.9r 8. 0 b. 8.r 6.9r 8. 0 c. 8.r 6.9r 8. 0 d. 8.r 6.9r GEMETRY A rectangle is 6 centimeters longer than it is wide. Find the possible dimensions if the area of the rectangle is more than 6 square centimeters. The width should be greater than cm and the length shoud be greater than 8 cm. FUND-RAISING For Eercises 6 8, use the following information. The girls softball team is sponsoring a fund-raising trip to see a professional baseball game. The charter a 60-passenger bus for $55. In order to make a profit, the will charge $5 per person if all seats on the bus are sold, but for each empt seat, the will increase the price b $.50 per person. 6. Write a quadratic function giving the softball team s profit P(n) from this fund-raiser as a function of the number of passengers n. 7. What is the minimum number of passengers needed in order for the softball team not to lose mone? 6 8. What is the maimum profit the team can make with this fund-raiser, and how man passengers will it take to achieve this maimum? $.50; 5 passengers 9. CRITICAL THINKING Graph the intersection of the graphs of and. See margin. 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can ou find the time a trampolinist spends above a certain height? Include the following in our answer: a quadratic inequalit that describes the time the performer spends more than 0 feet above the ground, and two approaches to solving this quadratic inequalit. Helping You Remember. A quadratic inequalit in two variables ma have the form a b c, a b c, a b c, or a b c. Describe a wa to remember which region to shade b looking at the inequalit smbol and without using a test point. Sample answer: If the smbol is or, shade the region above the parabola. If the smbol is or, shade the region below the parabola. For each inequalit, make a sketch of the related graph and find the solutions rounded to the nearest hundredth or 6 6 Chapter 6 Quadratic Functions and Inequalities

58 Standardized Test Practice Etending the Lesson 5. Which is a reasonable estimate of the area under the curve from 0 to 8? C A B C D 9 square units 58 square units 6 square units square units 5. If ( )( ) is positive, then A A C or. B or.. D. SLVE ABSLUTE VALUE INEQUALITIES BY GRAPHING Similar to quadratic inequalities, ou can solve absolute value inequalities b graphing. Graph the related absolute value function for each inequalit using a graphing calculator. For and, identif the values, if an, for which the graph lies below the -ais. For and, identif the values, if an, for which the graph lies above the -ais { all reals, } { 7 7} { 9 or } {.5 or.5} no real solutions {. 0.} Assess pen-ended Assessment Speaking Have students eplain how to test points in the coordinate plane in order to determine which region represents the solution to a quadratic inequalit. Also ask them to eplain how to analze the graph of a quadratic equation in order to determine the solution set for a quadratic inequalit. Assessment ptions Quiz (Lesson 6-7) is available on p. 70 of the Chapter 6 Resource Masters. Maintain Your Skills Mied Review 59. ( ) 8; (, 8), ; up 60. ( ) ; (, 0), ; down 6. ( 6) ; ( 6, 0), 6; up 6. 5 i Answers Write each equation in verte form. Then identif the verte, ais of smmetr, and direction of opening. (Lesson 6-6) Solve each equation using the method of our choice. Find eact solutions. (Lesson 6-5) , 8 6 Simplif. (Lesson 5-) 65. a b a b ab a 7b 65. (a b ab 5a 6b ) (a b 7ab b 7a) 66. ( ) (7 9 ) ( ) 68. (5a )( a) 5a a Find each product, if possible. (Lesson -) [ ] [ 5 6] 7. LAW ENFRCEMENT Thirt-four states classif drivers having at least a 0. blood alcohol content (BAC) as intoicated. An infrared device measures a person s BAC through an analsis of his or her breath. A certain detector measures BAC to within If a person s actual blood alcohol content is 0.08, write and solve an absolute value equation to describe the range of BACs that might register on this device. (Lesson -6) ; Lesson 6-7 Graphing and Solving Quadratic Inequalities 5 a. 0.98,.8; The owner will break even if he charges $0.98 or $.8 per square foot. b r.8; The owner will make a profit if the rent is between $0.98 and $.8. c.. r.5; If rent is set between $. and $.5 per sq ft, the profit will be greater than $0,000. d. r. or r.5; If rent is set between $0 and $. or above $.5 per sq ft, the profit will be less than $0,000. Answers Answers should include the following. 6t t.75 0 ne method of solving this inequalit is to graph the related quadratic function h(t ) 6t t The interval(s) at which the graph is above the -ais represents the times when the trampolinist is above 0 feet. A second method of solving this inequalit would be find the roots of the related quadratic equation 6t t and then test points in the three intervals determined b these roots to see if the satisf the inequalit. The interval(s) at which the inequalit is satisfied represent the times when the trampolinist is above 0 feet. Lesson 6-7 Graphing and Solving Quadratic Inequalities 5

59 Stud Guide and Review Vocabular and Concept Check This alphabetical list of vocabular terms in Chapter 6 includes a page reference where each term was introduced. Assessment A vocabular test/review for Chapter 6 is available on p. 68 of the Chapter 6 Resource Masters. Lesson-b-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabular PuzzleMaker ELL The Vocabular PuzzleMaker software improves students mathematics vocabular using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts (5 questions) Round Skills ( questions) Round Problem Solving ( questions) Stud Guide and Review Vocabular and Concept Check ais of smmetr (p. 87) completing the square (p. 07) constant term (p. 86) discriminant (p. 6) linear term (p. 86) maimum value (p. 88) minimum value (p. 88) Choose the letter of the term that best matches each phrase.. the graph of an quadratic function f. process used to create a perfect square trinomial b. the line passing through the verte of a parabola and dividing the parabola into two mirror images a. a function described b an equation of the form f() a b c, where a 0 h 5. the solutions of an equation i 6. a( h) k j 7. in the Quadratic Formula, the epression under the radical sign, b ac c b b ac 8. a g 6- See pages Eample 6 Chapter 6 Quadratic Functions and Inequalities For more information about Foldables, see Teaching Mathematics with Foldables. TM parabola (p. 86) quadratic equation (p. 9) Quadratic Formula (p. ) quadratic function (p. 86) quadratic inequalit (p. 9) quadratic term (p. 86) roots (p. 9) Graphing Quadratic Functions Concept Summar The graph of a b c, a 0, opens up, and the function has a minimum value when a 0, and opens down, and the function has a maimum value when a 0. Find the maimum or minimum value of f(). Since a 0, the graph opens down and the function has a maimum value. The maimum value of the function is the -coordinate of the verte. The -coordinate of the verte is or. Find ( ) the -coordinate b evaluating the function for. f() riginal function f() () () or 8 Replace with. Therefore, the maimum value of the function is 8. Square Root Propert (p. 06) verte (p. 87) verte form (p. ) Zero Product Propert (p. 0) zeros (p. 9) -intercept: c a. ais of smmetr b. completing the square c. discriminant d. constant term e. linear term f. parabola g. Quadratic Formula h. quadratic function i. roots j. verte form f() verte ais of smmetr: b a a b c 8 f() 8 Discuss with students how the might recognize a ke concept that needs to be included in the Foldable. Ask them to include a transition sentence or two in their notes that relates one topic to the net. Suggest that the use this discussion as the review their Foldable to add, delete, or reorganize material in order to make it more useful to them. Encourage students to refer to their Foldables while completing the Stud Guide and Review and to use them in preparing for the Chapter Test. 6 Chapter 6 Quadratic Functions and Inequalities

60 Chapter 6 Stud Guide and Review Stud Guide and Review 6- See pages Eercises 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. (See Eample on pages 87 and 88.) 9. f() f() 5. f() 8 7. f() 9. f(). f() See margin. Determine whether each function has a maimum or a minimum value. Then find the maimum or minimum value of each function. (See Eample on pages 88 and 89.) 5. f() 5 6. f() 7. f() 7 min.; 8 9 ma.; 5 ma.; 7 6 Solving Quadratic Equations b Graphing Concept Summar The solutions, or roots, of a quadratic equation are the zeros of the related quadratic function. You can find the zeros of a quadratic function b finding the -intercepts of its graph. A quadratic equation can have one real solution, two real solutions, or no real solution. ne Real Solution f() Two Real Solutions f() No Real Solution f() Answers 0c. f() f () (, 6) a. 7; ; b. f() c. f() 8 Eample Solve 5 0 b graphing. 5 The equation of the ais of smmetr is or 5 (). f() The zeros of the related function are and. Therefore, the solutions of the equation are and. Eercises Solve each equation b graphing. If eact roots cannot be found, state the consecutive integers between which the roots are located. (See Eamples on pages 9 and 95.) 9., , , ( ) 5 0, 8. between and ; between 8 and 7. between and ; between and 0 Chapter 6 Stud Guide and Review 7 f() 5 f() 8 f () 8 7 (, 9) a. 9; ; b. f() c. f() (, 9) 8 f () 9 9a. 0; ; 9c. 9b. f() f() 6 f() 6 0 (, ) a. 5; ; 0b. f() (continued on the net page) Chapter 6 Stud Guide and Review 7

61 Stud Guide and Review Chapter 6 Stud Guide and Review Answers a. ; ; b. f() c. f () (, ) f() 6- See pages Eample Solving Quadratic Equations b Factoring Concept Summar Zero Product Propert: For an real numbers a and b, if ab 0, then either a 0, b 0, or both a and b 0. Solve b factoring riginal equation ( )( 5) 0 Factor the trinomial. 0 or 5 0 Zero Product Propert 5 The solution set is { 5, }. Eercises Solve each equation b factoring. (See Eamples on pages 0 and 0.) { } {, } 7. c 8c ,. {, 8} 7. {, } Write a quadratic equation with the given root(s). Write the equation in the form a b c, where a, b, and c are integers. (See Eample on page 0.) 0., 5. 0, 7., 7 0 a. 6; ; b. f() See pages Completing the Square Concept Summar To complete the square for an quadratic epression b: Step Find one half of b, the coefficient of. Step Square the result in Step. Step Add the result of Step to b. b b b c f (). ; Eample Solve b completing the square Notice that is not a perfect square. 0 9 Rewrite so the left side is of the form b Since 0 5, add 5 to each side. ( 5) 6 Write the left side as a perfect square b factoring. 5 8 Square Root Propert 5 8 or 5 8 Rewrite as two equations. The solution set is {, }. f () 9 6 (, ) 6., i 7 Eercises Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. (See Eample on page 07.). c. c 5. 7 c 9 ; ; ( 7) Solve each equation b completing the square. (See Eamples 6 on pages 08 and 09.) n n Chapter 6 Quadratic Functions and Inequalities 8 Chapter 6 Quadratic Functions and Inequalities

62 Chapter 6 Stud Guide and Review Stud Guide and Review 6-5 See pages 9. The Quadratic Formula and the Discriminant Concept Summar b b ac Quadratic Formula: a where a 0 Solve b using the Quadratic Formula. b b ac a Quadratic Formula ( 5) ( 5) ( 66) () () Replace a with, b with 5, and c with Simplif. 5 7 or 5 7 Write as two equations. 6 The solution set is {, 6}. Eercises Complete parts a c 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. (See Eamples on pages 6.) 9. See margin Answers 9a. 9b. comple 9c. i 6 0a. 0 0b. irrational 0c. 6 a. 7 b. irrational 7 c See pages 8. Eample Analzing Graphs of Quadratic Functions Concept Summar As the values of h and k change, the graph of ( h) k is the graph of translated h units left if h is negative or h units right if h is positive. k units up if k is positive or k units down if k is negative. Consider the equation a( h) k. If a 0, the graph opens up; if a 0 the graph opens down. If a, the graph is narrower than the graph of. If a, the graph is wider than the graph of. Write the quadratic function in verte form. Then identif the verte, ais of smmetr, and direction of opening. riginal equation ( ) Group a b and factor, dividing b a. ( 9) (9) ( 7) 5 Complete the square b adding. Balance this with a subtraction of (9). Write 7 as a perfect square. So, a, h 7, and k 5. The verte is at ( 7, 5), and the ais of smmetr is 7. Since a is positive, the graph opens up. Chapter 6 Stud Guide and Review 9 Chapter 6 Stud Guide and Review 9

63 Stud Guide and Review Etra Practice, see pages Mied Problem Solving, see page 867. Answers. (, ); ; down ;, ; 7 ; up. ( ) 8; (, 8); ; down 5. ( ) See pages 9 5. Eample Eercises Write each equation in verte form, if not alread in that form. Then identif the verte, ais of smmetr, and direction of opening. (See Eamples and on pages and.). See margin.. 6( ) Graph each function. (See Eamples on pages and.) 5 7. See margin. 5. ( ) Write an equation for the parabola with the given verte that passes through the given point. (See Eample on page 5.) 8. verte: (, ) 9. verte: (, ) 50. verte: (, 5) point: (, ) point: ( 6, ) point: (0, ) ( ) ( ) ( ) 5 Graphing and Solving Quadratic Inequalities Concept Summar Graph quadratic inequalities in two variables as follows. Step Step Graph the related quadratic equation, a b c. Decide if the parabola should be solid or dashed. Test a point (, ) inside the parabola. Check to see if this point is a solution of the inequalit. Step If (, ) is a solution, shade the region inside the parabola. If (, ) is not a solution, shade the region outside the parabola. To solve a quadratic inequalit in one variable, graph the related quadratic function. Identif the values for which the graph lies below the -ais for and. Identif the values for which the graph lies above the -ais for and. Solve 0 0 b graphing. Find the roots of the related equation. 0 0 Related equation 0 ( 5)( ) Factor. 5 0 or 0 Zero Product Propert 5 Solve each equation. Sketch the graph of the parabola that has -intercepts at 5 and. The graph should open up since a 0. The graph lies below the -ais between 5 and. Therefore, the solution set is { 5 } Eercises Graph each inequalit See (See Eample on pages 9 and 0.) margin Solve each inequalit. (See Eamples,, and 5 on pages 0.) See pp. A F Chapter 6 Quadratic Functions and Inequalities Chapter 6 Quadratic Functions and Inequalities

64 Practice Test Practice Test Vocabular and Concepts Choose the word or term that best completes each statement.. The -coordinate of the verte of the graph of a b c is the (maimum, minimum) value obtained b the function when a is positive.. (The Square Root Propert, Completing the square) can be used to solve an quadratic equation. Skills and Applications 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. 5. See pp. A F.. f() 5. f() 8 5. f() 7 Determine whether each function has a maimum or a minimum value. Then find the maimum or minimum value of each function. 6. f() 6 9 min.; 0 7. f() min.; 6 8. f() ma.; 9. Write a quadratic equation with roots and 5. Write the equation in the form a b c 0, where a, b, and c are integers. 0 0 Solve each equation using the method of our choice. Find eact solutions See margin BALLNING At a hot-air balloon festival, ou throw a weighted marker straight down from an altitude of 50 feet toward a bull s ee below. The initial velocit of the marker when it leaves our hand is 8 feet per second. Find how long it will take the marker to hit the target b solving the equation 6t 8t about.7 s Write each equation in verte form, if not alread in that form. Then identif the verte, ais of smmetr, and direction of opening. 0. See pp. A F. 0. ( ) Graph each inequalit. 5. See pp. A F Solve each inequalit. 6. { 7 5} 7 8. See pp. A F. 6. ( 5)( 7) PETS A rectangular turtle pen is 6 feet long b feet wide. The pen is enlarged b increasing the length and width b an equal amount in order to double its area. What are the dimensions of the new pen? 8 ft b 6 ft 0. STANDARDIZED TEST PRACTICE Which of the following is the sum of both solutions of the equation 8 8 0? B A 6 B 8 C D Portfolio Suggestion Chapter 6 Practice Test Introduction In this chapter quadratic equations have been graphed and solved using man different methods, often following a process that involved numerous steps. Ask Students Select an item from this chapter that shows our best work, including a graph, and place it in our portfolio. Eplain wh ou believe it to be our best work and how ou came to choose this particular piece of work. Assessment ptions Vocabular Test A vocabular test/review for Chapter 6 can be found on p. 68 of the Chapter 6 Resource Masters. Chapter Tests There are si Chapter 6 Tests and an pen- Ended Assessment task available in the Chapter 6 Resource Masters. Chapter 6 Tests Form Tpe Level Pages MC basic A MC average B MC average C FR average 6 6 D FR average 6 6 FR advanced pen-ended Assessment Performance tasks for Chapter 6 can be found on p. 67 of the Chapter 6 Resource Masters. A sample scoring rubric for these tasks appears on p. A8. TestCheck and Worksheet Builder This networkable software has three modules for assessment. Worksheet Builder to make worksheets and tests. Student Module to take tests on-screen. Management Sstem to keep student records. Answers MC = multiple-choice questions FR = free-response questions , 6., 7.,., 5 9., 5., , , Chapter 6 Practice Test

65 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequentl given standardized tests. A practice answer sheet for these two pages can be found on p. A of the Chapter 6 Resource Masters. Standardized Test Practice NAME DATE PERID Standardized Test Practice 6 Student Recording Sheet (Use with pages Sheet, ofp. the Student AEdition.) Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 7 A B C D 9 A B C D A B C D 5 A B C D 8 A B C D 0 A B C D A B C D 6 A B C D Part Short Response/Grid In Solve the problem and write our answer in the blank. For Questions 0, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol Part Multiple Choice Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. In a class of 0 students, half are girls and ride the bus to school. If of the girls do not ride the bus to school, how man bos in this class ride the bus to school? C A B C D 5. In the figure below, the measures of m n p? D A 90 B 80 C 70 D 60 m 6 6. If 0, then 6 6? B A B C 8 D 6 7. If and p are both greater than zero and p p 0, then what is the value of p in terms of? D A B C D 8. For all positive integers n, n n. Which of the following equals? C A B 8 C 6 D / / / / / / Part Quantitative Comparison / / / / / / / / Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 5 A B C D 7 A B C D A B C D A B C D 6 A B C D 8 A B C D Answers p. f the points (, ), (, ), (, ), (, ), and (, ), which three lie on the same side of the line 0? C A (, ), (, ), (, ) B (, ), (, ), (, ) C (, ), (, ), (, ) n 9. Which number is the sum of both solutions of the equation 8 0? C A 6 B C D 6 0. ne of the roots of the polnomial 6 k 0 0 is 5. What is the value of k? C D (, ), (, ), (, ) A B Additional Practice See pp. 7 7 in the Chapter 6 Resource Masters for additional standardized test practice.. If k is an integer, then which of the following must also be integers? B I. 5k 5 II. 5 k 5k k 5 k 5 k A I onl B II onl C I and II D II and III 5. Which of the following is a factor of 7 8? D A B C D 8 C D 7 Test-Taking Tip Questions 8,,, 6,, and 7 Be sure to use the information that describes the variables in an standardized test item. For eample, if an item sas that 0, check to be sure that our solution for is not a negative number. Chapter 6 Quadratic Functions and Inequalities Log n for Test Practice The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit or TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra End-of-Course tests can be found on this CD-RM. Chapter 6 Quadratic Functions and Inequalities

66 Aligned and verified b Part Short Response/Grid In Part Quantitative Comparison Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. If n is a three-digit number that can be epressed as the product of three consecutive even integers, what is one possible value of n? 9, 80, or 960. If and are different positive integers and 6, what is one possible value of 5? 0,, 6, or 8. If a circle of radius inches has its radius decreased b 6 inches, b what percent is its area decreased? 75%. What is the least positive integer k for which k is the cube of an integer? 8 5. If AB BC in the figure, what is the -coordinate of point B? 7 C(, ) B(, ) A(, ) Compare the quantit in Column A and the quantit in Column B. Then determine whether: A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. Column A. s 0 s increased b 00% of s the length of side A C the perimeter of a rectangle with area 8 units Column B. In ABC, side A B has length 8, and side B C has length.. s 0 the perimeter of a rectangle with area 0 units C D D 6. In the figure, if is the center of the circle, what is the value of? C 0 7. Let a b be defined as the sum of all integers greater than a and less than b. For eample, or 5. What is the value of (75 90) (76 89)? If and 6, what is the value of? 7 9. B what amount does the sum of the roots eceed the product of the roots of the equation ( 7)( ) 0? 5 0. If 6 and 9, what is the greatest possible value of ( )? 8 5. t 5 9 t p q p q the measure of the measure of side side A B D A Chapter 6 Standardized Test Practice Chapter 6 Standardized Test Practice