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240 Chapter 4 Quadratic Functions and Factoring

1. VOCABULARY Copy and complete: The graph of a quadratic function iscalled a(n) ? .

2. Describe how to determine whether a quadratic function has aminimum value or a maximum value.

USING A TABLE Copy and complete the table of values for the function.

3. y5 4x2 4. y523x2

x 22 21 0 1 2

y ? ? ? ? ?

x 22 21 0 1 2

y ? ? ? ? ?

5. y5 1}2x2 6. y521

}3x2

x 24 22 0 2 4

y ? ? ? ? ?

x 26 23 0 3 6

y ? ? ? ? ?

MAKING A GRAPH Graph the function. Compare the graph with the graph ofy5 x2.

7. y5 3x2 8. y5 5x2 9. y522x2

10. y52x2 11. f(x)5 1}3x2 12. g(x)521

}4x2

13. y5 5x21 1 14. y5 4x2

1 1 15. f(x)52x21 2

16. g(x)522x22 5 17. f(x)5 3

}4x22 5 18. g(x)521

}5x22 2

ERROR ANALYSIS Describe and correct the error in analyzing the graph ofy5 4x2

1 24x2 7.

19.The x-coordinate of the vertex is:

x 5 b}2a5

24}2(4)

5 3

20.The y-intercept of the

graph is the value of c,

which is 7.

MAKING A GRAPH Graph the function. Label the vertex and axis of symmetry.

21. y5 x21 2x1 1 22. y5 3x2

2 6x1 4 23. y524x21 8x1 2

24. y522x22 6x1 3 25. g(x)52x2

2 2x2 1 26. f(x)526x22 4x2 5

27. y5 2}3x22 3x1 6 28. y523

}4x22 4x2 1 29. g(x)523

}5x21 2x1 2

30. f(x)5 1}2x21 x2 3 31. y5 8

}5x22 4x1 5 32. y525

}3x22 x2 4

4.1 EXERCISES

EXAMPLE 1

on p. 236for Exs. 3–12

EXAMPLE 2

on p. 237for Exs. 13–18

EXAMPLE 3

on p. 238for Exs. 21–32

HOMEWORKKEY

5WORKED-OUT SOLUTIONSon p. WS1 for Exs. 15, 37, and 57

5 TAKS PRACTICE AND REASONINGExs. 39, 40, 43, 53, 58, 60, 62, and 63

5MULTIPLE REPRESENTATIONSEx. 59

SKILL PRACTICE

WRITING

4.1 Graph Quadratic Functions in Standard Form 241

MINIMUMS OR MAXIMUMS Tell whether the function has a minimum value or amaximum value. Then find the minimum or maximum value.

33. y526x22 1 34. y5 9x2

1 7 35. f(x)5 2x21 8x1 7

36. g(x)523x21 18x2 5 37. f(x)5 3

}2x21 6x1 4 38. y521

}4x22 7x1 2

39. What is the effect on the graph of the functiony5 x2

1 2 when it is changed to y5 x22 3?

A The graph widens. B The graph narrows.

C The graph opens down. D The vertex moves down the y-axis.

40. Which function has the widest graph?

A y5 2x2 B y5 x2 C y5 0.5x2 D y52x2

IDENTIFYING COEFFICIENTS In Exercises 41 and 42, identify the values of a, b,and c for the quadratic function.

41. The path of a basketball thrown at an angle of 458 can be modeled byy520.02x2

1 x1 6.

42. The path of a shot put released at an angle of 358 can be modeled byy520.01x2

1 0.7x1 6.

43. Write three different quadratic functions whose graphshave the line x5 4 as an axis of symmetry but have different y-intercepts.

MATCHING In Exercises 44–46, match the equation with its graph.

44. y5 0.5x22 2x 45. y5 0.5x2

1 3 46. y5 0.5x22 2x1 3

A.

x

y

1

1

(0, 3)

(2, 5)

B.

(2, 1)x

y

1

1

(0, 3)

C.

(2,22)

y

(0, 0)x1

21


47. f(x)5 0.1x21 2 48. g(x)520.5x2

2 5 49. y5 0.3x21 3x2 1

50. y5 0.25x22 1.5x1 3 51. f(x)5 4.2x2

1 6x2 1 52. g(x)5 1.75x22 2.5

53. The points (2, 3) and (24, 3) lie on the graph of aquadratic function. Explain how these points can be used to find an equationof the axis of symmetry. Then write an equation of the axis of symmetry.

54. CHALLENGE For the graph of y5 ax21 bx1 c, show that the y-coordinate of

the vertex is2b2

}4a1 c.

EXAMPLE 4

on p. 239for Exs. 33–38

358

y

x

TAKS REASONING

TAKS REASONING

TAKS REASONING

TAKS REASONI G N

2425MULTIPLEREPRESENTATIONS

55. ONLINE MUSIC An online music store sells about 4000 songs each day when itcharges $1 per song. For each $.05 increase in price, about 80 fewer songs perday are sold. Use the verbal model and quadratic function to find how thestore can maximize daily revenue.

Revenue(dollars)

5Price

(dollars/song)p

Sales(songs)

R(x) 5 (11 0.05x) p (40002 80x)

56. DIGITAL CAMERAS An electronics store sells about 70 of a new model ofdigital camera per month at a price of $320 each. For each $20 decreasein price, about 5 more cameras per month are sold. Write a function thatmodels the situation. Then tell how the store can maximize monthly revenuefrom sales of the camera.

57. GOLDEN GATE BRIDGE Each cable joining the two towers on the Golden GateBridge can be modeled by the function

y5 1}9000

x22

7}15x1 500

where x and y are measured in feet. What is the height h above the road ofa cable at its lowest point?

58. A woodland jumping mouse hops along a parabolicpat 2x2

1 1.3x where x is the mouse’s horizontal position(in feet) and y is the corresponding height (in feet). Can the mouse jump overa fence that is 3 feet high? Explain.

59. MULTIPLE REPRESENTATIONS A community theater sells about150 tickets to a play each week when it charges $20 per ticket. For each$1 decrease in price, about 10 more tickets per week are sold. The theaterhas fixed expenses of $1500 per week.

a. Writing a Model Write a verbal model and a quadratic function torepresent the theater’s weekly profit.

b. Making a Table Make a table of values for the quadratic function.

c. Drawing a Graph Use the table to graph the quadratic function. Thenuse the graph to find how the theater can maximize weekly profit.

PROBLEM SOLVING

EXAMPLE 5

on p. 239for Exs. 55–58

5WORKED-OUT SOLUTIONSon p. WS1

5 TAKS PRACTICEAND REASONING

TAKS REASONING

h given by y520.

243

60. In 1971, astronaut Alan Shepard hit a golf ball onthe moon. The path of a golf ball hit at an angle of 458 and with a speed of100 feet per second can be modeled by

y52g}10,000

x21 x

where x is the ball’s horizontal position (in feet), y is the correspondingheight (in feet), and g is the acceleration due to gravity (in feet per secondsquared).

a. Model Use the information in the diagram to write functions for thepaths of a golf ball hit on Earth and a golf ball hit on the moon.

b. Graphing Calculator Graph the functions from part (a) on a graphingcalculator. How far does the golf ball travel on Earth? on the moon?

c. Interpret Compare the distances traveled by a golf ball on Earth and onthe moon. Your answer should include the following:

• a calculation of the ratio of the distances traveled

• a discussion of how the distances and values of g are related

61. CHALLENGE Lifeguards at a beachwant to rope off a rectangularswimming section. They have P feetof rope with buoys. In terms of P,what is the maximum area that theswimming section can have?

GRAPHINGCALCULATOR

In part (b), usethe calculator’szero featureto answer thequestions.

EXTRA PRACTICE for Lesson 4.1, p. 1013 ONLINE QUIZ at classzone.com

TAKS REASONING

62. TAKS PRACTICE Liz’s high score in a video game is 1200 points less thanthree times her friend’s high score. Let x represent her friend’s high score.Which expression can be used to determine Liz’s high score? TAKS Obj. 2

A 12002 3x B x2 1200}

3C x

}32 1200 D 3x2 1200

63. TAKS PRACTICE The total cost, c, of a school banquet is given byc5 25n1 1400, where n is the total number of students attending thebanquet. The total cost of the banquet was $9900. How many studentsattended the banquet? TAKS Obj. 4

F 177 G 340 H 396 J 452

MIXED REVIEW FOR TAKS

REVIEW

Lesson 1.2;

TAKS Workbook

REVIEW

Lesson 1.3;

TAKS Workbook

TAKS PRACTICE at classzone.com

4.2 Graph Quadratic Functions in Vertex or Intercept Form 249

4.2 EXERCISES

1. VOCABULARY Copy and complete: A quadratic function in the formy5 a(x2 h)2

1 k is in ? form.

Explain how to find a quadratic function’s maximum value orminimum value when the function is given in intercept form.

GRAPHING WITH VERTEX FORM Graph the function. Label the vertex and axis ofsymmetry.

3. y5 (x2 3)2 4. y5 (x1 4)2 5. f(x)52(x1 3)21 5

6. y5 3(x2 7)22 1 7. g(x)524(x2 2)2

1 4 8. y5 2(x1 1)22 3

9. f(x)522(x2 1)22 5 10. y521

}4

(x1 2)21 1 11. y5 1

}2

(x2 3)21 2

12. What is the vertex of the graph of the functiony5 3(x1 2)2

2 5?

A (2,25) B (22,25) C (25, 2) D (5,22)

GRAPHING WITH INTERCEPT FORM Graph the function. Label the vertex, axis ofsymmetry, and x-intercepts.

13. y5 (x1 3)(x2 3) 14. y5 (x1 1)(x2 3) 15. y5 3(x1 2)(x1 6)

16. f(x) 5 2(x2 5)(x2 1) 17. y52(x2 4)(x1 6) 18. g(x)524(x1 3)(x1 7)

19. y5 (x1 1)(x1 2) 20. f(x)522(x2 3)(x1 4) 21. y5 4(x2 7)(x1 2)

22. What is the vertex of the graph of the functiony52(x2 6)(x1 4)?

A (1, 25) B (21, 21) C (26, 4) D (6,24)

23. ERROR ANALYSIS Describe and correctthe error in analyzing the graph of thefunction y5 5(x2 2)(x1 3).

WRITING IN STANDARD FORM Write the quadratic function in standard form.

24. y5 (x1 4)(x1 3) 25. y5 (x2 5)(x1 3) 26. h(x)5 4(x1 1)(x2 6)

27. y523(x2 2)(x2 4) 28. f(x)5 (x1 5)22 2 29. y5 (x2 3)2

1 6

30. g(x)52(x1 6)21 10 31. y5 5(x1 3)2

2 4 32. f(x)5 12(x2 1)21 4

MINIMUM OR MAXIMUM VALUES Find the minimum value or the maximumvalue of the function.

33. y5 3(x2 3)22 4 34. g(x)524(x1 6)2

2 12 35. y5 15(x2 25)21 130

36. f(x)5 3(x1 10)(x2 8) 37. y52(x2 36)(x1 18) 38. y5212x(x2 9)

39. y5 8x(x1 15) 40. y5 2(x2 3)(x2 6) 41. g(x)525(x1 9)(x2 4)

EXAMPLE 1


EXAMPLE 3

on p. 247for Exs. 13–23

EXAMPLES5 and 6

on p. 248for Exs. 24–32

The x-intercepts of the graph

are 22 and 3.

HOMEWORKKEY


5 TAKS PRACTICE AND REASONINGExs. 12, 22, 49, 54, 55, 57, and 58

SKILL PRACTICE

G

TAKS REASONING

TAKS REASONING

WRITIN2.

250

x

y

42. GRAPHING CALCULATOR Consider the function y5 a(x2 h)21 k where

a5 1, h5 3, and k522. Predict the effect of each change in a, h, or kdescribed in parts (a)–(c). Use a graphing calculator to check your predictionby graphing the original and revised functions in the same coordinate plane.

a. a changes to23 b. h changes to 21 c. k changes to 2


43. y5 5(x2 2.25)22 2.75 44. g(x)528(x1 3.2)2

1 6.4 45. y520.25(x2 5.2)21 8.5

46. y522}3 1x2 1

}2 2

21

4}5

47. f(x)523}4

(x1 5)(x1 8) 48. g(x)5 5}2 1x2 4

}3 21x2 2

}5 2

49. Write two different quadratic functions in interceptform whose graphs have axis of symmetry x5 3.

50. CHALLENGE Write y5 a(x2 h)21 k and y5 a(x2 p)(x2 q) in standard

form. Knowing the vertex of the graph of y5 ax21 bx1 c occurs at x52 b

}2a

,

show that the vertex of the graph of y5 a(x2 h)21 k occurs at x5 h and that

the vertex of the graph of y5 a(x2 p)(x2 q) occurs at x5p1 q}

2.

PROBLEM SOLVING

EXAMPLES2 and 4

on pp. 246–247for Exs. 51–54

surface of

football field

surface of

football field

Not drawn to scaleNot drawn to scale x

y


51. BIOLOGY The function y520.03(x2 14)21 6 models the jump of a

red kangaroo where x is the horizontal distance (in feet) and y is thecorresponding height (in feet). What is the kangaroo’s maximum height?How long is the kangaroo’s jump?

52. CIVIL ENGINEERING The arch of the Gateshead Millennium Bridge forms aparabola with equation y520.016(x2 52.5)2

1 45 where x is the horizontaldistance (in meters) from the arch’s left end and y is the distance (in meters)from the base of the arch. What is the width of the arch?

53. MULTI-STEP PROBLEM Although a football field appears to be flat, itssurface is actually shaped like a parabola so that rain runs off to bothsides. The cross section of a field with synthetic turf can be modeled by

y520.000234x(x2 160)

where x and y are measured in feet.

a. What is the field’s width?

b. What is the maximum height of the field’s surface?

TAKS REASONING


3

EX

251

54. ump on a pogostick with a conventional spring can bemodeled by y520.5(x2 6)2

1 18, and ajump on a pogo stick with a bow spring canbe modeled by y521.17(x2 6)2

1 42, wherex and y are measured in inches. Comparethe maximum heights of the jumps on thetwo pogo sticks. Which constants in thefunctions affect the maximum heights ofthe jumps? Which do not?

55. A kernel of popcorn contains water that expandswhen the kernel is heated, causing it to pop. The equations below givethe “popping volume” y (in cubic centimeters per gram) of popcorn withmoisture content x (as a percent of the popcorn’s weight).

Hot-air popping: y520.761(x2 5.52)(x2 22.6)

Hot-oil popping: y520.652(x2 5.35)(x2 21.8)

a. Interpret For hot-air popping, what moisture content maximizes poppingvolume? What is the maximum volume?

b. Interpret For hot-oil popping, what moisture content maximizes poppingvolume? What is the maximum volume?

c. Graphing Calculator Graph the functions in the same coordinate plane.What are the domain and range of each function in this situation? Explainhow you determined the domain and range.

56. CHALLENGE Flying fish use their pectoral fins like airplanewings to glide through the air. Suppose a flying fishreaches a maximum height of 5 feet after flying ahorizontal distance of 33 feet. Write a quadraticfunction y5 a(x2 h)2

1 k that models the flightpath, assuming the fish leaves the water at (0, 0).Describe how changing the value of a, h, or kaffects the flight path.

ONLINE QUIZ at classzone.com

Horizontal position (in.)

2 4 6 8 10 120

bowspring

conventionalspringV

ertical position (in.) 40

30

20

10

0x

y

TA

57. TAKS PRACTICE A salesperson wants to analyze the time he spendsdriving to visit clients. In a typical week, the salesperson drives 870 milesduring a period of 22 hours. His average speed is 65 miles per hour on thehighway and 30 miles per hour in the city. About how many hours a weekdoes the salesperson spend driving in the city? TAKS Obj. 10

A 6 h B 8.2 h C 13.9 h D 16 h

58. TAKS PRACTICE What is the approximatearea of the shaded region? TAKS Obj. 8

F 21.5 cm2

G 42.9 cm2

H 121.4 cm2

J 150 cm2


REVIEW

Lesson 1.5;

TAKS Workbook

REVIEW

TAKS Preparation

p. 470;

TAKS Workbook


5 cm5 cm

KS REASONING

TAKS REASONING A j

EXTRA PRACTICE for Lesson 4.2, p. 101

4.3 Solve x2 1 bx 1 c 5 0 by Factoring 255

EXAMPLE 5 Find the zeros of quadratic functions

Find the zeros of the function by rewriting the function in intercept form.

a. y5 x22 x2 12 b. y5 x2

1 12x1 36

Solution

a. y5 x22 x2 12 Write original function.

5 (x1 3)(x2 4) Factor.

The zeros of the function are23 and 4.

CHECK Graph y5 x22 x2 12. The graph

passes through (23, 0) and (4, 0).

b. y5 x21 12x1 36 Write original function.

5 (x1 6)(x1 6) Factor.

The zero of the function is26.

CHECK Graph y5 x21 12x1 36. The graph

passes through (26, 0).

! GUIDED PRACTICE for Example 5

Find the zeros of the function by rewriting the function in intercept form.

10. y5 x21 5x2 14 11. y5 x2

2 7x2 30 12. f(x)5 x22 10x1 25

EXAMPLE 1


1. VOCABULARY What is a zero of a function y5 f(x)?

2. Explain the difference between a monomial, a binomial, and atrinomial. Give an example of each type of expression.

FACTORING Factor the expression. If the expression cannot be factored, say so.

3. x21 6x1 5 4. x2

2 7x1 10 5. a22 13a1 22

6. r21 15r1 56 7. p2

1 2p1 4 8. q22 11q1 28

9. b21 3b2 40 10. x2

2 4x2 12 11. x22 7x2 18

12. c22 9c2 18 13. x2

1 9x2 36 14. m21 8m2 65

4.3 EXERCISES

Zero

X=-3 Y=0

Zero

X=-6 Y=0

UNDERSTANDREPRESENTATIONS

If a real number k isa zero of the functiony 5 ax2 1 bx 1 c, thenk is an x-interceptof this function’sgraph and k is also aroot of the equationax2 1 bx 1 c 5 0.

HOMEWORKKEY




SKILL PRACTICE

WRITING

256

FACTORING WITH SPECIAL PATTERNS Factor the expression.

15. x22 36 16. b2

2 81 17. x22 24x1 144

18. t22 16t1 64 19. x21 8x1 16 20. c2

1 28c1 196

21. n21 14n1 49 22. s2

2 26s1 169 23. z22 121

SOLVING EQUATIONS Solve the equation.

24. x22 8x1 12 5 0 25. x2

2 11x1 30 5 0 26. x21 2x2 35 5 0

27. a22 49 5 0 28. b2

2 6b1 9 5 0 29. c21 5c1 4 5 0

30. n22 6n5 0 31. t21 10t1 25 5 0 32. w2

2 16w1 48 5 0

33. z22 3z5 54 34. r2

1 2r5 80 35. u2529u

36. m25 7m 37. 14x2 49 5 x2 38. 23y1 28 5 y2

ERROR ANALYSIS Describe and correct the error in solving the equation.

39. 40.

41. # What are the roots of the equation x21 2x2 63 5 0?

A 7, 29 B 27, 29 C 27, 9 D 7, 9

WRITING EQUATIONS Write an equation that you can solve to find the value of x.

42. A rectangular picnic site measures 24 feet by 10 feet. You want to double the site’s area by adding the same distance x to the length and the width.

43. A rectangular performing platform in a park measures 10 feet by 12 feet. You want to triple the platform’s area by adding the same distance x to the length and the width.

FINDING ZEROS Find the zeros of the function by rewriting the function in intercept form.

44. y5 x21 6x1 8 45. y5 x2

2 8x1 16 46. y5 x22 4x2 32

47. y5 x21 7x2 30 48. f(x)5 x2

1 11x 49. g(x)5 x22 8x

50. y5 x22 64 51. y5 x2

2 25 52. f(x)5 x22 12x2 45

53. g(x)5 x21 19x1 84 54. y5 x2

1 22x1 121 55. y5 x21 2x1 1

56. What are the zeros of f(x)5 x21 6x2 55?

A 211, 25 B 211, 5 C 25, 11 D 5, 11

57. REASONING Write a quadratic equation of the form x21 bx1 c5 0 that has

roots 8 and 11.

58. # For what integers b can the expression x21 bx1 7 be

factored? Explain.

EXAMPLE 2

on p. 253for Exs. 15–23

EXAMPLE 5

on p. 255for Exs. 44–55

EXAMPLE 3

on p. 254for Exs. 24–41

x2 1 7x 1 6 5 14

(x 1 6)(x 1 1) 5 14

x 1 6 5 14 or x 1 1 5 14

x 5 8 or x 5 13


x2 2 x 2 6 5 0

(x 2 2)(x 1 3) 5 0

x 2 2 5 0 or x 1 3 5 0

x 5 2 or x 5 23

EXAMPLE 4

on p. 254for Exs. 42–43

5 TAKS PRACTICE AND REASONING

TAKS REASONING

TAKS REASONING

TAKS REASONING

4.3 Solve x2 1 bx 1 c 5 0 by Factoring 257

GEOMETRY Find the value of x.

59. Area of rectangle5 36 60. Area of rectangle5 84

x1 5

x

x1 7

x1 2

61. Area of triangle5 42 62. Area of trapezoid5 32

2x1 8

x1 3

x1 2

x1 6

x

63. Write a quadratic function with zeros that areequidistant from 10 on a number line.

64. CHALLENGE Is there a formula for factoring the sum of two squares? You willinvestigate this question in parts (a) and (b).

a. Consider the sum of two squares x21 16. If this sum can be factored,

then there are integers m and n such that x21 165 (x1m)(x1 n). Write

two equations that m and n must satisfy.

b. Show that there are no integers m and n that satisfy both equationsyou wrote in part (a). What can you conclude?

EXAMPLE 4

on p. 254for Exs. 65–67

65. SKATE PARK A city’s skate park is a rectangle 100 feet long by50 feet wide. The city wants to triple the area of the skate park byadding the same distance x to the length and the width. Writeand solve an equation to find the value of x. What are the newdimensions of the skate park?

66. ZOO A rectangular enclosure at a zoo is 35 feet long by 18 feet wide. The zoowants to double the area of the enclosure by adding the same distance x tothe length and the width. Write and solve an equation to find the value of x.What are the new dimensions of the enclosure?

67. MULTI-STEP PROBLEM A museum has a café with arectangular patio. The museum wants to add 464 squarefeet to the area of the patio by expanding the existingpatio as shown.

a. Find the area of the existing patio.

b. Write a verbal model and an equation that you canuse to find the value of x.

c. Solve your equation. By what distance x should thelength and the width of the patio be expanded?

PROBLEM SOLVING

TAKS REASONING

258

68. MULTIPLE REPRESENTATIONS Use the diagram shown.

a. Writing an Expression Write a quadratic trinomial that representsthe area of the diagram.

b. Describing a Model Factor the expression from part (a). Explain howthe diagram models the factorization.

c. Drawing a Diagram Draw a diagram that models the factorizationx21 8x1 155 (x1 5)(x1 3).

69. SCHOOL FAIR At last year’s school fair, an 18 foot by 15 foot rectangularsection of land was roped off for a dunking booth. The length and widthof the section will each be increased by x feet for this year’s fair in order totriple the original area. Write and solve an equation to find the value of x.What is the length of rope needed to enclose the new section?

70. RECREATION CENTER A rectangular deck for a recreation center is 21 feetlong by 20 feet wide. Its area is to be halved by subtracting the same distancex from the length and the width. Write and solve an equation to find thevalue of x. What are the deck’s new dimensions?

71. A square garden has sides that are 10 feet long. Agardener wants to double the area of the garden by adding the same distancex to the length and the width. Write an equation that x must satisfy. Can yousolve the equation you wrote by factoring? Explain why or why not.

72. CHALLENGE A grocery store wants to doublethe area of its parking lot by expanding theexisting lot as shown. By what distance xshould the lot be expanded?


1 1 1x

x

1

1

75 ft

75 ft x

x Expanded part of lot

Old lot

165 ft

300 ft

Grocery store

TAKS REASONING

73. TAKS PRACTICE What is the slope of theline shown? TAKS Obj. 3

A 25}4

B 24}5

C 4}5

D 5}4

74. TAKS PRACTICE Which of the following best describes the graphs of theequations below? TAKS Obj. 7

y5 3x2 2

24y5 x1 8

F The lines have the same x-intercept.

G The lines have the same y-intercept.

H The lines are perpendicular to each other.

J The lines are parallel to each other.

MIXED REVIEW FOR TAKSTAKS PRACTICE at classzone.com

REVIEW

Lesson 2.2;

TAKS Workbook

REVIEW

Lesson 2.3;

TAKS Workbook

2 x

y

121222325

1

2

3

5

6

4.4 Solve ax21 bx 1 c 5 0 by Factoring 263

4.4 EXERCISES

EXAMPLE 3

on p. 260for Exs. 13–21

EXAMPLE 4

on p. 260for Exs. 22–31

EXAMPLES1 and 2


EXAMPLE 5

on p. 261for Exs. 32–40

EXAMPLE 7

on p. 262for Exs. 41–49

1. VOCABULARY What is the greatest common monomial factor of the terms ofthe expression 12x2

1 8x1 20?

Explain how the values of a and c in ax21 bx1 c help you

determine whether you can use a perfect square trinomial factoring pattern.

FACTORING Factor the expression. If the expression cannot be factored, say so.

3. 2x21 5x1 3 4. 3n2

1 7n1 4 5. 4r21 5r1 1

6. 6p21 5p1 1 7. 11z2

1 2z2 9 8. 15x22 2x2 8

9. 4y22 5y2 4 10. 14m2

1m2 3 11. 9d22 13d2 10

12. Which factorization of 5x21 14x2 3 is correct?

A (5x2 3)(x1 1) B (5x1 1)(x2 3)

C 5(x2 1)(x1 3) D (5x2 1)(x1 3)

FACTORING WITH SPECIAL PATTERNS Factor the expression.

13. 9x22 1 14. 4r2

2 25 15. 49n22 16

16. 16s21 8s1 1 17. 49x2

1 70x1 25 18. 64w21 144w1 81

19. 9p22 12p1 4 20. 25t2

2 30t1 9 21. 36x22 84x1 49

FACTORING MONOMIALS FIRST Factor the expression.

22. 12x22 4x2 40 23. 18z2

1 36z1 16 24. 32v22 2

25. 6u22 24u 26. 12m2

2 36m1 27 27. 20x21 124x1 24

28. 21x22 77x2 28 29. 236n2

1 48n2 15 30. 28y21 28y2 60

31. ERROR ANALYSIS Describe and correctthe error in factoring the expression.


32. 16x22 15 0 33. 11q2

2 445 0 34. 14s22 21s5 0

35. 45n21 10n5 0 36. 4x2

2 20x1 255 0 37. 4p21 12p1 95 0

38. 15x21 7x2 25 0 39. 6r2

2 7r2 55 0 40. 36z21 96z1 155 0

FINDING ZEROS Find the zeros of the function by rewriting the function inintercept form.

41. y5 4x22 19x2 5 42. g(x)5 3x2

2 8x1 5 43. y5 5x22 27x2 18

44. f(x)5 3x22 3x 45. y5 11x2

2 19x2 6 46. y5 16x22 2x2 5

47. y5 15x22 5x2 20 48. y5 18x2

2 6x2 4 49. g(x)5 12x21 5x2 7

4x2 2 36 5 4(x2 2 36)

5 4(x 1 6)(x 2 6)

HOMEWORKKEY


5 TAKS PRACTICE AND REASONINGExs. 12, 64, 65, 67, 69, and 70

SKILL PRACTICE

TAKS REASONING

WRITIN2. G

264

EXAMPLE 6

on p. 261for Exs. 62–63


50. Area of square5 36 51. Area of rectangle5 30 52. Area of triangle5 115

2x3x1 1

x

5x2 2

2x


53. 2x22 4x2 852x2

1 x 54. 24x21 8x1 25 52 6x 55. 18x2

2 22x5 28

56. 13x21 21x525x2

1 22 57. x5 4x22 15x 58. (x1 8)2

5 162 x21 9x

CHALLENGE Factor the expression.

59. 2x32 5x2

1 3x 60. 8x42 8x3

2 6x2 61. 9x32 4x

EXAMPLE 7

on p. 262for Exs. 64–65

PROBLEM SOLVING

62. ARTS AND CRAFTS You have a rectangular stained glass window thatmeasures 2 feet by 1 foot. You have 4 square feet of glass with which to makea border of uniform width around the window. What should the width of theborder be?

63. URBAN PLANNING You have just planted arectangular flower bed of red roses in a city park.You want to plant a border of yellow roses aroundthe flower bed as shown. Because you bought thesame number of red and yellow roses, the areasof the border and flower bed will be equal. Whatshould the width of the border of yellow roses be?

64. A surfboard shop sells 45 surfboards per month whenit charges $500 per surfboard. For each $20 decrease in price, the storesells 5 more surfboards per month. How much should the shop charge persurfboard in order to maximize monthly revenue?

A $340 B $492 C $508 D $660

65. A restaurant sells about 330 sandwiches each day at aprice of $6 each. For each $.25 decrease in price, 15 more sandwiches are soldper day. How much should the restaurant charge to maximize daily revenue?Explain each step of your solution. What is the maximum daily revenue?

66. PAINTINGS You are placing a mat around a 25 inch by 21 inchpainting as shown. You want the mat to be twice as wide tothe left and right of the painting as it is at the top and bottomof the painting. You have 714 square inches of mat that youcan use. How wide should the mat be to the left and right ofthe painting? at the top and bottom of the painting?



TAKS REASONING

TAKS REASONING

265

67. EXTENDED RESPONSE A U.S. Postal Service guidelinestates that for a rectangular package like the one shown,the sum of the length and the girth cannot exceed108 inches. Suppose that for one such package, thelength is 36 inches and the girth is as large as possible.

a. What is the girth of the package?

b. Write an expression for the package’s width w in terms of h.Write an equation giving the package’s volume V in terms of h.

c. What height and width maximize the volume of the package?What is the maximum volume? Explain how you found it.

68. CHALLENGE Recall from geometry the theorem aboutthe products of the lengths of segments of two chordsthat intersect in the interior of a circle. Use thistheorem to find the value of x in the diagram.

QUIZ for Lessons 4.1–4.4

Graph the function. Label the vertex and axis of symmetry. (p. 236)

1. y5 x22 6x1 14 2. y5 2x2

1 8x1 15 3. f(x)523x21 6x2 5

Write the quadratic function in standard form. (p. 245)

4. y5 (x2 4)(x2 8) 5. g(x)522(x1 3)(x2 7) 6. y5 5(x1 6)22 2

Solve the equation.

7. x21 9x1 205 0 (p. 252) 8. n2

2 11n1 245 0 (p. 252) 9. z22 3z2 405 0 (p. 252)

10. 5s22 14s2 35 0 (p. 259) 11. 7a2

2 30a1 85 0 (p. 259) 12. 4x21 20x1 255 0 (p. 259)

13. DVD PLAYERS A store sells about 50 of a new model of DVD player per monthat a price of $140 each. For each $10 decrease in price, about 5 more DVDplayers per month are sold. How much should the store charge in order tomaximize monthly revenue? What is the maximum monthly revenue? (p. 259)

3x1 2

x1 1

5x2 4

2x


TAKS REASONING

69. TAKS PRACTICE A pizza is divided into 12 equal slices asshown. The diameter of the pizza is 16 inches. What is theapproximate area of one slice of pizza? TAKS Obj. 8

A 15.47 in.2 B 16.76 in.2

C 21.21 in.2 D 67.02 in.2

70. TAKS PRACTICE While shopping at Store A, Sam finds a television on salefor $210. His friend tells him that the same television at Store B is on sale for$161. About what percent of the cost of the television at Store A does Sam saveby buying the television at Store B? TAKS Obj. 9

F 20% G 23% H 30% J 77%


REVIEW

Skills Review

Handbook p. 992;

TAKS Workbook

REVIEW

TAKS Preparation

p. 146;

TAKS Workbook

4.5 Solve Quadratic Equations by Finding Square Roots 269

EXAMPLE 5 Model a dropped object with a quadratic function

SCIENCE COMPETITION For a science competition,students must design a container that prevents an eggfrom breaking when dropped from a height of 50 feet.How long does the container take to hit the ground?

Solution

h 5216t21 h

0Write height function.

0 5216t21 50 Substitute 0 for h and 50 for h0.

250 5216t2 Subtract 50 from each side.

50}165 t2 Divide each side by216.

6Î}50}165 t Take square roots of each side.

61.8 ø t Use a calculator.

c Reject the negative solution,21.8, because time must be positive.The container will fall for about 1.8 seconds before it hits the ground.

at classzone.com

# GUIDED PRACTICE for Example 5

20. WHAT IF? In Example 5, suppose the egg container is dropped from a heightof 30 feet. How long does the container take to hit the ground?

ANOTHER WAY

For alternative methodsfor solving the problemin Example 5, turnto page 272 for theProblem SolvingWorkshop.

1. VOCABULARY In the expression Ï}

72 , what is 72 called?

2. WRITING Explain what it means to “rationalize the denominator”of a quotient containing square roots.

SIMPLIFYING RADICAL EXPRESSIONS Simplify the expression.

3. Ï}

28 4. Ï}

192 5. Ï}

150 6. Ï}

3 p Ï}

27

7. 4Ï}

6 p Ï}

6 8. 5Ï}

24 p 3Ï}

10 9. Î}5}16

10. Î}35}36

11. 8}

Ï}

312. 7}

Ï}

1213. Î}18

}11

14. Î}13}28

15. 2}

12 Ï}

316. 1}

51 Ï}

617. Ï

}

2}

41 Ï}

518. 31 Ï

}

7}

22 Ï}

10

4.5 EXERCISES

EXAMPLES1 and 2

on pp. 266–267for Exs. 3–20

After a successful egg drop

HOMEWORKKEY


5 TAKS PRACTICE AND REASONINGExs. 19, 34, 35, 36, 40, 41, 44, and 45

SKILL PRACTICE

WRITING

270

19. What is a completely simplified expression for Ï}

108 ?

A 2Ï}

27 B 3Ï}

12 C 6Ï}

3 D 10Ï}

8

ERROR ANALYSIS Describe and correct the error in simplifying the expressionor solving the equation.

20.Ï}

96 5 Ï}

4 p Ï}

24

5 2Ï}

24

21.5x2 5 405

x2 5 81

x 5 9

SOLVING QUADRATIC EQUATIONS Solve the equation.

22. s25 169 23. a2

5 50 24. x25 84

25. 6z25 150 26. 4p2

5 448 27. 23w252213

28. 7r22 105 25 29. x2

}252 6522 30. t2

}201 85 15

31. 4(x2 1)25 8 32. 7(x2 4)2

2 185 10 33. 2(x1 2)22 55 8

34. What are the solutions of 3(x1 2)21 45 13?

A 25, 1 B 21, 5 C 226 Ï}

3 D 26 Ï}

3

35. Describe two different methods for solving the equation

x22 45 0. Include the steps for each method.

36. Write an equation of the form x25 s that has (a) two

real solutions, (b) exactly one real solution, and (c) no real solutions.

37. CHALLENGE Solve the equation a(x1 b)25 c in terms of a, b, and c.

EXAMPLE 5

on p. 269for Exs. 38–39

EXAMPLES3 and 4

on pp. 267–268for Exs. 21–34

38. CLIFF DIVING A cliff diver dives off a cliff 40 feetabove water. Write an equation giving the diver’sheight h (in feet) above the water after t seconds.How long is the diver in the air?

39. ASTRONOMY On any planet, the height h (in feet) of a falling object t seconds

after it is dropped can be modeled by h52g}2t21 h

0 where h

0 is the object’s

initial height (in feet) and g is the acceleration (in feet per second squared)due to the planet’s gravity. For each planet in the table, find the time it takesfor a rock dropped from a height of 150 feet to hit the surface.

Planet Earth Mars Jupiter Saturn Pluto

g (ft/sec2) 32 12 76 30 2


PROBLEM SOLVING


TAKS REASONING

TAKS REASONING

TAKS REASONING

TAKS REASONING

13 271

40. The equation h5 0.019s2 gives the height h (in feet) ofthe largest ocean waves when the wind speed is s knots. Compare the windspeeds required to generate 5 foot waves and 20 foot waves.

41. You want to transform a square gravel parkinglot with 10 foot sides into a circular lot. You want the circle to have thesame area as the square so that you do not have to buy any additionalgravel.

a. Model Write an equation you can use to find the radius r of thecircular lot.

b. Solve What should the radius of the circular lot be?

c. Generalize In general, if a square has sides of length s, what is the radiusr of a circle with the same area? Justify your answer algebraically.

42. BICYCLING The air resistance R (in pounds)on a racing cyclist is given by the equationR5 0.00829s2 where s is the bicycle’s speed(in miles per hour).

a. What is the speed of a racing cyclist whoexperiences 5 pounds of air resistance?

b. What happens to the air resistance if thecyclist’s speed doubles? Justify your answeralgebraically.

43. CHALLENGE For a swimming pool with a rectangular base, Torricelli’s lawimplies that the height h of water in the pool t seconds after it begins

draining is given by h5 1Ï}

h02

2πd2Ï}

3}

lwt 2

2

where l and w are the pool’s

length and width, d is the diameter of the drain, and h0 is the water’s initial

height. (All measurements are in inches.) In terms of l, w, d, and h0, what is

the time required to drain the pool when it is completely filled?


r

10 ft

10 ft

TAKS REASONING

TAKS REASONING

44. TAKS PRACTICE The graph of whichinequality is shown? TAKS Obj. 1

A y < 2x2 3

B y > 2x2 3

C y ≤ 2x2 3

D y ≥ 2x2 3

45. TAKS PRACTICE Which two lines are perpendicular? TAKS Obj. 7

F 3x1 y521 and x1 3y5224

G 3x2 y5 12 and 3x1 y5 15

H 3x1 y521 and2x1 3y5 6

J 3x2 y5 12 and x2 3y5 9


x

y

2212223 3

1

2

21

22

23

REVIEW

Lesson 2.8;

Taks Workbook

REVIEW

Lesson 2.2;

Taks Workbook



1. VOCABULARY What is the difference between a binomial and a trinomial?

2. Describe what completing the square means for an expressionx21 bx.

SOLVING BY SQUARE ROOTS Solve the equation by finding square roots.

3. x21 4x1 45 9 4. x2

1 10x1 255 64 5. n21 16n1 645 36

6. m22 2m1 15 144 7. x2

2 22x1 1215 13 8. x22 18x1 815 5

9. t21 8t1 165 45 10. 4u2

1 4u1 15 75 11. 9x22 12x1 4523

12. What are the solutions of x22 4x 1 4521?

A 26 i B 226 i C 23,21 D 1, 3

FINDING C Find the value of c that makes the expression a perfect squaretrinomial. Then write the expression as the square of a binomial.

13. x21 6x1 c 14. x2

1 12x1 c 15. x22 24x1 c

16. x22 30x1 c 17. x2

2 2x1 c 18. x21 50x1 c

19. x21 7x1 c 20. x2

2 13x1 c 21. x22 x1 c

COMPLETING THE SQUARE Solve the equation by completing the square.

22. x21 4x5 10 23. x2

1 8x521 24. x21 6x2 35 0

25. x21 12x1 185 0 26. x2

2 18x1 865 0 27. x22 2x1 255 0

28. 2k21 16k5212 29. 3x2

1 42x5224 30. 4x22 40x2 125 0

31. 3s21 6s1 95 0 32. 7t2

1 28t1 565 0 33. 6r21 6r1 125 0

34. What are the solutions of x21 10x 1 8525?

A 56 2Ï}

3 B 56 4Ï}

3 C 256 2Ï}

3 D 256 4Ï}

3


35. Area of rectangle5 50 36. Area of parallelogram5 48

x1 10

x

x1 6

x

37. Area of triangle5 40 38. Area of trapezoid5 20

x1 4

x

x1 9

3x2 1

x

4.7 EXERCISES

EXAMPLE 1


EXAMPLE 2

on p. 285for Exs. 13–21

EXAMPLE 5

on p. 286for Exs. 35–38

EXAMPLES3 and 4

on pp. 285–286for Exs. 22–34

TAKS REASONING

HOMEWORKKEY




SKILL PRACTICE

WRITIN

of the formG

TAKS REASONING

4.7 Complete the Square 289

FINDING THE VERTEX In Exercises 39 and 40, use completing the square to find the vertex of the given function’s graph. Then tell what the vertex represents.

39. At Buckingham Fountain in Chicago, the water’s height h (in feet) above the main nozzle can be modeled by h5216t2

1 89.6t where t is the time (in seconds) since the water has left the nozzle.

40. When you walk x meters per minute, your rate y of energy use (in calories per minute) can be modeled by y5 0.0085x2

2 1.5x1 120.

WRITING IN VERTEX FORM Write the quadratic function in vertex form. Then identify the vertex.

41. y5 x22 8x1 19 42. y5 x2

2 4x2 1 43. y5 x21 12x1 37

44. y5 x21 20x1 90 45. f(x)5 x2

2 3x1 4 46. g(x)5 x21 7x1 2

47. y5 2x21 24x1 25 48. y5 5x2

1 10x1 7 49. y5 2x22 28x1 99


50. x2 1 10x 1 13 5 0

x2 1 10x 5 213

x2 1 10x 1 25 5 213 1 25

(x 1 5)2 5 12

x 1 5 5 6Ï}

12

x 5 25 6 Ï}

12

x 5 25 6 4 Ï}

3

51. 4x2 1 24x 2 11 5 0

4(x2 1 6x) 5 11

4(x2 1 6x 1 9) 5 11 1 9

4(x 1 3)2 5 20

(x 1 3)2 5 5

x 1 3 5 6Ï}

5

x 5 23 6 Ï}

5

COMPLETING THE SQUARE Solve the equation by completing the square.

52. x21 9x1 20 5 0 53. x2

1 3x 1 14 5 0 54. 7q21 10q5 2q2

1 155

55. 3x21 x5 2x2 6 56. 0.1x2

2 x1 9 5 0.2x 57. 0.4v21 0.7v5 0.3v2 2

58. Write a quadratic equation with real-number solutions that can be solved by completing the square but not by factoring.

59. In this exercise, you will investigate the graphical effect of completing the square.

a. Graph each pair of functions in the same coordinate plane.

y5 x21 2x y5 x2

1 4x y5 x22 6x

y5 (x1 1)2 y5 (x1 2)2 y5 (x2 3)2

b. Compare the graphs of y5 x21 bx and y5 1x1 b}

2 2

2. What happens to

the graph of y5 x21 bx when you complete the square?

60. REASONING For what value(s) of k does x21 bx1 1 b}2 2

25 k have

exactly 1 real solution? 2 real solutions? 2 imaginary solutions?

61. CHALLENGE Solve x21 bx1 c5 0 by completing the square. Your answer

will be an expression for x in terms of b and c.

EXAMPLES6 and 7

on p. 287for Exs. 41–49

Buckingham Fountain

125 ft

TAKS REASONING

TAKS REASONING

290

62. DRUM MAJOR While marching, a drum major tosses a baton into the air andcatches it. The height h (in feet) of the baton after t seconds can be modeledby h5216t2

1 32t1 6. Find the maximum height of the baton.

63. VOLLEYBALL The height h (in feet) of a volleyball t seconds after it is hit canbe modeled by h5216t2

1 48t1 4. Find the volleyball’s maximum height.

64. SKATEBOARD REVENUE A skateboard shopsells about 50 skateboards per week for theprice advertised. For each $1 decrease inprice, about 1 more skateboard per week issold. The shop’s revenue can be modeled byy5 (702 x)(501 x). Use vertex form to findhow the shop can maximize weekly revenue.

65. VIDEO GAME REVENUE A store sells about 40 video game systems eachmonth when it charges $200 per system. For each $10 increase in price,about 1 less system per month is sold. The store’s revenue can be modeledby y5 (2001 10x)(402 x). Use vertex form to find how the store canmaximize monthly revenue.

66. MULTIPLE REPRESENTATIONS The path of a ball thrown by a softballplayer can be modeled by the function

y520.0110x21 1.23x1 5.50

where x is the softball’s horizontal position (in feet) and y is thecorresponding height (in feet).

a. Rewriting a Function Write the given function in vertex form.

b. Making a Table Make a table of values for the function. Include values ofx from 0 to 120 in increments of 10.

c. Drawing a Graph Use your table to graph the function. What is themaximum height of the softball? How far does it travel?

67. Your school is adding a rectangular outdoor eatingsection along part of a 70 foot side of the school. The eating section will beenclosed by a fence along its three open sides.The school has 120 feet of fencing andplans to use 1500 square feet of landfor the eating section.

a. Write an equation for the areaof the eating section.

b. Solve the equation. Explainwhy you must reject one ofthe solutions.

c. What are the dimensions ofthe eating section?

PROBLEM SOLVING

EXAMPLE 7

on p. 287for Exs. 62–65


70 ft

120 – 2x

x

x

Eatingsection

5MULTIPLEREPRESENTATIONS


TAKS REASONING

291

68. CHALLENGE In your pottery class, you are given alump of clay with a volume of 200 cubic centimeters andare asked to make a cylindrical pencil holder. The pencilholder should be 9 centimeters high and have an innerradius of 3 centimeters. What thickness x should yourpencil holder have if you want to use all of the clay?

Solve the equation.

1. 4x25 64 (p. 266) 2. 3(p2 1)2

5 15 (p. 266) 3. 16(m1 5)25 8 (p. 266)

4. 22z25 424 (p. 275) 5. s2

1 125 9 (p. 275) 6. 7x22 4526 (p. 275)

Write the expression as a complex number in standard form. (p. 275)

7. (52 3i)1 (221 5i) 8. (221 9i)2 (71 8i) 9. 3i(72 9i)

10. (82 3i)(262 10i) 11. 4i}262 11i

12.32 2i}281 5i

Write the quadratic function in vertex form. Then identify the vertex. (p. 284)

13. y5 x22 4x1 9 14. y5 x2

1 14x1 45 15. f(x)5 x22 10x1 17

16. g(x)5 x22 2x2 7 17. y5 x2

1 x1 1 18. y5 x21 9x1 19

19. FALLING OBJECT A student drops a ball from a school roof 45 feet aboveground. How long is the ball in the air? (p. 266)


GEOMETRYREVIEW

The volume ofclay equals thedifference of thevolumes of twocylinders.

3 cmx cm

3 cm x cm

x cm

x cm

9 cm


Top view Side view

69. TAKS PRACTICE If quadrilateralMNPQ isreflected in the line y5 3, in which quadrant willthe image of point N appear? TAKS Obj. 7

A Quadrant I B Quadrant II

C Quadrant III D Quadrant IV

70. TAKS PRACTICE A hose adds 120 gallons of water to a swimming poolin 1.5 hours. How many hours will it take for the hose to fill a differentswimming pool that holds 600 gallons of water? TAKS Obj. 9

F 5 h G 6.25 h H 7.5 h J 8 h


REVIEW

TAKS Preparation

p. 674;

TAKS Workbook

REVIEW

TAKS Preparation

p. 146;

TAKS Workbook


x

y

1 2222324 21 3 4

1

MM NN

ŒŒ PP



4.8 EXERCISES

1. VOCABULARY Copy and complete: You can use the ? of a quadratic equation to determine the equation’s number and type of solutions.

2. ! WRITING Describe a real-life situation in which you can use the model h5216t21 v

0t1 h

0but not the model h5216t21 h

0.

EQUATIONS IN STANDARD FORM Use the quadratic formula to solve the equation.

3. x22 4x2 5 5 0 4. x22 6x1 7 5 0 5. t21 8t1 19 5 0

6. x22 16x1 7 5 0 7. 8w22 8w1 2 5 0 8. 5p2

2 10p1 24 5 0

9. 4x22 8x1 1 5 0 10. 6u21 4u1 11 5 0 11. 3r22 8r2 9 5 0

12. What are the complex solutions of the equation 2x22 16x1 50 5 0?

A 4 1 3i, 4 2 3i B 4 1 12i, 4 2 12i

C 16 1 3i, 16 2 3i D 16 1 12i, 16 2 12i

EQUATIONS NOT IN STANDARD FORM Use the quadratic formula to solve the equation.

13. 3w22 12w5212 14. x21 6x5215 15. s25214 2 3s

16. 23y25 6y2 10 17. 32 8v2 5v25 2v 18. 7x2 5 1 12x2523x

19. 4x21 3 5 x22 7x 20. 62 2t25 9t1 15 21. 4 1 9n2 3n25 2 2 n

SOLVING USING TWO METHODS Solve the equation using the quadratic formula. Then solve the equation by factoring to check your solution(s).

22. z21 15z1 24 5232 23. x22 5x1 10 5 4 24. m21 5m2 99 5 3m

25. s22 s2 3 5 s 26. r22 4r1 8 5 5r 27. 3x21 7x2 24 5 13x

28. 45x21 57x1 1 5 5 29. 5p21 40p1 100 5 25 30. 9n2

2 42n2 162 5 21n

USING THE DISCRIMINANT Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.

31. x22 8x1 16 5 0 32. s21 7s1 11 5 0 33. 8p21 8p1 3 5 0

34. 24w21 w2 14 5 0 35. 5x21 20x1 21 5 0 36. 8z2 10 5 z22 7z1 3

37. 8n22 4n1 2 5 5n2 11 38. 5x21 16x5 11x2 3x2 39. 7r22 5 5 2r1 9r2

SOLVING QUADRATIC EQUATIONS Solve the equation using any method.

40. 16t22 7t5 17t2 9 41. 7x2 3x25 85 1 2x21 2x 42. 4(x2 1)25 6x1 2

43. 25 2 16v25 12v(v1 5) 44. 3}2y22 6y5 3}

4y2 9 45. 3x21 9}

2x2 4 5 5x1 3}

4

46. 1.1(3.4x2 2.3)25 15.5 47. 19.25 528.5(2r2 1.75)2 48. 4.5 5 1.5(3.25 2 s)2

EXAMPLES1, 2, and 3

on pp. 292–293for Exs. 3–30

EXAMPLE 4

on p. 294for Exs. 31–39

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSon p. WS1 for Exs. 19, 39, and 71

5 TAKS PRACTICE AND REASONINGExs. 12, 51, 55, 62, 69, 72, 73, 75, and 76

SKILL PRACTICE

WRITING

TAKS REASONING

4.8 Use the Quadratic Formula and the Discriminant 297


49.3x2

1 6x 1 15 5 0

x 526 6 Ï}}

622 4(3)(15)

}}}}}}}}}}2(3)

526 6 Ï}

2144}}}}}}

6

526 6 12}}}}

6

5 1 or 23

50.x2

1 6x 1 8 5 2

x 526 6 Ï}}

622 4(1)(8)

}}}}}}}}}2(1)

526 6 Ï

}

4}}}}}

2

526 6 2}}}}

2

5 22 or 24

51. For a quadratic equation ax21 bx1 c5 0 with two real

solutions, show that the mean of the solutions is2 b}}2a

. How is this fact

related to the symmetry of the graph of y5 ax21 bx1 c?

VISUAL THINKING In Exercises 52–54, the graph of a quadratic functiony 5 ax2

1 bx 1 c is shown. Tell whether the discriminant of ax21 bx 1 c 5 0 is

positive, negative, or zero.

52.

x

y 53.

x

y 54.

x

y

55. What is the value of c if the discriminant of2x21 5x1 c5 0 is223?

A 223 B 26 C 6 D 14

THE CONSTANT TERM Use the discriminant to find all values of c for which theequation has (a) two real solutions, (b) one real solution, and (c) two imaginarysolutions.

56. x22 4x1 c5 0 57. x21 8x1 c5 0 58. 2x21 16x1 c5 0

59. 3x21 24x1 c5 0 60. 24x22 10x1 c5 0 61. x22 x1 c5 0

62. ! Write a quadratic equation in standard form that hasa discriminant of210.

WRITING EQUATIONS Write a quadratic equation in the form ax21 bx 1 c 5 0

such that c 5 4 and the equation has the given solutions.

63. 24 and 3 64. 24}3 and21 65. 211 i and212 i

66. REASONING Show that there is no quadratic equation ax21 bx1 c5 0 suchthat a, b, and c are real numbers and 3i and22i are solutions.

67. CHALLENGE Derive the quadratic formula by completing the square to solvethe general quadratic equation ax21 bx1 c5 0.

TAKS REASONING

TAKS REASONING

TAKS REASONING

298

PROBLEM SOLVING

EXAMPLE 5

on p. 295for Exs. 68–69

5 WORKED-OUT SOLUTIONSon p. WS1

68. FOOTBALL In a football game, a defensive player jumps up to block a pass bythe opposing team’s quarterback. The player bats the ball downward withhis hand at an initial vertical velocity of250 feet per second when the ball is7 feet above the ground. How long do the defensive player’s teammates haveto intercept the ball before it hits the ground?

69. ! For the period 1990–2002, the number S (in thousands)of cellular telephone subscribers in the United States can be modeled byS5 858t21 1412t1 4982 where t is the number of years since 1990. In whatyear did the number of subscribers reach 50 million?

A 1991 B 1992 C 1996 D 2000

70. MULTI-STEP PROBLEM A stunt motorcyclist makes a jump from one ramp20 feet off the ground to another ramp 20 feet off the ground. The jump

between the ramps can be modeled by y52 1}}640x21 1

}4x1 20 where x is the

horizontal distance (in feet) and y is the height above the ground (in feet).

a. What is the motorcycle’s height r when it lands on the ramp?

b. What is the distance d between the ramps?

c. What is the horizontal distance h the motorcycle has traveled when itreaches its maximum height?

d. What is the motorcycle’s maximum height k above the ground?

71. BIOLOGY The number S of ant species in Kyle Canyon, Nevada, can bemodeled by the function S520.000013E2

1 0.042E2 21 where E is theelevation (in meters). Predict the elevation(s) at which you would expect tofind 10 species of ants.

72. ! A city planner wants to createadjacent sections for athletics and picnics in the yardof a youth center. The sections will be rectangularand will be surrounded by fencing as shown. Thereis 900 feet of fencing available. Each section shouldhave an area of 12,000 square feet.

a. Show that w5 3002 4}3l.

b. Find the possible dimensions of each section.

TAKS REASONING

TAKS REASONING


13 299

73. You can model the position (x, y) of a movingob rametric equations. Such equations give x and y interms of a third variable t that represents time. For example, suppose thatwhen a basketball player attempts a free throw, the path of the basketballcan be modeled by the parametric equations

x5 20t

y5216t21 21t1 6

where x and y are measured in feet, t is measured in seconds, and the player’sfeet are at (0, 0).

a. Evaluate Make a table of values giving the position (x, y) of thebasketball after 0, 0.25, 0.5, 0.75, and 1 second.

b. Graph Use your table from part (a) to graph the parametric equations.

c. Solve The position of the basketball rim is (15, 10). The top of thebackboard is (15, 12). Does the player make the free throw? Explain.

74. CHALLENGE The Stratosphere Tower in Las Vegas is 921 feet talland has a “needle” at its top that extends even higher into the air.A thrill ride called the Big Shot catapults riders 160 feet up theneedle and then lets them fall back to the launching pad.

a. The height h (in feet) of a rider on the Big Shot can bemodeled by h5216t21 v

0t1 921 where t is the elapsed

time (in seconds) after launch and v0 is the initial vertical

velocity (in feet per second). Find v0 using the fact that the

maximum value of h is 921 1 1605 1081 feet.

b. A brochure for the Big Shot states that the ride up theneedle takes two seconds. Compare this time with the timegiven by the model h5216t21 v

0t1 921 where v

0 is the

value you found in part (a). Discuss the model’s accuracy.


Big Shot ride

75. TAKS PRACTICE In the figure shown,}AB isparallel to}ED . Which equation can be used to findthe value of x? TAKS Obj. 6

A 5x1 2255 360 B 5x1 2355 540

C 7x1 2355 360 D 7x1 2255 540

76. TAKS PRACTICE Music recital tickets are $4 for students and $6 for adults.A total of 725 tickets are sold and $3650 is collected. Which pair of equationscan be used to determine the number of students, s, and the number ofadults, a, who attended the music recital? TAKS Obj. 4

F s1 a5 725 G s1 a5 7254s1 6a5 3650 6s1 4a5 3650

H s2 a5 725 J 4s1 6a5 7254s2 6a5 3650 s1 a5 3650


REVIEW

TAKS Preparation

p. 408;

TAKS Workbook

REVIEW

Lesson 3.2;

TAKS Workbook


DE

B

C

A

2x8

(3x2 5)8

(2x1 50)8


TAKS REASONING

ject using a pair of pa


4.9 EXERCISES

1. VOCABULARY Give an example of a quadratic inequality in one variableand an example of a quadratic inequality in two variables.

2. ! WRITING Explain how to solve x21 6x2 8 < 0 using a table, by graphing,

and algebraically.

MATCHING INEQUALITIES WITH GRAPHS Match the inequality with its graph.

3. y ≤ x21 4x1 3 4. y >2x2

1 4x2 3 5. y < x22 4x1 3

A.

x

y

2

1

B.

x

y

2

1

C.

x

y

1

3

GRAPHING QUADRATIC INEQUALITIES Graph the inequality.

6. y <2x2 7. y ≥ 4x2 8. y > x22 9

9. y ≤ x21 5x 10. y < x2

1 4x2 5 11. y > x21 7x1 12

12. y ≤2x21 3x1 10 13. y ≥ 2x2

1 5x2 7 14. y ≥22x21 9x2 4

15. y < 4x22 3x2 5 16. y > 0.1x2

2 x1 1.2 17. y ≤22}3x21 3x1 1

ERROR ANALYSIS Describe and correct the error in graphing y ≥ x21 2.

18. 19.

GRAPHING SYSTEMS Graph the system of inequalities.

20. y ≥ 2x2 21. y >25x2 22. y ≥ x22 4

y <2x21 1 y > 3x2

2 2 y ≤22x21 7x1 4

23. y ≤2x21 4x2 4 24. y > 3x2

1 3x2 5 25. y ≥ x22 3x2 6

y < 2x21 x2 8 y <2x2

1 5x1 10 y ≥ 2x21 7x1 6

SOLVING USING A TABLE Solve the inequality using a table.

26. x22 5x < 0 27. x2

1 2x2 3 > 0 28. x21 3x ≤ 10

29. x22 2x ≥ 8 30. 2x2

1 15x2 50 > 0 31. x22 10x <216

32. x22 4x > 12 33. 3x2

2 6x2 2 ≤ 7 34. 2x22 6x2 9 ≥ 11

EXAMPLE 1


EXAMPLE 3

on p. 301for Exs. 20–25

EXAMPLE 4

on p. 302for Exs. 26–34

x

y

1

1

x

y

1

1

HOMEWORK

KEY5WORKED-OUT SOLUTIONS

on p. WS1 for Exs. 17, 39, and 73

5 TAKS PRACTICE AND REASONING

Exs. 44, 45, 68, 73, 78, and 79

5MULTIPLE REPRESENTATIONS

Ex. 74

SKILL PRACTICE

WRITING

4.9 Graph and Solve Quadratic Inequalities 305

SOLVING BY GRAPHING Solve the inequality by graphing.

35. x22 6x < 0 36. x2

1 8x ≤27 37. x22 4x1 2 > 0

38. x21 6x1 3 > 0 39. 3x2

1 2x2 8 ≤ 0 40. 3x21 5x2 3 < 1

41. 26x21 19x ≥ 10 42. 21

}2x21 4x ≥ 1 43. 4x2

2 10x2 7 < 10

44. What is the solution of 3x22 x2 4 > 0?

A x < 21 or x > 4}3

B 21 < x < 4}3

C x < 24}3

or x > 1 D 1 < x < 4}3

45. What is the solution of 2x21 9x ≤ 56?

A x ≤28 or x ≥ 3.5 B 28 ≤ x ≤ 3.5

C x ≤ 0 or x ≥ 4.5 D 0 ≤ x ≤ 4.5

SOLVING ALGEBRAICALLY Solve the inequality algebraically.

46. 4x2 < 25 47. x21 10x1 9 < 0 48. x2

2 11x ≥228

49. 3x22 13x > 10 50. 2x2

2 5x2 3 ≤ 0 51. 4x21 8x2 21 ≥ 0

52. 24x22 x1 3 ≤ 0 53. 5x2

2 6x2 2 ≤ 0 54. 23x21 10x > 22

55. 22x22 7x ≥ 4 56. 3x2

1 1 < 15x 57. 6x22 5 > 8x

58. GRAPHING CALCULATOR In this exercise, you will use a different graphical method to solve Example 6 on page 303.

a. Enter the equations y5 7.51x22 16.4x1 35.0 and y5 100 into a

graphing calculator.

b. Graph the equations from part (a) for 0 ≤ x ≤ 9 and 0 ≤ y ≤ 300.

c. Use the intersect feature to find the point where the graphs intersect.

d. During what years was the number of participating teams greater than 100? Explain your reasoning.

CHOOSING A METHOD Solve the inequality using any method.

59. 8x22 3x1 1 < 10 60. 4x2

1 11x1 3 ≥23 61. 2x22 2x2 1 > 2

62. 23x21 4x2 5 ≤ 2 63. x2

2 7x1 4 > 5x2 2 64. 2x21 9x2 1 ≥23x1 1

65. 3x22 2x1 1 ≤2x2

1 1 66. 5x21 x2 7 < 3x2

2 4x 67. 6x22 5x1 2 < 23x2

1 x

68. Write a quadratic inequality in one variable that has a solution of x < 22 or x > 5.

69. CHALLENGE The area A of the region bounded by a

parabola and a horizontal line is given by A5 2}3bh

where b and h are as defined in the diagram. Find the area of the region determined by each pair of inequalities.

a. y ≤2x21 4x b. y ≥ x2

2 4x2 5 y ≥ 0 y ≤ 3

EXAMPLE 5

on p. 302for Exs. 35–43

EXAMPLE 7

on p. 303 for Exs. 46–57

x

y

b

h

TAK

TAKS REASONING

TAKS REASONING

S REASONING

306

70. ENGINEERING A wire rope can safely support a weight W (in pounds)provided W ≤ 8000d2 where d is the rope’s diameter (in inches). Graphthe inequality.

71. WOODWORKING A hardwood shelf in a wooden bookcase can safely supporta weight W (in pounds) provided W ≤ 115x2 where x is the shelf’s thickness (ininches). Graph the inequality.

72. ARCHITECTURE The arch of the Sydney Harbor Bridge in Sydney, Australia,can be modeled by y 520.00211x2

1 1.06x where x is the distance (in meters)from the left pylons and y is the height (in meters) of the arch above thewater. For what distances x is the arch above the road?

73. The length L (in millimeters) of the larvae of the blackporgy fish can be modeled by

L(x)5 0.00170x21 0.145x1 2.35, 0 ≤ x ≤ 40

where x is the age (in days) of the larvae. Write and solve an inequality to findat what ages a larvae’s length tends to be greater than 10 millimeters. Explainhow the given domain affects the solution.

74. MULTIPLE REPRESENTATIONS A study found that a driver’s reaction timeA(x) to audio stimuli and his or her reaction time V(x) to visual stimuli (bothin milliseconds) can be modeled by

A(x)5 0.0051x22 0.319x1 15, 16 ≤ x ≤ 70

V(x)5 0.005x22 0.23x1 22, 16 ≤ x ≤ 70

where x is the driver’s age (in years).

a. Writing an Inequality Write an inequality that you can use to find thex-values for which A(x) is less than V(x).

b. Making a Table Use a table to find the solution of the inequalityfrom part (a). Your table should contain x-values from 16 to 70 inincrements of 6.

c. Drawing a Graph Check the solution you found in part (b) by usinga graphing calculator to solve the inequality A(x) < V(x) graphically.Describe how you used the domain 16 ≤ x ≤ 70 to determine a reasonablesolution.

d. Interpret Based on your results from parts (b) and (c), do you think adriver would react more quickly to a traffic light changing from green toyellow or to the siren of an approaching ambulance? Explain.

EXAMPLE 6

on p. 303for Exs. 72–74

EXAMPLE 2

on p. 301for Exs. 70–71


PROBLEM SOLVING

y

x

pylon

52 m

5 MULTIPLEREPRESENTATIONS


TAKS REASONING

307

75. SOCCER The path of a soccer ball kicked from the ground can be modeled by

y520.0540x21 1.43x

where x is the horizontal distance (in feet) from where the ball was kickedand y is the corresponding height (in feet).

a. A soccer goal is 8 feet high. Write and solve an inequality to find at whatvalues of x the ball is low enough to go into the goal.

b. A soccer player kicks the ball toward the goal from a distance of 15 feetaway. No one is blocking the goal. Will the player score a goal? Explainyour reasoning.

76. MULTI-STEP PROBLEM A truck that is 11 feet tall and7 feet wide is traveling under an arch. The arch canbe modeled by

y520.0625x21 1.25x1 5.75

where x and y are measured in feet.

a. Will the truck fit under the arch? Explain yourreasoning.

b. What is the maximum width that a truck 11 feettall can have and still make it under the arch?

c. What is the maximum height that a truck 7 feetwide can have and still make it under the arch?

77. CHALLENGE For clear blue ice on lakes and ponds, the maximum weight w(in tons) that the ice can support is given by

w(x)5 0.1x22 0.5x2 5

where x is the thickness of the ice (in inches).

a. Calculate What thicknesses of ice can support a weight of 20 tons?

b. Interpret Explain how you can use the graph of w(x) to determine theminimum x-value in the domain for which the function gives meaningfulresults.


78. TAKS PRACTICE Rachel is a cross-country runner.Her coach recorded the data shown at the right duringa timed practice run. If Rachel continues to run at thesame rate, what is the approximate distance she willrun in 25 minutes? TAKS Obj. 3

A 4.2 km B 5 km

C 6 km D 10 km

79. TAKS PRACTICE Which set of dimensions correspondsto a pyramid similar to the one shown? TAKS Obj. 8

F w5 1 unit, l5 2 units, h5 4 units

G w5 2 units, l5 3 units, h5 6 units

H w5 3 units, l5 4 units, h5 8 units

J w5 4 units, l5 6 units, h5 12 units


REVIEW

Lesson 2.5;

TAKS Workbook

REVIEW

TAKS Preparation

p. 902;

TAKS Workbook

Time(minutes)

Distance(kilometers)

6 1.2

12 2.4

15 3

h 5 24

w 5 9l 5 12


1. VOCABULARY Copy and complete: When you perform quadratic regressionon a set of data, the quadratic model obtained is called the ? .

2. Describe how to write an equation of a parabola if you knowthree points on the parabola that are not the vertex or x-intercepts.

WRITING IN VERTEX FORM Write a quadratic function in vertex form for theparabola shown.

3. 4. 5.

WRITING IN VERTEX FORM Write a quadratic function in vertex form whosegraph has the given vertex and passes through the given point.

6. vertex: (24, 1) 7. vertex: (1, 6) 8. vertex: (5,24)point: (22, 5) point: (21, 2) point: (1, 20)

9. vertex: (23, 3) 10. vertex: (5, 0) 11. vertex: (24,22)point: (1,21) point: (2,227) point: (0, 30)

12. vertex: (2, 1) 13. vertex: (21,24) 14. vertex: (3, 5)point: (4,22) point: (2,21) point: (7,23)

15. The vertex of a parabola is (5, 23) and another point onthe parabola is (1, 5). Which point is also on the parabola?

A (0, 3) B (21, 9) C (21, 15) D (7, 7)

16. The x-intercepts of a parabola are 4 and 7 and anotherpoint on the parabola is (2, 220). Which point is also on the parabola?

A (1, 21) B (8,24) C (5,240) D (5, 4)

WRITING IN INTERCEPT FORM Write a quadratic function in intercept form forthe parabola shown.

17.

x

y

1

1

(0, 6)

22 3

18.

21

(23, 3)

x

y

1

26 24

19.

x

y

1

1

(1,24)

323

4.10 EXERCISES

EXAMPLE 1


EXAMPLE 2

on p. 309for Exs. 16–26

x

y

1

1vertex

(3, 2)

(5, 6)

(21,21)

vertex

(22, 1)

x

y

1

1x

y

2

1

vertex

(21,23)

(1,21)

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSon p. WS1 for Exs. 19, 35, and 49


5 MULTIPLE REPRESENTATIONSEx. 50

SKILL PRACTICE

WRITING

TAKS REASONING

TAKS REASONING

4.10 Write Quadratic Functions and Models 313

WRITING IN INTERCEPT FORM Write a quadratic function in intercept formwhose graph has the given x-intercepts and passes through the given point.

20. x-intercepts: 2, 5 21. x-intercepts:23, 0 22. x-intercepts:21, 4point: (4,22) point: (2, 10) point: (2, 4)

23. x-intercepts: 3, 7 24. x-intercepts:25,21 25. x-intercepts:26, 3point: (6,29) point: (27,224) point: (0,29)

ERROR ANALYSIS Describe and correct the error in writing a quadratic functionwhose graph has the given x-intercepts or vertex and passes through the givenpoint.

26. x-intercepts: 4,23; point: (5,25) 27. vertex: (2, 3); point: (1, 5)

y 5 a(x 2 5)(x 1 5)

23 5 a(4 2 5)(4 1 5)

23 5 29a

1}3

5 a, so y 51}3 (x 2 5)(x 1 5)

y 5 a(x 2 2)(x 2 3)

5 5 a(1 2 2)(1 2 3)

5 5 2a

5}2

5 a, so y 55}2 (x 2 2)(x 2 3)

WRITING IN STANDARD FORM Write a quadratic function in standard form forthe parabola shown.

28.

x

y

4

23

(2,21)

(1,26)

(4,23)

29.

x

y

2

6

(26,22) (24,22)

(23, 4)

30.

x

y

2

2

(24,26)

(0,22)

(2, 6)

WRITING IN STANDARD FORM Write a quadratic function in standard form forthe parabola that passes through the given points.

31. (24,23), (0,22), (1, 7) 32. (22,24), (0,210), (3,27) 33. (22, 4), (0, 5), (1,211)

34. (21,21), (1, 11), (3, 7) 35. (21, 9), (1, 1), (3, 17) 36. (26,21), (23,24), (3, 8)

37. (22,213), (2, 3), (4, 5) 38. (26, 29), (24, 12), (2,23) 39. (23,22), (3, 10), (6,22)

WRITING QUADRATIC FUNCTIONS Write a quadratic function whose graph hasthe given characteristics.

40. passes through: 41. x-intercepts:211, 3 42. vertex: (4.5, 7.25)(20.5,21), (2, 8), (11, 25) passes through: (1, 2192) passes through: (7, 23)

43. Draw a parabola that passes through (22, 3). Write afunction for the parabola in standard form, intercept form, and vertex form.

44. Suppose you are given a set of data pairs (x, y). Describehow you can use ratios to determine whether the data can be modeled by aquadratic function of the form y5 ax2.

45. CHALLENGE Find a function of the form y5 ax21 bx1 c whose graph passes

through (1,24), (23,216), and (7, 14). Explain what the model tells you aboutthe points.

EXAMPLE 3

on p. 310for Exs. 28–39

TAKS REASONING

TAKS REASONING

314

EXAMPLE 4

on p. 311for Exs. 48–50

46. ANTENNA DISH Three points on the parabola formed bythe cross section of an antenna dish are (0, 4), (2, 3.25),and (5, 3.0625). Write a quadratic function that modelsthe cross section.

47. FOOTBALL Two points on the parabolic path of a kicked football are (0, 0)and the vertex (20, 15). Write a quadratic function that models the path.

48. MULTI-STEP PROBLEM The bar graph shows theaverage number of hours per person per year spenton the Internet in the United States for the years1997–2001.

a. Use a graphing calculator to create a scatter plot.

b. Use the quadratic regression feature of thecalculator to find the best-fitting quadraticmodel for the data.

c. Use your model from part (b) to predict theaverage number of hours a person will spend onthe Internet in 2010.

49. RUNNING The table shows how wind affects a runner’s performancein the 200 meter dash. Positive wind speeds correspond to tailwinds,and negative wind speeds correspond to headwinds. The change t infinishing time is the difference beween the runner’s time whenthe wind speed is s and the runner’s time when there is no wind.

Wind speed (m/sec), s 26 24 22 0 2 4 6

Change in finishingtime (sec), t 2.28 1.42 0.67 0 20.57 21.05 21.42

a. Use a graphing calculator to find the best-fitting quadratic model.

b. Predict the change in finishing time when the wind speed is 10 m/sec.

50. MULTIPLE REPRESENTATIONS The table shows the number of U.S.

households (in millions) with color televisions from 1970 through 2000.

Years since 1970 0 5 10 15 20 25 30

Households withcolor TVs (millions) 21 47 63 78 90 94 101

a. Drawing a Graph Make a scatter plot of the data. Draw the parabola thatyou think best fits the data.

b. Writing a Function Estimate the coordinates of three points on theparabola. Use the points to write a quadratic function for the data.

c. Making a Table Use your function from part (b) to make a table of datafor the years listed in the original table above. Compare the numbers ofhouseholds given by your function with the numbers in the original table.


PROBLEM SOLVING

EXAMPLES1 and 3

on pp. 309–310for Exs. 46–47

Yearly Time on the Internet

54

34

82

106

134

Hours per person

1998 1999 2000 2001

100

50

01997

150

x

y

1

1

5 MULTIPLEREPRESENTATIONS


315

51. MULTIPLE CHOICE The Garabit Viaduct in France has a parabolic arch aspart of its support. Three points on the parabola that models the arch are(0, 0), (40, 38.2), and (165, 0) where x and y are measured in meters. Whichpoint is also on the parabola?

A (10,211.84) B (26.74, 25) C (80, 51.95) D (125, 45)

52. CHALLENGE Let R be the maximum number of regions into whicha circle can be divided using n chords. For example, the diagramshows that R5 4 when n5 2. Copy and complete the table. Thenwrite a quadratic model giving R as a function of n.

n 0 1 2 3 4 5 6

R ? ? 4 ? ? ? ?


Use the quadratic formula to solve the equation. (p. 292)

1. x22 4x1 55 0 2. 2x2

2 8x1 15 0 3. 3x21 5x1 45 0

Graph the inequality. (p. 300)

4. y <23x2 5. y >2x21 2x 6. y ≥2x2

1 2x1 3

Solve the inequality. (p. 300)

7. 0 ≥ x21 5 8. 12 ≤ x2

2 7x 9. 2x21 2 >2 5x

Write a quadratic function whose graph has the given characteristics. (p. 309)

10. vertex: (5, 7) 11. x-intercepts:23, 5 12. passes through:passes through: (3, 11) passes through: (7,240) (21, 2), (4,223), (2,27)

13. SPORTS A person throws a baseball into the air with an initial verticalvelocity of 30 feet per second and then lets the ball hit the ground. The ballis released 5 feet above the ground. How long is the ball in the air? (p. 292)

2

1

4

3


TAKS REASONING

53. TAKS PRACTICE Charlie receives some money for his birthday. He depositsone third of the money in the bank. He purchases a concert ticket for $45.Then he spends half of the remaining money on dinner. Charlie has $8.50 left.How much money did he receive for his birthday? TAKS Obj. 10

A $80 B $93 C $118 D $124

54. TAKS PRACTICE Which equation represents a line that is parallel to the linethat passes through (24, 9) and (5,23)? TAKS Obj. 7

F 24x1 3y5 29 G 2x1 3y5 9

H 4x1 3y5212 J 2x2 3y5 11


REVIEW

Skills Review

Handbook p. 998;

TAKS Workbook

REVIEW

Lesson 2.3;

TAKS Workbook


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