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(1)Quadratic and Exponential Functions Chapter Overview and Pacing PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. Graphing Quadratic Functions (pp. 524–532) • Graph quadratic functions. • Find the equation of the axis of symmetry and the coordinates of the vertex of a parabola. Follow-Up: Use a graphing calculator to explore families of quadratic graphs.. 2. Solving Quadratic Equations by Graphing (pp. 533–538) • Solve quadratic equations by graphing. • Estimate solutions of quadratic equations by graphing.. 2. 2. 1. 0.5. Solving Quadratic Equations by Completing the Square (pp. 539-545) • Solve quadratic equations by finding the square root. • Solve quadratic equations by completing the square. Follow-Up: Use a graphing calculator to graph quadratic functions in vertex form.. 2. 2 (with 10-3 Follow-Up). 1. 1 (with 10-3 Follow-Up). Solving Quadratic Equations by Using the Quadratic Formula (pp. 546–553) • Solve quadratic equations by using the Quadratic Formula. • Use the discriminant to determine the number of solutions for a quadratic equation. Follow-Up: Use a graphing calculator to solve quadratic-linear systems.. 2. 2 1.5 1 (with 10-4 (with 10-4 (with 10-4 Follow-Up) Follow-Up) Follow-Up). Exponential Functions (pp. 554–560) • Graph exponential functions. • Identify data that displays exponential behavior.. 2. 2. 1. 1. Growth and Decay (pp. 561–565) • Solve problems involving exponential growth. • Solve problems involving exponential decay.. 2. 2. 1. 1. Geometric Sequences (pp. 567–573) • Recognize and extend geometric sequences. • Find geometric means. Follow-Up: Use a table to investigate rates of change.. 2. 2 (with 10-7 Follow-Up). 1. 1 (with 10-7 Follow-Up). Study Guide and Practice Test (pp. 574–579) Standardized Test Practice (pp. 580–581). 1. 1. 0.5. 0.5. Chapter Assessment. 1. 1. 0.5. 0.5. 16. 16. 9. 8. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 522A Chapter 10 Quadratic and Exponential Functions. 2 1.5 1.5 (with 10-1 (with 10-1 (with 10-1 Follow-Up) Follow-Up) Follow-Up).

(2) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 579–580. 581–582. 583. 584. 585–586. 587–588. 589. 590. 591–592. 593–594. 595. 596. 597–598. 599–600. 601. 602. 603–604. 605–606. 607. 608. 609–610. 611–612. 613. 614. 636. 615–616. 617–618. 619. 620. 636. 635. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 10 RESOURCE MASTERS. 9–12, 47–48. 10-1. 10-1. 76. 10-2. 10-2. 77. 10-3. 10-3. 29. (Follow-Up: graphing calculator). SC 20. 78. 10-4. 10-4. 30, 31. graphing calculator (Follow-Up: graphing calculator). GCS 42. 79. 10-5. 10-5. SM 77–80. 80. 10-6. 10-6. GCS 41. 81. 10-7. 10-7. 621–634, 638–640. 28. Materials. 75. SC 19. 635, 637. See pages T12–T13.. grid paper, graphing calculator (Follow-Up: graphing calculator) graphing calculator. grid paper, piece of string, graphing calculator. (Follow-Up: grid paper). 82. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 10 Quadratic and Exponential Functions 522B.

(3) Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 4, students graphed ordered pairs on a coordinate plane. They also analyzed arithmetic sequences. Students factored perfect square trinomials in Chapter 9.. Graphing Quadratic Functions The standard form of a quadratic function is y ax 2 bx c, where a 0. If a 0, there would be no x2 term, and therefore the equation would be linear. The graph of a quadratic function is a symmetrical curve called a parabola. If a is positive, the graph opens upward and the vertex is the minimum of the function. If a is negative, the graph opens downward and the vertex is the maximum of the function. When a graph of a parabola is folded so that the two sides exactly match, the fold line is called the axis of symmetry. The vertex is the only point of a parabola that lies on its axis of symmetry. The equation for the b 2a. b 2a. axis of symmetry is x . The value is also the x-coordinate of the vertex. Substitute this value into the equation to find the y-coordinate of the vertex.. This Chapter This chapter introduces students to quadratic functions by having them graph the functions and determine the axis of symmetry and vertex. Students solve quadratic equations by graphing, completing the square, and using the Quadratic Formula. They use the determinant to determine the number of real roots of quadratic equations. Students graph and translate exponential functions and then relate them to growth and decay problems. Finally, students analyze geometric sequences.. Future Connections The Quadratic Formula can be used to solve any second-degree polynomial equation in future studies of math. Exponential functions, such as growth and decay, are applied in many fields, including science and finance.. 522C. Chapter 10 Quadratic and Exponential Functions. Solving Quadratic Equations by Graphing Graphing can be used to find the solutions of a quadratic equation. The solutions of a quadratic equation are also called the roots of the equation. They are the x-intercepts of the graph of the related function. Quadratic equations can have two real roots, a double real root, or no real roots. There are two real roots when the vertex is on one side of the x-axis and the curve extends to the other side. There is a double real root when the vertex is on the x-axis. There are no real roots when the entire parabola is on one side of the x-axis. Sometimes the roots are not integers and must be estimated from the graph.. Solving Quadratic Equations by Completing the Square One way to solve some equations is to take the square root of each side. To do this, the quadratic expression on one side of the equation must be a perfect square. There are very few quadratic equations that fit this description. For many quadratic equations, a process called completing the square is used while solving the equation. To complete the square of a quadratic expression x 2 bx, find half of b and square it. Add this amount to the original expression. When solving an equation using this process, first add the square of half of b to each side of the equation. Solve by factoring and taking the square root of each side. If the coefficient of x 2 is not 1, first divide each term by this coefficient before completing the square..

(4) Solving Quadratic Equations by Using the Quadratic Formula Completing the square is the basis for the Quadratic Formula. Any quadratic equation can be solved using the Quadratic Formula. The coefficients a and b, and the constant c (from the standard form ax2 bx c 0), are substituted into the formula b b2 4ac . x . Then the expression is simplified 2a to determine the solutions. The discriminant is the value b2 4ac that is inside the radical in the Quadratic Formula. This value can be used to determine the number of roots, or solutions, a quadratic equation has. A positive discriminant indicates two real roots, while a negative discriminant indicates that there are no real roots. If the discriminant is 0, there is one real root.. Exponential Functions An exponential function has a variable as an exponent. The base of the exponent must be greater than 0, but not equal to 1. The base cannot equal 1 since 1 to any power equals 1. One way to graph an exponential function is to use ordered pairs. If the base is greater than 1, the graph rises faster and faster as the x values increase. If the base is less than 1, the graph falls more slowly as the x values increase. The graphs of exponential functions can be translated by numbers other than the base and exponent. The y-intercept is changed if a constant is multiplied by the original expression. Adding a constant to the original expression translates the graph up or down depending on whether the constant is positive or negative. Two ways to identify exponential functions are to look at the graph and to look for a pattern in the data. In the data, domain values at regular intervals have corresponding range values that have a common factor, not a common difference.. Growth and Decay The general formula for exponential growth is y C(1 + r)t. The original amount C increases by the same percent r over a given period of time t. Compound interest is one application of exponential growth. Exponential decay is a variation of exponential growth. Instead of the original amount increasing by the same percent over a given period of time, it decreases. Depreciation is an application of exponential decay.. Geometric Sequences Recall that in an arithmetic sequence, the terms increase or decrease by a constant value called the common difference. In a geometric sequence, each term is found by multiplying the previous term by the same number, called the common ratio. The common ratio is found by dividing a term by the previous term. If all the terms have the same sign, the common ratio is positive. If the terms alternate signs, the common ratio is negative. Once you determine the common ratio, you can use the formula an a1 r n 1 to find the nth term. In this equation, a1 is the first term and r is the common ratio. In other words, to find a specific term, multiply the first term of the sequence by the common ratio raised to a power one less than the number of the specific term. A term or terms between two other terms in a geometric sequence are called geometric means. Use the geometric sequence formula to find the geometric means. Solve for r. Once the common ratio is known, multiply the previous term by r to find the missing geometric mean or means.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Graphing Quadratic Functions (Lesson 28) • Graphing Technology: Parent and Family Graphs (Lesson 29) • More on Axis of Symmetry and Vertices (Lesson 30) • Solving Quadratic Equations by Graphing (Lesson 31) • Solving Quadratic Equations by Completing the Square (Lesson 39) • Solving Quadratic Equations by Using the Quadratic Formula (Lesson 32) • Exponential Functions (Lesson 33) • Growth and Decay (Lesson 34) Chapter 10 Quadratic and Exponential Functions 522D.

(5) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 523, 530, 538, 544, 552, 560, 565 Practice Quiz 1, p. 544 Practice Quiz 2, p. 560. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 9–12, 47–48 Quizzes, CRM pp. 635–636 Mid-Chapter Test, CRM p. 637 Study Guide and Intervention, CRM pp. 579–580, 585–586, 591–592, 597–598, 603–604, 609–610, 615–616. Mixed Review. pp. 530, 538, 544, 552, 560, 565, 572. Cumulative Review, CRM p. 638. Error Analysis. Find the Error, pp. 550, 558 Common Misconceptions, p. 534. Find the Error, TWE pp. 550, 558 Unlocking Misconceptions, TWE pp. 534, 540 Tips for New Teachers, TWE pp. 547, 548. Standardized Test Practice. pp. 527, 528, 530, 538, 543, 552, 560, 565, 572, 579, 580–581. TWE pp. 580–581 Standardized Test Practice, CRM pp. 639–640. Open-Ended Assessment. Writing in Math, pp. 530, 537, 543, 552, 560, 565, 572 Open Ended, pp. 528, 536, 542, 550, 557, 563, 570 Standardized Test, p. 581. Modeling: TWE pp. 530, 544, 560, 572 Speaking: TWE p. 538 Writing: TWE pp. 552, 565 Open-Ended Assessment, CRM p. 633. Chapter Assessment. Study Guide, pp. 574–578 Practice Test, p. 579. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 621–626 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 627–632 Vocabulary Test/Review, CRM p. 634. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key 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 flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 522E. Chapter 10 Quadratic and Exponential Functions.

(6) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson. AlgePASS Lesson. 10-1. 28 Graphing Quadratic Equations. 10-3. 29 Effects of Parameter Changes on Quadratic Functions. 10-4. 30 Solving Quadratic Equations Using the Quadratic Formula. 10-4. 31 Solving Word Problems Using Quadratic Equations. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 75–82 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 523 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 528, 535, 542, 550, 557, 563, 570) • Reading Mathematics, p. 566 • Writing in Math questions in every lesson, pp. 530, 537, 543, 552, 560, 565, 572 • Reading Study Tip, p. 525 • WebQuest, pp. 537, 572 Teacher Wraparound Edition • Foldables Study Organizer, pp. 523, 574 • Study Notebook suggestions, pp. 528, 536, 542, 550, 558, 563, 566, 570, 573 • Modeling activities, pp. 530, 544, 560, 572 • Speaking activities, p. 538 • Writing activities, pp. 552, 565 • Differentiated Instruction, (Verbal/Linguistic), p. 547 • ELL Resources, pp. 522, 529, 537, 543, 547, 551, 559, 564, 566, 571, 574 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 10 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 10 Resource Masters, pp. 583, 589, 595, 601, 607, 613, 619) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 6.3, 6.7, 6.8. For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 10 Quadratic and Exponential Functions 522F.

(7) Quadratic and Exponential Functions. Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. • Lesson 10-1 Graph quadratic functions. • Lessons 10-2 through 10-4 Solve quadratic equations. • Lesson 10-5 Graph exponential functions. • Lesson 10-6 Solve problems involving exponential growth and exponential decay. • Lesson 10-7 Recognize and extend geometric sequences.. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson.. Quadratic functions and equations are used to solve problems about fireworks, to simulate the flight of golf balls in computer games, to describe arches, to determine hang time in football, and to help with water management. Exponential functions are used to describe changes in population, to solve compound interest problems, and to determine concentration of chemicals in a body of water after a spill. Exponential decay is one type of exponential function. Carbon dating uses exponential decay to determine the age of fossils and dinosaurs. You will learn about carbon dating in Lesson 10-6.. Lesson 10-1 10-1 Follow-Up 10-2 10-3 10-3 Follow-Up 10-4 10-4 Follow-Up 10-5 10-6 10-7 10-7 Follow-Up. NCTM Standards. • • • • •. parabola (p. 524) completing the square (p. 539) Quadratic Formula (p. 546) exponential function (p. 554) geometric sequence (p. 567). Local Objectives. 2, 6, 8, 9, 10 2, 6, 7, 8 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 7, 8 2, 6, 8, 9, 10 2, 6 2, 6, 8, 9, 10 2, 6, 8, 9, 10 1, 2, 3, 6, 8, 9, 10 1, 2, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 522. Key Vocabulary. 522 Chapter 10 Quadratic and Exponential Functions. Vocabulary Builder. ELL. The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 10 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 10 test.. Chapter 10 Quadratic and Exponential Functions.

(8) Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 10. For Lesson 10-1. This section provides a review of the basic concepts needed before beginning Chapter 10. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 9–12 and 47–48.. Graph Functions. Use a table of values to graph each equation.. (For review, see Lesson 5-3.). 1–8. See pp. 581A– 581H.. 1. y x 5. 2. y 2x 3. 3. y 0.5x 1. 4. y 3x 2. 5. 2x 3y 12. 6. 5y 10 2x. 7. x 2y 6. 8. 3x 2y 9. For Lesson 10-3. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson.. Perfect Square Trinomials. Determine whether each trinomial is a perfect square trinomial. If so, factor it. (For review, see Lesson 9-6.). 9.. t2. 12t 36. 13. 9b2 6b 1. yes; (3b 1)2. 9. yes; (t 6)2. 10.. a2. 10. yes; (a 7)2. 14a 49. 11. m2 18m 81 no12. y2 8y 12 no. 14. 6x2 4x 1 no. 15. 4p2 12p 9. 16. 16s2 24s 9. yes; (2p 3)2. yes; (4s 3)2. For Lesson 10-7. Arithmetic Sequences. Find the next three terms of each arithmetic sequence. (For review, see Lesson 4-7.) 17. 5, 9, 13, 17, … 21, 25, 29. 18. 12, 5, 2, 9, … 16, 23, 30. 19. 4, 1, 2, 5, … 8, 11, 14. 20. 24, 32, 40, 48, … 56, 64, 72. 21. 1, 6, 11, 16, … 21, 26, 31. 22. 27, 20, 13, 6, … 1, 8, 15. 23. 5.3, 6.0, 6.7, 7.4, … 8.1, 8.8, 9.5. 24. 9.1, 8.8, 8.5, 8.2, … 7.9, 7.6, 7.3. For Lesson. Prerequisite Skill. 10-2 10-3. Finding x-Intercepts (p. 530) Factoring Perfect Square Trinomials (p. 538) Finding Square Roots (p. 544) Evaluating Expressions with Exponents (p. 552) Evaluating Expressions with Exponents (p. 560) Finding Terms in Arithmetic Sequences (p. 565). 10-4 10-5 10-6 10-7. Make this Foldable to help you organize information on quadratic and exponential functions. Begin with four sheets of grid paper. Fold in Half. Tape Unfold each sheet and tape to form one long piece.. Fold each sheet in half along the width.. Label Label each page with the lesson number as shown. Refold to form a booklet.. 1 10-. 102. 3 10-. 104. 5 10-. 106. 7 10-. 108. As you read and study the chapter, write notes and examples for each lesson on each page of the journal.. Reading and Writing. Chapter 10. Quadratic and Exponential Functions. 523. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Organization of Data: Sequencing Information Students use their Foldable to take notes, define terms, record concepts, and write examples. Ask students to note the order in which the concepts are presented in this chapter. Ask them to write about why the concepts and computations were presented in that sequence. If students have difficulty seeing the logic in this sequence, have them outline the key concepts in their own order, and justify their reasoning in writing. Chapter 10 Quadratic and Exponential Functions 523.

(9) Lesson Notes. 5-Minute Check Transparency 10-1 Use as a quiz or review of Chapter 9. Mathematical Background notes are available for this lesson on p. 522C.. • Graph quadratic functions. • Find the equation of the axis of symmetry and the coordinates of the vertex of a parabola.. Vocabulary • • • • • • •. quadratic function parabola minimum maximum vertex symmetry axis of symmetry. Building on Prior Knowledge Students were first introduced to nonlinear functions in Chapter 8. In this lesson, students will learn about one kind of nonlinear function, a quadratic function. Students will learn how to graph quadratic functions.. Height of Rocket. can you coordinate a fireworks display with recorded music? 80. The Sky Concert in Peoria, Illinois, is a 4th of July fireworks display set to music. If a rocket (firework) is launched with an initial velocity of 39.2 meters per second at a height of 1.6 meters above the ground, the equation h 4.9t2 39.2t 1.6 represents the rocket’s height h in meters after t seconds. The rocket will explode at approximately the highest point.. Height (meters). 1 Focus. Graphing Quadratic Functions. 60 40 20. 0. 2. 4 6 Time (seconds). GRAPH QUADRATIC FUNCTIONS The function describing the height of the rocket is an example of a quadratic function. A quadratic function can be written in the form y ax2 bx c, where a 0. This form of the quadratic function is called the standard form. Notice that this polynomial has degree 2 and the exponents are positive. The graph of a quadratic function is called a parabola.. Quadratic Function • Words. can you coordinate a fireworks display with recorded music? Ask students: • Which is more important to the music planners, the height of the firework when it explodes, or the time at which it explodes? Explain. The time is more important if the planners want to coordinate the explosions with music. • Why must the planners still know the height? The firework explodes at approximately its highest point. When the highest point is found, the value of t at this point is the time at which the firework will explode.. A quadratic function can be described by an equation of the form y ax2 bx c, where a 0.. • Models. y. O. y. x. x. O. Example 1 Graph Opens Upward Use a table of values to graph y 2x2 4x 5. Graph these ordered pairs and connect them with a smooth curve.. y. x. y. 2. 11. 10. 1. 1. 6. 0. 5. 1. 7. 2. 5. 3. 1. 4. 11. 2 4. 2. O. Resource Manager Workbook and Reproducible Masters Parent and Student Study Guide Workbook, p. 75 Teaching Algebra With Manipulatives Masters, pp. 1, 176. 2. 4x. 4. 524 Chapter 10 Quadratic and Exponential Functions. Chapter 10 Resource Masters • Study Guide and Intervention, pp. 579–580 • Skills Practice, p. 581 • Practice, p. 582 • Reading to Learn Mathematics, p. 583 • Enrichment, p. 584. 8. Transparencies 5-Minute Check Transparency 10-1 Answer Key Transparencies. Technology AlgePASS: Tutorial Plus, Lesson 28 Interactive Chalkboard. y 2x 2 4 x 5.

(10) Consider the standard form y ax2 bx c. Notice that the value of a in Example 1 is positive and the curve opens upward. The lowest point, or minimum, of the graph is located at (1, 7).. 2 Teach GRAPH QUADRATIC FUNCTIONS. Example 2 Graph Opens Downward Use a table of values to graph y x2 4x 1.. x. y. 1. 6. 0. 1. 1. 2. 2. 3. 3. 2. 4. 1. 5. 6. Graph these ordered pairs and connect them with a smooth curve.. y. In-Class Examples x. O. Teaching Tip. Tell students that their sketches of parabolas do not have to be perfect. However, students should not “connect the dots” with straight lines. The important thing is for the curve to pass through the graphed ordered pairs.. y x 2 4 x 1. Study Tip Notice that the value of a in Example 2 is negative and the curve opens downward. The highest point, or maximum, of the graph is located at (2, 3). The maximum or minimum point of a parabola is called the vertex.. Reading Math The plural of vertex is vertices. In math, vertex has several meanings. For example, there are the vertex of an angle, the vertices of a polygon, and the vertex of a parabola.. 1 Use a table of values to graph y x2 x 2.. SYMMETRY AND VERTICES. x. Parabolas possess a geometric property called symmetry. Symmetrical figures are those in which the figure can be folded in half so that each half matches the other exactly.. Model y. 4. A parabola is symmetrical. The vertex of the parabola lies on the line that divides the parabola into two matching halves.. The fold line in the activity above is called the axis of symmetry for the parabola. Each point on the parabola that is on one side of the axis of symmetry has a corresponding point on the parabola on the other side of the axis. The vertex is the only point on the parabola that is on the axis of symmetry. In the graph of y x2 x 6, the axis of 1 2. 1 2. 1 4. x y 2 8 1 0 0 4 1 4 2 0 3 8. O. x. y. x. y y 2x 26 2x 4. –2. O. 4x. –4 –8. x. O. axis of symmetry x1. Interactive. 2. Chalkboard. Notice the relationship between the values a and b and the equation of the axis of symmetry.. www.algebra1.com/extra_examples. O. y 2x2 2x 4.. y x2 x 6. symmetry is x . The vertex is , 6.. y x2 x 2. 2 Use a table of values to graph. Make a Conjecture. 1. What is the vertex of the parabola? (3, 1) 2. Write an equation of the fold line. x 3 3. Which point on the parabola lies on the fold line? (3, 1) 4. Write a few sentences to describe the symmetry of a parabola based on your findings in this activity.. y. y. 2 4 1 0 0 2 1 2 2 0 3 4. Symmetry of Parabolas • Graph y x2 6x 8 on grid paper. • Hold your paper up to the light and fold the parabola in half so that the two sides match exactly. • Unfold the paper.. Power Point®. Lesson 10-1 Graphing Quadratic Functions. Algebra Activity Materials grid paper • Tell students to make sure the sides of the parabola match before they crease the paper. • Make sure students understand the connection between the fold line (axis of symmetry) and the vertex. The vertex of a parabola always lies on the axis of symmetry.. PowerPoint® Presentations 525. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools Lesson 10-1 Graphing Quadratic Functions 525.

(11) SYMMETRY AND VERTICES. In-Class Example. Power Point®. Teaching Tip. When students use symmetry to graph parabolas, they need only find a few points, and then reflect those points across the line of symmetry. You may want to suggest that students occasionally check their reflected points by substituting them into the original equation.. Equation of the Axis of Symmetry of a Parabola. TEACHING TIP The equation for the axis of symmetry can be derived by completing the square. This skill is taught in Lesson 10-3.. • Words. The equation of the axis of symmetry for the graph of y ax2 bx c, where a 0,. • Model. y. b 2a. is x .. x. O. b. x 2a. You can determine information about a parabola from its equation.. Example 3 Vertex and Axis of Symmetry Consider the graph of y 3x2 6x 4.. 3 Consider the graph of y 2x2 8x 2.. a. Write the equation of the axis of symmetry. In y 3x2 6x 4, a 3 and b 6.. a. Write the equation of the axis of symmetry. x 2. b 2a 6 x or 1 2(3). x . b. Find the coordinates of the vertex. The coordinates of the vertex are (2, 6). c. Identify the vertex as a maximum or minimum. Since the coefficient of the x 2 term is negative, the parabola opens downward and the vertex is a maximum point. d. Graph the function. y y 2x 2 8x 2. Equation for the axis of symmetry of a parabola a 3 and b 6. The equation of the axis of symmetry is x 1.. Study Tip Coordinates of Vertex Notice that you can find the x-coordinate by knowing the axis of symmetry. However, to find the y-coordinate, you must substitute the value of x into the quadratic equation.. b. Find the coordinates of the vertex. Since the equation of the axis of symmetry is x 1 and the vertex lies on the axis, the x-coordinate for the vertex is 1. y 3x2 6x 4. Original equation. y 3(1)2 6(1) 4 x 1 y 3 6 4. Simplify.. y7. Add.. The vertex is at (1, 7). c. Identify the vertex as a maximum or minimum. Since the coefficient of the x2 term is negative, the parabola opens downward and the vertex is a maximum point. d. Graph the function.. O. x. You can use the symmetry of the parabola to help you draw its graph. On a coordinate plane, graph the vertex and the axis of symmetry. Choose a value for x other than 1. For example, choose 1 and find the y-coordinate that satisfies the equation. y 3x2 6x 4 y. 3(1)2. Original equation. 6(1) 4 Let x 1.. y 5. Simplify.. Graph (1, 5). Since the graph is symmetrical about its axis of symmetry x 1, you can find another point on the other side of the axis of symmetry. The point at (1, 5) is 2 units to the right of the axis. Go 2 units to the left of the axis and plot the point (3, 5). Repeat this for several other points. Then sketch the parabola. 526. y 3 x 2 6 x 4 x1. (3, 5). 2. Chapter 10 Quadratic and Exponential Functions. Interpersonal Place students in small groups. Since there are several tasks involved in graphing quadratic functions, have the group members decide which of the tasks they should complete in order to graph a given function. For example, one member can be responsible for finding the equation for the axis of symmetry, another can substitute values in order to determine points on the graph, and a third member can graph the points and draw the curve of the parabola. Chapter 10 Quadratic and Exponential Functions. x. O. Differentiated Instruction. 526. y. (1, 7). 2. (1, 5).

(12) CHECK Does (3, 5) satisfy the equation? y 3x2 6x 4. In-Class Example. Original equation. 5 ⱨ 3(3)2 6(3) 4. y 5 and x 3. 5 5 . Simplify.. Teaching Tip. Point out to students that although this equation does not look exactly like the other quadratic equations that they have graphed so far in this lesson, the equation still has a degree of 2 so it is quadratic. Furthermore, when they multiply x 1 by itself, the equation takes on the more traditional form.. The ordered pair (3, 5) satisfies y 3x2 4x 5, and the point is on the graph.. Standardized Example 4 Match Equations and Graphs Test Practice Multiple-Choice Test Item Which is the graph of y 1 (x 1)2? y. A. Power Point®. 4 Using the item choices from B. Example 4 in the Student Edition, which graph corresponds to the graph of y x2 2x 2? D. y. x. O. x. O. y. C. y. D O. O. x. x. Read the Test Item You are given a quadratic function, and you are asked to choose the graph that corresponds to it. Solve the Test Item First write the equation in standard form.. Test-Taking Tip Sometimes you can answer a question by eliminating the incorrect choices. For example, in this test question, choices A and B are eliminated because their axes of symmetry are not x 1.. y 1 (x 1)2. Original equation. y 1 x2 2x 1. (x 1)2 x2 2x 1. y11. x2. 2x 1 1. y x2 2x. Subtract 1 from each side. Simplify.. Then find the axis of symmetry of the graph of y x2 2x. b 2a 2 x or 1 2(1). x . Equation for the axis of symmetry a 1 and b 2. The axis of symmetry is x 1. Look at the graphs. Since only choices C and D have this as their axis of symmetry, you can eliminate choices A and B. Since the coefficient of the x2 term is positive, the graph opens upward. Eliminate choice D. The answer is C. Lesson 10-1 Graphing Quadratic Functions. 527. Example 4 When the answer choices for a test question are graphs, look for obvious clues that would make a graph incorrect. The text mentioned eliminating two of the graphs that had the incorrect axis of symmetry. You can also eliminate two of the parabolas that open in the wrong direction. Since the coefficient of the x2 term is positive, you know that the parabola opens upward and B and D are incorrect.. Standardized Test Practice. Lesson 10-1 Graphing Quadratic Functions 527.

(13) 3 Practice/Apply. Concept Check 1–3. See margin.. 1. Compare and contrast a parabola with a maximum and a parabola with a minimum. 2. OPEN ENDED Draw two different parabolas with a vertex of (2, 1).. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 10. • include sample graphs with annotations about how to graph a quadratic equation. • include any other item(s) that they find helpful in mastering the skills in this lesson.. 3. Explain how the axis of symmetry can help you graph a quadratic function.. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4, 5 6–8 9. 1, 2 3 4. Use a table of values to graph each function. 4–5. See pp. 581A–581H. 4. y x2 5. 5. y x2 4x 5. Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. 6–8. See pp. 581A–581H for graphs. 6. y x2 4x 9. x 2; (2, 13); min. Standardized Test Practice. 7. y x2 5x 6 1 2. 9. Which is the graph of y x2 1? B y. A. x. O. x. y. C. y. D. About the Exercises …. O. Organization by Objective • Graph Quadratic Functions: 10–15 • Symmetry and Vertices: 16–51. 18. x 0; (0, 0); min 19. x 0; (0, 0); max 20. x 0; (0, 2); min 21. x 0; (0, 5); max 22. x 1; (1, 4); max. Odd/Even Assignments Exercises 10–37 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercises 49, 54–59 require a graphing calculator.. ★ indicates increased difficulty. Basic: 11–29 odd, 37–40, 50–53, 60–80 Average: 11–37 odd, 39–43, 50–53, 60–80 (optional: 54–59) Advanced: 10–38 even, 44–74 (optional: 75–80). y. B. O. Assignment Guide. 8. y (x 2)2 1. x 2.5; (2.5, 12.25); max x 2; (2, 1); max. x. O. x. Practice and Apply Homework Help For Exercises 10–15 16–49 52, 53. See Examples 1, 2 3 4. Extra Practice See page 841.. 23. x 3; (3, 24); max 24. x 7; (7, 36); min 25. x 1; (1, 17); min 26. x 3; (3, 29); min. Use a table of values to graph each function. 10–15. See pp. 581A–581H. 10. y x2 3 13. y . x2. 11. y x2 7. 4x 3. 14. y . 3x2. 12. y x2 2x 8. 6x 4. 15. y 3x2 6x 1. 16. What is the equation of the axis of symmetry of the graph of 1 y 3x2 2x 5? x 3 17. Find the equation of the axis of symmetry of the graph of y 4x2 5x 16. 5 x 8 Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function. 18–35. See pp. 581A–581H for graphs. 18. y 4x2. 19. y 2x2. 21. y . x2. 24. y . x2. 27. y . 3x2. 5. 14x 13 6x 4. x 1; (1, 1); min. 528 Chapter 10 Quadratic and Exponential Functions. 22. y . x2. 25. y . x2. 20. y x2 2. 2x 3. 2x 18. 28. y 5 16x . 2x2. x 4; (4, 37); max. 23. y x2 6x 15 26. y 2x2 12x 11 29. y 9 8x 2x2. x 2; (2, 1); min. Answers 1. Both types of parabolas are U shaped. A parabola with a maximum opens downward, and its corresponding equation has a negative coefficient for the x2 term. A parabola with a minimum opens upward, and its corresponding equation has a positive coefficient for the x2-term. 528. 2. Sample answer:. Chapter 10 Quadratic and Exponential Functions. y. O. x. 3. If you locate several points of the graph on one side of the axis of symmetry, you can locate corresponding points on the other side of the axis of symmetry to help graph the equation..

(14) ★ 31. y 2(x 4)2 3 ★ 32. y 2 x2 10x 25. NAME ______________________________________________ DATE. p. 579 (shown) and p. 580 Graphing Quadratic Functions. ★ 33. y 1 3x2 12x 12 ★ 34. y 5 13(x 2)2 ★ 35. y 1 23(x 1)2 x 2; (2, 1); min x 2; (2, 5); min x 1; (1, 1); min. Graph Quadratic Functions. 36. The vertex of a parabola is at (4, 3). If one x-intercept is 11, what is the other x-intercept? 3. a function described by an equation of the form f (x) ax 2 bx c, where a 0. Quadratic Function. Example: y 2x 2 3x 8. The degree of a quadratic function is 2, and the exponents are positive. Graphs of quadratic functions have a general shape called a parabola. A parabola opens upward and has a minimum point when the value of a is positive, and a parabola opens downward and has a maximum point when the value of a is negative.. 37. What is the equation of the axis of symmetry of a parabola if its x-intercepts are 6 and 4? x 1 38. SPORTS A diver follows a path that is in the shape of a parabola. Suppose the diver’s foot reaches 1 meter above the height of the diving board at the maximum height of the dive. At that time, the diver’s foot is also 1 meter horizontally from the edge of the diving board. What is the distance of the diver’s foot from the diving board as the diver descends past the diving board? Explain.. ____________ PERIOD _____. Study Guide andIntervention Intervention, 10-1 Study Guide and. Example 1 Use a table of values to graph y x2 4x 1.. 1m 1m ?. x. y. 1. 6. Example 2 Use a table of values to graph y x2 6x 7.. y. y. x. y. 6. 7. 5. 2. 4. 1. 3. 2. 0. 1. 1. 2. 2. 3. 3. 2. 2. 1. 4. 1. 1. 2. 0. 7. O. x. Graph the ordered pairs in the table and connect them with a smooth curve.. O. x. Graph the ordered pairs in the table and connect them with a smooth curve.. Exercises Use a table of values to graph each function. 1. y x2 2. 2 m; parabolas are symmetric.. 2. y x2 4 y. 3. y x2 3x 2. y. y. O. ENTERTAINMENT For Exercises 39 and 40, use the following information. A carnival game involves striking a lever that forces a weight up a tube. If the weight reaches 20 feet to ring the bell, the contestant wins a prize. The equation h 16t2 32t 3 gives the height of the weight if the initial velocity is 32 feet per second.. x. winner O. O. x. NAME ______________________________________________ DATE. x. ____________ PERIOD _____. Skills Practice, 10-1 Practice (Average). p. 581 and Practice, p. 582Functions (shown) Graphing Quadratic. Use a table of values to graph each function.. 39. Find the maximum height of the weight. 19 ft. 1. y x2 2. 2. y x2 6x 3. y. 40. Will a prize be won? no. 3. y 2x2 8x 5. y. O. y. O. x. x O. PETS For Exercises 41–43, use the following information. Miriam has 40 meters of fencing to build a pen for her dog. 41. A x(20 x) or A x2 20x. 20 x. 4. y x2 3. 6. y 2x2 8x 1. x 2; (2, 5); max. y. x 2; (2, 7); min y. y. O O. x. 20 x PHYSICS For Exercises 7–9, use the following information. Miranda throws a set of keys up to her brother, who is standing on a third-story balcony with his hands 38 feet above the ground. If Miranda throws the keys with an initial velocity of 40 feet per second, the equation h 16t2 40t 5 gives the height h of the keys after t seconds.. ARCHITECTURE For Exercises 44–46, use the following information. The shape of the Gateway Arch in St. Louis, Missouri, is a catenary curve. It resembles a parabola with the equation h 0.00635x2 4.0005x 0.07875, where h is the height in feet and x is the distance from one base in feet.. 7. How long does it take the keys to reach their highest point? 1.25 s 8. How high do the keys reach? 30 ft 9. Will her brother be able to catch the keys? Explain. No, the keys will be 8 ft short. of their target. BASEBALL For Exercises 10–12, use the following information. A player hits a baseball at a 45° angle with the ground with an initial velocity of 80 feet per second from a height of three feet above the ground. The equation h 0.005x2 x 3 gives the path of the ball, where h is the height and x is the horizontal distance the ball travels.. ★ 44. What is the equation of the axis of symmetry? x 315 ★ 45. What is the distance from one end of the arch to the other? 630 ft ★ 46. What is the maximum height of the arch? 630 ft. 10. What is the equation of the axis of symmetry? x 100 11. What is the maximum height reached by the baseball? 53 ft 12. An outfielder catches the ball three feet above the ground. How far has the ball traveled horizontally when the outfielder catches it? 200 ft NAME ______________________________________________ DATE. 10-1 Reading to Learn Mathematics Pre-Activity. ★ 47. Use what you know about parabolas and their minimum values to estimate the year in which the average age of brides was the youngest. 1959 ★ 48. Estimate the average age of the brides during that year. about 20 years old ★ 49. Use a graphing calculator to check your estimates. See margin.. 80 meters 4 seconds after being launched.. Reading the Lesson quadratic. 1. The standard form for a. function is y ax2 bx c. For the. function y 2x2 5x 3, the value of a is. 3. the value of c is. 2. , the value of b is. 49.. 10-1 Enrichment Enrichment,. B.. y. y. x. b. The highest point of graph A is located at. minimum. When a figure is moved to a new position without undergoing any rotation, then the figure is said to have been translated to that position.. (1, 4) (1, 2). . This point is the . This point is the. (maximum/minimum) point of the graph.. y. 3. The maximum or minimum point of a parabola is called the the parabola.. The graph of a quadratic equation in the form y (x b)2 c is a translation of the graph of y x2.. vertex. you fold the parabola is the. Start with y x2.. through the. vertex. axis of symmetry. of the parabola. This line goes. of the parabola.. 5. For a quadratic function y ax2 bx c, the parabola opens upward if a. y (x 4)2. It opens downward if a. Then slide up 3 units. x. O. y (x 4)2 3. 2. y . of. 4. If you fold a parabola along a line to get two halves that match exactly, the line where. Slide to the right 4 units.. 1. .. (maximum/minimum) point of the graph.. c. The lowest point of graph B is located at. Translating Quadratic Graphs. 1. y . parabola. a. Each graph is a curve called a. maximum. x2. x. ____________ PERIOD _____. p. 584. These equations have the form y . , and. 529 O. Answer. 5. .. 2. The graphs of two quadratic functions are shown below. Complete each statement about the graphs.. O. NAME ______________________________________________ DATE. ELL. How can you coordinate a fireworks display with recorded music? Read the introduction to Lesson 10-1 at the top of page 524 in your textbook. According to the graph, at what height does the rocket explode and in how many seconds after being launched? It explodes at a height of. A.. Lesson 10-1 Graphing Quadratic Functions. ____________ PERIOD _____. Reading to Learn Mathematics, p. 583 Graphing Quadratic Functions. BRIDES For Exercises 47–49, use the following information. The equation a 0.003x2 0.115x 21.3 models the average ages of women when they first married since the year 1940. In this equation, a represents the average age and x represents the years since 1940.. www.algebra1.com/self_check_quiz. x. x O. 43. What is the greatest possible area of the pen? 100 m2. Source: World Book Encyclopedia. 5. y 2x2 8x 3. x 0; (0, 3); max. x. 42. What value of x will result in the greatest area? 10 m. The Gateway Arch is part of a tribute to Thomas Jefferson, the Louisiana Purchase, and the pioneers who settled the West. Each year about 2.5 million people visit the arch.. x. Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function.. x. 41. Use the diagram at the right to write an equation for the area A of the pen.. Architecture. Lesson 10-1. ★ 30. y 3(x 1)2 20. x2. c. Graph each equation.. x2. 2. 3. y . x2. . . 0.. 0.. Helping You Remember 6. Look up the word vertex in a dictionary. You will find that it comes from the Latin word vertere, which means to turn. How can you use the idea of “to turn” to remember what the vertex of a parabola is? Sample answer: The vertex of a parabola is the. 2. point at which the parabola turns upward or downward.. y. Lesson 10-1 Graphing Quadratic Functions 529. Lesson 10-1. 30. x 1; (1, 20); min 31. x 4; (4, 3); max 32. x 5; (5, 2); min.

(15) 50. CRITICAL THINKING Write a quadratic equation that represents a graph with. 4 Assess. 3 8. an axis of symmetry with equation x . Sample answer: y 4x2 3x 5 51. WRITING IN MATH. Open-Ended Assessment Modeling Model parabolas with string using a coordinate grid on an overhead projector, making sure that some graphs are not symmetric. For example, move a point one unit up, down, to the left, or right, so that the parabola looks pretty close, but is not symmetric. See if students can use what they know about the symmetry of parabolas to spot your mistakes. Have volunteers correct the parabolas.. Getting Ready for Lesson 10-2 PREREQUISITE SKILL Students will learn how to solve quadratic equations by graphing in Lesson 10-2. In order to solve quadratic equations by graphing, students will need to be able to identify the x-intercepts of the graphs. Use Exercises 75–80 to determine your students’ familiarity with identifying the x-intercepts of linear graphs.. How can you coordinate a fireworks display with recorded music? Include the following in your answer: • an explanation of how to determine when the rocket will explode, and • an explanation of how to determine the height of the rocket when it explodes.. Standardized Test Practice. 530. 52. Which equation corresponds to the graph at the right? A. y. y x2 4x 5 y x2 4x 5 y x2 4x 5 y x2 4x 5. A B C D. x. O. 53. Which equation does not represent a quadratic function? D y (x 3)2. A. Graphing Calculator 54. minimum; (5, 0) 55. maximum; (2, 7) 56. maximum; (2, 7). y 3x2. B. C. y 6x2 1. yx5. D. 54–59. See pp. 581A–581H for graphs. MAXIMUM OR MINIMUM Graph each function. Determine whether the vertex is a maximum or a minimum and give the ordered pair for the vertex. 54. y x2 10x 25. 55. y x2 4x 3. 56. y 2x2 8x 1. 57. y 2x2 40x 214. 58. y 0.25x2 4x 2. 59. y 0.5x2 2x 3. minimum; (8, 18). minimum; (10, 14). maximum; (2, 5). Maintain Your Skills Mixed Review. Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. (Lessons 9-5 and 9-6) 61. (a 11)2 60. x2 6x 9 prime 63.. 4q2. 9. (2q 3)(2q 3). 61. a2 22a 121 2a2. 64.. Find each sum or difference.. 25 prime. 62. 4m2 4m 1 (2m 1)2 65. 1 16g2. (1 4g)(1 4g). (Lesson 8-5). 66. (13x 9y) 11y 13x 20y. 67. (7p2 p 7) (p2 11). 6p2 p 18. 68. RECREATION At a recreation and sports facility, 3 members and 3 nonmembers pay a total of $180 to take an aerobics class. A group of 5 members and 3 nonmembers pay $210 to take the same class. How much does it cost members and nonmembers to take an aerobics class? (Lesson 7-3). Answer 51. In order to coordinate a firework with recorded music, you must know when and how high it will explode. Answers should include the following. • The rocket will explode when the rocket reaches the vertex or 39.2 when t which is 2(4.9) 4 seconds. • The height of the rocket when it explodes is the height when t 4. Therefore, h 4.9(42) 39.2(4) 1.6 or 80 meters.. Answer the question that was posed at the beginning of the lesson. See margin.. $15 for members, $45 for nonmembers Solve each inequality. Then check your solution. (Lesson 6-2) 69. 12b 144. {bb 12}. 70. 5w 125. {ww 25}. . 8 3r 2 71. rr 4 3 9. . Write an equation of the line that passes through each point with the given slope. (Lesson 5-4) 74. y 3x 12 2 3 72. (2, 13), m 4 73. (2, 7), m 0 74. (4, 6), m . y 4x 5. Getting Ready for the Next Lesson 530. 2. PREREQUISITE SKILL Find the x-intercept of the graph of each equation. (To review finding x-intercepts, see Lesson 4-5.). 75. 3x 4y 24 8. 76. 2x 5y 14 7. 77. 2x 4y 7 3.5. 78. 7y 6x 42 7. 79. 2y 4x 10 2.5. 80. 3x 7y 9 0 3. Chapter 10 Quadratic and Exponential Functions. Chapter 10 Quadratic and Exponential Functions. y 7.

(16) Graphing Calculator Investigation. A Follow-Up of Lesson 10-1. A Follow-Up of Lesson 10-1. Families of Quadratic Graphs. Getting Started. Recall that a family of graphs is a group of graphs that have at least one characteristic in common. On page 278, families of linear graphs were introduced. Families of quadratic graphs often fall into two categories—those that have the same vertex and those that have the same shape.. Know Your Calculator Remind students that the key squares the quantity but does not enter x2 into the equation. To enter 3x2, press 3 X,T,,n .. In each of the following families, the parent function is y x2. Graphing calculators make it easy to study the characteristics of these families of parabolas.. Graph each group of equations on the same screen. Use the standard viewing window. Compare and contrast the graphs. KEYSTROKES: Review graphing equations on pages 224 and 225. a. y x2, y 2x2, y 4x2. Teach. b. y x2, y 0.5x2, y 0.2x2 y 4x 2. yx. y 2x 2. y 0.5x 2. yx. y 0.2x 2. 2. Each graph opens upward and has its vertex at the origin. The graphs of y 2x2 and y 4x2 are narrower than the graph of y x2.. • Remind students that to set the calculator to the standard viewing window, press ZOOM 6. • To help students remember how the value of a in y ax2 affects the shape of the graph, suggest that students sketch some of the parabolas from these examples, along with the equations, in their study notebooks. • Students can use the TRACE feature on their calculator to help them identify the different parabolas on the screen. After pressing TRACE , students can use the arrow keys to move the cursor around on the graphs. The up and down arrows switch the cursor between graphs. The left and right arrows move the cursor along the individual graphs. The graph on which the cursor lies is identified in the top lefthand corner of the screen.. 2. Each graph opens upward and has its vertex at the origin. The graphs of y 0.5x2 and y 0.2x2 are wider than the graph of y x2.. How does the value of a in y ax2 affect the shape of the graph? c. y x2, y x2 3, y x2 2, y x2 4. d. y x2, y (x 3)2, y (x 2)2, y (x 4)2. y x2 3 y x2 y x2. y x2 2 y (x 4)2. y x2 4. y (x 2)2. Each graph opens upward and has the same shape as y x2. However, each parabola has a different vertex, located along the y-axis. How does the value of the constant affect the position of the graph?. y (x 3)2. Each graph opens upward and has the same shape as y x2. However, each parabola has a different vertex located along the x-axis. How is the location of the vertex related to the equation of the graph?. www.algebra1.com/other_calculator_keystrokes. Graphing Calculator Investigation Families of Quadratic Graphs. 531. Graphing Calculator Investigation Families of Quadratic Graphs 531.

(17) Graphing Calculator Investigation When analyzing or comparing the shapes of various graphs on different screens, it is important to compare the graphs using the same window with the same scale factors. Suppose you graph the same equation using a different window for each. How will the appearance of the graph change?. Assess • To change the scale of the viewing window, remind students to press WINDOW and change the appropriate settings, but not the Xres setting. • Have students use the TRACE function to see that even though the parabolas look different, the graph still has the same vertex and x-intercepts. Ask students to summarize how changes in the appearance and location of the graph are reflected in the equation. In the equation y ax2 bx c, changes in the value of a affect the width of the graph. Changes in c affect the vertical position of the graph. In the equation y (x c)2, changes in c affect the horizontal position of the graph.. Graph y x 2 7 in each viewing window. What conclusions can you draw about the appearance of a graph in the window used? a. standard viewing window. b. [10, 10] scl: 1 by [200, 200] scl: 50. c. [50, 50] scl: 5 by [10, 10] scl: 1. d. [0.5, 0.5] scl: 0.1 by [10, 10] scl: 1. ]. [. ]. The window greatly affects the appearance of the parabola. Without knowing the window, graph b might be of the family y ax2, where 0 a 1. Graph c looks like a member of y ax2 7, where a 1. Graph d looks more like a line. However, all are graphs of the same equation.. Answers 1.. Exercises Graph each family of equations on the same screen. Compare and contrast the graphs. 1–4. See margin. 1. y x2 2. y x2 3. y x2 4. y x2 y 3x2 y 0.6x2 y (x 5)2 y x2 7 y 6x2 y 0.4x2 y (x 4)2 y x2 5. All of the graphs open downward from the origin. y 3x 2 is narrower than y x 2, and y 6x 2 is the narrowest.. Use the families of graphs on page 531 and Exercises 1– 4 above to predict the appearance of the graph of each equation. Then draw the graph. 5–8. See pp. 581A–581H. 5. y 0.1x2 6. y (x 1)2 7. y 4x2 8. y x2 6 Describe how each change in y x2 would affect the graph of y x2. Be sure to consider all values of a, h, and k. 9–12. See pp. 581A–581H. 9. y ax2 10. y (x h)2 11. y x2 k 12. y (x h)2 k. 2.. 532 Investigating Slope-Intercept Form 532 Chapter 10 Quadratic and Exponential Functions. All of the graphs open downward from the origin. y 0.6x 2 is wider than y x 2, and y 0.4x 2 is the widest.. 3.. 4.. All of the graphs open downward, have the same shape, and have vertices along the x-axis. However, each vertex is different. 532. Chapter 10 Quadratic and Exponential Functions. All of the graphs open downward, have the same shape, and have vertices along the y-axis. However, each vertex is different..

(18) Solving Quadratic Equations by Graphing. Lesson Notes. • Solve quadratic equations by graphing. • Estimate solutions of quadratic equations by graphing.. Vocabulary • quadratic equation • roots • zeros. can quadratic equations be used in computer simulations?. 5-Minute Check Transparency 10-2 Use as a quiz or review of Lesson 10-1.. A golf ball follows a path much like a parabola. Because of this property, quadratic functions can be used to simulate parts of a computer golf game. One of the x-intercepts of the quadratic function represents the location where the ball will hit the ground.. Mathematical Background notes are available for this lesson on p. 522C.. SOLVE BY GRAPHING. Recall that a quadratic function has standard form f(x) ax2 bx c. In a quadratic equation , the value of the related quadratic function is 0. So for the quadratic equation 0 x2 2x 3, the related quadratic function is f(x) x2 2x 3. You have used factoring to solve equations like x2 2x 3 0. You can also use graphing to determine the solutions of equations like this. The solutions of a quadratic equation are called the roots of the equation. The roots of a quadratic equation can be found by finding the x-intercepts or zeros of the related quadratic function.. Example 1 Two Roots Solve x2 6x 7 0 by graphing. Graph the related function f(x) x2 6x 7. The equation of the axis of 6 2(1). symmetry is x or x 3. When x equals 3, f(x) equals (3)2 6(3) 7 or 16. So, the coordinates of the vertex are (3, 16). Make a table of values to find other points to sketch the graph. x. 1 Focus. f (x ). f(x) 4. 8. 9. 6. 7. 4. 15. 3. 16. 2. 15. 8. 0. 7. 12. 2. 9. 8. 6. 4. 2. O. 2x. 4. 16. To solve x2 6x 7 0, you need to know where the value of f(x) is 0. This occurs at the x-intercepts. The x-intercepts of the parabola appear to be 7 and 1. (continued on the next page) Lesson 10-2 Solving Quadratic Equations by Graphing 533. Building on Prior Knowledge In Chapter 9, students learned how to solve quadratic trinomial equations using factoring. In this lesson, students will apply what they already know about solving quadratic equations to check the solutions they found by graphing. can quadratic equations be used in computer simulations? Ask students: • If one of the x-intercepts represents where the ball hits the ground, what represents the ground? the x-axis • Suppose the green is uphill from the tee. How would this affect the value of the x-coordinate of the location where the ball lands? How would a downhill shot affect the ball’s x-coordinate? Assuming that the ground at the tee is represented by the x-axis, the y-coordinate of the landing spot would be positive if the shot was uphill, and negative if the shot was downhill.. Resource Manager Workbook and Reproducible Masters Chapter 10 Resource Masters • Study Guide and Intervention, pp. 585–586 • Skills Practice, p. 587 • Practice, p. 588 • Reading to Learn Mathematics, p. 589 • Enrichment, p. 590 • Assessment, p. 635. Parent and Student Study Guide Workbook, p. 76 School-to-Career Masters, p. 19. Transparencies 5-Minute Check Transparency 10-2 Answer Key Transparencies. Technology Interactive Chalkboard. Lesson x-x Lesson Title 533.

(19) CHECK Solve by factoring. x2 6x 7 0 (x 7)(x 1) 0 x70 or x 1 0 x 7 x1 . 2 Teach SOLVE BY GRAPHING. In-Class Examples. Power Point®. Factor. Zero Product Property Solve for x.. The solutions of the equation are 7 and 1.. 1 Solve x2 3x 10 0 by. Quadratic equations always have two roots. However, these roots are not always two distinct numbers. Sometimes the two roots are the same number.. graphing. {2, 5} 12. Original equation. f (x). Example 2 A Double Root Solve b2 4b 4 by graphing.. 6. First rewrite the equation so one side is equal to zero. –12. –6. O. 6. b2 4b 4. 12 x. b2 f (x ) x 2 3x 10 –12. b2 4b 4 0. Tell students to look at the table of values they create before graphing the function. If either the greatest or least y value is zero, then the vertex is on the x-axis and the solution is a double root.. 6x 9 by graphing. {3} x2. f (x). O. x f (x ) x 2 6x 9. Original equation Add 4 to each side. Simplify.. Graph the related function f(b) b2 4b 4.. Teaching Tip. 2 Solve. 4b 4 4 4. Study Tip Common Misconception Although solutions found by graphing may appear to be exact, you cannot be sure that they are exact. Solutions need to be verified by substituting into the equation and checking, or by using the algebraic methods that you will learn in this chapter.. Teaching Tip. Again, the table of values helps reveal whether a function has no real roots. If all of the y values are positive, first decreasing then increasing, or if all are negative, first increasing then decreasing, then the graph of the function does not cross the x-axis, and there are no real roots.. b. f(b). 4. 4. 3. 1. 2. 0. 1. 1. 0. 4. f (b ). Notice that the vertex of the parabola is the b-intercept. Thus, one solution is 2. What is the other solution? Try solving the equation by factoring. b2 4b 4 0 (b 2)(b 2) 0 b20 or b 2 0 b 2 b 2. Original equation Factor. Zero Product Property Solve for b.. There are two identical factors for the quadratic function, so there is only one root, called a double root. The solution is 2. Thus far, you have seen that quadratic equations can have two real roots or one double real root. Can a quadratic equation have no real roots?. Example 3 No Real Roots Solve x2 x 4 0 by graphing. Graph the related function f(x) x2 x 4. The graph has no x-intercept. Thus, there are no real number solutions for this equation. The symbol , indicating an empty set, is often used to represent no real solutions.. x. f(x). 1. 6. 0. 4. 1. 4. 2. 6. f (x ). f (x ) x 2 x 4 O. 3 Solve x2 2x 3 0 by graphing.. 534. Chapter 10 Quadratic and Exponential Functions. f (x). Unlocking Misconceptions f (x ) x 2 2x 3 O. x. [empty set, or no real roots]. 534. b. O. f (b) b 2 4b 4. Chapter 10 Quadratic and Exponential Functions. Students may incorrectly assume that the vertex of a parabola lies on coordinates that are integers. Point out that in Example 3, the least y value for the function is somewhat less than 4, since these are the least integer y values. Also the graph clearly shows that the vertex lies somewhat below the y value of 4.. x.

(20) ESTIMATE SOLUTIONS In Examples 1 and 2, the roots of the equation were integers. Usually the roots of a quadratic equation are not integers. In these cases, use estimation to approximate the roots of the equation.. In-Class Examples. Example 4 Rational Roots Solve 6n 7 0 by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.. graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.. Graph the related function f(n) n2 6n 7. n. f(n) 7. 5. 2. 4. 1. 3. 2. 2. 1. 1. 2. 0. 7. Notice that the value of the function changes from negative to positive between the n values of 5 and 4 and between 2 and 1.. f (n ). f (x ) f (n ) n 2 6n 7. O. O. n. One root is between 0 and 1, and the other root is between 3 and 4.. Example 5 Estimate Solutions to Solve a Problem FOOTBALL When a football player punts a football, he hopes for a long “hang time.” Hang time is the total amount of time the ball stays in the air. A time longer than 4.5 seconds is considered good. If a punter kicks the ball with an upward velocity of 80 feet per second and his foot meets the ball 2 feet off the ground, the function y 16t2 80t 2 represents the height of the ball y in feet after t seconds. What is the hang time of the ball?. On September 21, 1969, Steve O’Neal set a National Football League record by punting the ball 98 yards. Source: The Guinness Book of Records. Concept Check. You need to find the solution of the equation 0 16t2 80t 2. Use a graphing calculator to graph the related function y 16t2 80t 2. The x-intercept is about 5. Therefore, the hang time is about 5 seconds. [2, 7] scl: 1 by [20, 120] scl: 10. f (x ). O. Teaching Tip. Students need to set the viewing window using the specifications given in order to see the parabola properly. Suggest that students use the CALC function to find the x-intercept. Press 2nd [CALC] 2. Then set the left bound just above the x-axis, and the right bound just below the x-axis, and hit ENTER again to have the calculator guess the intercept.. 5 MODEL ROCKETS Shelly. Since 5 seconds is greater than 4.5 seconds, this kick would be considered to have good hang time.. 1. State the real roots of the quadratic equation whose related function is graphed at the right. 3, 1. x. f (x ) x 2 4x 2. The n-intercepts of the graph are between 5 and 4 and between 2 and 1. So, one root is between 5 and 4, and the other root is between 2 and 1.. Football. Power Point®. 4 Solve x2 4x 2 0 by. n2. 6. ESTIMATE SOLUTIONS. x. built a model rocket for her science project. The equation y 16t2 250t models the flight of the rocket, launched from ground level at a velocity of 250 feet per second, where y is the height of the rocket in feet after t seconds. For how many seconds was Shelly’s rocket in the air? between 15 and 16 seconds. 2. Write the related quadratic function for the equation 7x2 2x 8.. f(x) 7x2 2x 8 www.algebra1.com/extra_examples. Lesson 10-2 Solving Quadratic Equations by Graphing. 535. Differentiated Instruction Logical Have students revisit Example 4. Have them first estimate between which integers the roots lie. Then suggest that students substitute fractional values for x in the quadratic equation to estimate the fractional value that produces a y value closest to zero. Students can check their estimates by using a graphing calculator to calculate the roots of the given function.. Lesson 10-2 Solving Quadratic Equations by Graphing 535.

(21) 3. OPEN ENDED Draw a graph to show a counterexample to the following statement. All quadratic equations have two different solutions. See margin.. 3 Practice/Apply Guided Practice. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 10. • include examples of how to solve quadratic equations by graphing. • include any other item(s) that they find helpful in mastering the skills in this lesson.. GUIDED PRACTICE KEY Exercises. Examples. 4–6 7–9 10. 1–3 4 5. Application. Homework Help For Exercises 11–20 21–34 35–46. See Examples 1–3 4 5. Organization by Objective • Solve by Graphing: 11–20, 35–40 • Estimate Solutions: 21–34, 41–46. 3. Sample answer:. 536. 8. x2 16 0 4, 4. 7. t2 9t 5 0. 9 t 8, 1 t 0. 9. w2 3w 5. 2 w 1, 4 w 5. 10. NUMBER THEORY Two numbers have a sum of 4 and a product of 12. Use a quadratic equation to determine the two numbers. 2, 6. Solve each equation by graphing. 11–16. See pp. 581A–581H for graphs. 11. c2 5c 24 0 3, 8 12. 5n2 2n 6 0 14.. b2. 12b 36 0 6. 15.. x2. 2x 5 0. 13. x2 6x 9 0 3 16. r2 4r 12 0 6, 2. 17. The roots of a quadratic equation are 2 and 6. The minimum point of the graph of its related function is at (4, 2). Sketch the graph of the function. See pp. 581A–581H. 18. The roots of a quadratic equation are 6 and 0. The maximum point of the graph of its related function is at (3, 4). Sketch the graph of the function. See pp. 581A–581H.. Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie. 21. a2 12 0 9s 12 0. 24.. 3s2. 27.. a2. 8a 4. 30.. p2. 16 8p. 22. n2 7 0. 23. 2c2 20c 32 0. 25.. x2. 6x 6 0. 26. y2 4y 1 0. 28.. x2. 6x 7. 29. m2 10m 21. 31.. 12n2. 21–32. See pp. 581A–581H.. 26n 30. 32. 4x2 35 4x. 33. One root of a quadratic equation is between 4 and 3, and the other root is between 1 and 2. The maximum point of the graph of the related function is at (1, 6). Sketch the graph of the function. See pp. 581A–581H.. Design The Winter Palace and the rest of the State Hermitage Museum in St. Petersburg, Russia, house 322 art galleries with about three million pieces of art.. 34. One root of a quadratic equation is between 1 and 0, and the other root is between 6 and 7. The minimum point of the graph of the related function is at (3, 5). Sketch the graph of the function. See pp. 581A–581H.. DESIGN For Exercises 35–39, use the following information. An art gallery has walls that are sculptured Source: The Guinness Book with arches that can be represented by the of Records quadratic function f(x) x2 4x 12, where x is in feet. The wall space under each arch is to be painted a different color from the arch itself. 35. See pp. 581A–581H 35. Graph the quadratic function and determine its x-intercepts.. for graph; 6, 2.. 36. What is the length of the segment along the floor of each arch? 8 ft. Chapter 10 Quadratic and Exponential Functions. 48. Since quadratic functions can be used to model a golf ball after it is hit, solving the related quadratic equation will determine where the ball hits the ground. Answers should include the following. • In the golf problem, one intercept represents the ball’s original location and the other intercept represents where the ball hits the ground. • Using the quadratic function y 0.0015x 2 0.3x, the ball will hit the ground 200 yd from the starting point.. y. O. Solve each equation by graphing. If integral roots cannot be found, estimate the roots by stating the consecutive integers between which the roots lie.. 20. NUMBER THEORY Use a quadratic equation to find two numbers whose sum is 5 and whose product is 24. 3, 8. 536. Answer. 6. c2 3 0. 19. NUMBER THEORY The sum of two numbers is 9, and their product is 20. Use a quadratic equation to determine the two numbers. 4, 5. About the Exercises …. Basic: 11–33 odd, 35–37, 39, 47–50, 53–68 Average: 11–33 odd, 35–39, 41, 42, 47–50, 53–68 (optional: 51, 52) Advanced: 12–34 even, 35, 36, 38, 40–62 (optional: 63–68). 5. a2 10a 25 0 5. Practice and Apply. See page 842.. Assignment Guide. 4. x2 7x 6 0 1, 6. ★ indicates increased difficulty. Extra Practice. Odd/Even Assignments Exercises 11–34 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercises 51–52 require a graphing calculator.. Solve each equation by graphing. 4–9. See pp. 581A–581H for graphs.. x. Chapter 10 Quadratic and Exponential Functions.

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