Problem 1

Got It?

Lesson 4-3 Modeling With Quadratic Functions 209Lesson 4-3 Modeling With Quadratic Functions 209

How do you use the 3 given points?Use them to write a system of 3 equations. Solve the system to get a, b, and c.

Objective To model data with quadratic functions

When you know the vertex and a point on a parabola, you can use vertex form to write an equation of the parabola. If you do not know the vertex, you can use standard form and any three points of the parabola to fi nd an equation.

Essential Understanding Th ree noncollinear points, no two of which are in line vertically, are on the graph of exactly one quadratic function.

You and a friend are tossing a ball back and forth. You each toss and catch the ball at waist level, 3 feet high. What equation, in standard form, models the path of the ball? Explain your reasoning.

4-3 Modeling With Quadratic Functions

Writing an Equation of a Parabola

A parabola contains the points (0, 0), (21, 22), and (1, 6). What is the equation of this parabola in standard form?

Substitute the (x, y) values into y 5 ax2 1 bx 1 c to write a system of equations.

Use (0, 0). Use (21, 22). Use (1, 6).

y 5 ax2 1 bx 1 c y 5 ax2 1 bx 1 c y 5 ax2 1 bx 1 c

0 5 a(0)2 1 b(0) 1 c 22 5 a(21)2 1 b(21) 1 c 6 5 a(1)2 1 b(1) 1 c

0 5 c 22 5 a 2 b 1 c 6 5 a 1 b 1 c

Since c 5 0, the resulting system has two variables. e

a 2 b 5 22

a 1 b 5 6

Substitute a 5 2, b 5 4, and c 5 0 into standard form: y 5 2x2 1 4x 1 0. y 5 2x2 1 4x is the equation of the parabola that contains the given points.

1. What is the equation of a parabola containing the points (0, 0), (1, 22), and (21, 24)?

Use elimination.a 2 and b 4.

Try making a sketch of the path of the ball based on what you know about projectile motion.

oss ,

of of

10 ft

6 ft

Content StandardsF.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Also F.IF.4

MATHEMATICAL PRACTICES

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Problem 2

Got It?

210 Chapter 4 Quadratic Functions and Equations210 Chapter 4 Quadratic Functions and Equations

Comparing Quadratic Models

Physics Campers at an aerospace camp launch rockets on the last day of camp. Th e path of Rocket 1 is modeled by the equation h 5 216t2 1 150t 1 1 where t is time in seconds and h is the distance from the ground. Th e path of Rocket 2 is modeled by the graph at the right. Which rocket fl ew higher?

Find the maximum height of each rocket by using the models of their paths.

Rocket 1

Th e maximum height of Rocket 1 is at the vertex of the parabola.

2b

2a, f Q2 b2aR Use the vertex formula.

2150

2(216), f Q2 150

2(216)R a 5 216, b 5 150

(4.7, 352.6) Simplify.

Th e maximum height of Rocket 1 is 352.6 feet.

Rocket 2

Th e maximum height of Rocket 2 is at the vertex of the parabola.

You can use the graph to fi nd the approximate maximum height of the rocket.

400500600700800

200300

1000

2 4 86 10 14 1612Time (s)

Hei

ght

(ft)

y

x

(6, 580)

Th e maximum height of Rocket 2 is at about 580 feet.

Rocket 2 fl ew higher than Rocket 1.

2. a. Which rocket stayed in the air longer? b. What is the reasonable domain and range for each quadratic model? c. Reasoning Describe what the domains tell you about each of the models

and why the domains for the models are diff erent.

400500600700800

200300

1000

2 4 86 10 14 1612Time (s)

Hei

ght

(ft)

y

x

What property of the quadratic tells you how high the rocket fl ew?The parabolas model the rockets’ paths, so the maximums of each parabola describe how high the rockets fl ew.

STEM

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Problem 3

Got It?

Lesson 4-3 Modeling With Quadratic Functions 211

When more than three data points suggest a quadratic function, you can use the quadratic regression feature of a graphing calculator to fi nd a quadratic model.

Using Quadratic Regression

Th e table shows a meteorologist’s predicted temperatures for an October day in Sacramento, California.

A What is a quadratic model for this data?

Step 1 Enter the data.Use the 24-hour clock to represent times after noon.

Step 2 Use QuadReg. Step 3 Graph the data and the function.

A quadratic model for temperature is y 5 20.469x2 1 14.716x 2 36.121.

B Use your model to predict the high temperature for the day. At what time does the high temperature occur?

Use the Maximum feature or tables.

Predict the high temperature for the day to be 79.48F at about 3:42 p.m.

3. Th e table shows a meteorologist’s predicted temperatures for a summer day in Denver, Colorado. What is a quadratic model for this data? Predict the high temperature for the day. At what time does the high temperature occur?

8 A.M.

10 A.M.

12 P.M.

2 P.M.

4 P.M.

6 P.M.

Time

52

64

72

78

81

76

PredictedTemperature ( F)

Sacramento, CA

526472788176- - - - - -

L2L1

L3

381012141618- - - - - -

- - - - - - L3

How do you write times using a 24-hour clock?Add 12 to the number of hours past noon. So, 2 P.M. is 14:00 in the 24-hour clock.

QuadReg y = ax2 + bx + c a = –.46875 b = 14.71607143 c = –36.12142857 R2 = .9919573999

A quadratic modelis reasonable.

MaximumX=15.697148 Y=79.378709

15.415.515.615.715.815.916

79.33779.3679.37479.37979.37479.35979.336

Y1 79.3787053571

X Y1

16 represents 4 P.M.The maximum occursat approximately 15.7,or about 3:42 P.M.

6 A.M.

9 A.M.

12 P.M.

3 P.M.

6 P.M.

9 P.M.

Time

63

76

86

89

85

76

PredictedTemperature ( F)

Denver, CO

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212 Chapter 4 Quadratic Functions and Equations

Lesson CheckDo you know HOW?Find a quadratic function that includes each set of values.

1. (1, 0), (2, 23), (3, 210)

2. 2

3.5

1

3.5 27.5

0

7.5

1 2

15.5

x

y

3. 2

41.5

1

25.5

0

13.5

1

5.5

x

y

Do you UNDERSTAND? 4. Compare and Contrast How do you know whether

to perform a linear regression or a quadratic regression for a given set of data?

5. Reasoning Explain how you can determine which of the quadratic functions in Exercise 1 and Exercise 2 attains the greatest values.

6. Error Analysis Your classmate says he can write the equation of a quadratic function that passes through the points (3, 4), (5, 22), and (3, 0). Explain his error.

Practice and Problem-Solving Exercises

Find an equation in standard form of the parabola passing through the points.

7. (1, 22), (2, 22), (3, 24) 8. (1, 22), (2, 24), (3, 24) 9. (21, 6), (1, 4), (2, 9)

10. (1, 1), (21, 23), (23, 1) 11. (3, 26), (1, 22), (6, 3) 12. (22, 9), (24, 5), (1, 0)

13. 14. 15.

16. A player throws a basketball toward a hoop. Th e basketball follows a parabolic path that can be modeled by the equation y 5 20.125x2 1 1.84x 1 6. Th e table models the parabolic path of another basketball thrown from somewhere else on the court.

If the center of the hoop is located at (12, 10), will each ball pass through the hoop?

17. Physics A man throws a ball off the top of a building and records the height of the ball at diff erent times, as shown in the table.

a. Find a quadratic model for the data. b. Use the model to estimate the height of the ball at 2.5 seconds. c. What is the ball’s maximum height?

PracticeA See Problem 1.

x f(x)

1

1

2 8

1

3

x f(x)

1

1

2 8

17

17

x f(x)

1

1

2 4

4

2

See Problems 2 and 3.

x y

2

4

10 12

10

12

Time (s) Height (ft)

0

Height of a Ball

1

2

3

46

63

48

1

STEM

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

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Lesson 4-3 Modeling With Quadratic Functions 213

Determine whether a quadratic model exists for each set of values. If so, write the model.

18. f(22) 5 16, f(0) 5 0, f(1) 5 4 19. f(0) 5 5, f(2) 5 3, f(21) 5 0

20. f(21) 5 24, f(1) 5 22, f(2) 5 21 21. f(22) 5 7, f(0) 5 1, f(2) 5 0

22. a. Geometry Copy and complete the table. It shows the total number of segments whose endpoints are chosen from x points, no three of which are collinear.

b. Write a quadratic model for the data. c. Predict the number of segments that can be drawn using 10 points.

23. Think About a Plan Th e table shows the height of a column of water as it drains from its container. Use a quadratic model of this data to estimate the water level at 30 seconds.

• What system of equations can you use to solve this problem? • How can you determine if your answer is reasonable?

24. A parabola contains the points (21, 8), (0, 4), and (1, 2). Name another point also on the parabola.

25. a. Postal Rates Find a quadratic model for the data. Use 1981 as year 0.

b. Describe a reasonable domain and range for your model. (Hint: Th is is a discrete, real situation.)

c. Estimation Estimate when fi rst-class postage was 37 cents. d. Use your model to predict when fi rst-class postage will be 50 cents. Explain why

your prediction may not be valid.

26. Road Safety Th e table and graph below give the stopping distance for an automobile under certain road conditions.

a. Compare the models. What are the reasonable domain and range for each road condition?

b. Writing Explain what that means about stopping distances under certain road conditions.

ApplyB

Number of points, x

Number of segments, y

2

1

3

3

0

20

40

ElapsedTime (s)

120

83

50

WaterLevel (mm)

Water Levels

Year

Price (cents)

1981

18

1991

29

1995

32

1999

33

2001

34

2006

39

Price of First-Class Stamp

2007

41

2008

42

SOURCE: United States Postal Service

Speed (mi/h)

Stopping Distance on Dry Roadway (ft)

20

17

30

38

40

67

50

105

55

127

100125150175200

5075

250

10 30 50 70Speed (mi/h)

Stop

ping

Dis

tanc

eon

Wet

Roa

dway

(ft) y

x

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214 Chapter 4 Quadratic Functions and Equations

C 27. a. A parabola contains the points (0, 24), (2, 4), and (4, 4). Find the vertex. b. Reasoning What is the minimum number of data points you need to fi nd a

single quadratic model for a data set? Explain.

28. A model for the height of an arrow shot into the air is h(t) 5 216t2 1 72t 1 5, where t is time and h is height. Without graphing, answer the following questions.

a. What can you learn by fi nding the graph’s intercept with the h-axis? b. What can you learn by fi nding the graph’s intercept(s) with the t-axis?

ChallengeC

Mixed Review

Graph each function.

33. y 5 x2 2 6x 2 3 34. y 5 2x2 1 9x 2 4 35. y 5 3x2 2 4x 1 1

Solve each system by elimination.

36. e

x 1 y 5 7

5x2 y 5 5 37. e

2x 2 3y 5 214

3x 2 y 5 7 38. e

x2 3y 5 2

x2 2y 5 1

For Exercises 39–40, y varies directly with x.

39. If y 5 2 when x 5 5, fi nd y when x 5 2. 40. If y 5 22 when x 5 4, fi nd y when x 5 7.

Get Ready! To prepare for Lesson 4-4, do Exercises 41–43.

Simplify by combining like terms.

41. x2 1 x 1 4x 2 1 42. 6x2 2 4(3)x 1 2x 2 3 43. 4x2 2 2(5 2 x) 2 3x

See Lesson 4-2.

See Lesson 3-2.

See Lesson 2-2.

See Lesson 1-3.

Standardized Test Prep

29. Th e graph of a quadratic function has vertex (23, 22). What is the axis of symmetry?

x 5 23 x 5 3 y 5 22 y 5 2

30. Which function is NOT a quadratic function?

y 5 (x 2 1)(x 2 2) y 5 3x 2 x2

y 5 x2 1 2x 2 3 y 5 2x2 1 x(x 2 3)

31. Which is the composition f (g(x)), if f (x) 5 2x 2 3 and g(x) 5 7 1 5x?

f (g(x)) 5 4x 1 4 f(g(x)) 5 25x 2 8

f (g(x)) 5 4x 2 10 f (g(x)) 5 25x 2 10

32. Mark has 42 coins consisting of dimes and quarters. Th e total value of his coins is $6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem.

SAT/ACT

ExtendedResponse

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