Chapter 11 Quadratic Functions

Transcription

1 Chapter 11 Quadratic Functions Mathematical Overview The relationship among parabolas, quadratic functions, and quadratic equations is investigated through activities that eplore both the geometric and algebraic properties of parabolas. Students look at the characteristics of three special cases of the quadratic function in standard form ( a 2 b c, where a 0), and at using graphs, factoring, and the quadratic formula as methods for solving quadratic equations. This chapter eamines how parabolic shapes focus energ, how quadratic functions can be used to model real-world situations, and how the solutions to quadratic equations can help answer questions about real-world quadratic models. Lesson Summaries Lesson 11.1 Investigation: Parabolas A Geometric Look In this Investigation, students use focus-directri paper to investigate the geometric properties of parabolas that make it possible for parabolic-shaped objects to focus sound, light, radio waves, etc. to a single point. Students eamine the location of the focus, directri, ais of smmetr, verte, points on the graph, and whether the graph is increasing or decreasing. With the information the gather, students write an equation of the form 2 4p for the parabola. Students etend their knowledge of transformations to include writing equations of the form ( h) 2 4p( k) for parabolas. Lesson 11.2 Activit: Parabolas An Algebraic Look In this Activit, students construct phsical models of a suspension bridge cable on a coordinate grid. The answer questions about the geometric characteristics of parabolas introduced in Lesson Students then write an equation that models the curve formed when weights are added to the chains in their model. The make connections between the equation and its graph. The also use their algebraic skills to determine that the equation is a quadratic function that represents a smooth U-shaped curve called a parabola. Lesson 11.3 Investigation: Graphing Quadratic Functions This Investigation is designed to help students make the connection between a quadratic function and its graph. Students are given a function in standard form a 2 b c, where a 0. The locate the verte, create a table of values, and then sketch the graph. The effects that the constants a, b, and c have on a graph are eplored b looking at three special cases: a 2, a 2 c, and a 2 b. The Investigation concludes with a summar of important concepts and skills that can be used to help sketch the graph of an quadratic function. Lesson 11.4 R.A.P. In this lesson, students Review And Practice solving problems that require the use of skills and concepts taught in previous math levels. The skills reviewed in this lesson are skills that are needed as a basis for solving problems throughout this course. Lesson 11.5 Polnomial Arithmetic In this lesson, students etend their stud of monomials from Chapter 10 to adding, subtracting, and multipling algebraic epressions that are monomials or sums of monomials (polnomials). Students use the acronm F.O.I.L. and visual models known as area models to assist them when multipling binomials. Lessons 11.6 Investigation: Solving Quadratic Equations and 11.7 Investigation: The Quadratic Formula In Lesson 11.6, students solve quadratic equations both graphicall and algebraicall. The eamine how the intercepts of the graph of a quadratic function relate to the factors in the factored form of the function set equal to 0. In Lesson 11.7, students use the quadratic formula to solve quadratic equations. The investigate the discriminant to find the connections among the value of the discriminant, the number of roots of the equation, and the graph of the related function. Lesson 11.8 Modeling with Quadratic Functions In this lesson, students use quadratic models to describe real-world situations involving free-fall motion and the relationship between time and the height of an object. Chapter 11 Etension: Eploring Quadratic Data In the first section of this Etension, students eamine distance-versus-time graphs of data collected b a motion detector. In the second section, students use first and second differences to distinguish between linear and quadratic data given in tables. 373a

2 Lesson Guide Lesson/Objectives Chapter 11 Opener: Is It a Parabola? recognize the usefulness of parabolas in the dail lives of people Investigation: Parabolas A Geometric Look graph a parabola given the focus and directri. write an equation of a parabola when the distance from the verte to the focus is known. identif the verte and ais of smmetr of a parabola. use an equation of a parabola to model a real-world situation Activit: Parabolas An Algebraic Look identif the mathematical properties of a model of a suspension bridge cable. find an equation of a parabola that passes through the origin given a point on the graph and the other -intercept Investigation: Graphing Quadratic Functions find the coordinates of the verte of a graph of a quadratic function. sketch the graph of a quadratic function. identif whether a parabola opens upward or downward b eamining the function. identif the range and domain of a given quadratic function R.A.P. solve problems that require previousl learned concepts and skills Polnomial Arithmetic add and subtract polnomials. multipl a polnomial b a monomial. multipl two binomials Investigation: Solving Quadratic Equations use graphs to solve quadratic equations. solve quadratic equations b factoring. make connections between the roots of a quadratic equation and the zeros of the graph of the related quadratic function Investigation: The Quadratic Formula use the quadratic formula to solve quadratic equations. select and defend a method of solving a quadratic equation Modeling with Quadratic Functions use a quadratic function to model a real-world situation. Chapter 11 Etension: Eploring Quadratic Data investigate distance-versus-time graphs. use first and second differences to identif quadratic functions. Per pair: focus-directri paper or Handout 11A (several copies) ruler grid paper Materials Per group: 1 -inch grid paper 4 bulletin board or cardboard straight pins or tacks paper clips lightweight chain weights (e.g., large metal washers) Per pair: grid paper graphing calculator or computer Per student: graphing calculator Per group: graphing calculator Per student: graphing calculator Optional: color pencils Optional: Handout 11B Optional: TRM table shell for Question 1, Part (b) TRM for Eercises 28, 29 Optional: grid paper color pencils algebra tiles Optional: grid paper TRM for Eercise 18 Optional: grid paper Optional: TRM table shells for Questions 13, 15, 17, 19 motion detector Pacing Guide Da 1 Da 2 Da 3 Da 4 Da 5 Da 6 Da 7 Da 8 Da 9 Da 10 Da 11 Da 12 Da 13 Basic p. 374, project review Standard p. 374, project review etension Block p. 374, , , , 11.8 project, review etension Supplement Support See the Book Companion Website at and the Teacher s Resource Materials (TRM) for additional resources. 373b

3 CHAPTER 11 Quadratic Functions CHAPTER 11 Quadratic Functions CONTENTS Chapter Opener: Is It a Parabola? 374 Lesson 11.1 INVESTIGATION: Parabolas A Geometric Look 375 Lesson 11.2 ACTIVITY: Parabolas An Algebraic Look 382 Lesson 11.3 INVESTIGATION: Graphing Quadratic Functions 386 Lesson 11.4 R.A.P. 390 Lesson 11.5 Polnomial Arithmetic 392 Lesson 11.6 INVESTIGATION: Solving Quadratic Equations 398 Lesson 11.7 INVESTIGATION: The Quadratic Formula 404 Lesson 11.8 Modeling with Quadratic Functions 409 Modeling Project: Designing a Hero s Fall 416 Chapter Review 417 Etension: Eploring Quadratic Data

4 CHAPTER 11 OPENER 5e Engage Lesson Objective recognize the usefulness of parabolas in the dail lives of people. Vocabular none Description This reading helps students understand the applications of the reflective properties of parabolas. After students have finished reading the Chapter Opener, have them discuss the difference between the two diagrams on the page. Then ask how the could get a person who is standing a long distance awa to hear what the are saing without moving closer to that person. If the respond that the could cup their hands around their mouths, ask them how the think the should shape their hands. Light Source Is It a Parabola? You ma alread know that a mathematical curve called a parabola is used in the design of structures such as suspension bridges and in the modeling of projectile motion. But did ou know that the same curve has a reflection or focusing propert? When a parabola is rotated around its line of smmetr, it forms a three-dimensional surface. This surface is used in the design of car headlights and satellite dishes because of its reflection propert. The reflection propert of the parabola can be shown using a parabolic surface that is a mirror as shown in the figure to the left. If a light is placed at a particular point, all of the light ras will reflect off the mirror in lines parallel to the line of smmetr. Car headlights, flashlights, and searchlights demonstrate this propert. This reflection propert also works in reverse, as shown in the figure below. If energ such as light or sound comes to a parabolic receiver, such as a satellite dish, it is reflected b the surface of the receiver to a single point. Applications of this propert are used in communication sstems, radar sstems, reflector microphones found on the sidelines of football games, telescopes, and solar furnaces. Receiver Another application that is familiar to most people is the cupping of our hands around our ears in order to hear distant sounds. Believe it or not, the ideal shape for our hands is a parabola! In this chapter, ou will eplore both the geometric and algebraic properties of this special curve. 374 Chapter 11 QUADRATIC FUNCTIONS 374

5 Lesson 11.1 INVESTIGATION: Parabolas A Geometric Look In addition to focusing sound and light, parabola-shaped objects can focus radio waves and other forms of energ. In this lesson, ou will eamine the geometric properties of parabolas that make this possible. As ou saw in the Chapter Opener, the shape of a parabola makes it ideal for applications that involve reflecting and focusing. Telescopes, satellite dishes, car headlights, cameras, and even flashlights are just a few of the things that depend on the reflection (or focusing) propert of parabolas. The abilit of a parabola to focus energ is a direct result of its special geometric properties. A parabola can be defined as the set of points in a plane that are equidistant from a given line and a given point not on the line. The given line is called the directri of the parabola, and the given point is called the focus. parabola focus directri In this Investigation, ou will create a graph of a set of points in a plane that are equidistant from a given line and a given point not on the line, and eamine some of the special characteristics of our parabola. 1. Eamine a piece of focus-directri paper. What do ou notice about the paper? PARABOLAS A GEOMETRIC LOOK Lesson Lesson 11.1 Investigation Answers 1. Sample answer: It has several parallel, horizontal lines and there are several concentric circles with their centers at the point shown on the paper. The distance between the circles is equal to the distance between the parallel lines. 5e Eplore Lesson Objectives graph a parabola given the focus and directri. write an equation of a parabola when the distance from the verte to the focus is known. identif the verte and ais of smmetr of a parabola. use an equation of a parabola to model a real-world situation. Vocabular ais of smmetr directri focus parabola verte (of a parabola) Materials List focus-directri paper or Handout 11A rulers grid paper LESSON 11.1 Description This lesson works best as a twoda lesson with students working in pairs. On Da 1 have students work through the Investigation, Questions In this lesson, students are given the geometric definition of a parabola. Then during the Investigation, the use focus-directri paper (Handout 11A) to create several parabolas and investigate their properties. Once all groups are finished, debrief the results of the Investigation and go over the Eample given. Wrapping Up the Investigation: On Da 2, have students work with a partner on the eercises. Make sure that plent of focus-directri paper is available for students who wish to eplore. 375

6 LESSON See students work. 3. The distance from the focus to the point is one unit and the distance from the directri to the point is one unit, so the distances are equal. 2. On our piece of focus-directri paper, draw a horizontal line that is two units below the point at the center of the circles. Label the center point focus and the line directri. You will use this focus and directri to create a parabola. 3. In the figure shown below, a second point is shown in red. What can ou sa about the distance between this point and the focus and the distance between this point and the directri? Drawing a graph similar to the one below ma be helpful for students who are having difficultl locating points on the parabola. focus directri 4. Use the circles and lines on the paper to help ou locate at least ten more points that are equall distant from the focus and the directri. Find points that are to the left and to the right of the focus. (Recall that the distance between a point and a line is the length of the perpendicular from the given point to the line.) 5. Connect the points with a smooth curve. Since all points on our curve are equidistant from a given point (the focus) and a given line (the directri) all of the points ou plotted lie on the graph of a parabola. 6. Describe our graph. 7 a. Does the graph show line smmetr? If so, describe the line of smmetr. b. For a parabola, the line of smmetr of the graph is called the ais of smmetr. Draw the line of smmetr for our graph and label it ais of smmetr. 8. Now draw a horizontal ais and vertical ais on our paper. Place the origin where our ais of smmetr intersects the parabola. This lowest point on this graph is the minimum value and is called the verte of the parabola. Label the horizontal ais and the vertical ais. 9. Does our graph represent a function? Eplain. In Question 4, students ma struggle finding points on the graph of the parabola. If so, begin b giving them hints. For eample, have them use the circles to locate points that are two units from the focus. Then have them use the lines to locate points that are two units from the directri. Finall ask them to locate the two points that are two units from the focus and two units from the directri. 376 Chapter 11 QUADRATIC FUNCTIONS 4. Sample graph: focus 5. Sample graph: 6. Sample answer: It is U shaped and opens upward. 7a. Yes; the line of smmetr passes through the focus and is perpendicular to the directri. 7b 9. See answers on page 377. directri focus directri 376

7 10 a. Eamine onl the portion of the graph where 0. Look at the curve from left to right. Is the graph increasing or decreasing? b. Now eamine onl the portion of the graph where 0. Look from left to right at the curve. Is the graph increasing or decreasing? 11 a. What happens to the height of the graph as increases without bound? (Hint: to answer this question, look at the graph to see what happens to it as ou move farther and farther to the right along the horizontal ais.) b. What happens to the height of the graph as decreases without bound? (Hint: to answer this question, look at the graph to see what happens to it as ou move farther and farther to the left along the horizontal ais.) If a parabola has a verte at the origin, a focus at (0, p), and a directri p, the standard-form equation of the parabola is 2 4 p. Notice that the value of p can be positive or negative as p represents the directed distance from the verte to the focus. 12. Find an equation for our parabola in Question a. Use another piece of focus-directi paper. In this construction, draw the directri two units above the focus at the center of the circles. Then construct the parabola with the given focus and directri. b. Again draw a horizontal ais and vertical ais on our paper with the origin at the verte of our parabola. 14. Does the parabola open up or down? 15. What are the coordinates of the verte of this parabola? 16. Describe when the graph is increasing and when it is decreasing. (Hint: be sure to look at the curve from left to right as ou describe the curve.) 17. What is the directed distance from the verte to the focus? 18. What is the value of p? 19. Write an equation for the parabola. LESSON 11.1 In Question 10a and 10b, students will likel need help understanding the concept of increasing/decreasing and alwas looking from left to right because increasing/decreasing refers to what does as increases. It ma be helpful for some students to use a piece of paper to cover the portion of the graph that the are not eamining. 10a. increasing 10b. decreasing 11a. It increases. 11b. It increases. 12. p 1, so the equation is 2 4(1) or a. Sample graph: directri focus 13b. Sample graph: PARABOLAS A GEOMETRIC LOOK Lesson directri 7b. Sample graph: 8. Sample graph: focus ais of smmetr focus directri verte focus directri 9. Yes, for each input value, there is eactl one output value. 14. down 15. (0, 0) 16. When 0, the graph is increasing. When 0, the graph is decreasing

8 LESSON 11.1 ADDITIONAL EXAMPLE The verte of a parabola is at the origin and its focus is located at the point (0, 7). Find equations for the directri and the parabola. Then graph the parabola. Directri: 7; Parabola: 2 28 Sample graph: 5 focus 2 = 28 Find equations of the directri and the parabola if the focus of the parabola is the point (0, 3) and the verte is the origin. Graph the parabola. Solution: First sketch the given information. From that information and the definition of a parabola, ou know that the directri is a horizontal line 3 units above the verte, since the focus is 3 units below the verte. So, an equation of the directri is 3. You also know that the directed distance from the verte to the focus is 3. So, p 3, and an equation of the parabola is 2 4 p 2 4( 3) 2 12 To graph the parabola, it is helpful to locate a few points on the parabola = focus 4 = Practice for Lesson 11.1 In Eercise 1, students eplore what happens to the shape of a parabola as the focus and directri are moved. If students struggle with answering these questions, have them use focus-directri paper to create parabolas as described. 1 a. The directri of the parabola in the first part of the Investigation was below the focus. What happens as ou move the focus closer to the directri? (If ou are not sure, use focus-directri paper and tr placing the directri 1 unit awa from the focus, rather than 2 units. What happens to the parabola?) b. What happens if ou move the directri farther awa from the focus? (For eample, place the directri 4 units awa from the focus, rather than 1 or 2 units.) c. How can ou create a parabola so that it opens to the right? (Hint: tr drawing a graph.) Practice for Lesson 11.1 Answers 1a. Sample answer: It appears to be more narrow. 1b. As ou move the focus awa from the directri, the parabola appears to open wider. 1c. Construct a vertical directri and place the focus to the right of it. 378 Chapter 11 QUADRATIC FUNCTIONS 378

9 d. How can ou create a parabola so that it opens to the left? e. Do the graphs in Parts (c) and (d) represent functions? Eplain. 2. Think about a parabola with its verte at the origin and the -ais as its ais of smmetr. If the parabola is shifted h units horizontall and k units verticall, then the result is a parabola with its verte at (h, k) and an ais of smmetr parallel to the -ais. The standard form of an equation for this parabola is ( h) 2 4p( k). p p ais of smmetr focus (h, k) directri a. What are the coordinates of the focus? b. What is the equation of the ais of smmetr? c. What is the equation of the directri? 3. Find equations of the directri and the parabola. 10 LESSON 11.1 If students struggle with Eercise 2, remind them of their work in Lesson 9.5 on Translations in a Coordinate Plane. If necessar, take time to review that section of Lesson 9.5 with the students. This Eercise provides a foundation for Eercises 5, 8, and 9. 1d. Construct a vertical directri and place the focus to the left of it. 1e. No, both graphs show more than one output value for some input values. Neither passes the vertical line test. 2a. (h, k p) or (h, p k) 2b. h 2c. p k or k ( p) 3. p 5; directri: 5; equation: 2 4(5); focus (0, 5) PARABOLAS A GEOMETRIC LOOK Lesson

10 LESSON Find the focus and an equation of the parabola p 4; focus: (0, 4); equation: 2 4( 4); h 3, k 4, p 6; directri: 10; equation: ( ( 3)) 2 4(6)( ( 4)); ( 3) 2 24( 4) directri: = Find an equation of the directri and an equation of the parabola. 10 ais of smmetr 5 focus ( 3, 2) verte ( 3, 4) Chapter 11 QUADRATIC FUNCTIONS 380

11 For Eercises 6 9, find the verte, the focus, and the equation of the directri for each parabola. Then draw the graph ( 1) 2 4( 3) 9. ( 2) 2 2( 5) 10. State whether the following statements are true or false. Then eplain our reasoning. Statement A: All parabolas are graphs of functions. Statement B: All equations of the form 2 4p are functions and can be represented b parabolas. 11. A satellite dish has a diameter of 20 inches and a depth of 2 inches. Where should the receiver be located? Eplain. LESSON verte: (0, 0); focus: (0, 2); directri: Eercise 11 If students struggle getting started with this eercise, have them draw a cross-section of the satellite dish on a coordinate plane. If the verte is placed at the origin, the point (10, 2) will be on the parabola. Using that information and the equation 2 4p, the can find the value of p, the distance from the verte to the focus. 7. verte: (0, 0); focus: (0, 4); directri: PARABOLAS A GEOMETRIC LOOK Lesson verte: (1, 3); focus: (1, 4); directri: Statement A is false. Sample answer: As we saw in this Investigation, some parabolas can open to the right or left. These are not functions. Statement B is true. Sample answer: The graphs of 2 4p are parabolas that open up or down and are functions. 11. Sample solution: ( 10, 2) in in. (10, 2) verte: ( 2, 5); focus: ( 2, 5.5); directri: Since the point (10, 2) is on the graph, (p)(2) and p inches. The receiver 8 should be located at the focus which is outside the satellite dish, 12.5 inches from the verte of the parabola

12 LESSON e Engage Lesson Objectives identif the mathematical properties of a model of a suspension bridge cable. find an equation of a parabola that passes through the origin given a point on the graph and the other -intercept. Vocabular quadratic function Materials List Per group: 1 -inch grid paper 4 bulletin board or cardboard tacks or pushpins lightweight chain paper clips weights such as large metal washers or a cop of Handout 11B Description Preparation: Have students work in groups of two or three. Provide each group with the materials needed for the Activit. If it is not possible to provide students with the needed materials, give each group a cop of Handout 11B. Lesson 11.2 ACTIVITY: Parabolas An Algebraic Look One of the most common tpes of bridges, especiall for spanning large distances over rivers and harbors, is the suspension bridge. There are man such bridges throughout the world, but one of the most famous is the Golden Gate Bridge in San Francisco. In this lesson, ou will create models of curved suspension bridge cables and eplore their algebraic form. In this Activit, ou will construct a small model of a suspension bridge cable and then eplore its mathematical properties. The properties are essentiall the same for all suspension bridges. 1. Use the following directions to create a model of a suspension bridge. Attach a sheet of grid paper to a piece of cardboard or a bulletin board. Choose a point near the upper-left corner of the paper as the origin. Then draw - and -aes through that point. Note that onl Quadrant IV of the coordinate plane appears on our paper. Label the aes with an convenient scale. Use the same scale on each ais. Use a pin or tack to attach one end of a chain at the origin. Then attach another link of the chain somewhere on the -ais so that the chain hangs in the shape of a suspension bridge cable. During the Activit: In this Activit, students use simple materials to create models of suspension bridge cables. Once the models are created, the answer questions about the parabolas formed b their weighted chains. Some of the questions posed in the Activit focus on the phsical characteristics of parabolas that were introduced in Lesson New to this lesson is the concept of finding an equation that can be used to model the curve. It is not intended for students to learn the form a( d ) of a quadratic function, nor is the derivation of this equation discussed in this lesson. The purpose of giving the students the equation 382 Chapter 11 QUADRATIC FUNCTIONS 382 Comap2e_Modeling_Ch11.indd 382 is to allow them to easil write an equation for a parabola and then make connections between the equation and its graph. Focus attention on Questions 5 and 6. These questions reinforce the concept that if a point lies on a graph, then it satisfies the equation of that graph and vice versa. After Question 7, a definition of quadratic functions is introduced. Continue to reinforce the concept that all quadratic functions can be modeled b a parabola, but not all parabolas can be modeled b a quadratic function. Closing the Activit: Have students compare their simulated suspension cables and the equations that model them with those of other groups. If it is not possible to have students create a model of a suspension bridge cable as described in Question 1, provide them with a cop of Handout 11B, and then have them read through Question 1 and begin the Activit with Question 2. Lesson 11.2 Activit Answers 1. See students models. 2/3/12 2:24 PM

13 To complete our model, use small hooks, such as opened paper clips, to place identical weights at equall spaced horizontal intervals along the curved chain. Caution: These weights will not be equall spaced along the chain. 2. When a chain or cable supports a load that is evenl spaced in the horizontal direction, it assumes the approimate shape of a parabola. a. Identif the two points where our parabola intersects the -ais. b. What is the verte of our parabola? c. Find an equation of the ais of smmetr. 3. Does the graph that is modeled b our chain represent a function? Eplain. 4. The equation a( d) represents a parabola that crosses the -ais at the origin and at the point (d, 0). Since our curve goes through the origin and another point on the -ais, ou can use this form to find an equation for our chain s parabolic shape. Find the equation of our parabola. (Hint: To find the value of a, substitute the coordinates of the verte and the value of d in the equation a( d).) 5. You can check to see if our function actuall describes the curved shape of our chain. Evaluate the function when 10, 15, and 20. Check to see if these (, ) pairs are on our chain. 6. Does the point (4, 10) lie on our chain? Use our equation from Question 4 to justif our answer. 7. Use the Distributive Propert to rewrite our equation from Question 4. CONNECTIONS LESSON 11.2 A true parabola will result onl from a uniform horizontal loading of a cable, that is, one where the load is continuousl applied to the cable at a constant horizontal rate. For a real suspension bridge, as well as for the model in this lesson, the load is not continuous but is applied at fied horizontal intervals. Therefore, the shape is onl approimatel parabolic. A cable or chain supporting no load at all has a slightl different shape, called a catenar, which has a mathematical form different from that of a parabola. The catenar curve is not discussed in this course as it involves the hperbolic cosine function. A quadratic function is a function that can be represented b the equation a 2 b c, where a, b, and c are real numbers and a 0. The graphs of quadratic functions are parabolas. 8. Does our model represent a quadratic function? Eplain. When students hang the weights on the chain, point out that this is intended to simulate the hanging of a road using vertical cables. The change in the shape of the chain after the weights are hung is probabl insignificant because a parabola and catenar are almost identical on this scale. PARABOLAS AN ALGEBRAIC LOOK Lesson a. A sample answer using 1 4 -inch grid paper: (0, 0) and (42, 0) 2b. Sample answer: (21, 13) 2c. Sample answer: Yes, for each input value, there is eactl one output value. It also passes the vertical line test. 4. Sample answer: For the points (0, 0), (42, 0) and (21, 13), d 42, 21, 13. So, a and the equation of the parabolic shape is ( 42). 5. The (, ) pairs should be close to points on the chain. 6. Sample answer: No, the point does not lie on the chain, because the ordered pair (4, 10) is not a solution to the equation ( 42). 7. Sample answer: Yes, it represents a function and its equation can be written in the form a 2 b c. 383

14 LESSON 11.2 ADDITIONAL EXAMPLE Find an equation of the graph of a quadratic function that crosses the -ais at (0, 0) and (12, 0) and passes through the point (4, 16). 0.5( 12) Find an equation of the graph of a quadratic function that crosses the -ais at (0, 0) and ( 8, 0) and passes through the point ( 4, 12). Solution: First sketch a graph of the parabola using the given information You know that the graph passes through the origin and that d 8. Use the point ( 4, 12) to find a. a( d ) 12 a( 4)( 4 ( 8)) 12 a( 4)(4) 12 16a 3 4 a So, an equation of the graph is 3 ( ( 8)) or 4 3 ( 8) Chapter 11 QUADRATIC FUNCTIONS 384 9a. Sample graphs: 12 = 2 = 3 = 4 = 6 6 = 5 9b. Sample answer: All but are curves. Ever graph passes through the origin and (1, 1). The functions with even powers are U-shaped, pass through the point ( 1, 1), and have the -ais as an ais of smmetr. The functions with odd powers are not U-shaped, and the all pass through ( 1, 1). 9c. Sample answer: The graph should be a U-shaped curve that passes through (0, 0), (1, 1), and ( 1, 1). Its ais of smmetr will be the -ais, and it will be narrower than the other graphs.

15 Practice for Lesson 11.2 LESSON 11.2 For Eercises 1 6, use the information shown in the figure below. Eercises 1 6 are designed to reinforce the concepts, skills, and vocabular introduced in the Activit. 1. What is the equation of the parabola in the form a( d)? 2. Use the Distributive Propert to rewrite our equation from Eercise 1 in the form a 2 b. 3. Does our equation represent a quadratic function? Eplain. 4. Use one of our equations to find the coordinates of the verte of the parabola. 5. Describe where the ais of smmetr in the parabola is located. Find its equation. 6. Which of these points lie on the parabola? (4, 4) (8, 6) (14, 7) (18, 5.4) (20, 4) 7 a. Find an equation of a quadratic function that passes through the following points: the origin, (5, 0), and (2, 6). b. Does the graph open upward or downward? c. What do ou notice about the coefficient of the 2 term of our equation? 8. The equation of a quadratic function that passes through the origin is 5( 25). a. Suppose the graph of the original function is translated 2 units to the right. Find an equation of the new function. b. Suppose the graph of the original function is translated 3 units to the left. Find an equation of the new function. c. Suppose the graph of the original function is translated h units horizontall. Find an equation of the new function. 9. It is ver difficult to just look at a graph and tell whether it is or is not a parabola because not all U-shaped graphs are parabolas. a. On the same set of aes, graph the functions, 2, 3, 4, and 5. b. What do ou notice about these graphs? c. Using our observations, how would ou epect the graph of 6 to look? PARABOLAS AN ALGEBRAIC LOOK Lesson Practice for Lesson 11.2 Answers 1. From the figure, d 24 (the distance between the origin and the second point on the -ais). Then substituting the coordinates of the point (10, 7) in the equation a( 24), the value of a can be calculated. Since a 0.05, the equation is 0.05( 24). Eercise 8 If students have difficult with the translations, suggest that the refer back to Lesson 9.5. If students ask if the translation is to the left or right in Part (c), point out that the do not need to assign a sign to h as values for h will be negative if the translation is to the left and positive if the translation is to the right. Eercise 9 How students create the graphs in this eercise does not matter. The ma use calculators, computers, or create them b hand. It is their observations that are important. 6. (4, 4), (14, 7), and (18, 5.4) lie on the parabola. 7a. ( 5) 7b. downward 7c. It is negative. 8a. 5( 2)(( 2) 25) 5( 2)( 27) 8b. 5( ( 3))(( ( 3)) 25) 5( 3)( 22) 8c. 5( h)(( h) 25) 5( h)( h 25) or 5( h)( (25 h)) 9a c. See answers on page Yes, it is of the form a 2 b c with a 0.05, b 1.2, and c The ais of smmetr is the line 12. So, the -coordinate of the verte is 12. Use that value in the equation to find the -coordinate. The verte is located at (12, 7.2). 5. It is a vertical line passing through the verte (12, 7.2). An equation of the ais is

16 386 LESSON e Eplore Lesson Objectives find the coordinates of the verte of a graph of a quadratic function. sketch the graph of a quadratic function. identif whether a parabola opens upward or downward b eamining the function. identif the range and domain of a given quadratic function. Vocabular standard form (of a quadratic function) Materials grid paper Description This Investigation works best if students work in pairs. Question 1 asks students to use a table of values to sketch the graph of a quadratic function. In all questions beond that one, students will need graphing calculators or computers as the investigate the questions. This Investigation is designed to help students make the connections between a function and its graph. In order to help students accomplish this goal, the are given the epression b for the -coordinate of the 2a verte. Verification of this appears in Eercise 17 of Lesson A summar of important concepts and skills is provided for the student at the end of the Investigation. Wrapping Up the Investigation: After going through the answers to the questions in the Investigation, focus on the summar provided at the end. Ask students how this summar can be used to help them sketch the graph of an given quadratic function. Lesson 11.3 Investigation Answers 1a. b 2 1; The 2a 2(1) -coordinate of the verte is ( 1) 2 2( 1) 4 5. So, the coordinates of the verte are ( 1, 5). Lesson 11.3 Note You will be asked to show wh this is true in Lesson 11.7, Eercise Chapter 11 1b INVESTIGATION: Graphing Quadratic Functions In Lesson 11.2, ou eplored several quadratic functions and found that their graphs are smooth, U-shaped curves called parabolas. In this lesson, ou will continue to eplore the graphs of quadratic functions. Recall that a quadratic function is a function that can be written in the standard form a 2 b c, where a 0. When sketching the graph of a quadratic function, it is helpful to first locate the verte. For a function written in standard form a 2 b c, the -coordinate of the verte is b 2a. Once ou have found the coordinates of the verte, create a table of values with several -values less than the -value of the verte and several greater than the -value of the verte. 1. Consider the function a. Find the coordinates of the verte. b. Use the function to complete this table. QUADRATIC FUNCTIONS verte c. Sketch the graph b plotting the points in our table and connecting them with a smooth curve. In the following questions, ou will investigate the effects that the constants a, b, and c, have on the graphs of quadratic functions. You will consider three special cases of the graphs of a 2 b c. In order to eplore the characteristics of these graphs, it is helpful to use either a graphing calculator or a computer. 1c. Sample graph:

17 CASE 1: a 2 (when b 0 and c 0) 2. Consider the graphs of quadratic functions whose general form is a 2. For an value of a, what are the coordinates of the verte? Eplain. 3 a. Using a calculator or computer, graph a 2 on the same set of aes using 5, 3, 1, and 0.5 for the values of a. What do ou observe as the values of a become smaller? b. Name three things that these four parabolas have in common. 4. Now graph a 2 on the same set of aes using 5, 5, 3, and 3 for the values of a. What do ou observe when a 0? What do ou observe when a 0? 5. Without graphing, how can ou tell whether the verte is a maimum or a minimum? 6. What is the domain of the function a 2? 7 a. What is the range of the function a 2 when a 0? b. What is the range of the function a 2 when a 0? CASE 2: a 2 c (when b 0) 8. Investigate the effect that the value of a has on the graph of a 2 3 b graphing a 2 3 for various values of a on the same set of aes. a. What do all of the graphs have in common? b. What happens to the graph when a 0? c. What happens to the graph when a 0? 9. Investigate the effect that the value of c has on the graph b graphing 2 2 c for various values of c on the same set of aes. (Hint: Be sure to use values of c that are greater than 0, equal to 0, and less than 0.) a. What do ou observe about the general shape of each of the graphs? b. What do ou observe about the ais of smmetr for each graph? c. What do ou observe about the vertices? 10 a. What is the -coordinate of the verte of the graph of a 2 c for an value of a or c? b. What is the -coordinate of the verte? 11. When a is positive, how man -intercepts does the graph of a 2 c have? (Hint: Be sure to consider values of c that are greater than 0, equal to 0, and less than 0.) GRAPHING QUADRATIC FUNCTIONS Lesson a. The -coordinate of the verte is 0. 10b. The -coordinate of the verte is c. 11. When a is positive and c 0, there are two -intercepts, and when c 0, there is one -intercept, and when c 0, there are no -intercepts. LESSON 11.3 In Questions 2 14, students need graphing calculators or computers so the can eplore man different graphs quickl and make conjectures from what the see. Question 5 If students do not understand the term maimum or minimum, point out that a verte is a maimum if it is the highest point, the point with the greatest -value, of the graph. The verte is a minimum if it is the lowest point, the point with the least -value, of the graph. 2. (0, 0); When b 0, the -coordinate of the verte is b 0, and when 0, 2a 0. 3a. As a gets smaller, the parabola appears wider. 3b. The verte and the ais of smmetr are the same for each parabola and each parabola opens up. 4. When a 0, the parabola opens upward. When a 0, it opens downward. 5. If a 0, the parabola opens upward and the verte is a minimum. If a 0, it opens downward and the verte is a maimum. 6. all real numbers 7a. When a 0, the range is all real numbers greater than or equal to 0. 7b. When a 0, the range is all real numbers less than or equal to 0. 8a. The verte of each graph is the same (0, 3). The ais of smmetr for each graph is the -ais. 8b. The parabola opens upward. 8c. The parabola opens downward. 9a. The general shapes of the graphs are the same. 9b. The ais of smmetr is the -ais for each graph. 9c. The verte of each graph lies on the -ais. 387

18 LESSON When a is negative and c 0, there are two -intercepts, and when c 0, there is one -intercept, and when c 0, there are no -intercepts. 13. There is onl one -intercept and its coordinates are (0, c). 14a. The graph opens upward when a 0. It opens downward when a 0. 14b. Sample answer: The ais of smmetr varies, but it is not the -ais in an of the graphs. 14c. In each graph, one of the -intercepts is the origin. Practice for Lesson 11.3 Answers 1. Sample graph: 6 2. Sample graph: When a is negative, how man -intercepts does the graph of a 2 c have? (Hint: Be sure to consider values of c that are greater than 0, equal to 0, and less than 0.) 13. How man -intercepts does the graph of a 2 c have? What are the coordinates? CASE 3: a 2 b (when c 0) 14. Consider the graphs of quadratic functions whose general form is a 2 b. Graph the function for various non-zero values of a and b. a. When does the graph open upward? When does it open downward? b. For each graph, what do ou notice about its ais of smmetr? c. For each graph, what do ou notice about the -intercepts? In this Investigation, ou found that the graphs of quadratic functions in the form a 2 b c have the following properties: There is a single maimum or minimum point called the verte that lies on the ais of smmetr. The -coordinate of that point is b 2a. The graph of a quadratic function is a U-shaped curve called a parabola. When a is positive, the parabola opens upward. When a is negative, it opens downward. The domain of a quadratic function includes all real numbers. For parabolas that open upward, the range includes all numbers greater than or equal to the -coordinate of the verte. For those that open downward, the range includes all numbers less than or equal to the -coordinate of the verte. All quadratic functions have eactl one -intercept, but there ma be zero, one, or two -intercepts. Practice for Lesson For Eercises 1 3, sketch the graph of the function Chapter 11 QUADRATIC FUNCTIONS 4 In Eercises 1 3, the sketching should be done without the help of technolog. 3. Sample graph: a. It opens upward because a 0. 7b. (2, 3); It is a minimum since the graph opens upward. 7c. the vertical line 2 7d. The domain is all real numbers. The range is all real numbers greater than or equal to 3. 7e. Sample graph: 2 8a. Answers will var. Some things that should be predicted are that the graphs have different aes of smmetr and that neither of these aes is the -ais. The have different vertices. The verte of is a minimum, and its graph opens upward. The verte of is a maimum, and its graph opens downward (1, 2); maimum 5. (0, 5); minimum 6. (2, 8); maimum

19 For Eercises 4 6, find the verte of the function and determine whether it is a maimum or minimum Consider the graph of Before graphing this function, use the observations ou made about graphs of quadratic functions in the Investigation to answer the questions in Parts (a) (d). a. Does the graph open upward or downward? Eplain how ou know. b. What are the coordinates of the verte of the graph? Is it a maimum or minimum? Eplain how ou know. c. What is the equation of the ais of smmetr of the graph? d. What is the domain of the function? What is the range? e. Graph the function. 8 a. Without graphing the functions and , predict how the differ. b. Graph each function to see if our prediction in Part (a) is correct. 9. A model rocket rises verticall so that its height h above the ground (in feet) is given b h 16t 2 300t, with time t measured in seconds. a. In how man seconds after the rocket is launched will it reach its maimum height? b. What is the maimum height that the rocket will reach before it begins its descent? 10. Man functions have graphs that are transformations of graphs of simpler functions. For eample, the graph of 2 3 is a vertical shift of 2 upward b three units. For Parts (a) (d), compare the graph of the given function to the graph of 2. a. 2 2 b. 2 c. ( 3) 2 d. ( 5) Compare the graph of the given function to the graph of 2. Let c be a positive real number. a. ( c) 2 b. ( c) 2 c. 2 c d. 2 c COMMON ERROR LESSON 11.3 Eercise 6 Some students will use 8 for a and 2 for b when finding the verte of the parabola. Remind them that a is the coefficient of the 2 term and b is the coefficient of the term in the standard form of a quadratic function regardless of the order of the terms. Eercises 7 and 8 Encourage students to answer the questions before graphing the function, as one of the goals of this lesson is to be able to predict what the graph of a quadratic function might look like without actuall drawing the graph. Eercise 10 If students have difficult recognizing the translations of the parent function 2, encourage them to graph both 2 and the given function on the same set of aes. Eercise 11 This eercise is designed to help students generalize what the discovered in Eercise 10. 9b. h 16(9.4) 2 300(9.4) 1,406 feet 10a. It is the graph of 2 shifted downward two units. 10b. It is the graph of 2 reflected in the -ais. 10c. It is the graph of 2 shifted three units to the right. 10d. It is the graph of 2 shifted five units to the left and four units up. 11a. It is the graph of 2 shifted c units to the right. 11b. It is the graph of 2 shifted c units to the left. GRAPHING QUADRATIC FUNCTIONS Lesson c. It is the graph of 2 shifted upward c units. 11d. It is the graph of 2 shifted downward c units. 8b. Sample graphs: 20 = = a. The t-coordinate of the verte of the graph is b 300 2a 2( 16) So, the maimum height is reached at about 9.4 seconds. 389

20 LESSON e Evaluate Lesson Objective solve problems that require previousl learned concepts and skills. Eercise Reference Eercises 1 2: Lesson 11.1 Eercise 3: Lesson 9.5 Eercise 4: Lesson 9.1 Eercises 5 10: Appendi A Eercises 11 13: Appendi C Eercises 14 16: Appendi M Eercises 17 22: Appendi H Eercises 23 25: Appendi D Eercises 26 27: Lesson 8.3 Eercises 28 29: Lesson 9.1 Eercises 30 32: Lesson 10.2 Eercise 33: Lesson 6.3 Lesson 11.4 R.A.P. Fill in the blank. 1. The minimum point of a parabola that opens upward is called the of the parabola. 2. The line of smmetr of the graph of a parabola is called the. Choose the correct answer. 3. Triangle XYZ has vertices X(4, 3), Y(2, 1), and Z( 1, 3). What are the coordinates of the image X Y Z if triangle XYZ is reflected in the -ais? A. X(4, 3), Y(2, 1), and Z( 1, 3) B. X( 4, 3), Y( 2, 1), and Z(1, 3) C. X( 4, 3), Y( 2, 1), and Z(1, 3) D. X(3, 4), Y(1, 2), and Z( 3, 1) 4. How man lines of smmetr does the rectangle have? A. one B. two C. four D. none Multipl or divide. Write our answer in simplest form Lesson 11.4 R.A.P. Answers 1. verte 2. ais of smmetr 3. A 4. B Chapter 11 Comap2e_Modeling_Ch11.indd 390 For Eercises 11 13, use the diagram to the left. 11. What is the ratio of the number of blue marbles to the number of red marbles? 12. What is the ratio of the number of red marbles to the total number of marbles? 13. What is the ratio of the number of ellow marbles to the number of marbles that are not ellow? Simplif the radical QUADRATIC FUNCTIONS 2/3/12 2:25 PM

21 Evaluate the epression for the given value of the variable , (6 b), b c c 2 25, c ( 1) 2, Solve t t 3, t 6 t v 2 2 2v, v n 4 n d For Eercises 26 27, consider the following sstem of equations Is (1, 4) a solution of the sstem? Eplain. 27. Is it the onl solution? Eplain. For Eercises 28 29, use the figure below. 28. Does the figure have line smmetr? If so, how man lines of smmetr does it have? If not, eplain wh not. 29. Does the figure have rotational smmetr? If so, find the angle of rotation. If not, eplain wh not. Simplif m g8 2 3h The data below show the amount of time in minutes, in the last 10 das, that a musician spent practicing his trombone. LESSON Yes, it satisfies both equations. 27. No, the sstem of equations is dependent. 28. Yes, there are si lines of smmetr. 29. Yes, the least angle of rotation is m g h 4 33a minutes 33b. lower etreme: 15 minutes; upper etreme: 80 minutes 33c. lower quartile: 30 minutes; upper quartile: 60 minutes 55, 45, 30, 20, 60, 55, 80, 60, 50, 15 a. What is the median of the data? b. Identif the lower and upper etremes. c. Identif the lower and upper quartiles. R.A.P. Lesson

22 LESSON e Eplain Lesson Objectives add and subtract polnomials. multipl a polnomial b a monomial. multipl two binomials. Vocabular binomial polnomial trinomial Description Lesson 11.5 Recall If the variable parts of two terms are eactl the same, the terms are called like terms. Polnomial Arithmetic In Chapter 10, ou worked with monomials, epressions that are the product of a coefficient and one or more variables raised to non-negative integer powers. In this lesson ou will add, subtract, and multipl algebraic epressions that are monomials or sums of monomials. A polnomial can be defined as a monomial or a sum of monomials. If a polnomial has two terms, such as 7a 6, it is called a binomial. If it has three terms, such as , it is called a trinomial. ADDING AND SUBTRACTING POLYNOMIALS To add and subtract two polnomial epressions, add or subtract the coefficients of the like terms. This lesson focuses on the skills needed to add, subtract, and multipl polnomials. The number of das spent on this lesson depends on the skill level of the students in the class. If students recall these skills from previous courses, use this as a one-da review and move on. If students struggle, ou ma want to consider spending two das teaching these skills. If that is the case, spend one da on adding, subtracting, and multipling a polnomial b a monomial. Spend the second da on multipling polnomials. ADDITIONAL EXAMPLE Add. (3t 2 t 5) (6t 2 1) 9t 2 t 4 Add. ( ) ( ) Solution: Group like terms. ( ) ( ) ( ) [7 ( 4)] ( 5 1) Combine like terms Subtract. ( ) ( 2 3 2) Solution: Add the opposite. ( ) ( 2 3 2) ( ) ( 2 3 2) Group like terms. [3 2 ( 2 )] ( 8 3) [( 7) ( 2)] Combine like terms ADDITIONAL EXAMPLE Subtract. ( 2 6 7) ( 2 2 7) Chapter 11 QUADRATIC FUNCTIONS 392

23 MULTIPLYING A POLYNOMIAL BY A MONOMIAL To multipl a polnomial b a monomial, ou can use the Distributive Propert and either a horizontal or vertical format. LESSON 11.5 Multipl. 2( ) Solution: Horizontal format: ADDITIONAL EXAMPLE Multipl. 5n(2n 4 3n 2 1) 10n 5 15n 3 5n Distributive Propert 2( ) 2(3 3 ) 2(4) 2( 5) Multipl Vertical format: Distributive Propert 2 Multipl MULTIPLYING TWO BINOMIALS To multipl a binomial b a binomial, use the Distributive Propert twice and make sure that each term of one binomial is multiplied b each term of the other binomial. Either a horizontal or a vertical format can be used. Multipl. ( 2)( 5) Solution: Horizontal format: ADDITIONAL EXAMPLE Multipl. ( 7)( 3) Distributive Propert ( 2)( 5) ( 5) 2( 5) Distributive Propert Combine like terms Vertical format: 5 2 Multipl b Multipl b. 2 5 Combine like terms POLYNOMIAL ARITHMETIC Lesson

24 LESSON 11.5 ADDITIONAL EXAMPLE A parallelogram has a base of (n 1) cm and a height of (2n 3) cm. Write an epression to represent the area of the parallelogram. 2n 2 n 3 Write an epression in the form of a trinomial that represents the volume of this rectangular solid. Solution: Formula for the volume of a rectangular solid V lwh Substitution ( 2)( 5) Distributive Propert [( 5) 2( 5)] Distributive Propert [( ) (5) 2() 2(5)] Multipl. ( ) Man students will have learned to use the F.O.I.L. pattern to multipl binomials in previous courses. The acronm F.O.I.L. provides students with a wa of making sure that each term in one binomial is multiplied b each term of the other binomial. Continue to remind students that this pattern is a wa of making sure that the Distributive Propert is used twice correctl. Combine like terms. ( ) Distributive Propert To multipl binomials mentall, make sure that each term of one binomial is multiplied b each term of the other binomial. It ma be helpful to notice the F.O.I.L. pattern where the first terms of the two binomials are multiplied, then the outer terms, then the inner terms, and finall the last terms. ADDITIONAL EXAMPLE Multipl. (3 5)(4 5) Multipl. (2 5)(3 7) Solution: Distributive Propert (2 5)(3 7) 2(3 7) 5(3 7) Distributive Propert 2(3) 2(7) 5(3) 5(7) Combine like terms product of product of product of product of the first terms the outer terms the inner terms the last terms Combine like terms Chapter 11 QUADRATIC FUNCTIONS 394

25 USING AREA MODELS TO MULTIPLY BINOMIALS You can use area models to multipl two binomials. For eample, the area model below can be used to find ( 4)( 8) To find the area of the model, find the sum of the smaller areas ( 4)( 8) So, ( 4)( 8) Practice for Lesson 11.5 LESSON 11.5 Practice for Lesson 11.5 Answers a 2b 3c a 2 7ab 6b p 2 20p t 2 28t w 2 8w m 4 11m 2 n 15n h 2 26h a 2 4ab 48b 2 Find the sum or difference. 1. ( ) ( 3 2 4) 2. (2 3 1) (4 9) 3. (2a 4b 4c) (5a 2b c) 4. ( ) ( ) 5. ( 2 10) ( 2 3 7) (3 2 8) 6. (2a 2 5ab b 2 ) (4a 2 ab 5b 2 ) (ab a 2 ) Find the product. 7. 5(p 2 4p 2) 8. 3(2 2 1) 9. (2t 6)(3t 5) 10. (6 2w)(w 7) 11. (2 3)(2 ) 12. (2m 2 5n)(m 2 3n) 13. 2(3h 2)(4h 7) 14. 4(a 4b)(a 3b) POLYNOMIAL ARITHMETIC Lesson

26 LESSON 11.5 Be sure to assign Eercises 18 and 19 as the provide students with a visual model for multipling two binomials. These area models will be helpful to some students as the review factoring in Lesson Sample answer: and Sample answer: and a. 2(6 1) 2(3 1) 18 17b. (6 1)(3 1) a. Sample answer: b c. (2 1)( 3) a Write two polnomials whose sum is Write two polnomials whose difference is a. Write an epression to represent the perimeter of the rectangle to the left. 3 1 b. Write an epression to represent the area of the rectangle. 18. The area model shown in the figure below can be used to show the product of (2 1) and ( 3) ???? a. Complete the model b replacing each question mark with the area of the indicated rectangle. b. Use our model from Part (a) to find a polnomial epression for (2 1)( 3). c. Verif our results algebraicall. 19. You can also use an area model to help ou find the square of a binomial. Draw an area model for each epression. Find the total sum of the areas of the rectangles and write our sum as a trinomial. a. ( 5) 2 b. ( 8) 2 c. Look carefull at our answers to Parts (a) and (b). Use our observations to fill in the blanks in the following: (a b) 2 The square of a b is the square of plus twice the product of plus the square of. d. Find ( 7) 2. Use an area model if needed. e. What number would ou add to the epression below to make it a perfect square trinomial? Use an area model if needed Chapter 11 QUADRATIC FUNCTIONS b c. a 2 2ab b 2 ; a; a and b; b 19d e

27 20. If ou cut the corners out of a rectangular piece of cardboard and then fold up the flaps, ou can make a bo. LESSON in. 36 in. a. Once the cardboard is folded, what are the length, width, and height of the bo in terms of? b. What is the volume of the bo? c. Assuming that the bo has no lid, what is the outside surface area of the bo? 21. The owner of a lot that is 40 feet wide b 60 feet long must give up a strip of land from one of the shorter sides of her lot for street improvements. The strip that is to be removed must be of uniform width, but it is not et known how wide the strip will be. The owner will be paid $15 per square foot for the land. a. Draw a picture to model this situation. Label the width of the strip w. b. Write an epression for the land area to be taken b the street improvements. c. Write an epression for the amount of mone that the owner will be paid. d. Write an epression for the land area that the owner will have left. 22. Multipl ( 1)(2 2 4). Eercise 20c Point out that there is more than one wa to find the surface area of the bo. One wa is to find the sum of the areas of the five sides of the bo. Another wa is to consider the area of the original rectangular piece of cardboard and subtract the areas of the four corners that were removed before folding. 20a. l (36 2) in.; w (20 2) in.; h in. 20b. V (20 2)(36 2) in. 3 20c. SA (20 2)(36 2) 2(20 2) 2(36 2) in. 2 COMMON ERROR Eercise 22 Some students ma tr to use the F.O.I.L. pattern to multipl these two epressions. Point out that the F.O.I.L. pattern onl works for multipling binomials. To multipl other polnomials, the Distributive Propert must be used. 21a. Sample answer: POLYNOMIAL ARITHMETIC Lesson ft w 40 ft 21b. 40w square feet 21c. 15(40w) 600w dollars 21d. 40(60 w) 2,400 40w

28 LESSON e Eplore Lesson Objectives use graphs to solve quadratic equations. solve quadratic equations b factoring. make connections between the roots of a quadratic equation and the zeros of the graph of the related quadratic function. Vocabular quadratic equation roots Zero Product Propert zeros Materials List none Description This lesson ma require more than one da, depending on the students backgrounds in factoring. If ou choose to make this a two-da lesson, have students work through the first two sections of the Investigation on Da 1. Then on Da 2 have them work through the third section. If students need more help with factoring, ou ma want to spend time reviewing the different methods of factoring. In this lesson, students use their calculators to formalize the connection between the zeros of a function and the factors contained in the factored form of the related quadratic equation. The also use factoring to solve quadratic equations and formalize the Zero Product Propert. Wrapping Up the Investigation: Once students have completed answering all of the questions in the Investigation, debrief b having them share their answers. Then go over the final section of the lesson, the eamples, and the steps that are used to solve a quadratic equation b factoring. It might be helpful to go over the steps as ou work through an additional eample. Lesson 11.6 Recall Factoring a quadratic epression means finding two or more epressions whose product is the given quadratic epression. For eample, the epression 2 20 can be written in factored form as ( 5)( 4). 398 Chapter 11 Lesson 11.6 Investigation Answers 1. The t-coordinate of the verte is b 0 0. The 2a 32 h-coordinate is 400. So the coordinates of the verte are (0, 400). 2. Sample graph: INVESTIGATION: Solving Quadratic Equations In Lesson 11.2, quadratic functions were used to model the shapes of curved suspension bridge cables. You can also use quadratic functions to model paths of rockets and objects thrown into the air. In this lesson, ou will use graphs and algebraic methods to solve quadratic equations. In turn, these solutions will help ou answer questions about our quadratic models. A quadratic equation is an equation that can be written in the standard form a 2 b c 0, where a 0. SOLVING QUADRATIC EQUATIONS BY GRAPHING One wa to solve a quadratic equation is to graph its related function and then find the point or points where the function crosses the -ais. 1. Suppose that an object is dropped from the top of a 400-foot building. The height of the object, in feet, t seconds after it has been dropped can be modeled b the function h 16t What are the coordinates of the verte of the graph of this function? 2. Use a graphing calculator and a viewing window of [ 6, 6] [0, 500] to graph this function. Sketch the graph. 3. Use our graph to approimate the coordinates of the points where h The t-coordinates of the points where the function crosses the horizontal ais are the solutions of the quadratic equation 16t These solutions are also called roots of the equation 16t or zeros of the function 16t What are the roots of the equation? 5. Interpret a real-world meaning for each of these roots, if possible. SOLVING QUADRATIC EQUATIONS BY FACTORING A quadratic equation can also be solved algebraicall. One wa to do this is to rewrite the equation so that one side of the equation is equal to 0. Then factor the quadratic epression on the other side, if possible. 6. Suppose the height h in feet of a rocket after t seconds is given b the function h 16t 2 80t. Write a related equation in standard form that can be solved to find the times when the height h of the rocket is 64 feet above the ground. QUADRATIC FUNCTIONS 3. Sample answer: ( 5, 0) and (5, 0) 4. 5 and 5 5. When t 5, the height of the object 0. This means that the ball hits the ground after 5 seconds. There is no real-world meaning for t 5 because the time must be positive t 2 80t 64; 16t 2 80t [ 6, 6] [0, 500]

29 7. Use a graphing calculator to graph h 16t 2 80t 64. Sketch the graph and record our viewing window. 8. At which t-values does the graph cross the horizontal ais? 9. Use a graphing calculator to make a table of values for h 16t 2 80t 64. At which t-values does h equal 0? How do these values relate to the horizontal intercepts of our graph from Question 8? 10. The factored form of 16t 2 80t 64 0 is 16(t 4)(t 1) 0. Eplain how ou know that this is a true statement. 11. How do the intercepts in Questions 8 and 9 relate to the factors (t 4) and (t 1) in the factored form of the equation in Question 10? 12. What happens when ou substitute the t-value for each intercept into the factored form of the equation? 13. When is the rocket 64 feet above the ground? 14. Consider the function 0.5( 1)( 5), which is written in factored form. a. Use a graphing calculator to graph the function. b. Where does the graph cross the -ais? c. Substitute the -intercepts into the function. What -values do the ield? d. Wh do ou think that an -intercept is called a zero? 7. LESSON 11.6 In Questions 7 and 8, students ma struggle relating the function h 16t 2 80t 64, which is written in terms of the variables h and t, to the graph and table of the related function, , given in and on the calculator. If this is the case, help students make the connections among the variables b asking which variable is the independent variable and which is the dependent variable in each equation. 14a. The factored form of a quadratic function, a( r 1 )( r 2 ), is a useful wa to write a function because the values r 1 and r 2 are the -intercepts or zeros of the graph of the function. These values are also the roots of the related quadratic equation a( r 1 )( r 2 ) 0. In the factored form of a quadratic function, the onl things that change are the numbers. In general, it can be written as:? (? )(? ) The challenge is to replace the question marks with the correct values. 15. Write a quadratic function in factored form that has -intercepts at 4 and 2. Use our graphing calculator to graph the function, sketch the graph, and record our window. 16. Revise our answer to Question 15 so that the verte of the parabola is located at the point ( 1, 3.6). Eplain how ou determined our answer. 17. Write our answer from Question 16 in standard form ( a 2 b c). Then use our calculator to displa the graph of the function in standard form and the graph of the function in factored form at the same time. Compare the graphs. SOLVING QUADRATIC EQUATIONS Lesson intercept is substituted into the function, Sample answer: This graph shows 2( 4)( 2). 8. The graph crosses the horizontal ais when t 1 or t When t 1 and t 4, h 0 in the table. These are the same values where the graph crosses the horizontal ais. 10. Sample answer: If ou find the product of 16(t 4)(t 1), ou get 16t 2 80t Sample answer: The are the values that are found when the factors are set equal to 0 and the resulting equations are solved. 12. When t 4 is substituted into 16(t 4)(t 1) 0, it makes the equation true. When t 1 is substituted into the equation, it also makes the equation true. 13. one second after the rocket was launched and 4 seconds after launch 14b. The graph crosses the -ais when 1 and when 5. 14c. When 1, 0. When 5, 0. 14d. Sample answer: When the -value of the horizontal See answers beginning on page

30 LESSON 11.6 Point out to students that this propert can be etended to more than two factors. For eample, if abc 0 then a or b or c (or an two or all three) is equal to 0. FACTORING QUADRATIC EQUATIONS When ou used factoring to solve quadratic equations, ou ma have noticed that when an epression is in factored form and one of the factors is equal to 0, then the epression is equal to 0. For eample, 4(0) 0 5( 3 3) 0 4(8 8) 0 5( 2)(0) 0 This propert is called the Zero Product Propert. Zero Product Propert If a and b are real numbers and ab 0, then a or b (or both) is equal to 0. ADDITIONAL EXAMPLE Solve for. 6 or 2 Point out that when a product of two or more factors is equal to a non-zero number, the Zero Product Propert cannot be used. First the factors must be multiplied and then the equation must be rewritten so that one side of the equation is 0. Man students will find that a diagram can help them factor trinomials. Take time to talk about the diagram that is shown in Eample 1. Point out that when the are factoring a trinomial of the form 2 b c, the are looking for two integers, r and s, such that r s b and rs c. Solve for. Solution: Original equation Add 5 to each side Factor the trinomial. Use an area model if needed. f ( 5)( 1) 0 Zero Product Propert 5 0 or 1 0 Solve each equation for. 5 1 Check: Substitute 5 and 1 for in the original equation ? 6(5) 5 1 2? 6(1) 5? ? The roots of the equation are 5 and Chapter 11 QUADRATIC FUNCTIONS If algebra tiles are available, the can be used instead of the diagram to help students factor. Both the tiles and the diagram use the same reasoning. 400

31 LESSON 11.6 Solve for. 3 Solution: Original equation Factor the trinomial. Use an area model if needed. f (2 1)( 3) 0 Zero Product Propert or 3 0 Solve each equation for Check: Substitute 1 and 3 for in the original equation ? 0? 2(3)2 7(3) ?? The roots of the equation are 1 and To solve a quadratic equation b factoring: Write the equation in standard form, a 2 b c 0. Write the equation in factored form if possible. Set each factor equal to zero. Solve each of these equations. Since each of these roots makes at least one factor equal to 0, according to the Zero Product Propert, each of these roots is a solution to the equation. 6 3 ADDITIONAL EXAMPLE Solve for. 1 or 2 3 Students ma or ma not be familiar with the following method of factoring a trinomial of the form a 2 b c. Point out that when the are factoring a trinomial of the form a 2 b c, the should look for two numbers, r and s, such that r s b and rs ac. Then the should factor as shown: a 2 (a r)(a s) b c a One (or both) of the factors in the numerator will have common factors which will divide out the a in the denominator. This will ield the standard factored form. SOLVING QUADRATIC EQUATIONS Lesson

32 LESSON 11.6 Practice for Lesson 11.6 COMMON ERROR Eercise 1 Remind students that a zero of a function is a value of that makes the value of the function 0, not the value of the function when 0. 1 a. On the same set of aes, sketch three parabolas. Sketch one that has no zeros and label it A. Sketch a second one that has two zeros and label it B. Sketch a third one that has eactl one zero and label it C. b. Is it possible to sketch a parabola with more than two zeros? If so, make a sketch of it. 2. Complete the area model to the right? to show the factors of the trinomial What are the factors? Factor the polnomial. Use an area model if needed. 2? 3. 2t 2 5t 4. m 2 12m 20?? Solve the equation b graphing. Estimate the solution if necessar k d 2 9d 25 Solve the equation b factoring n 2 7n Knowing the roots of a quadratic equation can help ou factor the equation. For eample, if the roots to the equation are 2 and 5, then 2 or 5. So the factors must be ( 2) and ( 5). You can check our factors b multipling ( 2) b ( 5). The product confirms our factors. a. The roots to the equation are 4 and 1. Use this information to factor b. The roots to the equation are 6 and 2. Use this information to factor A flare is launched from a life raft. The function h 192t 16t 2, where h represents the height of the flare in feet after t seconds, can be used to model the path of the flare. a. When is the flare 512 feet above the raft? b. How long is the flare in the air? Eercises 7 9 Point out to students that even though their answers ma look eact on their calculator graphs, the cannot be sure of their solutions until the check them b substituting the values into the original equations. Practice for Lesson 11.6 Answers 1a. Sample sketch: C A B Chapter 11 QUADRATIC FUNCTIONS 402 1b. No, it is not possible to sketch a parabola with more than two zeros. 2. The factors are 2 and t(2t 5) 4. (m 2)(m 10) 5. ( 18)( 1) 6. 2( 5)( 2) 7. 7, , , , a. ( 4)( 1) 13b. ( 6)( 2) 14a. 192t 16t 2 512; The flare is 512 feet when t 4 seconds and again when t 8 seconds. 14b. 192t 16t 2 0; The flare is on the ground when t 0 and again when t 12. It is in the air for 12 seconds.

33 15. A heav brick is tossed into the air from a height of 48 feet. The function h 16t 2 32t 48 can be used to model the height h of the brick after t seconds. How long will it take for the brick to hit the ground? 16. Solve ( 2)( 8) 40 b factoring. 17. Some quadratic equations can be solved b taking the square root of each side of the equation. For eample, Original equation 3t 2 36 Divide both sides b 3. t 2 12 Take the square root of each side. t 12 Simplif. t 2 3 Approimate the roots. t 3.5 and t 3.5 a. Solve for. b. Solve ( 4) 2 7 for. Leave our answer in radical form. 18. You can also solve quadratic equations that do not factor into perfect squares b using an algebraic method called completing the square. In this method, a quadratic epression is made into a perfect square. For eample, Original equation Subtract 4 from both sides Add 9 to both sides of the equation to 2 make one side of the equation into a perfect square trinomial. Simplif Factor the trinomial. ( 3) 2 5 Take the square root of both sides. 3 5 Subtract 3 from both sides. 3 5 The solutions in radical form are 3 5 and 3 5. a. Solve b completing the square. Leave our answer in radical form, if necessar. b. Solve b completing the square. Leave our answer in radical form, if necessar. c. Solve b completing the square. Leave our answer in radical form, if necessar LESSON 11.6 Eercise 15 Point out that there are two solutions (3 and 1) to the equation 16t 2 32t Ask students wh there is onl one solution to the problem. COMMON ERROR Eercise 16 If students tr to solve this quadratic equation and get 32 and 38 for roots, remind them that the Zero Product Propert cannot be used on this equation as it is written. One side of the equation must first be 0. Also point out the importance of checking solutions in the original equation. Eercise 18 This eercise is optional. If our curriculum includes solving quadratics b completing the square, be sure to assign this eercise and go over it in class. Point out that the method of completing the square can be used to solve an quadratic equation. COMMON ERROR SOLVING QUADRATIC EQUATIONS Lesson Eercise 18 If students fail to find the number that is needed to complete the square, have them look at the diagram that is used to complete the square. Then have them generalize the method. To find the value that completes the square for the epression 2 b, take half the b and square the result seconds ; The roots are 2 and a. 2, 2 17b a. 4, 14 18b. 5, 1 18c

34 LESSON e Eplore Lesson Objectives use the quadratic formula to solve quadratic equations. select and defend a method of solving a quadratic equation. Vocabular discriminant quadratic formula Lesson 11.7 INVESTIGATION: The Quadratic Formula In Lesson 11.6, ou learned how to solve quadratic equations both algebraicall and graphicall. In this lesson, ou will learn to use the quadratic formula to solve quadratic equations. USING THE QUADRATIC FORMULA TO SOLVE QUADRATIC EQUATIONS Not all quadratic equations are easil factored. In fact, onl a small percent of all quadratic functions that are used to model real-world situations can be factored easil. The good news is that there is an algebraic method that can be used to find the eact roots of an quadratic equation. This method uses a formula known as the quadratic formula. Materials List none Description As a class, have students read through the first part of the lesson where the quadratic formula is given to students. (Note: the derivation of this formula is given in Eercise 18.) Then have them work through Eample 1. Take a few minutes to discuss the eample before having students work in small groups or pairs to complete the Investigation portion of this lesson. In the Investigation, students eplore the usefulness of knowing the value of the discriminant of a quadratic equation in determining the number and tpes of roots for an quadratic equation. Wrapping Up the Investigation: Once students have completed answering all of the questions in the Investigation, debrief b having them share their answers. Then go over the final section of the lesson. Work through Eample 2 as a class, taking time to talk about the different methods the have used in solving quadratic equations. 404 Chapter 11 Take time to look at the two solutions that arise from the quadratic formula. Rewrite each solution as the sum (or difference) of two fractions: b b 2 4ac and 2a 2a b b 2 4ac. Then ask students 2a 2a what the notice. The should notice that the first fraction in each solution ( b 2a ) is the -coordinate of the verte. The second fraction in each solution must then be the distance from the -coordinate of the verte to the -coordinate of each zero. The Quadratic Formula If a, b, and c are real numbers and a 0, then the solutions to a quadratic equation written in standard form (a 2 b c 0) are given b the formula b b 2 4ac. 2a Notice that this formula represents two solutions: b b 2 4ac and b b 2 4ac 2a 2a Use the quadratic formula to solve the equation If necessar, round answers to the nearest hundredth. Solution: Begin b writing the equation in standard form and identifing a, b, and c. Original equation Add 4 to both sides a 5 b 11 c 4 QUADRATIC FUNCTIONS COMMON ERROR Eample 1 Students often forget to write their equations in standard form before appling the quadratic formula. Take time to point out that in this eample, the equation was written in standard form before the values of a, b, and c were determined. ADDITIONAL EXAMPLE Use the quadratic formula to solve the equation If necessar, round answers to the nearest hundredth or

35 Note Recall that the square root of a negative number is not a real number. Quadratic formula Substitute values for a, b, and c. Simplif. Separate the solutions and evaluate THE DISCRIMINANT b b 2 4ac 2a (5)(4) 2(5) In the quadratic formula, the epression under the radical sign, b 2 4ac, is called the discriminant. In the following Investigation, ou will eplore the connections among the value of the discriminant of an equation, the number of roots of the equation, and the graph of the related function. 1 a. Use the quadratic formula to solve b. How man real roots does the equation have? c. Use a calculator to graph the related function. How man -intercepts does the graph have? d. Is the value of the discriminant greater than 0, less than 0, or equal to 0? 2 a. Use the quadratic formula to solve b. How man real roots does the equation have? c. Use a calculator to graph the related function. How man -intercepts does the graph have? d. Is the value of the discriminant greater than 0, less than 0, or equal to 0? 3 a. Use the quadratic equation to solve b. How man real roots does the equation have? c. Use a calculator to graph the related function. How man -intercepts does the graph have? d. Is the value of the discriminant greater than 0, less than 0, or equal to 0? 4. Summarize what ou learned about the value of the discriminant of a quadratic equation, the number of roots of the equation, and the graph of the related function. or The approimate roots of the equation are 0.46 and LESSON 11.7 Lesson 11.7 Investigation Answers 1a. 1 2, 3 1b. two 1c. The graph has two -intercepts. 1d. greater than 0 2a b. one 2c. The graph has one -intercept. 2d. equal to 0 3a. no real solutions 3b. none 3c. none 3d. less than 0 4. Sample answer: If the discriminant is positive, then the equation has two real roots, and the graph crosses the -ais twice. If the discriminant is 0, then the equation has one real root, and the graph touches the -ais at one point. If the discriminant is negative, the equation has no real roots and the graph does not intersect the -ais. THE QUADRATIC FORMULA Lesson

36 LESSON 11.7 ADDITIONAL EXAMPLE Solve b using the quadratic formula and b factoring. 3 or 3.5 Solve the equation 2t 2 3t 5 0 b first using the quadratic formula, and then solve it b factoring. Solution: Using the quadratic formula (a 2, b 3, and c 5): Quadratic formula t b b 2 4ac 2a Substitute values for a, b, and c. ( 3) ( 3) 2 4(2)( 5) 2(2) Simplif Separate the solutions and simplif. t 3 49 or t t The roots of the equation are 1 and 2.5. Using factoring: Original equation 2t 2 3t 5 0 Factor the trinomial. f Use an area model if needed. (2t 5)(t 1) 0 Zero Product Propert 2t 5 0 or t 1 0 Solve each equation for t. t 2.5 t 1 The roots of the equation are 1 and 2.5. Even though the quadratic formula can be used to solve an quadratic equation, it ma not alwas be the most efficient method. Below are some pros and cons for the different methods ou have used to solve a quadratic equation. 2t 5 2t 2 5t +1 2t 5 If students are familiar with the Completing the Square method (Lesson 11.6, Eercise 18) of solving quadratic equations, talk about adding a fourth row to this table. When discussing the pros and cons of this method, be sure that students understand that this method works for all quadratic equations. Point out that it works best for equations in which the coefficient of the firstdegree term is an even number. 406 Method Pros and Cons graphing This method can be used sometimes, but the solutions ma not be eact. Use this method when an approimate solution is good enough. factoring This method can be used when the factors are eas to find. This method is generall quicker than the others. quadratic formula This method alwas works, and it alwas gives accurate solutions. The drawback is that other methods ma be easier in some cases. Chapter 11 QUADRATIC FUNCTIONS 406

37 Practice for Lesson 11.7 Use the quadratic formula to solve the equation. If necessar, round answers to the nearest hundredth. 1. n 2 8n t 2 8t ( 2 2) 4 0 For the equation, state the value of the discriminant. Then state the number of real roots for the equation Use either factoring or the quadratic formula to solve the equation. Round to the nearest tenth if necessar. 9. 2t 2 12t r 2 10r Without graphing, determine the -intercepts of A stone is thrown from a catapult. The function h 16t 2 80t describes the height h of the stone as a function of time t. a. How high is the stone at 3.5 seconds? b. Write a related equation in standard form that can be solved to find the times when the stone will be 84 feet above the ground. c. Using an method ou prefer, solve our equation to find the times that the stone is 84 feet above the ground. Round our answer to the nearest tenth, if necessar. d. When will the stone be 36 feet high? 15. The main cable of a suspension bridge can be modeled b the function h , where h represents the height of the cable above the roadwa and measures the horizontal distance across the bridge starting from the left tower. h At what positions will the vertical support cables be 50 feet long? THE QUADRATIC FORMULA Lesson COMMON ERROR LESSON 11.7 Eercise 8 If students think that the discriminant is 48 instead of 80, the forgot to write the equation in standard form before determining the values of a, b, and c in the discriminant. Stress that these values can be determined onl after the equation is written in standard form. Practice for Lesson 11.7 Answers 1. 5, , , , ; 1 real root 6. 15, 0 real roots ; 2 real roots 8. 80; 2 real roots 9. 2, , , , The -intercepts are 2 and 8. 14a. 84 feet high 14b. 16t 2 80t 84 0 or 16t 2 80t c. 1.5 seconds and 3.5 seconds 14d. 0.5 seconds and again at 4.5 seconds ; ; is about 47 or

38 LESSON 11.7 In Eercise 16, challenge students to eamine the function and think about what the function means when the speed s is zero miles per hour. Then challenge them to find a suitable problem domain for this situation. Eercise 17 Students were told in Lesson 11.3 that the -coordinate of the verte of the graph of a quadratic function is b. This eercise is designed 2a to show students wh this is so. Eercise 18 Unless students are proficient in the skill of completing the square, this derivation of the quadratic formula ma be difficult to follow. If ou choose to assign this eercise, take the time to go over it as a class. The following area model ma be helpful to students who need help with completing the square. 2 @ 16. The function d 0.08s 2 2s 28 can be used to model the braking distance (in feet) of a certain car traveling at a given speed s (in miles per hour). a. What braking distance does this function predict for a speed of 27 miles per hour? b. If the vehicle takes 64 feet to brake to a complete stop, what speed does the function predict? 17. The quadratic formula b b 2 4ac gives the -intercepts of 2a the points where the graph of the quadratic equation in standard form crosses the -ais. Use this information and the fact that the -coordinate of the verte is midwa between the two zeros to determine the -coordinate of the verte of the graph. (Hint: Consider the two -intercepts, b b 2 4ac and b b 2 4ac.) 2a 2a 18. The quadratic formula can be derived b solving the general form of a quadratic equation. Cop the derivation below and suppl the missing reasons. Original equation a 2 b c 0 a. a 2 b c b. Simplif. Complete the square. Factor the left side, and simplif the right side. 2 b a a 2 b a a c a 2 b a c a b2 4a 2 b2 4a c 2 a b 2a 2 b2 4ac 4a 2 c. b 2a b 2 4ac 4a 2 d. b 2a b 2 4ac 4a 2 Simplif. b b 2 4ac 2a 2a Combine the fractions. b b 2 4ac 2a The coefficient of 2 is 1. So ou can complete the square μ on the left side of the equation b adding the square of 1_ b_ 2 a. @ b 408 Chapter 11 QUADRATIC FUNCTIONS 16a. about 32.3 feet 16b. about 37.1 miles per hour 17. The -intercept of the verte is midwa between the two -intercepts. So, it is the average of the two -intercepts. b b 2 4ac b b 2 4ac 2a 2a 2 b b 2 4ac b b 2 4ac 2a 2b 4a b 2a Sample answers: a. Subtract c from each side. b. Divide each side b a. c. Take the square root of each side. d. Subtract b from each side. 2a 408

39 Lesson 11.8 Note The acceleration g due to the force of gravit is a constant value that is equal to about 32 ft/s 2 in U.S. Customar units or about 9.8 m/s 2 in SI units. Modeling with Quadratic Functions In previous lessons, ou have used quadratic functions to model such things as suspension bridge cables and the motion of an object thrown into the air. In this lesson, ou will eamine these and other applications of quadratic functions. If an object is dropped downward and has little air resistance, it will continue to speed up due to the force of gravit. Consider the velocit of an object to be positive when it is moving upward, negative when it is moving downward, and zero when it is not moving. Problems of this tpe are generall referred to as free-fall problems, in which the relationship between the height of an object and time is modeled with a quadratic function. In general, the height of an object h is given b acceleration due to gravit h 1 initial height 2 gt2 v 0 t h 0 initial velocit Consider these three functions. h 1 2 (32)t2 45t 12 h 1 2 (9.8)t2 2t 112 h 1 2 (32)t2 450 In the first function, an object whose height is given in feet is thrown or shot upward with an initial velocit of 45 ft/s from a height of 12 feet above the ground. In the second, the object, whose height is given in meters, is thrown downward with a velocit of 2 m/s from a height of 112 meters. In the third function, the object is simpl dropped from a height of 450 feet. 5e Lesson Objective use a quadratic function to model a real-world situation. Vocabular none LESSON 11.8 Eplain Description This lesson focuses on using quadratic models to describe realworld situations. Free-fall motion is described and a function that can be used to epress the height of an object in terms of time is eplained. Have students carefull read the Note on this page. Be sure to point out that the value of the constant g tells ou the units to use in free-fall problems. It is important for students to understand that the functions that are used to model freefall motion describe the height of the object vs time, and that the graph of the function does not show the path of the object. In free-fall situations, the actual motion of the object is one-dimensional because the object onl moves up and down. MODELING WITH QUADRATIC FUNCTIONS Lesson

40 LESSON 11.8 A model rocket rises verticall so that its height in feet above the ground is given b h 300t 16t 2. Have students work through Eample 1 on their own, using the quadratic formula to confirm that the solutions are correct. Also take time to interpret the two zeros in Part (b). ADDITIONAL EXAMPLE A juggler throws a ball into the air. Its height h in feet above the ground at time t is given b h 16t 2 20t 4. At what time will the ball hit the ground? It hits the ground about 1.4 seconds after it was thrown upward. a. At what time will the rocket return to the ground? b. At what time will the rocket be at a height of 1,000 feet? c. At what time will the rocket be at a height of 2,000 feet? Solution: a. At ground level, h 0. So, 300t 16t 2 0. This equation can be solved b factoring. Original equation 300t 16t 2 0 Factor. 4t(75 4t) 0 Zero Product Propert 4t 0 or 75 4t 0 Solve each equation for t. t 0 4t 75 t So, the rocket is on the ground at 0 seconds when it is launched. It returns to the ground after seconds. b. To find when the rocket is at a height of 1,000 feet, solve the equation 300t 16t 2 1,000 or 16t 2 300t 1, Using the quadratic formula, the solutions are t 4.34 seconds and t seconds. c. To find when the rocket is at a height of 2,000 feet, solve the equation 300t 16t 2 2,000 or 16t 2 300t 2, But the discriminant of this equation is negative. b 2 4ac ( 300) 2 4(16)(2,000) 38,000 This indicates that there are no real solutions to the equation. The rocket will never reach a height of 2,000 feet. 410 Chapter 11 QUADRATIC FUNCTIONS 410

41 LESSON 11.8 A model rocket blasts off from the ground and peaks in 3 seconds at 144 feet. a. When will the rocket hit the ground? b. Write a function for the height h feet of the rocket at time t seconds. c. How high will the rocket be 2 seconds after launch? d. When will the rocket again be at the same height as in Part (c)? Eplain. Solution: a. Since it takes 3 seconds for the rocket to reach its peak, it will be on the ground again 3 seconds after it peaks. So, the rocket will hit the ground 6 seconds after it blasts off. b. The roots of the function are 0 and 6, so, h a(t 0)(t 6). It is possible to find the value of a since the ordered pair (3, 144), makes the equation true. h a(t 0)(t 6) 144 a(3 0)(3 6) 144 9a 16 a Therefore, the function is h 16(t 0)(t 6) or h 16t 2 96t. c. h 16t 2 96t 16(2) 2 96(2) 128 So, 2 seconds after the rocket is launched, it will be 128 feet above the ground. d. Substitute 128 for h in h 16t 2 96t, and solve for t. h 16t 2 96t t 2 96t 0 16t t (t 2 6t 8) 0 16(t 2)(t 4) t 2 0 or t 4 0 t 2 t 4 The rocket will be 128 feet above the ground again at 4 seconds after launch. After students have worked through Eample 2, take time to reinforce how to find the equation of a parabola when the two zeros and a point are known. If students questioned the equation a( d) that was introduced in Lesson 11.2, this is a good time to discuss where it comes from. ADDITIONAL EXAMPLE A parabola crosses the -ais at the origin and has a minimum at (4, 12). a. What is the other -intercept? b. Write an equation of the parabola. a. (8, 0) b. 3 4 ( 8) or MODELING WITH QUADRATIC FUNCTIONS Lesson

42 LESSON 11.8 Eercises 1 3 If students question the units the should use in their descriptions, have them look at the coefficient of the t 2 term. If the coefficient is 1 (32) or 16, the know that the 2 are working in the Customar sstem because the acceleration of gravit in that sstem is 32 ft/s 2. If the coefficient is 1 (9.8) or 4.9, the know that the are 2 working in the metric sstem because the acceleration of gravit in that sstem is 9.8 m/s 2. If students have difficult getting started with Eercises 4 and 5, suggest that the revisit Eample 2 to see how to find the equation of a parabola when both zeros and a third point on the parabola is known. Practice for Lesson 11.8 For Eercises 1 3, describe the vertical motion of the object that is modeled b the given function. 1. h 16t h 4.9t 2 55t 1 3. h 16t 2 2t Write an equation for the parabola that passes through the point (2, 7) and has a verte at the origin. 5. Write an equation for the parabola that crosses the -ais at the points (0, 0) and (20, 0) and passes through the point ( 2, 22). 6. A golf ball hit b a professional plaer from an elevated tee follows a path given b the equation h until it hits the fairwa. For this function, represents the horizontal distance of the ball from the tee (in ards) and h is the height of the ball (in ards) measured from the level of the flat fairwa. a. What is the height of the tee? b. What is the height of the ball above the fairwa after it has traveled 100 ards horizontall from the tee? c. How far does the ball travel horizontall before it hits the fairwa? 7. The Golden Gate Bridge in San Francisco is a suspension bridge in which the two towers are about 1,280 meters apart. The two main suspension cables are attached to the tops of the towers 213 meters above the mean high-water level in the ba. The lowest point of each cable is 67 meters above the water. 1,280 meters Eercise 6 Point out to students that the equation in this problem describes the actual path (height vs distance) of an object. Previous problems that involved free fall have described one-dimensional up-and-down motion (height vs time). Practice for Lesson 11.8 Answers 1. Sample answer: An object is dropped from the height of 650 feet. 2. Sample answer: An object is projected upward into the air from a location that is 1 meter above the ground. The initial velocit is 55 m/s. 3. Sample answer: An object is thrown downward from a location that is 800 feet above the ground. The initial velocit is 2 feet per second. 4. Since the verte is at the origin, the function is of the form a 2. Using the point (2, 7), a So, an equation for the parabola is Chapter meters Tower 5. The zeros of the function are 0 and 20. So, a( 0)( 20). It is possible to find the value of a since the ordered pair ( 2, 22), makes the equation true. Because a 0.5, the function is 0.5( 0)( 20) or a. 2.5 ards 6b. When 100, h 62.5 ards. Cable QUADRATIC FUNCTIONS 0 67 meters Tower As ou saw in the Activit in Lesson 11.2, the main cables of a suspension bridge hang in the shape of a parabola. Find an equation for the shape of one of the cables of the Golden Gate Bridge. (Assume that the origin is placed at the lowest point of the cable, as shown in the figure.) 6c. The ball hits the fairwa when h 0. Solving for ields two solutions: 2.75 and The positive result is the onl root that makes sense in this contet. So, the ball travels about 303 ards horizontall. 7. If the origin is placed at the verte of the parabola, the equation is of the form a 2. One end of the cable is at the point (640, 146). So, a(640) 2 146, and a Hence, a function that models one of the cables is

43 8. The freshman class is deciding whether or not to raise the price of tickets to the spring dance. The function p d 20d 2 can be used to estimate the total profit p if the ticket price is raised d dollars. a. What profit is epected if the ticket price is not raised? b. What is the least amount the ticket price can be raised if the want to make a profit of $155? 9. A ball is shot straight up from a cannon. After 6 seconds, its maimum height is 576 feet. a. Write a function that can be used to model the height of the ball. b. Use our function to find the height of the ball 4 seconds after launch. c. At what time will the ball be 320 feet above the ground? 10. The figure below shows a computer circuit board that is 10 cm b 8 cm. The circuit components are to be printed onto a rectangular area in the interior of the board, leaving a border of uniform width. If the interior must have an area of 72 cm 2, determine the width of the border. w 10 cm w 72 cm 2 8 cm 11. A football is kicked into the air. It hits the ground 60 ards awa from where it was kicked. At its highest point, the ball is 7 ards above the ground. Assume the relationship between height and distance is quadratic. a. Sketch a graph of the relationship between height and distance. b. Label the verte with its coordinates. What does this point mean in this situation? c. Write a function that can be used to model the relationship between the height of the ball and the distance downfield. d. Assume that the ball is being kicked for a 51-ard field goal where the height of the crossbar is 10 feet. Will the ball make it through the goal posts? Eplain. COMMON ERROR 8a. Letting d 0, the profit is $75. 8b d 20d 2 ; d 1 or d 4. The least amount is d 1. Profit will be above $155 for d between 1 and 4, and below $155 again for d 4. 9a. h 16(t)(t 12) or h 16t 2 192t 9b. 512 feet 9c. at 2 and 10 seconds 10. Using (10 2w)(8 2w) 72 and solving for w, the border width is about 0.23 cm. 11a. Height (ards) LESSON 11.8 Eercise 10 If students attempt to solve the equation (10 2w)(8 2w) 72 without multipling the binomials and then writing the equation in standard form, remind them of the Zero Product Propert and point out that the propert onl works when the product of two factors is equal to 0. MODELING WITH QUADRATIC FUNCTIONS Lesson Distance (ards) 11b. Verte: (30, 7); At 30 ards downfield from where the ball is kicked, the ball is 7 ards above the plaing field. 11c. h a(d 0)(d 60); Using the point (30, 7), a 7. So the 900 function is h d d or h (d2 60d). 11d. Yes. At a distance of 51 ards downfield, it will be 3.57 ards, or feet, off the ground. 413

44 LESSON The Gatewa Arch in St. Louis, Missouri, is 630 feet high and 630 feet wide at its outermost etremes. Even though its shape is actuall a curve called a catenar, it can be approimated b a parabola. 630 Eercise 12 As students work through the problems in the Chapter Review, the ma notice that Eercise 12 in that review is ver similar to this one. Take this opportunit to compare and contrast both the problems and the equations. When students notice that the equations are not the same, ask them wh that is true. Point out that making wise decisions about where to place the origin is often helpful when creating algebraic models Find an equation for a parabola that approimates the outside edge of the arch if the origin of its graph is located on the ground at the leftmost edge of the arch. 13. The table below shows the distances required for a car to stop when braking at different speeds. Eercise 13 This eercise provides the opportunit for students to use quadratic regression to determine a quadratic function that fits given data. If quadratic regression is not part of our curriculum, this eercise ma be omitted. Speed (mph) Stopping Distance (ft) The roots of the equation are 0 and 630. So, a( 0)( 630). It is possible to find the value of a since the ordered pair (315, 630) makes the equation true. Because a , the equation is ( 0)( 630) or a. 414 Chapter 11 QUADRATIC FUNCTIONS a. Use a graphing calculator to make a scatter plot of the data. [0, 100] [0, 500] 414

45 b. Use quadratic regression on the data to find a quadratic function that best fits the data. Write the regression equation epressing stopping distance d as a function of speed s. c. Use the model found in Part (b) to predict the stopping distance for a speed of 75 miles per hour. 14. Consider the following sstem of equations: a. What kind of equation is 4 14? b. What kind of equation is 2 1? c. Solve the sstem of equations. d. Suppose that one equation of a sstem is linear and the other is quadratic. Sketch three graphical situations, one that shows no real solutions, one that shows one real solution, and one that shows two real solutions to the sstem. LESSON 11.8 Eercise 13b Point out to students that the can use a calculator to perform a quadratic regression in the same wa the performed linear regression in Lesson 7.6. The onl difference is that the must select QuadReg from the STAT CALC menu. Eercise 14 If students struggle getting started solving this sstem, remind them of the methods the used in Chapter 8. Point out that the could use substitution (solve the first equation for or solve the second equation for ) or elimination (multipl the second equation b 4 and add the two equations) to solve the sstem. MODELING WITH QUADRATIC FUNCTIONS Lesson b. d s s c. d (75) (75) mph 14a. linear 14b. quadratic 14c. (2, 3), ( 9 4, ) 14d. Sample answers: No real solutions: Two real solutions: One real solution:

46 MODELING PROJECT 5e Elaborate Materials List none Description In this project students use what the know about quadratic functions to model a free-fall situation. This project can be done in small groups or b individual students. Sample Answers This sample answer is based on the following assumptions: The height of the building is 30 feet. The stagecoach is 10 feet long and 9 feet high. The stagecoach will maintain a constant speed of 20 mph at least 50 feet before and during the fall. 1. The hero s initial height is 30 feet and the acceleration due to gravit is 32 ft/s 2. Substitute these values into the function that represents free-fall motion. h 1 2 gt2 h 0 h 1 2 (32)t2 30 h 16t The height of the stagecoach roof is 9 feet. So, solve the equation 9 16t The hero will reach the height of the stagecoach in seconds. 3. about 29.3 feet per second 4. Since the hero will be 9 feet above the ground in seconds, the mark should be set so that it will take the stagecoach seconds to get to the drop point. d (1.146 s)(29.3 ft/sec) feet So, the mark should be set at feet before the drop point. Modeling Project 416 CHAPTER 11 Chapter 11 Designing a Hero s Fall In this project ou will design a stunt for a Western movie in which the hero falls off a rooftop onto the roof of a stagecoach being drawn b a team of horses. In order to make our plans, ou will need to choose the height of the building and the dimensions (length and height) of the stagecoach. Assume that the stagecoach travels at a speed of 20 miles per hour. In order to make our stunt work so that the hero does not get hurt, it is important for the person to know when to jump. 1. What function will ou use to model the motion of the hero during the fall? Eplain our reasoning. 2. How long will it take the hero to fall from the top of the building to the top of the stagecoach? 3. In this situation, height is calculated in feet, and time is calculated in seconds. However, the speed of the stagecoach is 20 miles per hour. Convert the speed of the stagecoach to feet per second. 4. The hero should begin the fall the instant the center of the stagecoach (not including the horses) reaches a mark in the road. Where should ou place this mark so that the hero will land safel on top of the stagecoach? Recall that when an object falls freel from rest, its height can be modeled b the function h 1 2 gt2 h 0 where h is the height of the falling object, g is the acceleration due to gravit (a constant), t is time, and h 0 is the initial height of the object. QUADRATIC FUNCTIONS 416

47 Chapter 11 Review You Should Be Able to: CHAPTER REVIEW Lesson 11.1 graph a parabola given the focus and directri. Lesson 11.4 solve problems that require previousl learned concepts and skills. 5e Evaluate write an equation of a parabola when the distance from the verte to the focus is known. identif the verte and ais of smmetr of a parabola. use an equation of a parabola to model a real-world situation. Lesson 11.5 add and subtract polnomials. multipl a polnomial b a monomial. multipl two binomials. Lesson 11.6 Lesson 11.2 identif the mathematical properties of a model of a suspension bridge cable. find an equation of a parabola that passes through the origin given a point on the graph and the other -intercept. use graphs to solve quadratic equations. solve quadratic equations b factoring. make connections between the roots of a quadratic equation and the zeros of the graph of the related quadratic function. Lesson 11.3 find the coordinates of the verte of a graph of a quadratic function. sketch the graph of a quadratic function. identif whether a parabola opens upward or downward b eamining the function. identif the range and domain of a given quadratic function. Lesson 11.7 use the quadratic formula to solve quadratic equations. select and defend a method of solving a quadratic equation. Lesson 11.8 use a quadratic function to model a real-world situation. Ke Vocabular parabola (p. 375) quadratic function (p. 383) directri (p. 375) standard form (of a quadratic function) (p. 386) focus (p. 375) polnomial (p. 392) ais of smmetr (p. 376) binomial (p. 392) verte (of a parabola) (p. 376) trinomial (p. 392) CHAPTER REVIEW Chapter

48 CHAPTER REVIEW Chapter 11 Test Review Answers 1. directri, focus 2. parabola 3. downward 4. discriminant 5. verte: (0, 0); focus: (0, 3); directri: 3 6. B 7. B 8. Sample graph: ( 1 2, 0 ); The verte is a minimum. 10a. It opens downward because a 0. 10b. (1, 2); It is a maimum since the graph opens downward. 10c. the vertical line 1 10d. The domain is all real numbers. The range is all real numbers less than or equal to quadratic equation (p. 398) roots (p. 398) zeros (p. 398) Chapter 11 Test Review Chapter 11 QUADRATIC FUNCTIONS Zero Product Propert (p. 400) quadratic formula (p. 404) discriminant (p. 405) Fill in the blank. 1. A parabola can be defined as the set of points in a plane that is equidistant from a given line called the and a given point called the. 2. The graph of a quadratic function is called a(n). 3. When a quadratic function is written in standard form and a is negative, the parabola opens. 4. The value of the number under the radical sign in the quadratic formula is called the. 5. Consider the graph of Find the verte, the focus, and the equation of the directri for the parabola. 6. How man real solutions does the equation have? A. 0 B. 1 C. 2 D. infinitel man 7. A parabola crosses the -ais at (0, 0) and (22, 0), and it opens upward. Which of these points could be the verte? A. (0, 22) B. (11, 15) C. (8, 11) D. (11, 18) 8. Sketch the graph of Find the verte of Is this verte a maimum or a minimum? 10. Consider the graph of Without graphing the function, answer the following questions. a. Does the graph open upward or downward? Eplain how ou know. b. What are the coordinates of the verte of the graph? Is it a maimum or a minimum? c. What is the equation of the ais of smmetr of the graph? d. What is the domain of the function? What is the range? 418

49 11. Write an epression to represent the area of the shaded region. 4 CHAPTER REVIEW The Gatewa Arch in St. Louis, Missouri is 630 feet high. The distance between the legs of the arch at ground level is 630 feet. Its shape is actuall a curve called a catenar, but it can be approimated b a parabola. a. Assume the origin of a coordinate sstem is located at the top of the arch. What are the coordinates of the point at the lower-right edge of the right leg? b. Find an equation for a parabola that approimates the arch. (Hint: Recall that a parabola with its verte at the origin can be modeled with the function a 2.) Solve b graphing. 14. Factor the polnomial. a. t 2 5t 24 b. 3m 2 27m Solve 2 30 b factoring. 16. A penn is dropped from the top of the Tower of the Americas in San Antonio, Teas. The function h 16t can be used to model the height of the penn (h) after t seconds. About how long will it take for the penn to reach the ground? 17. Use the quadratic formula to solve State the value of the discriminant of How man real roots does the equation have? 19. A to rocket is fired in the middle of a field. The relationship between the height of the rocket and time can be modeled b the function h 16t 2 256t, where h represents the height of the rocket in feet after t seconds. When is the rocket 540 feet above the ground? 11. (2)(4) 7 12a. (315, 630) 12b. Substituting the coordinates (315, 630) for and in the equation gives ou 630 a(315) 2, so a The equation is , 3 14a. (t 8)(t 3) 14b. 3(m 4)(m 5) 15. 5, t ; about 6.8 seconds 17. 2, ; 2 real roots 19. The rocket is 540 feet above the ground at 2.5 seconds and again at 13.5 seconds after launch. CHAPTER REVIEW Chapter

50 CHAPTER 11 EXTENSION 5e Elaborate Lesson Objectives investigate distance-versus-time graphs. use first and second differences to identif quadratic functions. Vocabular first difference second difference Materials List motion detector (optional) Description In the first section of this Etension, students should work in small groups as the eplore the graphs of data collected b a motion detector. The will interpret distance-vs-time graphs, answering questions about increasing or decreasing rates. In the second section of this lesson, students use first and second differences to eplore linear and quadratic functions. CHAPTER 11 Chapter Etension Eploring Quadratic Data In this Etension ou will look at distance-versus-time graphs. For these graphs, consider time as the independent variable and distance as the dependent variable. A motion detector can track a person walking toward or awa from it. The motion detector along with a graphing calculator can be used to create a distance-versus-time graph of a person s walk. INTERPRETING DISTANCE-VERSUS-TIME GRAPHS 1. Imagine a person walking awa from a motion detector. As time increases, what happens to the person s distance from the motion detector? 2. Would ou epect the distance d versus time t graph to be a straight line or a curved line? Eplain our answer. 3. Would ou epect the graph to increase or decrease from left to right? Eplain our answer. Consider Graphs A, B, and C below. Each is a distance-versus-time graph of a person s walk. Graph A Graph B Graph C 4. In which graph(s) was the person walking awa from the motion detector? Eplain our answer. 5. In which graph(s) was the person walking at a constant rate for at least part of the time? Eplain our answer. If a motion detector is available for students or classroom demonstrations, it will be both helpful and eciting for students to use it to collect data and recreate Graphs A, B, and C before answering Questions 4 6. Also have them collect data and create distancevs-time graphs for Questions 7 and Chapter Look at Graph A. Eplain the short horizontal segments on the graph. QUADRATIC FUNCTIONS Chapter 11 Etension Answers 1. The distance increases. 2. Sample answer: If the student walks at a constant rate, it will be a straight line. It will be curved if the student speeds up or slows down. 3. Sample answer: The graph should increase from left to right because as time increases, the distance from the motion detector increases. 4. A, B, and C; As time increased, the person s distance from the detector increased. 5. A and B; The graph shows a straight line which means that for ever second, the same amount of distance was covered. 6. Sample answer. The horizontal segments indicate that the person s distance from the motion detector was not changing during those periods of time. This means that the person paused before beginning the walk and then stopped walking at the end. 420

51 7. A person walked in front of a motion detector. His graph of distance versus time is shown below. Describe his walk. CHAPTER 11 EXTENSION 8. The graph below shows a person who walked in a straight line in front of the motion detector while the detector recorded her distance from it ever 0.1 seconds for 6 seconds. COMMON ERROR In Question 8, be sure that students understand that this graph does not require an up-and-down motion b the person walking in front of the motion detector. a. Based on the graph, did this person walk toward or awa from the motion detector? Eplain our answer. b. After she began moving, did she walk at a constant rate or did she speed up or slow down? Eplain our answer. c. What rules could ou give to a walker so that he or she could walk a graph like the one in the figure? DISTINGUISHING BETWEEN LINEAR AND QUADRATIC DATA IN TABLES Two people walked in front of motion detectors at different times. Table A on the net page shows some of the data from the first person s walk. Table B on the net page shows some of the data from the second person s walk. CHAPTER EXTENSION Chapter Sample answer: He began walking as soon as the motion detector started collecting data and then walked at a constant rate. He walked toward the motion detector because the graph shows that as time increased, distance decreased. 8a. Sample answer: At first the person walked toward the detector, then walked awa. Distance was decreasing for 3 seconds and then began to increase. 8b. Sample answer: At first, she walked at a fairl constant rate, then slowed down to change direction. After changing direction, she increased her rate. 8c. Sample answer: Start about 20 feet from the motion detector. Walk toward the detector at a rate of about 5 feet per second. Graduall slow down and then begin walking awa from the device. When walking awa, graduall increase our rate to 5 feet per second. 421

52 CHAPTER 11 EXTENSION Table A (first person s walk) Time (seconds) Distance (feet) Table B (second person s walk) Time (seconds) Distance (feet) Using first and second differences can help determine a function for quadratic data. This lesson can be etended b providing students with the following information. If the -interval is equal to 1 in a table of values and the second differences are constant, the value for a in the standard form of a quadratic function, a 2 b c, is half of the constant second differences. If the first -value is 0, then the first first-difference is equal to a b. The -value when 0 is the value of c Use our graphing calculator to create a distance-versus-time scatter plot of the first person s data. Record our window and sketch our graph. 10. Use our graphing calculator to create a distance-versus-time scatter plot of the second person s data. Record our window and sketch our graph. 11. Describe the shape of the graphs for the two walks. 12. What do ou think might be a cause of the differences in the shapes of the graphs? After discussing these points, challenge students to use first and second differences to find a function that fits the data in the table below As ou have seen in previous chapters, it is often helpful to look at the differences in consecutive table values and also eamine successive ratios. 13. Add a third column to Table A and label it First Differences. Determine the differences in the dependent variable from one table value to the net and enter them in the third column. These differences are often referred to as first differences. Table A (first person s walk) If students have difficult, point out that the column of second differences is a constant 4, therefore a 2. The value for c is 2 (b inspecting the table where 0). The first number in the first difference column is 7, and So the equation is Chapter 11 Time (seconds) QUADRATIC FUNCTIONS Distance (feet) First Differences Table A (first person s walk) Time (seconds) Distance (feet) First Differences Sample answer: The first graph appears to be linear. The second graph appears to be shaped like a parabola. 12. Sample answer: The first person walked awa from the motion detector at a constant rate. The second person did not walk at a constant rate and changed directions, first walking toward the detector and then awa from it