A1S11100.qxd 6/10/09 8:03 AM Page 502 Looking Ahead To .
A1S11100.qxd6/10/098:03 AMPage 502Looking Ahead to Chapter 11FocusIn Chapter 11, you will solve quadratic functions using methods such as factoring,extracting square roots, and the quadratic formula. You will use quadratic functions tosolve problems involving vertical motion and other real-world applications.Chapter Warm-upAnswer these questions to help you review skills that you will need in Chapter 11.Factor each expression completely.1. 2x2 5x 32. 2x2 9x 203. 9x2 254. 6x2 19x 105. x2 x 306. 4x2 8x 3Evaluate each radical.7. 248. 379. 10110. 163Read the problem scenario below.A water reservoir is being built in the shape of a semi-sphere. Calculate the amount ofwater that the reservoir will hold if it is built with the following dimensions. Be sure toinclude the correct units in your answers. Write your answers using a complete sentence.11. The diameter of the reservoir is 50 feet.11 2009 Carnegie Learning, Inc.12. The radius of the reservoir is 30 feet.13. The diameter of the reservoir is 100 feet.Key Termsgeneral form of a quadraticpolynomial p. 507Converse of the MultiplicationProperty of Zero p. 510square root p. 514positive square root p. 514504negative square root p. 514principal square root p. 514radical symbol p. 514radicand p. 514perfect square p. 515quadratic formula p. 519, 521Chapter 11 Modeling and Solving Quadratic Functionsdiscriminant p. 521factoring p. 521extracting square roots p. 521vertical motion model p. 523axis of symmetry p. 531
6/8/092:34 PMPage 503CHAPTER11Modeling and SolvingQuadratic Functions 2009 Carnegie Learning, Inc.A1S11100.qxdWhen a musician plucks a guitar string, the string vibrates and transmits its vibrationthrough the guitar. The sound is amplified and you hear a musical note. In Lesson 11.2,you will use an equation to find the tension of the string and wave speed of the vibrations.11.1 To Factor or Not to Factor11.4 Pumpkin CatapultSolving Quadratic Equations usingFactoring and Square Roots p. 50711.2 Guitar Strings and Other ThingsUsing a Vertical Motion Model p. 52311.5 Viewing the Night SkyUsing Quadratic Functions p. 53 1Square Roots and Radicals p. 51311.3 Kicking a Soccer BallUsing the Quadratic Formula to SolveQuadratic Equations p. 517Chapter 11 Modeling and Solving Quadratic Functions50511
4/11/0810:41 AMPage 492 2009 Carnegie Learning, Inc.A1S11101.qxd1150 6Chapter 11 Modeling and Solving Quadratic Functions
CH-11.1.qxd:A1S101016/12/0911:37 AM11.1Page 505To Factor or Not to FactorSolving Quadratic Equations using Factoring and Square RootsObjectivesIn this lesson,you will: Factor quadratictrinomials. Solve quadraticequations usingfactoring. Solve quadraticequations usingsquare roots.Key Terms general form of aquadratic trinomial Converse of theMultiplication Propertyof ZeroProblem 1Factoring QuadraticsThe general form of a quadratic trinomial is ax2 bx c, where a,b, and c are constants. Some quadratic trinomials with a 1 can befactored as x2 bx c (x r1 )(x r2 ) , where r1 and r1 are integers.1. Perform each multiplication.a. (x 1)(x 6) b. (x 1)(x 6) c. (x 3)(x 2) d. (x 3)(x 2) e. (x 1)(x 6) f. (x 1)(x 6) g. (x 3)(x 2) h. (x 3)(x 2) 2. How are the values of r1 and r2 related to the value of c?3. How are the values of r1 and r2 related to the value of b?The relationships that you describe in Questions 2 and 3 between thecoefficients of the original quadratic trinomial and the factors can beused to factor a quadratic trinomial of the form x2 bx c. 2009 Carnegie Learning, Inc.For example, the quadratic trinomial x2 4x 5 can be factored byperforming the following steps: List the factor pairs of the constant term c.For x2 4x 5, the constant term is 5. Factor pairs of 5are: 1 and 5, 1 and 5. Calculate the sum of each factor pair.11 1 5 41 ( 5) 4 Select the factor pair whose sum is equal to the coefficient of thex term, b.For x2 4x 5, the x term is 4x. The coefficient of this term is 4. So, select the factor pair whose sum is equal to 4. Thefactor pair of 1 and 5 has a sum of 4. Write the quadratic trinomial in factored form.x2 4x 5 (x 5)(x 1)Lesson 11.1 Solving Quadratic Equations using Factoring and Square Roots507
CH-11.1.qxd:A1S101016/12/0911:37 AMPage 506Problem 1Factoring Quadratics4. Factor x2 10x 24 by performing the following steps.a. List the factor pairs of the constant term c.b. Calculate the sum of each factor pair:c. Select the factor pair whose sum is equal to the coefficient ofthe x term, b.d. Write the quadratic trinomial in factored form.5. Factor each trinomial using this method.b. x2 5x 2411508Chapter 11 Probability 2009 Carnegie Learning, Inc.a. x2 10x 24
CH-11.1.qxd:A1S101016/12/0911:38 AMPage 507Problem 1Factoring Quadraticsc. x2 3x 28d. x2 12x 28e. x2 10x 25f. x2 25 2009 Carnegie Learning, Inc.g. x2 14x 4511h. x2 17x 52Lesson 11.1 Solving Quadratic Equations using Factoring and Square Roots509
CH-11.1.qxd:A1S101016/12/0911:38 AMPage 508Problem 2Solving Quadratic Equations byFactoringThe Converse of the Multiplication Property of Zero states that ifab 0, then either a 0, b 0, or both a and b are equal to 0. Aquadratic equation written in factored form can be solved by applyingthe Converse of the Multiplication Property of Zero, setting each factorequal to zero.For example, solve the quadratic equation x2 17x 52 0 byperforming the following steps. Perform transformations so one side of the equation is equal tozero.The quadratic equation x2 17x 52 0 already has one sideset equal to zero. Factor the quadratic expression.x2 17x 52 0(x 4)(x 13) 0 Apply the Converse of the Multiplication Property of Zero and solveeach resulting equation.x 4 0x 4 4 0 4x 4orororx 13 0x 13 13 0 13x 13 Check the solutions.(4) 2 17(4) 52 16 68 52 0(13) 2 17(13) 52 169 221 52 0Solutions: x 4 or x 131. Solve each quadratic equation by factoring.11b. x2 x 12 0510Chapter 11 Probability 2009 Carnegie Learning, Inc.a. x2 16x 15 0
CH-11.1.qxd:A1S101016/12/0911:38 AMPage 509Problem 2Solving Quadratic Equations byFactoringc. x2 12x 20 0d. x2 12x 36 0e. x2 x 42 0f. x2 16x 48 0 2009 Carnegie Learning, Inc.g. x2 9x 18 0A quadratic equation that is a difference of two squares can besolved by factoring. In addition, a quadratic equation that is adifference of two squares can be solved by isolating the squaredvariable and taking the square roots of both sides.112. Solve each quadratic equation using both methods.a. x2 81 0Lesson 11.1 Solving Quadratic Equations using Factoring and Square Roots511
CH-11.1.qxd:A1S101016/12/0911:38 AMPage 510Problem 2Solving Quadratic Equations byFactoringb. 9x2 25 0A quadratic equation that contains a perfect square trinomial canbe solved by factoring. It can also be solved by writing the perfectsquare trinomial as a factor squared, taking the square root ofboth sides, and solving for the variable.3. Solve each quadratic equation.a. x2 12x 36 0 2009 Carnegie Learning, Inc.b. 4x2 12x 9 011Be prepared to share your solutions and methods.512Chapter 11 Probability
4/11/0810:03 AMPage 38511.2Guitar Strings and Other ThingsSquare Roots and RadicalsObjectivesIn this lesson,you will: Evaluate the squareroot of a perfect square. Approximate asquare root.Key Terms square root positive square root negative square root principal square rootSCENARIOWhen you pluck a string on a guitar,the string vibrates and produces sound. When the stringvibrates, the vibrations are repeating waves of movementup and down, as shown below.If the guitar is not tuned properly, the correct notes will not beplayed, and the result may not sound musical. To tune a guitarproperly requires a change in the tension of the strings. The tensioncan be thought of as the amount of stretch on the string betweentwo fixed points. A string with the correct tension produces thecorrect wave speed, which in turn produces the correct sounds. radical symbol radicand perfect squareProblem 1Good VibrationsConsider a string that weighs approximately 0.0026 pound per inchand is 34 inches long. An equation that relates the wave speed v incycles per second and tension t in pounds is v2 t.A. Find the tension of the string if the wave speed is 9.5 cyclesper second. Show all your work and use a complete sentencein your answer.Take NoteA cycle of a wave is the 2009 Carnegie Learning, Inc.A1S10804.qxdmotion of the string upand then down one time.B. Find the tension of the string if the wave speed is 8.5 cyclesper second. Show all your work and use a complete sentencein your answer.11C. Find the tension of the string if the wave speed is 7.6 cyclesper second. Show all your work and use a complete sentencein your answer.D. What happens to the tension as the wave speed increases?Use a complete sentence in your answer.Lesson 11.2 Square Roots and Radicals51 3
A1S10804.qxd4/11/0810:03 AMPage 386Investigate Problem 11. Write an equation that you can use to find the wave speed of astring when the tension on the string is 81 pounds.What must the wave speed be? How do you know? Use acomplete sentence in your answer.2. Write an equation that you can use to find the wave speed of astring when the tension on the string is 36 pounds.What must the wave speed be? How do you know? Use acomplete sentence in your answer.3.Just the Math: Square Root Your answers to Questions 1and 2 are square roots of 81 and 36, respectively. Formally, youcan say that a number b is a square root of a if b2 a.Take NoteFinding the square rootof a number is the inverseSo, 9 is a square root of 81 because 92 81 and 6 is a squareroot of 36 because 62 36.Is there another number whose square is 81? If so, namethe number.operation of finding theIs there another number whose square is 36? If so, name thenumber.Every positive number has two square roots: a positive squareroot and a negative square root. So, you can see that thesquare roots of 81 are 9 and –9, and the square roots of 36 are6 and –6. The positive square root is called the principal squareroot. An expression such as 36 indicates that you should findthe principal, or positive, square root of 36.114. Complete each statement below.Take Note 4 25 100 49 The symbol, , is called theradical symbol. The numberunderneath a radical symbolis called the radicand.51 4Chapter 11 Modeling and Solving Quadratic Functions 2009 Carnegie Learning, Inc.square of a number.
4/30/0812:59 PMPage 387Investigate Problem 15. Each of the radicands in Question 4 is a perfect square.Can you explain why these numbers are called perfect squares?Use a complete sentence in your answer.6. Write an equation that you can use to find the wave speed of astring when the tension on the string is 42 pounds.What number represents the wave speed? Write your answer asa radical.Can you write this number as a positive integer? Why orwhy not? Use a complete sentence in your answer.7. Because 42 is not a perfect square, we have to approximatethe value of 42. To do this, we will use perfect squares.Complete the statements below.The perfect square that is closest to 42 and is less than 42 is.The perfect square that is closest to 42 and is greater than 42 is.So, 42 is between and and 42 is between and . 2009 Carnegie Learning, Inc.A1S10804.qxdEstimate 42 by choosing numbers between 6 and 7. Test eachnumber by finding its square and seeing how close it is to 42.6.42 6.52 Which number is closer to 42?11Take NoteRemember that the symbolSo, 42 .8. What is the wave speed of a string if the tension is 42 pounds?Use a complete sentence in your answer. means “is approximatelyequal to.”9. What happens to the wave speed as the tension increases?Use a complete sentence in your answer.Lesson 11.2 Square Roots and Radicals515
A1S10804.qxd4/11/0810:03 AMPage 388Investigate Problem 110. Approximate 13 to the nearest tenth. First complete eachstatement below. Show all your work. 13 13 13 11. Approximate 30 to the nearest tenth. First complete eachstatement below. Show all your work.12. Approximate 75 to the nearest tenth. First complete eachstatement below. Show all your work. 75 75 75 1151 6Chapter 11 Modeling and Solving Quadratic Functions 2009 Carnegie Learning, Inc. 30 30 30
4/17/088:04 AMPage 39711.3Kicking a Soccer BallUsing the Quadratic Formula to Solve Quadratic EquationsObjectivesIn this lesson,you will: Solve a quadraticequation by using thequadratic formula. Find the value of thediscriminant.Key Terms quadratic formula discriminantSCENARIOA friend of yours is working on a project thatinvolves the path of a soccer ball. She tells you that she hascollected data for several similar soccer kicks in a controlledenvironment (with no wind and minimum spin on the ball).She has modeled the general path of the ball using a quadraticfunction. You are interested in her model because you arestudying quadratic functions in your math class.Problem 1The Path of a Soccer BallYour friend’s model is y 0.01x2 0.6x where x is the horizontaldistance that the ball has traveled in meters and y is the verticaldistance that the ball has traveled in meters.A. Complete the table of values that shows the vertical andhorizontal distances that the ball has traveled. Copy thevalues into the table on the next page.Quantity NameUnitHorizontal distanceVertical distancemetersmetersExpression 2009 Carnegie Learning, Inc.A1S10806.qxd11B. Can you approximate from your table how far the ball traveledbefore it hit the ground? If so, describe the distance. Use acomplete sentence in your answer.Lesson 11.3 Using the Quadratic Formula to Solve Quadratic Equations517
A1S10806.qxd4/17/088:05 AMPage 398Problem 1Quantity HorizontalName distanceUnitsmetersC. Create a graph of the path of the ball on the grid below.First, choose your bounds and intervals. Be sure to labelyour graph clearly.Variable quantityLower boundUpper ldistanceThe Path of a Soccer Ball(units)D. Use your graph to determine the maximum height of the ball.Use a complete sentence in your answer.11E. What is the y-intercept of the graph? What does it represent inthis problem situation? Use a complete sentence in your answer.F. How far does the ball travel horizontally before it hits the ground?Use a complete sentence in your answer.518Chapter 11 Modeling and Solving Quadratic Functions 2009 Carnegie Learning, Inc.(label)
4/11/0810:07 AMPage 399Investigate Problem 11. In terms of the graph of the function, how can you interpret youranswer to part (F)? Use a complete sentence in your answer.2. Write an equation that you can use to algebraically find theanswer to the question in part (F).Can you visually determine the solutions to this equation?Can you solve this equation by using the methods that youlearned in the previous lesson?3.Just the Math: Quadratic Formula To solve theequation in Question 2, we can use the quadratic formula.The quadratic formula states that the solutions to theequation ax2 bx c 0 when a 0 are given byTake Notex The symbol means b b2 4ac.2a“plus or minus” and is aYou will see where this formula comes from in Chapter 13.compact way to write aFor instance, consider the equation 2x2 3x 1 0.solution. For instance,x m n is the compactnotation for x m n andWhat are the values of a, b, and c in this equation? Write youranswer using a complete sentence.x m n.To find the solutions of the equation, substitute the values fora, b, and c into the quadratic formula and simplify: 2009 Carnegie Learning, Inc.A1S10806.qxdx ( ) ( ) 2 4( ) ( )2( ) 3 4 3 11 4So, the solutions are x 3 13 1421 1 and x .44442Lesson 11.3 Using the Quadratic Formula to Solve Quadratic Equations5 19
A1S10806.qxd4/11/0810:07 AMPage 400Investigate Problem 1For each quadratic equation below, find the values of a, b, and c.Take Note5x2 6x 1 08x2 4x 6 010x2 1 0 x2 8x 2Remember that you musthave zero on one side of theequation in order to use thequadratic formula.4. Use the quadratic formula to find the horizontal distance that theball travels before it hits the ground. Show all your work. Use acomplete sentence to describe the answers that you find.5. Write an equation that you can use to find the horizontal distancethe ball has traveled when it reaches a height of five meters. 2009 Carnegie Learning, Inc.Solve the equation. Show all your work and use a completesentence in your answer.115 20Chapter 11 Modeling and Solving Quadratic Functions
4/11/0810:07 AMPage 401Investigate Problem 1Does your answer make sense? Use complete sentences toexplain your reasoning.6. For each quadratic equation below, find the value of b2 4ac.Show all your work. Write your answer as a radical.x2 7x 2 0 3x2 4x 8 08x2 8x 2 03x2 5x 2 0What can you conclude about the number of solutions of eachof the quadratic equations above? Use complete sentencesin your answer.The expression b2 4ac is called the discriminant of thequadratic formula. 2009 Carnegie Learning, Inc.A1S10806.qxdSummarySolving Quadratic EquationsIn this lesson and the previous lesson, you explored three differentmethods for solving a quadratic equation, depending on its form:11 An equation in the form (x a)(x b) 0 is solved bydetermining the value of x that makes each factor zero.This method is called factoring. An equation in the form x2 b is solved by recognizing thatx b and x b satisfy the equation x2 b. This methodis called extracting square roots. An equation in the form ax2 bx c 0 is solved by using thequadratic formula: x b b2 4ac.2aLesson 11.3 Using the Quadratic Formula to Solve Quadratic Equations5 21
4/11/0810:07 AMPage 402 2009 Carnegie Learning, Inc.A1S10806.qxd115 22Chapter 11 Modeling and Solving Quadratic Functions
4/11/0810:08 AMPage 40311.4Pumpkin CatapultUsing a Vertical Motion ModelObjectiveSCENARIOIn this lesson,you will: Write and use avertical motion model.Every year, the city of Millsboro, Delaware,holds a competition called the World Championship Punkin Chunkin,which is a pumpkin throwing competition. Participants build apumpkin catapult that hurls a pumpkin. The catapult that hurlsthe pumpkin the farthest is the winner.Key TermProblem 1 vertical motion modelTake NoteOften, in a model, whenthe independent variablerepresents time, the variablet is used instead of x.A Pumpkin CatapultYou can model the motion of a pumpkin that is released by a catapultby using the vertical motion model y 16t2 vt h, where t isthe time that the object has been moving in seconds, v is the initialvelocity (speed) of the object in feet per second, h is the initial heightof the object in feet, and y is the height of the object in feet at timet seconds.A. Suppose that a catapult is designed to hurl a pumpkin froma height of 30 feet at an initial velocity of 212 feet per second.Write a quadratic function that models the height of the pumpkinin terms of time.B. Write an equation that you can use to determine whenthe pumpkin will hit the ground. Then solve the equation.Show all your work. 2009 Carnegie Learning, Inc.A1S10807.qxd11Do both solutions have meaning in the problem situation? Usecomplete sentences to explain your reasoning.Lesson 11.4 Using a Vertical Motion Model52 3
A1S10807.qxd4/17/088:07 AMPage 404Problem 1A Pumpkin CatapultWhen does the pumpkin hit the ground? Use a completesentence in your answer.C. Complete the table of values that shows the height of thepumpkin in terms of time.Quantity NameTimeHeightUnitExpression02510121315D. Create a graph of the model to see the path of the pumpkinon the grid on the next page. First, choose your bounds andintervals. Be sure to label your graph clearly.Lower boundUpper boundInterval 2009 Carnegie Learning, Inc.Variable quantity11524Chapter 11 Modeling and Solving Quadratic Functions
4/11/0810:08 AMPage 405A Pumpkin Catapult(label)(units)Problem 1(label)(units)Investigate Problem 11. Does your answer to part (B) make sense in terms of the graph?Write your answer using a complete sentence. 2009 Carnegie Learning, Inc.A1S10807.qxd2. What is the height of the pumpkin three seconds after it islaunched from the catapult? Show all your work and usea complete sentence in your answer.11What is the height of the pumpkin eight seconds after it islaunched from the catapult? Show all your work and usea complete sentence in your answer.What is the height of the pumpkin 20 seconds after it is launchedfrom the catapult? Show all your work and use a completesentence in your answer.Lesson 11.4 Using a Vertical Motion Model52 5
A1S10807.qxd4/11/0810:08 AMPage 406Investigate Problem 1Do all of your answers to Question 2 make sense?Use a complete sentence to explain your reasoning.3. When is the pumpkin at its highest point? Show all your workand use a complete sentence in your answer.4. What is the maximum height of the pumpkin? Show all yourwork and use a complete sentence in your answer. 2009 Carnegie Learning, Inc.5. When is the pumpkin at a height of 500 feet? Show all your workand use a complete sentence in your answer.11Is your answer confirmed by your graph?52 6Chapter 11 Modeling and Solving Quadratic Functions
4/11/0810:08 AMPage 407Problem 2How Far Can the Pumpkin Go?In 2005, the winner in the catapult division of the WorldChampionship Punkin Chunkin hurled a pumpkin 2862.28 feet.A. A model for the path of a pumpkin being launched from thecatapult described in Problem 1 is y 0.00036x2 x 30,where x is the horizontal distance of the pumpkin in feet andy is the vertical distance of the pumpkin in feet. According tothe model, what is the pumpkin’s height when it has traveled500 feet horizontally? Show all your work and use a completesentence in your answer.B. What is the pumpkin’s height when it has traveled 1000 feethorizontally? Show all your work and use a complete sentencein your answer.C. What is the pumpkin’s height when it has traveled 2500 feethorizontally? Show all your work and use a complete sentencein your answer. 2009 Carnegie Learning, Inc.A1S10807.qxdD. What is the pumpkin’s height when it has traveled 3000 feethorizontally? Show all your work and use a complete sentencein your answer.11E. Do you think that this catapult has a chance of beating the2005 catapult winner? Use complete sentences to explainyour reasoning.Lesson 11.4 Using a Vertical Motion Model52 7
A1S10807.qxd4/11/0810:08 AMPage 408Investigate Problem 21. Algebraically determine the horizontal distance the pumpkintravels before it hits the ground. Does it beat the winner?Show all your work and use a complete sentence in your answer.2. Create a graph of the model to see the path of the pumpkinon the grid below. First, choose your bounds and intervals.Lower boundUpper bound(label)11(label)52 8Interval 2009 Carnegie Learning, Inc.(units)Variable quantityChapter 11 Modeling and Solving Quadratic Functions(units)
4/11/0810:08 AMPage 409Investigate Problem 23. Use the information from Problem 1 and this problem to findthe horizontal distance the pumpkin travels after 10 seconds.Show all your work and use a complete sentence in your answer. 2009 Carnegie Learning, Inc.A1S10807.qxd11Lesson 11.4 Using a Vertical Motion Model52 9
4/11/0810:08 AMPage 410 2009 Carnegie Learning, Inc.A1S10807.qxd11530Chapter 11 Modeling and Solving Quadratic Functions
4/30/0812:55 PMPage 41111.5Viewing the Night SkyUsing Quadratic FunctionsIn this lesson,you will: Analyze a quadraticfunction that modelsthe shape of an object.Key Terms axis of symmetrySCENARIOA telescope uses two lenses, an objective lens andan eyepiece, to enable you to magnify stars, planets, and otherobjects in the night sky. The objective lens is shaped like a parabola.Telescopes are described by the aperture (pronounced ap r-ch r ) andthe focal length. The aperture is the width of the objective lens. Thefocal length is a bit more complicated. It is the distance from thevertex (the lowest point) of the lens to a point, called the focal point,on the axis of symmetry. The focal point is the point at which lightrays coming into the telescope meet after they bounce off the lens.eObjectiveelight ray vertex domain rangefocal pointfocal lengthobjective lensapertureThe shape of the lens can be described by the quadratic function1 2x , where x is the number of units to the right of the axisy 4pof symmetry, y is the height of the lens, and p is the focal length. 2009 Carnegie Learning, Inc.A1S10808.qxdProblem 1Size of the LensA. The aperture of one telescope model is 6 inches and the focallength of the objective lens is 48 inches. Write the function thatrepresents the shape of the lens.11B. What is the axis of symmetry of the graph of the lens?Use a complete sentence in your answer.C. What point is the vertex of the lens? Use a complete sentence inyour answer.Lesson 11.5 Using Quadratic Functions531
A1S10808.qxd4/17/088:18 AMPage 412Problem 1Size of the LensD. What is the domain of this function in the problem situation?Use a complete sentence in your explanation.Investigate Problem 11. Complete the table of values that shows the shape of the lens.Use your domain as a guide for choosing the x-values.If necessary, round your answers to the nearest hundredth.Quantity NameHorizontal positionHeightUnitExpression11Use complete sentences to explain how you found your answerto Question 2.3. What is the maximum height of this lens? If necessary,round your answer to the nearest thousandth. Use a completesentence to explain how you found your answer.532Chapter 11 Modeling and Solving Quadratic Functions 2009 Carnegie Learning, Inc.2. What is the range of the function? Show all your work and use acomplete sentence in your answer.
4/11/0810:09 AMPage 413Investigate Problem 14. Create a graph of the quadratic function on the grid below.First, choose your bounds and intervals. Be sure to labelyour graph clearly.Lower boundUpper boundInterval(label)(units)Variable quantity 2009 Carnegie Learning, Inc.A1S10808.qxd(label)Problem 2(units)Increase the Focal LengthA. Now consider a different telescope. This telescope has the sameaperture as the one in part (A), but its focal length is 56 inches.Write the function that represents the lens shape.B. What is the axis of symmetry of the graph of the lens?Use a complete sentence in your answer.Lesson 11.5 Using Quadratic Functions53311
A1S10808.qxd4/17/088:22 AMPage 414Problem 2Increase the Focal LengthC. What is the vertex of the lens? Use a complete sentencein your answer.D. What is the domain of this function in the problem situation?Use a complete sentence in your explanation.Investigate Problem 21. Complete the table of values that shows the shape ofthe lens. Use your domain as a guide for choosing the x-values.If necessary, round your answers to the nearest thousandth.Quantity NameHorizontal positionHeightUnit2. What is the range of the function? Show all your work and use acomplete sentence in your answer.11Use complete sentences to explain how you found your answerto Question 2.534Chapter 11 Modeling and Solving Quadratic Functions 2009 Carnegie Learning, Inc.Expression
4/11/0810:09 AMPage 415Investigate Problem 23. What is the maximum height of this lens? If necessary,round your answer to the nearest thousandth. Use acomplete sentence to explain how you found your answer.4. Create a graph of the quadratic function on the grid below.First, choose your bounds and intervals. Be sure to labelyour graph clearly.Lower boundUpper boundInterval(label)(units)Variable quantity 2009 Carnegie Learning, Inc.A1S10808.qxd11(label)(units)Lesson 11.5 Using Quadratic Functions535
A1S10808.qxd4/11/0810:09 AMPage 416Investigate Problem 25. What is the difference in the widths of the lenses? What is thedifference in the heights of the lenses? Use complete sentencesin your answer.6. How does the focal length affect the shape of the lens?Use a complete sentence in your answer. 2009 Carnegie Learning, Inc.Use complete sentences to explain how you found your answerto Question 6.11536Chapter 11 Modeling and Solving Quadratic Functions
Aug 11, 2009 · square root p. 514 positive square root p. 514 negative square root p. 514 principal square root p. 514 radical symbol p. 514 radicand p. 514 perfect square p. 515 quadratic formula p. 519, 521 discriminant p. 521 factoring p. 521 extracting square roots p. 521 verti
fabulas.qxd:Leyendas_Becquer_CAT.qxd 26/4/10 15:22 Página 14. 15 No hay que imitar a la cigarra ya que siempre llega el in-vierno en la vida y nos falta lo que despreciamos otro tiempo. Pero tampoco se debe ser tan poco caritativo como la hormiga, porque compartir las cosas da mucho gusto.
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Analisis Real 2 Arezqi Tunggal Asmana, S.Pd., M.Pd. i ANALISIS REAL 2: TERJEMAHAN DAN PEMBAHASAN Dari: Introduction to Real Analysis (Fourt Edition) Oleh: Robert G. Bartle dan Donald R. Sherbert Oleh: Arezqi Tunggal Asmana, S.Pd., M.Pd. Bagi: Para Mahasiswa Para Guru atau Dosen Tahun 2018 . Analisis Real 2 Arezqi Tunggal Asmana, S.Pd., M.Pd. ii KATA PENGANTAR Puji syukur kami haturkan .
