square root, p. 266. â¢ complex number, p. 276. â¢ imaginary number, p. 276. â¢ completing the square, p. 284. â¢ quadratic formula, p. 292. â¢ d...

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Quadratic Functions and Factoring 4.1

Graph Quadratic Functions in Standard Form

4.2

Graph Quadratic Functions in Vertex or Intercept Form

4.3

Solve x 2 1 bx 1 c 5 0 by Factoring

4.4

Solve ax 2 1 bx 1 c 5 0 by Factoring

4.5

Solve Quadratic Equations by Finding Square Roots

4.6

Perform Operations with Complex Numbers

4.7

Complete the Square

4.8

Use the Quadratic Formula and the Discriminant

4.9

Graph and Solve Quadratic Inequalities

4.10 Write Quadratic Functions and Models

Before In previous chapters, you learned the following skills, which you’ll use in Chapter 4: evaluating expressions, graphing functions, and solving equations.

Prerequisite Skills VOCABULARY CHECK

y

Copy and complete the statement.

(0, 2)

1. The x-intercept of the line shown is ? .

1

(3, 0) 1

2. The y-intercept of the line shown is ? .

x

SKILLS CHECK Evaluate the expression when x 5 23. (Review p. 10 for 4.1, 4.7.) 3. 25x 2 1 1

4. x 2 2 x 2 8

5. (x 1 4)2

6. 23(x 2 7)2 1 2

Graph the function and label the vertex. (Review p. 123 for 4.2.) 7. y 5 x 1 2

8. y 5 x 2 3

9. y 5 22x

10. y 5 x 2 5 1 4

Solve the equation. (Review p. 18 for 4.3, 4.4.) 11. x 1 8 5 0

12. 3x 2 5 5 0

13. 2x 1 1 5 x

14. 4(x 2 3) 5 x 1 9

1SFSFRVJTJUFTLJMMTQSBDUJDFBUDMBTT[POFDPN

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Now In Chapter 4, you will apply the big ideas listed below and reviewed in the Chapter Summary on page 317. You will also use the key vocabulary listed below.

Big Ideas 1 Graphing and writing quadratic functions in several forms 2 Solving quadratic equations using a variety of methods 3 Performing operations with square roots and complex numbers KEY VOCABULARY • standard form of a quadratic function, p. 236

• root of an equation, p. 253 • zero of a function, p. 254

• completing the square, p. 284

• parabola, p. 236

• square root, p. 266

• quadratic formula, p. 292

• vertex form, p. 245

• complex number, p. 276

• discriminant, p. 294

• intercept form, p. 246

• imaginary number, p. 276

• best-fitting quadratic model, p. 311

• quadratic equation, p. 253

Why? You can use quadratic functions to model the heights of projectiles. For example, the height of a baseball hit by a batter can be modeled by a quadratic function.

Algebra The animation illustrated below for Example 7 on page 287 helps you answer this question: How does changing the ball speed and hitting angle affect the maximum height of a baseball?

4HEFUNCTIONISNOWINVERTEXFORMYnTn 2EMEMBERTHATTHEVERTEXOFTHEPARABOLAISATH K ANDTHATTHEMAXIMUMHEIGHTOFTHE BASEBALLINFLIGHTISK7HATISTHEMAXIMUMHEIGHTOFTHEBASEBALLINFEET -AXIMUMHEIGHTFEET

3TART

A quadratic function models the height of a baseball in flight.

#HECK!NSWER

Rewrite the function in vertex form to find the maximum height of the ball.

Algebra at classzone.com Other animations for Chapter 4: pages 238, 247, 269, 279, 300, and 317

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4.1 Before Now Why?

Key Vocabulary • quadratic function • parabola • vertex • axis of symmetry • minimum value • maximum value

Graph Quadratic Functions in Standard Form You graphed linear functions. You will graph quadratic functions. So you can model sports revenue, as in Example 5.

A quadratic function is a function that can be written in the standard form y 5 ax 2 1 bx 1 c where a Þ 0. The graph of a quadratic function is a parabola.

For Your Notebook

KEY CONCEPT Parent Function for Quadratic Functions

The parent function for the family of all quadratic functions is f(x) 5 x 2. The graph of f(x) 5 x 2 is the parabola shown below. The axis of symmetry divides the parabola into mirror images and passes through the vertex.

y

The lowest or highest point on a parabola is the vertex. The vertex for f (x) 5 x 2 is (0, 0).

y 5 x2

1

x

1

For f(x) 5 x 2, and for any quadratic function g(x) 5 ax 2 1 bx 1 c where b 5 0, the vertex lies on the y-axis and the axis of symmetry is x 5 0.

EXAMPLE 1

Graph a function of the form y 5 ax 2

Graph y 5 2x 2 . Compare the graph with the graph of y 5 x 2 . Solution

STEP 1 SKETCH A GRAPH Choose values of x on both sides of the axis of symmetry x 5 0.

Make a table of values for y 5 2x 2. x

22

21

0

1

2

y

8

2

0

2

8

y

y 5 x2 y 5 2x 2

STEP 2 Plot the points from the table. STEP 3 Draw a smooth curve through the points.

3

STEP 4 Compare the graphs of y 5 2x 2 and y 5 x 2 . Both open up and have the same vertex and axis of symmetry. The graph of y 5 2x 2 is narrower than the graph of y 5 x 2.

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1

x

Chapter 4 Quadratic Functions and Factoring

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Graph a function of the form y 5 ax 2 1 c

EXAMPLE 2

1 x 2 1 3. Compare the graph with the graph of y 5 x 2 . Graph y 5 2} 2

Solution

STEP 1

1 x 2 1 3. Make a table of values for y 5 2} 2

SKETCH A GRAPH Choose values of x that are multiples of 2 so that the values of y will be integers.

x

24

22

0

2

4

y

25

1

3

1

25

y

y 5 x2 1 1

STEP 2 Plot the points from the table.

x

y 5 2 12 x 2 1 3

STEP 3 Draw a smooth curve through the points. 1 x 2 1 3 and STEP 4 Compare the graphs of y 5 2} 2

y 5 x 2 . Both graphs have the same axis of 1 x 2 1 3 opens down and is symmetry. However, the graph of y 5 2} 2

wider than the graph of y 5 x 2. Also, its vertex is 3 units higher.

✓

GUIDED PRACTICE

for Examples 1 and 2

Graph the function. Compare the graph with the graph of y 5 x 2 . 1. y 5 24x 2

1 x2 1 2 3. f(x) 5 } 4

2. y 5 2x 2 2 5

GRAPHING ANY QUADRATIC FUNCTION You can use the following properties to graph any quadratic function y 5 ax 2 1 bx 1 c, including a function where b Þ 0.

For Your Notebook

KEY CONCEPT Properties of the Graph of y 5 ax 2 1 bx 1 c y 5 ax 2 1 bx 1 c , a > 0

y 5 ax 2 1 bx 1 c , a < 0 y

y

x52

b 2a

(0, c) x

x52

(0, c)

x

b 2a

Characteristics of the graph of y 5 ax 2 1 bx 1 c: • The graph opens up if a > 0 and opens down if a < 0. • The graph is narrower than the graph of y 5 x 2 if a > 1 and wider if a < 1. b and the vertex has x-coordinate 2 b . • The axis of symmetry is x 5 2} } 2a 2a

• The y-intercept is c. So, the point (0, c) is on the parabola.

4.1 Graph Quadratic Functions in Standard Form

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EXAMPLE 3

Graph a function of the form y 5 ax 2 1 bx 1 c

Graph y 5 2x 2 2 8x 1 6. Solution

STEP 1

Identify the coefficients of the function. The coefficients are a 5 2,

b 5 28, and c 5 6. Because a > 0, the parabola opens up.

STEP 2 Find the vertex. Calculate the x-coordinate. AVOID ERRORS

y

axis of symmetry

(28) 2(2)

b 52 x 5 2} }52 2a

Be sure to include the negative sign before the fraction when calculating the x-coordinate of the vertex.

x52

Then find the y-coordinate of the vertex. 1

y 5 2(2)2 2 8(2) 1 6 5 22

x

1

So, the vertex is (2, 22). Plot this point.

vertex (2, 22)

STEP 3 Draw the axis of symmetry x 5 2. STEP 4 Identify the y-intercept c, which is 6. Plot the point (0, 6). Then reflect this point in the axis of symmetry to plot another point, (4, 6).

y

STEP 5 Evaluate the function for another value of x, such as x 5 1. y 5 2(1)2 2 8(1) 1 6 5 0

1 5

Plot the point (1, 0) and its reflection (3, 0) in the axis of symmetry.

x

(2, 22)

STEP 6 Draw a parabola through the plotted points. "MHFCSB

✓

at classzone.com

GUIDED PRACTICE

for Example 3

Graph the function. Label the vertex and axis of symmetry. 4. y 5 x 2 2 2x 2 1

1 x 2 2 5x 1 2 6. f(x) 5 2} 3

5. y 5 2x 2 1 6x 1 3

For Your Notebook

KEY CONCEPT Minimum and Maximum Values Words Graphs

For y 5 ax 2 1 bx 1 c, the vertex’s y-coordinate is the minimum value of the function if a > 0 and the maximum value if a < 0. y

y

maximum

x

x

minimum a is positive

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a is negative

Chapter 4 Quadratic Functions and Factoring

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EXAMPLE 4

Find the minimum or maximum value

Tell whether the function y 5 3x 2 2 18x 1 20 has a minimum value or a maximum value. Then find the minimum or maximum value. Solution Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex. (218) 2(3)

b 52 x 5 2} }53 2a

y 5 3(3)2 2 18(3) 1 20 5 27

Minimum X=3

c The minimum value is y 5 27. You can check the answer on a graphing calculator.

EXAMPLE 5

Y=-7

Solve a multi-step problem

GO-CARTS A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for every $1 reduction in the price per racer. How can the owner of the go-cart track maximize weekly revenue?

Solution

STEP 1

Define the variables. Let x represent the price

reduction and R(x) represent the weekly revenue.

STEP 2 Write a verbal model. Then write and simplify a quadratic function. Revenue

5

(dollars)

INTERPRET FUNCTIONS Notice that a 5 220 < 0, so the revenue function has a maximum value.

Price (dollars/racer)

p

Attendance

p

(380 1 20x)

(racers)

R(x)

5

(35 2 x)

R(x)

5

13,300 1 700x 2 380x 2 20x 2

R(x)

5

220x 2 1 320x 1 13,300

STEP 3 Find the coordinates (x, R(x)) of the vertex. b 5 2 320 5 8 x 5 2} }

Find x-coordinate.

R(8) 5 220(8)2 1 320(8) 1 13,300 5 14,580

Evaluate R(8).

2a

2(220)

c The vertex is (8, 14,580), which means the owner should reduce the price per racer by $8 to increase the weekly revenue to $14,580.

✓

GUIDED PRACTICE

for Examples 4 and 5

7. Find the minimum value of y 5 4x 2 1 16x 2 3. 8. WHAT IF? In Example 5, suppose each $1 reduction in the price per racer

brings in 40 more racers per week. How can weekly revenue be maximized? 4.1 Graph Quadratic Functions in Standard Form

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4.1

EXERCISES

HOMEWORK KEY

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

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 39, 40, 43, 53, 58, and 60 5 MULTIPLE REPRESENTATIONS Ex. 59

SKILL PRACTICE 1. VOCABULARY Copy and complete: The graph of a quadratic function is

called a(n) ? . 2. ★ WRITING Describe how to determine whether a quadratic function has a

minimum value or a maximum value. EXAMPLE 1 on p. 236 for Exs. 3–12

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

3. y 5 4x 2

4. y 5 23x 2

x

22

21

0

1

2

x

22

21

0

1

2

y

?

?

?

?

?

y

?

?

?

?

?

1x2 5. y 5 } 2

1x2 6. y 5 2} 3

x

24

22

0

2

4

x

26

23

0

3

6

y

?

?

?

?

?

y

?

?

?

?

?

MAKING A GRAPH Graph the function. Compare the graph with the graph of

y 5 x 2. 7. y 5 3x 2

8. y 5 5x 2

9. y 5 22x 2

10. y 5 2x 2

1x2 11. f(x) 5 } 3

1x2 12. g(x) 5 2} 4

EXAMPLE 2

13. y 5 5x 2 1 1

14. y 5 4x 2 1 1

15. f(x) 5 2x 2 1 2

on p. 237 for Exs. 13–18

16. g(x) 5 22x 2 2 5

3x2 2 5 17. f(x) 5 } 4

1x2 2 2 18. g(x) 5 2} 5

ERROR ANALYSIS Describe and correct the error in analyzing the graph of y 5 4x 2 1 24x 2 7.

19.

20.

The x-coordinate of the vertex is: b 5 24 5 3 x5} } 2a

2(4)

The y-intercept of the graph is the value of c, which is 7.

EXAMPLE 3

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

on p. 238 for Exs. 21–32

21. y 5 x 2 1 2x 1 1

22. y 5 3x 2 2 6x 1 4

23. y 5 24x 2 1 8x 1 2

24. y 5 22x 2 2 6x 1 3

25. g(x) 5 2x 2 2 2x 2 1

26. f(x) 5 26x 2 2 4x 2 5

2 x 2 2 3x 1 6 27. y 5 } 3

3 x 2 2 4x 2 1 28. y 5 2} 4

3 x 2 1 2x 1 2 29. g(x) 5 2} 5

1x2 1 x 2 3 30. f(x) 5 } 2

8 x 2 2 4x 1 5 31. y 5 } 5

5x2 2 x 2 4 32. y 5 2} 3

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EXAMPLE 4

MINIMUMS OR MAXIMUMS Tell whether the function has a minimum value or a

on p. 239 for Exs. 33–38

maximum value. Then find the minimum or maximum value. 33. y 5 26x 2 2 1

34. y 5 9x 2 1 7

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

36. g(x) 5 23x 2 1 18x 2 5

3 x 2 1 6x 1 4 37. f(x) 5 } 2

1 x 2 2 7x 1 2 38. y 5 2} 4

39. ★ MULTIPLE CHOICE What is the effect on the graph of the function

y 5 x 2 1 2 when it is changed to y 5 x 2 2 3?

A The graph widens.

B The graph narrows.

C The graph opens down.

D The vertex moves down the y-axis.

40. ★ MULTIPLE CHOICE Which function has the widest graph?

A y 5 2x 2

B y 5 x2

C y 5 0.5x 2

D y 5 2x 2

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 by

y 5 20.02x 2 1 x 1 6. 42. The path of a shot put released at an angle of 358 can be modeled by

y 5 20.01x 2 1 0.7x 1 6. y

358 x

43. ★ OPEN-ENDED MATH Write three different quadratic functions whose graphs

have the line x 5 4 as an axis of symmetry but have different y-intercepts. MATCHING In Exercises 44–46, match the equation with its graph.

44. y 5 0.5x 2 2 2x A.

45. y 5 0.5x 2 1 3 B.

y

46. y 5 0.5x 2 2 2x 1 3 C.

y

y

(2, 5) (0, 0) (0, 3)

(0, 3)

21

1

1 1

x

(2, 1) 1

x

1

(2, 22) x

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

47. f(x) 5 0.1x 2 1 2

48. g(x) 5 20.5x 2 2 5

49. y 5 0.3x 2 1 3x 2 1

50. y 5 0.25x 2 2 1.5x 1 3

51. f(x) 5 4.2x 2 1 6x 2 1

52. g(x) 5 1.75x 2 2 2.5

53. ★ SHORT RESPONSE The points (2, 3) and (24, 3) lie on the graph of a

quadratic function. Explain how these points can be used to find an equation of the axis of symmetry. Then write an equation of the axis of symmetry. 54. CHALLENGE For the graph of y 5 ax 2 1 bx 1 c, show that the y-coordinate of b 2 1 c. the vertex is 2} 4a

4.1 Graph Quadratic Functions in Standard Form

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PROBLEM SOLVING EXAMPLE 5

55. ONLINE MUSIC An online music store sells about 4000 songs each day when it

charges $1 per song. For each $.05 increase in price, about 80 fewer songs per day are sold. Use the verbal model and quadratic function to find how the store can maximize daily revenue.

on p. 239 for Exs. 55–58

Revenue (dollars)

R(x)

5

5

Price

p

(dollars/song)

Sales (songs)

p (4000 2 80x)

(1 1 0.05x)

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

56. DIGITAL CAMERAS An electronics store sells about 70 of a new model of

digital camera per month at a price of $320 each. For each $20 decrease in price, about 5 more cameras per month are sold. Write a function that models the situation. Then tell how the store can maximize monthly revenue from sales of the camera. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

57. GOLDEN GATE BRIDGE Each cable joining the two towers on the Golden Gate

Bridge can be modeled by the function 1 x 2 2 7 x 1 500 y5} } 9000

15

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

FT H X

FT

58. ★ SHORT RESPONSE A woodland jumping mouse hops along a parabolic

path given by y 5 20.2x 2 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 over a fence that is 3 feet high? Explain.

59.

MULTIPLE REPRESENTATIONS A community theater sells about 150 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 theater has fixed expenses of $1500 per week.

a. Writing a Model Write a verbal model and a quadratic function to

represent 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. Then

use the graph to find how the theater can maximize weekly profit.

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5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

★

5 STANDARDIZED TEST PRACTICE

5 MULTIPLE REPRESENTATIONS

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60. ★ EXTENDED RESPONSE In 1971, astronaut Alan Shepard hit a golf ball on

the moon. The path of a golf ball hit at an angle of 458 and with a speed of 100 feet per second can be modeled by g 10,000

y 5 2}x 2 1 x where x is the ball’s horizontal position (in feet), y is the corresponding height (in feet), and g is the acceleration due to gravity (in feet per second squared). a. Model Use the information in the diagram to write functions for the

paths of a golf ball hit on Earth and a golf ball hit on the moon.

GRAPHING CALCULATOR

b. Graphing Calculator Graph the functions from part (a) on a graphing

In part (b), use the calculator’s zero feature to answer the questions.

c. Interpret Compare the distances traveled by a golf ball on Earth and on

calculator. How far does the golf ball travel on Earth? on the moon? the 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 beach

want to rope off a rectangular swimming section. They have P feet of rope with buoys. In terms of P, what is the maximum area that the swimming section can have?

*

W

W

MIXED REVIEW Solve the equation. (p. 18) 62. x 2 3 5 0

63. 3x 1 4 5 0

64. 29x 1 7 5 24x 2 5

65. 5x 2 2 5 22x 1 12

66. 0.7x 1 3 5 0.2x 2 2

67. 0.4x 5 20.5x 2 5

PREVIEW

Graph the function. (p. 123)

Prepare for Lesson 4.2 in Exs. 68–73.

68. y 5 x 2 5

69. y 5 2x 1 2

70. y 5 3x 2 1

71. y 5 24x 1 1

72. f(x) 5 2x 2 3 1 6

73. g(x) 5 25x 1 4 2 1

74. AVERAGE SPEED You are driving on a road trip. At 9:00 A.M., you are

340 miles west of Nashville. At 2:00 P.M., you are 70 miles west of Nashville. Find your average speed. (p. 82)

EXTRA PRACTICE for Lesson 4.1, p. 1013 4.1 GraphONLINE at classzone.com QuadraticQUIZ Functions in Standard Form

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Use after Lesson 4.1

classzone.com Keystrokes

4.1 Find Maximum and Minimum Values

QUESTION

EXAMPLE

How can you use a graphing calculator to find the maximum or minimum value of a function?

Find the maximum value of a function

Find the maximum value of y 5 22x 2 2 10x 2 5 and the value of x where it occurs.

STEP 1 Graph function

STEP 2 Choose left bound

Graph the given function and select the maximum feature.

Move the cursor to the left of the maximum point. Press .

CALCULATE 1:value 2:zero 3:minimum 4:maximum 5:intersect 6:dy/dx

Left Bound? X=-3.4042 Y=5.8646

STEP 3 Choose right bound

STEP 4 Find maximum

Move the cursor to the right of the maximum point. Press .

Put the cursor approximately on the maximum point. Press

Right Bound? X=-1.4893 Y=5.4572

Maximum X=-2.5

Y=7.5

c The maximum value of the function is y 5 7.5 and occurs at x 5 22.5.

PRACTICE Tell whether the function has a maximum value or a minimum value. Then find the maximum or minimum value and the value of x where it occurs.

244

1. y 5 x 2 2 6x 1 4

2. f (x) 5 x 2 2 3x 1 3

3. y 5 23x 2 1 9x 1 2

4. y 5 0.5x 2 1 0.8x 2 2

1 x 2 2 3x 1 2 5. h(x) 5 } 2

3 x 2 1 6x 2 5 6. y 5 2} 8

Chapter 4 Quadratic Functions and Factoring

.

4.2 Before Now Why?

Key Vocabulary • vertex form • intercept form

Graph Quadratic Functions in Vertex or Intercept Form You graphed quadratic functions in standard form. You will graph quadratic functions in vertex form or intercept form. So you can find the height of a jump, as in Ex. 51.

In Lesson 4.1, you learned that the standard form of a quadratic function is y 5 ax2 1 bx 1 c where a Þ 0. Another useful form of a quadratic function is the vertex form, y 5 a(x 2 h)2 1 k.

For Your Notebook

KEY CONCEPT Graph of Vertex Form y 5 a(x 2 h)2 1 k The graph of y 5 a(x 2 h)2 1 k is the parabola y 5 ax2 translated horizontally h units and vertically k units. Characteristics of the graph of y 5 a(x 2 h)2 1 k:

y

y 5 a(x 2 h) 2 1 k (h, k) k

y 5 ax 2

• The vertex is (h, k).

(0, 0)

x

h

• The axis of symmetry is x 5 h. • The graph opens up if a > 0 and down if a < 0.

EXAMPLE 1

Graph a quadratic function in vertex form

1 (x 1 2)2 1 5. Graph y 5 2} 4

Solution

STEP 1

1 , h 5 22, and Identify the constants a 5 2} 4

k 5 5. Because a < 0, the parabola opens down.

STEP 2 Plot the vertex (h, k) 5 (22, 5) and draw the axis of symmetry x 5 22.

y

vertex (22, 5) axis of symmetry x 5 22

1

STEP 3 Evaluate the function for two values of x.

1

1 (0 1 2)2 1 5 5 4 x 5 0: y 5 2} 4

x

y

(22, 5)

1 (2 1 2)2 1 5 5 1 x 5 2: y 5 2} 4

Plot the points (0, 4) and (2, 1) and their reflections in the axis of symmetry.

STEP 4 Draw a parabola through the plotted points.

2 x 1

4.2 Graph Quadratic Functions in Vertex or Intercept Form

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EXAMPLE 2

Use a quadratic model in vertex form

CIVIL ENGINEERING The Tacoma

Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable can be modeled by the function

Y

FT

FT

1 (x 2 1400)2 1 27 y5}

X

7000

D

where x and y are measured in feet. What is the distance d between the two towers?

.OTDRAWNTOSCALE

Solution

The vertex of the parabola is (1400, 27). So, a cable’s lowest point is 1400 feet from the left tower shown above. Because the heights of the two towers are the same, the symmetry of the parabola implies that the vertex is also 1400 feet from the right tower. So, the distance between the two towers is d 5 2(1400) 5 2800 feet.

✓

GUIDED PRACTICE

for Examples 1 and 2

Graph the function. Label the vertex and axis of symmetry. 1. y 5 (x 1 2)2 2 3

1 (x 2 3)2 2 4 3. f(x) 5 } 2

2. y 5 2(x 2 1)2 1 5

4. WHAT IF? Suppose an architect designs a bridge with cables that can be 1 (x 2 1400)2 1 27 where x and y are measured in feet. modeled by y 5 } 6500

Compare this function’s graph to the graph of the function in Example 2.

INTERCEPT FORM If the graph of a quadratic function has at least one x-intercept,

then the function can be represented in intercept form, y 5 a(x 2 p)(x 2 q).

For Your Notebook

KEY CONCEPT Graph of Intercept Form y 5 a(x 2 p)(x 2 q) Characteristics of the graph of y 5 a(x 2 p)(x 2 q): • The x-intercepts are p and q.

y

x5

• The axis of symmetry is halfway

p1q 2

between (p, 0) and (q, 0). It has p1q 2

equation x 5 }.

y 5 a(x 2 p)(x 2 q)

• The graph opens up if a > 0 and

(q, 0)

opens down if a < 0.

x

(p, 0)

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EXAMPLE 3

Graph a quadratic function in intercept form

Graph y 5 2(x 1 3)(x 2 1). Solution AVOID ERRORS Remember that the x-intercepts for a quadratic function written in the form y 5 a(x 2 p)(x 2 q) are p and q, not 2p and 2q.

STEP 1

Identify the x-intercepts. Because p 5 23

y

and q 5 1, the x-intercepts occur at the points (23, 0) and (1, 0).

STEP 2 Find the coordinates of the vertex. p1q 2

1

(23, 0)

21

(1, 0) x

23 1 1 5 21 x5}5} 2

y 5 2(21 1 3)(21 2 1) 5 28 So, the vertex is (21, 28).

(21, 28)

STEP 3 Draw a parabola through the vertex and the points where the x-intercepts occur. "MHFCSB

EXAMPLE 4

at classzone.com

Use a quadratic function in intercept form

FOOTBALL The path of a placekicked football can

be modeled by the function y 5 20.026x(x 2 46) where x is the horizontal distance (in yards) and y is the corresponding height (in yards). a. How far is the football kicked? b. What is the football’s maximum height?

Solution a. Rewrite the function as y 5 20.026(x 2 0)(x 2 46). Because p 5 0 and

q 5 46, you know the x-intercepts are 0 and 46. So, you can conclude that the football is kicked a distance of 46 yards. b. To find the football’s maximum height, calculate the coordinates of

the vertex. p1q 2

0 1 46 5 23 x5}5} 2

y 5 20.026(23)(23 2 46) ø 13.8 The maximum height is the y-coordinate of the vertex, or about 13.8 yards.

✓

GUIDED PRACTICE

for Examples 3 and 4

Graph the function. Label the vertex, axis of symmetry, and x-intercepts. 5. y 5 (x 2 3)(x 2 7)

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

7. y 5 2(x 1 1)(x 2 5)

8. WHAT IF? In Example 4, what is the maximum height of the football if the

football’s path can be modeled by the function y 5 20.025x(x 2 50)?

4.2 Graph Quadratic Functions in Vertex or Intercept Form

n2pe-0402.indd 247

247

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FOIL METHOD You can change quadratic functions from intercept form or vertex

form to standard form by multiplying algebraic expressions. One method for multiplying two expressions each containing two terms is FOIL.

For Your Notebook

KEY CONCEPT FOIL Method Words

To multiply two expressions that each contain two terms, add the products of the First terms, the Outer terms, the Inner terms, and the Last terms.

Example

F

O

I

L

2

(x 1 4)(x 1 7) 5 x 1 7x 1 4x 1 28 5 x2 1 11x 1 28

EXAMPLE 5

Change from intercept form to standard form

Write y 5 22(x 1 5)(x 2 8) in standard form.

REVIEW FOIL For help with using the FOIL method, see p. 985.

y 5 22(x 1 5)(x 2 8) 5 22(x2 2 8x 1 5x 2 40) 2

5 22(x 2 3x 2 40) 2

5 22x 1 6x 1 80

EXAMPLE 6

Write original function. Multiply using FOIL. Combine like terms. Distributive property

Change from vertex form to standard form

Write f(x) 5 4(x 2 1)2 1 9 in standard form. f (x) 5 4(x 2 1)2 1 9 5 4(x 2 1)(x 2 1) 1 9

Rewrite (x 2 1) 2 .

5 4(x2 2 x 2 x 1 1) 1 9

Multiply using FOIL.

2

5 4(x 2 2x 1 1) 1 9

Combine like terms.

5 4x2 2 8x 1 4 1 9

Distributive property

2

5 4x 2 8x 1 13

✓

Write original function.

GUIDED PRACTICE

Combine like terms.

for Examples 5 and 6

Write the quadratic function in standard form. 9. y 5 2(x 2 2)(x 2 7)

10. y 5 24(x 2 1)(x 1 3)

11. f(x) 5 2(x 1 5)(x 1 4)

12. y 5 27(x 2 6)(x 1 1)

2

248

n2pe-0402.indd 248

13. y 5 23(x 1 5) 2 1

14. g(x) 5 6(x 2 4)2 2 10

15. f(x) 5 2(x 1 2)2 1 4

16. y 5 2(x 2 3)2 1 9

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:04:36 AM

4.2

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 19, 29, and 53

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 12, 22, 49, 54, and 55

SKILL PRACTICE 1. VOCABULARY Copy and complete: A quadratic function in the form

y 5 a(x 2 h)2 1 k is in ? form. 2. ★ WRITING Explain how to find a quadratic function’s maximum value or

minimum value when the function is given in intercept form. EXAMPLE 1

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

on p. 245 for Exs. 3–12

symmetry. 3. y 5 (x 2 3)2

4. y 5 (x 1 4)2

5. f(x) 5 2(x 1 3)2 1 5

6. y 5 3(x 2 7)2 2 1

7. g(x) 5 24(x 2 2)2 1 4

8. y 5 2(x 1 1)2 2 3

1 (x 1 2)2 1 1 10. y 5 2} 4

9. f(x) 5 22(x 2 1)2 2 5

1 (x 2 3)2 1 2 11. y 5 } 2

12. ★ MULTIPLE CHOICE What is the vertex of the graph of the function

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

A (2, 25)

B (22, 25)

C (25, 2)

D (5, 22)

EXAMPLE 3

GRAPHING WITH INTERCEPT FORM Graph the function. Label the vertex, axis of

on p. 247 for Exs. 13–23

symmetry, and x-intercepts. 13. y 5 (x 1 3)(x 2 3)

14. y 5 (x 1 1)(x 2 3)

15. y 5 3(x 1 2)(x 1 6)

16. f(x) 5 2(x 2 5)(x 2 1)

17. y 5 2(x 2 4)(x 1 6)

18. g(x) 5 24(x 1 3)(x 1 7)

19. y 5 (x 1 1)(x 1 2)

20. f(x) 5 22(x 2 3)(x 1 4)

21. y 5 4(x 2 7)(x 1 2)

22. ★ MULTIPLE CHOICE What is the vertex of the graph of the function

y 5 2(x 2 6)(x 1 4)? A (1, 25)

B (21, 21)

C (26, 4)

D (6, 24)

23. ERROR ANALYSIS Describe and correct

the error in analyzing the graph of the function y 5 5(x 2 2)(x 1 3).

EXAMPLES 5 and 6 on p. 248 for Exs. 24–32

The x-intercepts of the graph are 22 and 3.

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

24. y 5 (x 1 4)(x 1 3)

25. y 5 (x 2 5)(x 1 3)

26. h(x) 5 4(x 1 1)(x 2 6)

27. y 5 23(x 2 2)(x 2 4)

28. f(x) 5 (x 1 5)2 2 2

29. y 5 (x 2 3)2 1 6

30. g(x) 5 2(x 1 6)2 1 10

31. y 5 5(x 1 3)2 2 4

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

MINIMUM OR MAXIMUM VALUES Find the minimum value or the maximum

value of the function. 33. y 5 3(x 2 3)2 2 4

34. g(x) 5 24(x 1 6)2 2 12

35. y 5 15(x 2 25)2 1 130

36. f(x) 5 3(x 1 10)(x 2 8)

37. y 5 2(x 2 36)(x 1 18)

38. y 5 212x(x 2 9)

39. y 5 8x(x 1 15)

40. y 5 2(x 2 3)(x 2 6)

41. g(x) 5 25(x 1 9)(x 2 4)

4.2 Graph Quadratic Functions in Vertex or Intercept Form

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42. GRAPHING CALCULATOR Consider the function y 5 a(x 2 h)2 1 k where

a 5 1, h 5 3, and k 5 22. Predict the effect of each change in a, h, or k described in parts (a)–(c). Use a graphing calculator to check your prediction by graphing the original and revised functions in the same coordinate plane. a. a changes to 23

b. h changes to 21

c. k changes to 2

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

43. y 5 5(x 2 2.25)2 2 2.75 2 x2 1 46. y 5 2} } 3 2

1

2

2

4 1} 5

44. g(x) 5 28(x 1 3.2)2 1 6.4

45. y 5 20.25(x 2 5.2)2 1 8.5

3 (x 1 5)(x 1 8) 47. f(x) 5 2} 4

5 x2 4 x2 2 48. g(x) 5 } } } 5 2 3

21

1

2

49. ★ OPEN-ENDED MATH Write two different quadratic functions in intercept

form whose graphs have axis of symmetry x 5 3. 50. CHALLENGE Write y 5 a(x 2 h)2 1 k and y 5 a(x 2 p)(x 2 q) in standard b, form. Knowing the vertex of the graph of y 5 ax2 1 bx 1 c occurs at x 5 2} 2a

show that the vertex of the graph of y 5 a(x 2 h)2 1 k occurs at x 5 h and that p1q 2

the vertex of the graph of y 5 a(x 2 p)(x 2 q) occurs at x 5 }.

PROBLEM SOLVING EXAMPLES 2 and 4 on pp. 246–247 for Exs. 51–54

51. BIOLOGY The function y 5 20.03(x 2 14)2 1 6 models the jump of a

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

y

x GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

52. CIVIL ENGINEERING The arch of the Gateshead Millennium Bridge forms a

parabola with equation y 5 20.016(x 2 52.5)2 1 45 where x is the horizontal distance (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? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

53. MULTI-STEP PROBLEM Although a football field appears to be flat, its

surface is actually shaped like a parabola so that rain runs off to both sides. The cross section of a field with synthetic turf can be modeled by y

y 5 20.000234x(x 2 160) where x and y are measured in feet.

surface of football field

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

250

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5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

★

Not drawn to scale

x

5 STANDARDIZED TEST PRACTICE

10/17/05 10:04:38 AM

stick with a conventional spring can be modeled by y 5 20.5(x 2 6)2 1 18, and a jump on a pogo stick with a bow spring can be modeled by y 5 21.17(x 2 6)2 1 42, where x and y are measured in inches. Compare the maximum heights of the jumps on the two pogo sticks. Which constants in the functions affect the maximum heights of the jumps? Which do not?

Vertical position (in.)

54. ★ SHORT RESPONSE A jump on a pogo

y 40

bow spring

30 20 10

conventional spring

0 0

2 4 6 8 10 12 x Horizontal position (in.)

55. ★ EXTENDED RESPONSE A kernel of popcorn contains water that expands

when the kernel is heated, causing it to pop. The equations below give the “popping volume” y (in cubic centimeters per gram) of popcorn with moisture content x (as a percent of the popcorn’s weight). Hot-air popping: y 5 20.761(x 2 5.52)(x 2 22.6) Hot-oil popping: y 5 20.652(x 2 5.35)(x 2 21.8) a. Interpret For hot-air popping, what moisture content maximizes popping

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

volume? 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? Explain how you determined the domain and range. 56. CHALLENGE Flying fish use their pectoral fins like airplane

wings to glide through the air. Suppose a flying fish reaches a maximum height of 5 feet after flying a horizontal distance of 33 feet. Write a quadratic function y 5 a(x 2 h)2 1 k that models the flight path, assuming the fish leaves the water at (0, 0). Describe how changing the value of a, h, or k affects the flight path.

MIXED REVIEW PREVIEW

Solve the equation.

Prepare for Lesson 4.3 in Exs. 57–64.

57. x 2 5 5 0 (p. 18)

58. 2x 1 3 5 0 (p. 18)

59. 23x 2 14 5 25x 2 7 (p. 18)

60. 25(3x 1 4) 5 17x 1 2 (p. 18)

61. x 2 9 5 16 (p. 51)

62. 4x 1 9 5 27 (p. 51)

63. 7 2 2x 5 1 (p. 51)

64. 3 2 5x 5 7 (p. 51)

Use the given matrices to perform the indicated operation, if possible. If not possible, state the reason. A5

F

G F G F G F

21 3 ,B5 2 25

2 26 ,C5 3 8

21 4 ,D5 22 3

G F

3 0 21 ,E5 6 1 4

1 22 4 3 21 5

G

65. 2A 1 B (p. 187)

66. 3(B 1 C) (p. 187)

67. D 2 4E (p. 187)

68. AB (p. 195)

69. A(B 2 C) (p. 195)

70. 4(CD) (p. 195)

EXTRA PRACTICE for Lesson 4.2, ONLINE QUIZ at classzone.com 4.2p. 1013 Graph Quadratic Functions in Vertex or Intercept Form

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251

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4.3 Before

Solve x 2 1 bx 1 c 5 0 by Factoring You graphed quadratic functions.

Now

You will solve quadratic equations.

Why?

So you can double the area of a picnic site, as in Ex. 42.

Key Vocabulary • monomial • binomial • trinomial • quadratic equation • root of an equation • zero of a function

A monomial is an expression that is either a number, a variable, or the product of a number and one or more variables. A binomial, such as x 1 4, is the sum of two monomials. A trinomial, such as x2 1 11x 1 28, is the sum of three monomials. You know how to use FOIL to write (x 1 4)(x 1 7) as x2 1 11x 1 28. You can use factoring to write a trinomial as a product of binomials. To factor x2 1 bx 1 c, find integers m and n such that: x2 1 bx 1 c 5 (x 1 m)(x 1 n) 5 x2 1 (m 1 n)x 1 mn So, the sum of m and n must equal b and the product of m and n must equal c.

EXAMPLE 1

Factor trinomials of the form x 2 1 bx 1 c

Factor the expression. a. x2 2 9x 1 20

b. x2 1 3x 2 12

Solution a. You want x2 2 9x 1 20 5 (x 1 m)(x 1 n) where mn 5 20 and m 1 n 5 29. AVOID ERRORS When factoring x2 1 bx 1 c where c > 0, you must choose factors x 1 m and x 1 n such that m and n have the same sign.

Factors of 20: m, n Sum of factors: m 1 n

1, 20

21, 220

2, 10

22, 210

4, 5

24, 25

21

221

12

212

9

29

c Notice that m 5 24 and n 5 25. So, x2 2 9x 1 20 5 (x 2 4)(x 2 5). b. You want x2 1 3x 2 12 5 (x 1 m)(x 1 n) where mn 5 212 and m 1 n 5 3. Factors of 212: m, n Sum of factors: m 1 n

21, 12

1, 212

22, 6

2, 26

23, 4

3, 24

11

211

4

24

1

21

c Notice that there are no factors m and n such that m 1 n 5 3. So, x2 1 3x 2 12 cannot be factored.

✓

GUIDED PRACTICE

for Example 1

Factor the expression. If the expression cannot be factored, say so. 1. x 2 2 3x 2 18

252

n2pe-0403.indd 252

2. n2 2 3n 1 9

3. r 2 1 2r 2 63

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:06:09 AM

FACTORING SPECIAL PRODUCTS Factoring quadratic expressions often involves trial and error. However, some expressions are easy to factor because they follow special patterns.

For Your Notebook

KEY CONCEPT Special Factoring Patterns Pattern Name

Pattern

Example

Difference of Two Squares

a 2 b 5 (a 1 b)(a 2 b)

x2 2 4 5 (x 1 2)(x 2 2)

Perfect Square Trinomial

a2 1 2ab 1 b2 5 (a 1 b)2

x2 1 6x 1 9 5 (x 1 3)2

a2 2 2ab 1 b2 5 (a 2 b)2

x2 2 4x 1 4 5 (x 2 2)2

EXAMPLE 2

2

2

Factor with special patterns

Factor the expression. a. x2 2 49 5 x2 2 72

Difference of two squares

5 (x 1 7)(x 2 7) b. d2 1 12d 1 36 5 d2 1 2(d)(6) 1 62

5 (d 1 6) 2

c. z 2 26z 1 169 5 z2 2 2(z)(13) 1 132

5 (z 2 13)

✓

GUIDED PRACTICE

Perfect square trinomial

2

Perfect square trinomial

2

for Example 2

Factor the expression. 4. x 2 2 9

5. q2 2 100

6. y 2 1 16y 1 64

7. w 2 2 18w 1 81

SOLVING QUADRATIC EQUATIONS You can use factoring to solve certain

quadratic equations. A quadratic equation i n one variable can be written in the form ax2 1 bx 1 c 5 0 where a ? 0. This is called the standard form of the equation. The solutions of a quadratic equation are called the roots of t he equation. If the left side of ax2 1 bx 1 c 5 0 can be factored, t hen the equation can be solved using the zero product property.

KEY CONCEPT

For Your Notebook

Zero Product Property Words

If the product of two expressions is zero, then one or both of the expressions equal zero.

Algebra

If A and B are expressions and AB 5 0, then A 5 0 or B 5 0.

Example

If (x 1 5)(x 1 2) 5 0, then x 1 5 5 0 or x 1 2 5 0. That is, x 5 25 or x 5 22.

4.3 Solve x 2 1 bx 1 c 5 0 by Factoring

n2pe-0403.indd 253

253

10/17/05 10:06:12 AM

★

EXAMPLE 3

Standardized Test Practice

What are the roots of the equation x 2 2 5x 2 36 5 0?

UNDERSTAND ANSWER CHOICES Sometimes a standardized test question may ask for the solution set of an equation. The answer choices will be given in the format {a, b}.

A 24, 29

B 4, 29

C 24, 9

D 4, 9

Solution x2 2 5x 2 36 5 0

Write original equation.

(x 2 9)(x 1 4) 5 0

Factor.

x 2 9 5 0 or x 1 4 5 0 x 5 9 or

Zero product property

x 5 24

Solve for x.

c The correct answer is C. A B C D

EXAMPLE 4

Use a quadratic equation as a model

NATURE PRESERVE A town has a nature preserve with a

rectangular field that measures 600 meters by 400 meters. The town wants to double the area of the field by adding land as shown. Find the new dimensions of the field. Solution New area (square meters)

5

2(600)(400)

5

New length (meters)

(600 1 x)

p

p

New width (meters)

(400 1 x)

480,000 5 240,000 1 1000x 1 x2 2

Multiply using FOIL.

0 5 x 1 1000x 2 240,000

Write in standard form.

0 5 (x 2 200)(x 1 1200)

Factor.

x 2 200 5 0

or

x 5 200 or

x 1 1200 5 0 x 5 21200

Zero product property Solve for x.

c Reject the negative value, 21200. The field’s length and width should each be increased by 200 meters. The new dimensions are 800 meters by 600 meters.

✓

GUIDED PRACTICE

for Examples 3 and 4

8. Solve the equation x 2 2 x 2 42 5 0. 9. WHAT IF? In Example 4, suppose the field initially measures

1000 meters by 300 meters. Find the new dimensions of the field.

ZEROS OF A FUNCTION In Lesson 4.2, you learned that the x-intercepts of the graph of y 5 a(x 2 p)(x 2 q) are p and q. Because the function’s value is zero when x 5 p and when x 5 q, the numbers p and q are also called zeros of the function.

254

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Chapter 4 Quadratic Functions and Factoring

10/17/05 10:06:13 AM

EXAMPLE 5

Find the zeros of quadratic functions

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

UNDERSTAND REPRESENTATIONS

a. y 5 x2 2 x 2 12

If a real number k is a zero of the function y 5 ax2 1 bx 1 c, then k is an x-intercept of this function’s graph and k is also a root of the equation ax2 1 bx 1 c 5 0.

b. y 5 x2 1 12x 1 36

Solution a. y 5 x2 2 x 2 12

5 (x 1 3)(x 2 4)

Write original function. Factor.

The zeros of the function are 23 and 4.

CHECK Graph y 5 x2 2 x 2 12. The graph passes through (23, 0) and (4, 0). b. y 5 x2 1 12x 1 36

5 (x 1 6)(x 1 6)

Zero X=-3

Y=0

Zero X=-6

Y=0

Write original function. Factor.

The zero of the function is 26.

CHECK Graph y 5 x2 1 12x 1 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. y 5 x2 1 5x 2 14

4.3

EXERCISES

11. y 5 x2 2 7x 2 30

HOMEWORK KEY

12. f(x) 5 x 2 2 10x 1 25

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 33, 47, and 67

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 41, 56, 58, 63, and 71 5 MULTIPLE REPRESENTATIONS Ex. 68

SKILL PRACTICE 1. VOCABULARY What is a zero of a function y 5 f (x)? 2. ★ WRITING Explain the difference between a monomial, a binomial, and a

trinomial. Give an example of each type of expression. EXAMPLE 1 on p. 252 for Exs. 3–14

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

3. x2 1 6x 1 5

4. x 2 2 7x 1 10

5. a2 2 13a 1 22

6. r 2 1 15r 1 56

7. p2 1 2p 1 4

8. q2 2 11q 1 28

9. b2 1 3b 2 40

10. x 2 2 4x 2 12

11. x2 2 7x 2 18

12. c 2 2 9c 2 18

13. x 2 1 9x 2 36

14. m2 1 8m 2 65

4.3 Solve x 2 1 bx 1 c 5 0 by Factoring

n2pe-0403.indd 255

255

10/17/05 10:06:14 AM

EXAMPLE 2

FACTORING WITH SPECIAL PATTERNS Factor the expression.

on p. 253 for Exs. 15–23

15. x 2 2 36

16. b2 2 81

17. x2 2 24x 1 144

18. t 2 2 16t 1 64

19. x 2 1 8x 1 16

20. c 2 1 28c 1 196

21. n2 1 14n 1 49

22. s 2 2 26s 1 169

23. z2 2 121

EXAMPLE 3

SOLVING EQUATIONS Solve the equation.

on p. 254 for Exs. 24–41

24. x 2 2 8x 1 12 5 0

25. x 2 2 11x 1 30 5 0

26. x2 1 2x 2 35 5 0

27. a2 2 49 5 0

28. b2 2 6b 1 9 5 0

29. c 2 1 5c 1 4 5 0

30. n2 2 6n 5 0

31. t 2 1 10t 1 25 5 0

32. w 2 2 16w 1 48 5 0

33. z2 2 3z 5 54

34. r 2 1 2r 5 80

35. u2 5 29u

36. m2 5 7m

37. 14x 2 49 5 x2

38. 23y 1 28 5 y 2

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

39.

40.

x2 2 x 2 6 5 0 (x 2 2)(x 1 3) 5 0

x2 1 7x 1 6 5 14 (x 1 6)(x 1 1) 5 14

x2250

or x 1 3 5 0

x52

x 1 6 5 14 or x 1 1 5 14

x 5 23

or

x58

or

x 5 13

41. ★ MULTIPLE CHOICE What are the roots of the equation x2 1 2x 2 63 5 0?

A 7, 29

B 27, 29

C 27, 9

D 7, 9

EXAMPLE 4

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

on p. 254 for Exs. 42–43

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. EXAMPLE 5

FINDING ZEROS Find the zeros of the function by rewriting the function in

on p. 255 for Exs. 44–55

intercept form. 44. y 5 x2 1 6x 1 8

45. y 5 x2 2 8x 1 16

46. y 5 x2 2 4x 2 32

47. y 5 x2 1 7x 2 30

48. f(x) 5 x 2 1 11x

49. g(x) 5 x2 2 8x

50. y 5 x2 2 64

51. y 5 x2 2 25

52. f(x) 5 x 2 2 12x 2 45

53. g(x) 5 x2 1 19x 1 84

54. y 5 x2 1 22x 1 121

55. y 5 x2 1 2x 1 1

56. ★ MULTIPLE CHOICE What are the zeros of f(x) 5 x 2 1 6x 2 55?

A 211, 25

B 211, 5

C 25, 11

D 5, 11

57. REASONING Write a quadratic equation of the form x 2 1 bx 1 c 5 0 that has

roots 8 and 11. 58. ★ SHORT RESPONSE For what integers b can the expression x 2 1 bx 1 7 be

factored? Explain.

256

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Chapter 4 Quadratic on p. WS1Functions and Factoring

★

5 STANDARDIZED TEST PRACTICE

10/17/05 10:06:15 AM

GEOMETRY Find the value of x.

59. Area of rectangle 5 36

60. Area of rectangle 5 84

x12

x x17

x15

61. Area of triangle 5 42

62. Area of trapezoid 5 32 x16

x13

x

2x 1 8

x12

63. ★ OPEN-ENDED MATH Write a quadratic function with zeros that are

equidistant from 10 on a number line. 64. CHALLENGE Is there a formula for factoring the sum of two squares? You will

investigate this question in parts (a) and (b). a. Consider the sum of two squares x2 1 16. If this sum can be factored,

then there are integers m and n such that x2 1 16 5 (x 1 m)(x 1 n). Write two equations that m and n must satisfy. b. Show that there are no integers m and n that satisfy both equations

you wrote in part (a). What can you conclude?

PROBLEM SOLVING EXAMPLE 4 on p. 254 for Exs. 65–67

65. SKATE PARK A city’s skate park is a rectangle 100 feet long by

50 feet wide. The city wants to triple the area of the skate park by adding the same distance x to the length and the width. Write and solve an equation to find the value of x. What are the new dimensions of the skate park? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

66. ZOO A rectangular enclosure at a zoo is 35 feet long by 18 feet wide. The zoo

wants to double the area of the enclosure by adding the same distance x to the length and the width. Write and solve an equation to find the value of x. What are the new dimensions of the enclosure? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

67. MULTI-STEP PROBLEM A museum has a café with a

rectangular patio. The museum wants to add 464 square feet to the area of the patio by expanding the existing patio as shown. a. Find the area of the existing patio.

X

b. Write a verbal model and an equation that you can

use to find the value of x. c. Solve your equation. By what distance x should the

%XISTING PATIO FT

FT

X

length and the width of the patio be expanded?

4.3 Solve x 2 1 bx 1 c 5 0 by Factoring

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68.

1 1 1

x

MULTIPLE REPRESENTATIONS Use the diagram shown.

a. Writing an Expression Write a quadratic trinomial that represents

the area of the diagram. b. Describing a Model Factor the expression from part (a). Explain how

the diagram models the factorization. c. Drawing a Diagram Draw a diagram that models the factorization

x 1 1

x2 1 8x 1 15 5 (x 1 5)(x 1 3). 69. SCHOOL FAIR At last year’s school fair, an 18 foot by 15 foot rectangular

section of land was roped off for a dunking booth. The length and width of the section will each be increased by x feet for this year’s fair in order to triple 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 feet

long by 20 feet wide. Its area is to be halved by subtracting the same distance x from the length and the width. Write and solve an equation to find the value of x. What are the deck’s new dimensions? 71. ★ SHORT RESPONSE A square garden has sides that are 10 feet long. A

gardener wants to double the area of the garden by adding the same distance x to the length and the width. Write an equation that x must satisfy. Can you solve the equation you wrote by factoring? Explain why or why not. 72. CHALLENGE A grocery store wants to double

300 ft

the area of its parking lot by expanding the existing lot as shown. By what distance x should the lot be expanded?

165 ft

75 ft x

Grocery store

75 ft

x

Old lot Expanded part of lot

MIXED REVIEW PREVIEW Prepare for Lesson 4.4 in Exs. 73–81.

Solve the equation. 73. 2x 2 1 5 0 (p. 18)

74. 3x 1 4 5 0 (p. 18)

75. 28x 1 7 5 0 (p. 18)

76. 6x 1 5 5 0 (p. 18)

77. 4x 2 5 5 0 (p. 18)

78. 3x 1 1 5 0 (p. 18)

79. x 2 6 5 7 (p. 51)

80. 2x 2 5 5 10 (p. 51)

81. 4 2 3x 5 8 (p. 51)

Graph the function.

84. y 5 x 2 4 2 4 (p. 123)

1 x 1 2 (p. 89) 83. f(x) 5 2} 4 1 85. y 5 }x 1 1 1 2 (p. 123) 2

86. y 5 22x 2 1 8x 1 7 (p. 236)

87. g(x) 5 22(x 1 1)2 2 4 (p. 245)

88. f(x) 5 (x 1 4)(x 2 2) (p. 245)

89. y 5 2(x 2 3)(x 2 7) (p. 245)

82. y 5 3x 2 1 (p. 89)

90. PARK DESIGN A city plans to place a playground in a triangular region of

a park. The vertices of the triangle are (0, 0), (14, 3), and (6, 25) where the coordinates are given in feet. Find the area of the triangular region. (p. 203)

258

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forFactoring Lesson 4.3, p. 1013 Chapter 4 EXTRA QuadraticPRACTICE Functions and

ONLINE QUIZ at classzone.com

10/17/05 10:06:20 AM

4.4 Before Now Why?

Solve ax 2 1 bx 1 c 5 0 by Factoring You used factoring to solve equations of the form x 2 1 bx 1 c 5 0. You will use factoring to solve equations of the form ax 2 1 bx 1 c 5 0. So you can maximize a shop’s revenue, as in Ex. 64.

Key Vocabulary

To factor ax2 1 bx 1 c when a ? 1, find integers k, l, m, and n such that:

• monomial, p. 252

ax2 1 bx 1 c 5 (kx 1 m)(lx 1 n) 5 klx2 1 (kn 1 lm)x 1 mn So, k and l must be factors of a, and m and n must be factors of c.

EXAMPLE 1

Factor ax 2 1 bx 1 c where c > 0

Factor 5x 2 2 17x 1 6. FACTOR EXPRESSIONS When factoring ax2 1 bx 1 c where a > 0, it is customary to choose factors kx 1 m and lx 1 n such that k and l are positive.

Solution You want 5x2 2 17x 1 6 5 (kx 1 m)(lx 1 n) where k and l are factors of 5 and m and n are factors of 6. You can assume that k and l are positive and k ≥ l. Because mn > 0, m and n have the same sign. So, m and n must both be negative because the coefficient of x, 217, is negative. 5, 1

5, 1

5, 1

5, 1

26, 21

21, 26

23, 22

22, 23

(kx 1 m)(lx 1 n)

(5x 2 6)(x 2 1)

(5x 2 1)(x 2 6)

(5x 2 3)(x 2 2)

(5x 2 2)(x 2 3)

ax 2 1 bx 1 c

5x2 2 11x 1 6

5x2 2 31x 1 6

5x2 2 13x 1 6

5x 2 2 17x 1 6

k, l m, n

c The correct factorization is 5x2 2 17x 1 6 5 (5x 2 2)(x 2 3).

EXAMPLE 2

Factor ax 2 1 bx 1 c where c < 0

Factor 3x 2 1 20x 2 7. Solution You want 3x2 1 20x 2 7 5 (kx 1 m)(lx 1 n) where k and l are factors of 3 and m and n are factors of 27. Because mn < 0, m and n have opposite signs. k, l m, n (kx 1 m)(lx 1 n) ax 2 1 bx 1 c

3, 1

3, 1

3, 1

3, 1

7, 21

21, 7

27, 1

1, 27

(3x 1 7)(x 2 1)

(3x 2 1)(x 1 7)

(3x 2 7)(x 1 1)

(3x 1 1)(x 2 7)

3x2 1 4x 2 7

3x 2 1 20x 2 7

3x2 2 4x 2 7

3x2 2 20x 2 7

c The correct factorization is 3x2 1 20x 2 7 5 (3x 2 1)(x 1 7). 4.4 Solve ax 2 1 bx 1 c 5 0 by Factoring

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✓

GUIDED PRACTICE

for Examples 1 and 2

Factor the expression. If the expression cannot be factored, say so. 1. 7x2 2 20x 2 3

2. 5z2 1 16z 1 3

3. 2w 2 1 w 1 3

4. 3x 2 1 5x 2 12

5. 4u2 1 12u 1 5

6. 4x 2 2 9x 1 2

FACTORING SPECIAL PRODUCTS If the values of a and c in ax 2 1 bx 1 c are perfect

squares, check to see whether you can use one of the special factoring patterns from Lesson 4.3 to factor the expression.

EXAMPLE 3

Factor with special patterns

Factor the expression. a. 9x2 2 64 5 (3x)2 2 82

Difference of two squares

5 (3x 1 8)(3x 2 8) b. 4y 2 1 20y 1 25 5 (2y)2 1 2(2y)(5) 1 52

5 (2y 1 5)

Perfect square trinomial

2

c. 36w 2 2 12w 1 1 5 (6w)2 2 2(6w)(1) 1 12

5 (6w 2 1)

✓

GUIDED PRACTICE

Perfect square trinomial

2

for Example 3

Factor the expression. 7. 16x 2 2 1

8. 9y 2 1 12y 1 4

10. 25s 2 2 80s 1 64

9. 4r 2 2 28r 1 49

11. 49z2 1 42z 1 9

12. 36n2 2 9

FACTORING OUT MONOMIALS When factoring an expression, first check to see

whether the terms have a common monomial factor.

EXAMPLE 4

Factor out monomials first

Factor the expression. AVOID ERRORS Be sure to factor out the common monomial from all of the terms of the expression, not just the first term.

✓

a. 5x 2 2 45 5 5(x2 2 9)

b. 6q2 2 14q 1 8 5 2(3q2 2 7q 1 4)

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

d. 12p 2 21p 1 3 5 3(4p2 2 7p 1 1)

c. 25z 1 20z 5 25z(z 2 4)

GUIDED PRACTICE

5 2(3q 2 4)(q 2 1) 2

for Example 4

Factor the expression.

260

n2pe-0404.indd 260

13. 3s 2 2 24

14. 8t 2 1 38t 2 10

15. 6x 2 1 24x 1 15

16. 12x 2 2 28x 2 24

17. 216n2 1 12n

18. 6z2 1 33z 1 36

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:07:24 AM

SOLVING QUADRATIC EQUATIONS As you saw in Lesson 4.3, if the left side of the

quadratic equation ax2 1 bx 1 c 5 0 can be factored, then the equation can be solved using the zero product property.

EXAMPLE 5

Solve quadratic equations

Solve (a) 3x 2 1 10x 2 8 5 0 and (b) 5p2 2 16p 1 15 5 4p 2 5. a.

3x2 1 10x 2 8 5 0

Write original equation.

(3x 2 2)(x 1 4) 5 0

Factor.

3x 2 2 5 0

or

2 x5} 3

or

x1450 x 5 24

b. 5p2 2 16p 1 15 5 4p 2 5

Write in standard form.

2

p 2 4p 1 4 5 0

Divide each side by 5.

2

(p 2 2) 5 0

If the square of an expression is zero, then the expression itself must be zero.

Factor.

p22 50

Zero product property

p52

EXAMPLE 6

Solve for x.

Write original equation.

5p2 2 20p 1 20 5 0 INTERPRET EQUATIONS

Zero product property

Solve for p.

Use a quadratic equation as a model

QUILTS You have made a rectangular quilt that is 5 feet by 4 feet. You want to use the remaining 10 square feet of fabric to add a decorative border of uniform width to the quilt. What should the width of the quilt’s border be?

X X X

X X X

Solution Write a verbal model. Then write an equation. Area of border

5

Area of quilt and border

2

Area of quilt

(square feet)

(square feet)

(square feet)

10

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

(5)(4)

10 5 20 1 18x 1 4x2 2 20

Multiply using FOIL.

2

Write in standard form.

2

0 5 2x 1 9x 2 5

Divide each side by 2.

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

Factor.

0 5 4x 1 18x 2 10

2x 2 1 5 0

or

1 x5} 2

or

x1550 x 5 25

Zero product property Solve for x.

1 ft, or 6 in. c Reject the negative value, 25. The border’s width should be } 2

4.4 Solve ax 2 1 bx 1 c 5 0 by Factoring

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FACTORING AND ZEROS To find the maximum or minimum value of a quadratic function, you can first use factoring to write the function in intercept form y 5 a(x 2 p)(x 2 q). Because the function’s vertex lies on the axis of symmetry

p1q 2

x 5 }, the maximum or minimum occurs at the average of the zeros p and q.

EXAMPLE 7

Solve a multi-step problem

MAGAZINES A monthly teen magazine has 28,000 subscribers when it charges $10 per annual subscription. For each $1 increase in price, the magazine loses about 2000 subscribers. How much should the magazine charge to maximize annual revenue? What is the maximum annual revenue?

Solution

STEP 1

Define the variables. Let x represent the price increase

and R(x) represent the annual revenue.

STEP 2 Write a verbal model. Then write and simplify a quadratic function. Annual revenue

5

Number of subscribers

p

Subscription price

(dollars)

(people)

(dollars/person)

R(x)

5 (28,000 2 2000x) p

(10 1 x)

R(x)

5 (22000x 1 28,000)(x 1 10)

R(x)

5 22000(x 2 14)(x 1 10)

STEP 3 Identify the zeros and find their average. Find how much each subscription should cost to maximize annual revenue. The zeros of the revenue function are 14 and 210. The average of the 14 1 (210) 2

zeros is } 5 2. To maximize revenue, each subscription should cost $10 1 $2 5 $12.

STEP 4 Find the maximum annual revenue. R(2) 5 22000(2 2 14)(2 1 10) 5 $288,000 c The magazine should charge $12 per subscription to maximize annual revenue. The maximum annual revenue is $288,000.

✓

GUIDED PRACTICE

for Examples 5, 6, and 7

Solve the equation. 19. 6x 2 2 3x 2 63 5 0

20. 12x 2 1 7x 1 2 5 x 1 8

21. 7x2 1 70x 1 175 5 0

22. WHAT IF? In Example 7, suppose the magazine initially charges $11 per

annual subscription. How much should the magazine charge to maximize annual revenue? What is the maximum annual revenue?

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4.4

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 27, 39, and 63

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 12, 64, 65, and 67

SKILL PRACTICE 1. VOCABULARY What is the greatest common monomial factor of the terms of

the expression 12x2 1 8x 1 20? 2. ★ WRITING Explain how the values of a and c in ax2 1 bx 1 c help you

determine whether you can use a perfect square trinomial factoring pattern. EXAMPLES 1 and 2 on p. 259 for Exs. 3–12

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

3. 2x2 1 5x 1 3

4. 3n2 1 7n 1 4

5. 4r 2 1 5r 1 1

6. 6p2 1 5p 1 1

7. 11z2 1 2z 2 9

8. 15x 2 2 2x 2 8

9. 4y 2 2 5y 2 4

10. 14m2 1 m 2 3

11. 9d2 2 13d 2 10

12. ★ MULTIPLE CHOICE Which factorization of 5x2 1 14x 2 3 is correct?

A (5x 2 3)(x 1 1)

B (5x 1 1)(x 2 3)

C 5(x 2 1)(x 1 3)

D (5x 2 1)(x 1 3)

EXAMPLE 3

FACTORING WITH SPECIAL PATTERNS Factor the expression.

on p. 260 for Exs. 13–21

13. 9x2 2 1

14. 4r 2 2 25

15. 49n2 2 16

16. 16s 2 1 8s 1 1

17. 49x2 1 70x 1 25

18. 64w 2 1 144w 1 81

19. 9p2 2 12p 1 4

20. 25t 2 2 30t 1 9

21. 36x2 2 84x 1 49

EXAMPLE 4

FACTORING MONOMIALS FIRST Factor the expression.

on p. 260 for Exs. 22–31

22. 12x 2 2 4x 2 40

23. 18z2 1 36z 1 16

24. 32v 2 2 2

25. 6u2 2 24u

26. 12m2 2 36m 1 27

27. 20x 2 1 124x 1 24

28. 21x 2 2 77x 2 28

29. 236n2 1 48n 2 15

30. 28y 2 1 28y 2 60

31. ERROR ANALYSIS Describe and correct

4x2 2 36 5 4(x2 2 36)

the error in factoring the expression.

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

EXAMPLE 5

SOLVING EQUATIONS Solve the equation.

on p. 261 for Exs. 32–40

32. 16x 2 2 1 5 0

33. 11q2 2 44 5 0

34. 14s 2 2 21s 5 0

35. 45n2 1 10n 5 0

36. 4x 2 2 20x 1 25 5 0

37. 4p2 1 12p 1 9 5 0

38. 15x 2 1 7x 2 2 5 0

39. 6r 2 2 7r 2 5 5 0

40. 36z2 1 96z 1 15 5 0

EXAMPLE 7

FINDING ZEROS Find the zeros of the function by rewriting the function in

on p. 262 for Exs. 41–49

intercept form. 41. y 5 4x 2 2 19x 2 5

42. g(x) 5 3x 2 2 8x 1 5

43. y 5 5x 2 2 27x 2 18

44. f(x) 5 3x2 2 3x

45. y 5 11x 2 2 19x 2 6

46. y 5 16x2 2 2x 2 5

47. y 5 15x 2 2 5x 2 20

48. y 5 18x2 2 6x 2 4

49. g(x) 5 12x2 1 5x 2 7

4.4 Solve ax 2 1 bx 1 c 5 0 by Factoring

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GEOMETRY Find the value of x.

50. Area of square 5 36

51. Area of rectangle 5 30

52. Area of triangle 5 115

x

2x

2x

3x 1 1

5x 2 2 SOLVING EQUATIONS Solve the equation.

53. 2x2 2 4x 2 8 5 2x2 1 x

54. 24x 2 1 8x 1 2 5 5 2 6x

55. 18x2 2 22x 5 28

56. 13x2 1 21x 5 25x 2 1 22

57. x 5 4x 2 2 15x

58. (x 1 8)2 5 16 2 x2 1 9x

CHALLENGE Factor the expression.

59. 2x 3 2 5x 2 1 3x

60. 8x4 2 8x 3 2 6x 2

61. 9x 3 2 4x

PROBLEM SOLVING EXAMPLE 6 on p. 261 for Exs. 62–63

62. ARTS AND CRAFTS You have a rectangular stained glass window that

measures 2 feet by 1 foot. You have 4 square feet of glass with which to make a border of uniform width around the window. What should the width of the border be? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

63. URBAN PLANNING You have just planted a

rectangular flower bed of red roses in a city park. You want to plant a border of yellow roses around the flower bed as shown. Because you bought the same number of red and yellow roses, the areas of the border and flower bed will be equal. What should the width of the border of yellow roses be?

FT FT

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

EXAMPLE 7 on p. 262 for Exs. 64–65

64. ★ MULTIPLE CHOICE A surfboard shop sells 45 surfboards per month when

it charges $500 per surfboard. For each $20 decrease in price, the store sells 5 more surfboards per month. How much should the shop charge per surfboard in order to maximize monthly revenue? A $340

B $492

C $508

D $660

65. ★ SHORT RESPONSE A restaurant sells about 330 sandwiches each day at a

price of $6 each. For each $.25 decrease in price, 15 more sandwiches are sold per day. How much should the restaurant charge to maximize daily revenue? Explain each step of your solution. What is the maximum daily revenue? X

66. PAINTINGS You are placing a mat around a 25 inch by 21 inch

painting as shown. You want the mat to be twice as wide to the left and right of the painting as it is at the top and bottom of the painting. You have 714 square inches of mat that you can use. How wide should the mat be to the left and right of the painting? at the top and bottom of the painting?

IN

X X

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5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

★

IN

X

5 STANDARDIZED TEST PRACTICE

10/17/05 10:07:29 AM

67. ★ EXTENDED RESPONSE A U.S. Postal Service guideline

states that for a rectangular package like the one shown, the sum of the length and the girth cannot exceed 108 inches. Suppose that for one such package, the length 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 about

the products of the lengths of segments of two chords that intersect in the interior of a circle. Use this theorem to find the value of x in the diagram.

3x 1 2 5x 2 4 2x

x11

MIXED REVIEW PREVIEW

Solve the equation. (p. 18)

Prepare for Lesson 4.5 in Exs. 69–74.

69. 11 1 12x 5 3(4x 1 7)

70. 6x 2 19 5 5(3 1 2x)

71. 29(5x 1 3) 5 9x 2 42

72. 6(x 2 7) 5 15(2x 2 4)

73. 9(x 2 3) 5 3(5x 2 17)

74. 4(3x 2 11) 5 3(11 2 x) 1 x

Solve the system of linear equations using Cramer’s rule. (p. 203) 75. 4x 1 9y 5 214

3x 1 5y 5 27

76. 8x 1 5y 5 22

77. 5x 2 8y 5 250

2x 1 3y 5 14

2x 2 3y 5 225

Graph the function. 78. y 5 x2 2 3x 2 18 (p. 236)

79. f(x) 5 2x 2 1 11x 1 5 (p. 236)

80. y 5 3(x 1 6)2 2 2 (p. 245)

81. g(x) 5 (x 1 4)(x 2 5) (p. 245)

QUIZ for Lessons 4.1–4.4 Graph the function. Label the vertex and axis of symmetry. (p. 236) 1. y 5 x2 2 6x 1 14

2. y 5 2x2 1 8x 1 15

3. f(x) 5 23x2 1 6x 2 5

Write the quadratic function in standard form. (p. 245) 5. g(x) 5 22(x 1 3)(x 2 7)

6. y 5 5(x 1 6)2 2 2

7. x 2 1 9x 1 20 5 0 (p. 252)

8. n2 2 11n 1 24 5 0 (p. 252)

9. z2 2 3z 2 40 5 0 (p. 252)

10. 5s 2 2 14s 2 3 5 0 (p. 259)

11. 7a2 2 30a 1 8 5 0 (p. 259)

4. y 5 (x 2 4)(x 2 8)

Solve the equation. 12. 4x 2 1 20x 1 25 5 0 (p. 259)

13. DVD PLAYERS A store sells about 50 of a new model of DVD player per month

at a price of $140 each. For each $10 decrease in price, about 5 more DVD players per month are sold. How much should the store charge in order to maximize monthly revenue? What is the maximum monthly revenue? (p. 259)

EXTRA PRACTICE for Lesson 4.4, p. 1013

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4.5 Before

Solve Quadratic Equations by Finding Square Roots You solved quadratic equations by factoring.

Now

You will solve quadratic equations by finding square roots.

Why?

So you can solve problems about astronomy, as in Ex. 39.

A number r is a square root of a number s if r 2 5 s. A positive number s has two } } square roots, written as Ï s and 2Ï s. For example, because 32 5 9 and (23)2 5 9, } } the two square roots of 9 are Ï 9 5 3 and 2Ï 9 5 23. The positive square root of a number is also called the principal square root.

Key Vocabulary • square root • radical • radicand • rationalizing the

}

}

The expression Ï s is called a radical. The symbol Ï is a radical sign, and the

denominator • conjugates

number s beneath the radical sign is the radicand of the expression.

For Your Notebook

KEY CONCEPT Properties of Square Roots (a > 0, b > 0) }

}

}

Ïab 5 Ïa p Ïb

Product Property

}

}

Example

Î 252 5 ÏÏ252 5 Ï52

}

}

}

}

Ï18 5 Ï9 p Ï2 5 3Ï2

}

Ïa Î}ab 5 } Ïb }

Quotient Property

}

Example

}

}

} }

}

SIMPLIFYING SQUARE ROOTS You can use the properties above to simplify

expressions containing square roots. A square-root expression is simplified if: • no radicand has a perfect-square factor other than 1, and • there is no radical in a denominator

EXAMPLE 1

Use properties of square roots

Simplify the expression. USE A CALCULATOR You can use a calculator } to approximate Ï s when s is not a perfect square. For example, } Ï80 ø 8.944.

✓

}

}

}

}

}

a. Ï 80 5 Ï 16 p Ï 5 5 4Ï 5 }

Î 814 5 ÏÏ814 5 29 }

c.

}

} }

GUIDED PRACTICE

}

}

}

}

}

b. Ï 6 p Ï 21 5 Ï 126 5 Ï 9 p Ï 14 5 3Ï 14 }

}

Î 167 5 ÏÏ167 5 Ï47 }

d.

}

}

} }

}

for Example 1

Simplify the expression. }

1. Ï 27

Î 649

}

2. Ï 98

}

5.

266

n2pe-0405.indd 266

}

Î 154

}

}

6.

}

}

3. Ï 10 p Ï 15

Î 1125

}

}

Î 3649

}

}

7.

}

4. Ï 8 p Ï 28 8.

}

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:09:19 AM

RATIONALIZING THE DENOMINATOR

Suppose the denominator of a fraction } } } has the form Ïb , a 1 Ïb , or a 2 Ïb where a and b are rational numbers. The table shows how to eliminate the radical from the denominator. This is called rationalizing the denominator. }

Form of the denominator

Multiply numerator and denominator by:

}

}

Ïb

Ïb }

}

a 1 Ïb

a 2 Ïb

}

}

a 2 Ïb

a 1 Ïb

}

The expressions a 1 Ï b and a 2 Ï b are called conjugates of each other. Their product is always a rational number.

EXAMPLE 2

Rationalize denominators of fractions

Î2

}

5 and (b) 3 . Simplify (a) } } } 7 1 Ï2

Solution a.

}

}

}

} }

}

Î 52 5 ÏÏ52

3 7 2 Ï2 3 b. } } p } } } 5 } 7 1 Ï2 7 2 Ï2 7 1 Ï2

}

}

Ï2

Ï2

}

Ï5 p Ï2 5} } } }

21 2 3Ï2 5} } }

49 2 7Ï2 1 7Ï 2 2 2 }

}

21 2 3Ï2 5}

Ï10 5}

47

2

SOLVING QUADRATIC EQUATIONS You can use square roots to solve some types

of quadratic equations. For example, if s > 0, then the equation x2 5 s has two } } real-number solutions: x 5 Ï s and x 5 2Ï s. These solutions are often written in } condensed form as x 5 6Ï s (read as “plus or minus the square root of s”).

EXAMPLE 3

Solve a quadratic equation

Solve 3x 2 1 5 5 41. 3x2 1 5 5 41

Write original equation.

3x2 5 36

Subtract 5 from each side.

2

x 5 12 AVOID ERRORS When solving an equation of the form x 2 5 s where s > 0, make sure to find both the positive and negative solutions.

Divide each side by 3. }

x 5 6Ï12 }

Take square roots of each side. }

x 5 6Ï 4 p Ï 3 }

x 5 62Ï 3

Product property Simplify.

}

}

c The solutions are 2Ï 3 and 22Ï 3 .

CHECK Check the solutions by substituting them into the original equation. 2

3x 1 5 5 41 }

2

3x 1 5 5 41 }

3(2Ï3 ) 1 5 0 41

3(22Ï 3 ) 1 5 0 41

3(12) 1 5 0 41

3(12) 1 5 0 41

2

41 5 41 ✓

2

41 5 41 ✓

4.5 Solve Quadratic Equations by Finding Square Roots

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10/17/05 10:09:22 AM

★

EXAMPLE 4

Standardized Test Practice

1 (z 1 3)2 5 7? What are the solutions of the equation } 5

A 238, 32

}

}

}

}

5

5

B 23 2 5Ï7 , 23 1 5Ï7

}

Ï35 Ï35 D 23 2 } , 23 1 }

}

C 23 2 Ï35 , 23 1 Ï 35 Solution 1 5

2 }(z 1 3) 5 7

Write original equation.

(z 1 3)2 5 35

Multiply each side by 5. }

z 1 3 5 6Ï35

Take square roots of each side. }

z 5 23 6 Ï35

Subtract 3 from each side. }

}

The solutions are 23 1 Ï 35 and 23 2 Ï35 . c The correct answer is C. A B C D

✓

GUIDED PRACTICE

for Examples 2, 3, and 4

Simplify the expression.

Î 65

}

9.

Î 98

}

10.

}

26 13. } } 7 2 Ï5

Î 1712

}

11.

}

2 14. } } 4 1 Ï 11

}

21 15. } } 9 1 Ï7

Î 1921

}

12.

}

4 16. } } 8 2 Ï3

Solve the equation. 17. 5x 2 5 80

18. z2 2 7 5 29

19. 3(x 2 2)2 5 40

MODELING DROPPED OBJECTS When an object is dropped, its height h (in feet) above the ground after t seconds can be modeled by the function

h 5 216t 2 1 h0 where h0 is the object’s initial height (in feet). The graph of h 5 216t 2 1 200, representing the height of an object dropped from an initial height of 200 feet, is shown at the right. The model h 5 216t 2 1 h0 assumes that the force of air resistance on the object is negligible. Also, this model works only on Earth. For planets with stronger or weaker gravity, different models are used (see Exercise 39).

268

n2pe-0405.indd 268

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:09:23 AM

EXAMPLE 5

Model a dropped object with a quadratic function

SCIENCE COMPETITION For a science competition,

students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground? Solution

ANOTHER WAY For alternative methods for solving the problem in Example 5, turn to page 272 for the Problem Solving Workshop.

h 5 216t 2 1 h0

Write height function.

2

Substitute 0 for h and 50 for h0.

2

Subtract 50 from each side.

0 5 216t 1 50 250 5 216t 50 16

2 }5t

Divide each side by 216.

Î 16

Take square roots of each side.

61.8 ø t

Use a calculator.

}

50 5 t 6 }

After a successful egg drop

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. "MHFCSB

✓

at classzone.com

GUIDED PRACTICE

for Example 5

20. WHAT IF? In Example 5, suppose the egg container is dropped from a height

of 30 feet. How long does the container take to hit the ground?

4.5

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 17, 27, and 41

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 19, 34, 35, 36, 40, and 41

SKILL PRACTICE }

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. EXAMPLES 1 and 2 on pp. 266–267 for Exs. 3–20

SIMPLIFYING RADICAL EXPRESSIONS Simplify the expression. }

}

3. Ï 28

}

4. Ï 192

5. Ï 150

8 11. } } Ï3

7 12. } } Ï 12

Î 165 13. Î 18 11

2 15. } } 1 2 Ï3

1 16. } } 5 1 Ï6

Ï2 17. } } 4 1 Ï5

}

}

7. 4Ï 6 p Ï 6

}

}

8. 5Ï 24 p 3Ï 10

}

9.

}

}

} }

}

}

6. Ï 3 p Ï 27

Î 3536 14. Î 13 28

}

10.

}

}

} }

3 1 Ï7 18. } } 2 2 Ï10

4.5 Solve Quadratic Equations by Finding Square Roots

n2pe-0405.indd 269

269

10/17/05 10:09:24 AM

}

19. ★ MULTIPLE CHOICE What is a completely simplified expression for Ï 108 ? }

}

A 2Ï27

}

B 3Ï12

}

C 6Ï 3

D 10Ï8

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

20.

}

21.

}

}

Ï 96 5 Ï 4 p Ï 24

5x2 5 405

}

x2 5 81

5 2Ï24

x 59 SOLVING QUADRATIC EQUATIONS Solve the equation.

EXAMPLES 3 and 4 on pp. 267–268 for Exs. 21–34

22. s 2 5 169

23. a2 5 50

24. x 2 5 84

25. 6z2 5 150

26. 4p2 5 448

27. 23w 2 5 2213

28. 7r 2 2 10 5 25

x 2 2 6 5 22 29. } 25

t 2 1 8 5 15 30. } 20

31. 4(x 2 1)2 5 8

32. 7(x 2 4)2 2 18 5 10

33. 2(x 1 2)2 2 5 5 8

34. ★ MULTIPLE CHOICE What are the solutions of 3(x 1 2)2 1 4 5 13?

A 25, 1

}

B 21, 5

}

C 22 6 Ï3

D 2 6 Ï3

35. ★ SHORT RESPONSE Describe two different methods for solving the equation x2 2 4 5 0. Include the steps for each method. 36. ★ OPEN-ENDED MATH Write an equation of the form x 2 5 s that has (a) two

real solutions, (b) exactly one real solution, and (c) no real solutions. 37. CHALLENGE Solve the equation a(x 1 b)2 5 c in terms of a, b, and c.

PROBLEM SOLVING EXAMPLE 5 on p. 269 for Exs. 38–39

38. CLIFF DIVING A cliff diver dives off a cliff 40 feet

above water. Write an equation giving the diver’s height h (in feet) above the water after t seconds. How long is the diver in the air? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

39. ASTRONOMY On any planet, the height h (in feet) of a falling object t seconds g after it is dropped can be modeled by h 5 2}t 2 1 h0 where h0 is the object’s 2

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 takes for a rock dropped from a height of 150 feet to hit the surface. Planet g (ft/sec2)

Earth

Mars

Jupiter

Saturn

Pluto

32

12

76

30

2

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

270

n2pe-0405.indd 270

5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

★

5 STANDARDIZED TEST PRACTICE

10/17/05 10:09:25 AM

40. ★ SHORT RESPONSE The equation h 5 0.019s 2 gives the height h (in feet) of

the largest ocean waves when the wind speed is s knots. Compare the wind speeds required to generate 5 foot waves and 20 foot waves. 41. ★ EXTENDED RESPONSE You want to transform a square gravel parking

lot with 10 foot sides into a circular lot. You want the circle to have the same area as the square so that you do not have to buy any additional gravel.

10 ft 10 ft

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

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

c. Generalize In general, if a square has sides of length s, what is the radius

r 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 equation R 5 0.00829s2 where s is the bicycle’s speed (in miles per hour). a. What is the speed of a racing cyclist who

experiences 5 pounds of air resistance? b. What happens to the air resistance if the

cyclist’s speed doubles? Justify your answer algebraically. 43. CHALLENGE For a swimming pool with a rectangular base, Torricelli’s law

implies that the height h of water in the pool t seconds after it begins draining is given by h 5

1

}

2 2πd2Ï3 Ïh0 2 }t where l and w are the pool’s }

lw

2

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?

MIXED REVIEW PREVIEW

Evaluate the power. (p. 10)

Prepare for Lesson 4.6 in Exs. 44–51.

44. (25)2

45. (24)2

46. (28)2

47. (213)2

48. 232

49. 2112

50. 2152

51. 272

Solve the equation or inequality. 52. x 2 8 5 2 (p. 18)

53. 3x 1 4 5 13 (p. 18)

54. 2x 2 1 5 6x 1 3 (p. 18)

55. x 1 9 > 5 (p. 41)

56. 27x 2 15 ≥ 6 (p. 41)

57. 3 2 6x ≤ 23 2 10x (p. 41)

58. x 1 12 5 5 (p. 51)

59. 22 1 3x 5 10 (p. 51)

60.

1

}2 x 1 9 ≥ 4 (p. 51)

In Exercises 61 and 62, (a) draw a scatter plot of the data, (b) approximate the best-fitting line, and (c) estimate y when x 5 20. (p. 113) 61.

x

24

23

0

2

5

y

5

9

28

33

39

62.

x

1

2

3

4

5

y

120

91

58

31

5

EXTRA PRACTICE for Lesson 4.5, p.4.5 1013 ONLINE QUIZ classzone.com Solve Quadratic Equations by at Finding Square Roots

n2pe-0405.indd 271

271

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Using

ALTERNATIVE METHODS

LESSON 4.5 Another Way to Solve Example 5, page 269 MULTIPLE REPRESENTATIONS In Example 5 on page 269, you solved a quadratic

equation by finding square roots. You can also solve a quadratic equation using a table or a graph.

PROBLEM

METHOD 1

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

Using a Table One alternative approach is to write a quadratic equation and then

use a table of values to solve the equation. You can use a graphing calculator to make the table.

STEP 1 Write an equation that models the situation using the height function h 5 216t 2 1 h0.

h 5 216t 2 1 h0 2

0 5 216t 1 50

Write height function. Substitute 0 for h and 50 for h0.

STEP 2 Enter the function y 5 216x2 1 50 into a graphing calculator. Note that time is now represented by x and height is now represented by y.

Y1=-16X2+50 Y2= Y3= Y4= Y5= Y6= Y7=

STEP 3 Make a table of values for the function. Set the table so that the x-values start at 0 and increase in increments of 0.1.

TABLE SETUP TblStart=0 Tbl=.1 Indpnt: Auto Ask Depend: Auto Ask

STEP 4 Scroll through the table to find the time x at which the height y of the container is 0 feet. The table shows that y 5 0 between x 5 1.7 and x 5 1.8 because y has a change of sign.

X 1.5 1.6 1.7 1.8 1.9

Y1 14 9.04 3.76 -1.84 -7.76

X=1.8

c The container hits the ground between 1.7 and 1.8 seconds after it is dropped.

272

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Chapter 4 Quadratic Functions and Factoring

10/17/05 10:09:27 AM

METHOD 2

Using a Graph Another approach is to write a quadratic equation and then use a

graph to solve the equation. You can use a graphing calculator to make the graph.

STEP 1 Write an equation that models the situation using the height function h 5 216t 2 1 h0.

h 5 216t 2 1 h0

Write height function.

0 5 216t 2 1 50

Substitute 0 for h and 50 for h0.

STEP 2 Enter the function y 5 216x2 1 50 into a graphing calculator. Note that time is now represented by x and height is now represented by y.

Y1=-16X2+50 Y2= Y3= Y4= Y5= Y6= Y7=

STEP 3 Graph the height function. Adjust the viewing window so that you can see the point where the graph crosses the positive x-axis. Find the positive x-value for which y 5 0 using the zero feature. The graph shows that y 5 0 when x ø 1.8. c The container hits the ground about 1.8 seconds after it is dropped.

Zero X=1.767767 Y=0

P R AC T I C E SOLVING EQUATIONS Solve the quadratic equation using a table and using a graph.

1. 2x2 2 12x 1 10 5 0 2. x 2 1 7x 1 12 5 0 3. 9x2 2 30x 1 25 5 0 4. 7x2 2 3 5 0 2

5. x 1 3x 2 6 5 0 6. WHAT IF? How long does it take for an egg

container to hit the ground when dropped from a height of 100 feet? Find the answer using a table and using a graph. 7. WIND PRESSURE The pressure P (in pounds

per square foot) from wind blowing at s miles per hour is given by P 5 0.00256s2. What wind speed produces a pressure of 30 lb/ft 2 ? Solve this problem using a table and using a graph.

8. BIRDS A bird flying at a height of 30 feet carries

a shellfish. The bird drops the shellfish to break it and get the food inside. How long does it take for the shellfish to hit the ground? Find the answer using a table and using a graph. 9. DROPPED OBJECT You are dropping a ball

from a window 29 feet above the ground to your friend who will catch it 4 feet above the ground. How long is the ball in the air before your friend catches it? Solve this problem using a table and using a graph. 10. REASONING Explain how to use the table

feature of a graphing calculator to approximate the solution of the problem on page 272 to the nearest hundredth of a second. Use this procedure to find the approximate solution.

Using Alternative Methods

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MIXED REVIEW of Problem Solving

STATE TEST PRACTICE

classzone.com

Lessons 4.1–4.5 1. MULTI-STEP PROBLEM A pinecone falls from a

tree branch that is 20 feet above the ground. a. Write a function that models the height of

the pinecone as it falls. b. Graph the function. c. After how many seconds does the pinecone

4. SHORT RESPONSE You are creating a metal

border of uniform width for a wall mirror that is 20 inches by 24 inches. You have 416 square inches of metal to use. Write and solve an equation to find the border’s width. Does doubling the width require exactly twice as much metal? Explain.

hit the ground? d. What are the domain and range of the

function? IN

2. MULTI-STEP PROBLEM Some harbor police

departments have fire-fighting boats with water cannons. The boats are used to fight fires that occur within the harbor. IN 5. EXTENDED RESPONSE A pizza shop sells about

80 slices of pizza each day during lunch when it charges $2 per slice. For each $.25 increase in price, about 5 fewer slices are sold each day during lunch. a. Write a function that gives the pizza shop’s

revenue R if there are x price increases. b. What value of x maximizes R? Explain the a. The function y 5 20.0035x(x 2 143.9)

models the path of water shot by a water cannon where x is the horizontal distance (in feet) and y is the corresponding height (in feet). What are the domain and range of this function? b. How far does the water cannon shoot? c. What is the maximum height of the water? 3. EXTENDED RESPONSE The diagonal of the

screen on a laptop computer measures 15 inches. The ratio of the screen’s width w to its height h is 4 : 3. a. Write an expression for w in terms of h.

meaning of your answer in this situation. c. Repeat parts (a) and (b) using the equivalent

assumption that each $.25 decrease in price results in 5 more slices being sold. Why do you obtain a negative x-value in part (b)? 6. GRIDDED ANSWER You have a rectangular

vegetable garden that measures 42 feet by 8 feet. You want to double the area of the garden by expanding the length and width as shown. Find the value of x. x

42 ft 8 ft

b. Use the Pythagorean theorem and the result

from part (a) to write an equation that you can use to find h.

x

c. Solve the equation from part (b). Explain

why you must reject one of the solutions. d. What are the height, width, and area of the

laptop screen?

274

n2pe-0405.indd 274

7. OPEN-ENDED Write three different quadratic

functions in standard form whose graphs have a vertex of (23, 2).

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:09:40 AM

4.6 Before

Perform Operations with Complex Numbers You performed operations with real numbers.

Now

You will perform operations with complex numbers.

Why?

So you can solve problems involving fractals, as in Exs. 70–73.

Key Vocabulary • imaginary unit i • complex number • imaginary number • complex conjugates • complex plane • absolute value of a

complex number

Not all quadratic equations have real-number solutions. For example, x 2 5 21 has no real-number solutions because the square of any real number x is never a negative number. To overcome this problem, mathematicians created an expanded system of } numbers using the imaginary unit i, defined as i 5 Ï 21. Note that i 2 5 21. The imaginary unit i can be used to write the square root of any negative number.

For Your Notebook

KEY CONCEPT The Square Root of a Negative Number Property

Example }

}

1. If r is a positive real number, then Ï 2r 5 i Ï r. }

2. By Property (1), it follows that (i Ï r ) 5 2r.

EXAMPLE 1

2

}

}

Ï23 5 i Ï3 }

(i Ï3 )2 5 i 2 p 3 5 23

Solve a quadratic equation

Solve 2x 2 1 11 5 237. 2x 2 1 11 5 237

Write original equation.

2

2x 5 248

Subtract 11 from each side.

x 2 5 224

Divide each side by 2.

}

x 5 6Ï224

Take square roots of each side.

}

x 5 6i Ï 24

Write in terms of i.

}

x 5 62i Ï 6

Simplify radical. }

}

c The solutions are 2i Ï 6 and 22i Ï 6.

✓

GUIDED PRACTICE

for Example 1

Solve the equation. 1. x 2 5 213

2. x 2 5 238

3. x 2 1 11 5 3

4. x 2 2 8 5 236

5. 3x 2 2 7 5 231

6. 5x 2 1 33 5 3

4.6 Perform Operations with Complex Numbers

n2pe-0406.indd 275

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10/17/05 10:10:45 AM

COMPLEX NUMBERS A complex number written in standard form is a number a 1 bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part.

Complex Numbers (a 1 bi) Real Numbers (a 1 0i )

Imaginary Numbers (a 1 bi, b Þ 0) 2 1 3i

If b ? 0, then a 1 bi is an imaginary number. If a 5 0 and b ? 0, then a 1 bi is a pure imaginary number. The diagram shows how different types of complex numbers are related.

21

p

Two complex numbers a 1 bi and c 1 di are equal if and only if a 5 c and b 5 d. For example, if x 1 yi 5 5 2 3i, then x 5 5 and y 5 23.

5 2

2

5 2 5i

Pure Imaginary Numbers (0 1 bi, b Þ 0) 24i

6i

For Your Notebook

KEY CONCEPT Sums and Differences of Complex Numbers

To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Sum of complex numbers:

(a 1 bi) 1 (c 1 di) 5 (a 1 c) 1 (b 1 d)i

Difference of complex numbers:

(a 1 bi) 2 (c 1 di) 5 (a 2 c) 1 (b 2 d)i

EXAMPLE 2

Add and subtract complex numbers

Write the expression as a complex number in standard form. a. (8 2 i) 1 (5 1 4i)

b. (7 2 6i) 2 (3 2 6i)

c. 10 2 (6 1 7i) 1 4i

Solution a. (8 2 i) 1 (5 1 4i) 5 (8 1 5) 1 (21 1 4)i

5 13 1 3i b. (7 2 6i) 2 (3 2 6i) 5 (7 2 3) 1 (26 1 6)i

GUIDED PRACTICE

Write in standard form. Definition of complex subtraction

5 4 1 0i

Simplify.

54

Write in standard form.

c. 10 2 (6 1 7i) 1 4i 5 [(10 2 6) 2 7i] 1 4i

✓

Definition of complex addition

Definition of complex subtraction

5 (4 2 7i) 1 4i

Simplify.

5 4 1 (27 1 4)i

Definition of complex addition

5 4 2 3i

Write in standard form.

for Example 2

Write the expression as a complex number in standard form. 7. (9 2 i) 1 (26 1 7i)

276

n2pe-0406.indd 276

8. (3 1 7i) 2 (8 2 2i)

9. 24 2 (1 1 i) 2 (5 1 9i)

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:10:48 AM

EXAMPLE 3

Use addition of complex numbers in real life

ELECTRICITY Circuit components such as resistors, inductors, and capacitors all oppose the flow of current. This opposition is called resistance for resistors and reactance for inductors and capacitors. A circuit’s total opposition to current flow is impedance. All of these quantities are measured in ohms (Ω). READING Note that while a component’s resistance or reactance is a real number, its impedance is a complex number.

Resistor

Inductor

Capacitor

Resistance or reactance

R

L

C

Impedance

R

Li

2Ci

Component and symbol

5Ω 3Ω

4Ω

Alternating current source

The table shows the relationship between a component’s resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled. The impedance for a series circuit is the sum of the impedances for the individual components. Find the impedance of the circuit shown above. Solution The resistor has a resistance of 5 ohms, so its impedance is 5 ohms. The inductor has a reactance of 3 ohms, so its impedance is 3i ohms. The capacitor has a reactance of 4 ohms, so its impedance is 24i ohms. Impedance of circuit 5 5 1 3i 1 (24i) 552i

Add the individual impedances. Simplify.

c The impedance of the circuit is 5 2 i ohms.

MULTIPLYING COMPLEX NUMBERS To multiply two complex numbers, use the distributive property or the FOIL method just as you do when multiplying real numbers or algebraic expressions.

EXAMPLE 4

Multiply complex numbers

Write the expression as a complex number in standard form. a. 4i(26 1 i)

b. (9 2 2i)(24 1 7i)

Solution a. 4i(26 1 i) 5 224i 1 4i 2 AVOID ERRORS When simplifying an expression that involves complex numbers, be sure to simplify i 2 to 21.

Distributive property

5 224i 1 4(21)

Use i 2 5 21.

5 224i 2 4

Simplify.

5 24 2 24i

Write in standard form.

b. (9 2 2i)(24 1 7i) 5 236 1 63i 1 8i 2 14i 2

Multiply using FOIL.

5 236 1 71i 2 14(21)

Simplify and use i 2 5 21.

5 236 1 71i 1 14

Simplify.

5 222 1 71i

Write in standard form.

4.6 Perform Operations with Complex Numbers

n2pe-0406.indd 277

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COMPLEX CONJUGATES Two complex numbers of the form a 1 bi and a 2 bi are called complex conjugates. The product of complex conjugates is always a real number. For example, (2 1 4i)(2 2 4i) 5 4 2 8i 1 8i 1 16 5 20. You can use this fact to write the quotient of two complex numbers in standard form.

EXAMPLE 5

Divide complex numbers

7 1 5i in standard form. Write the quotient } 1 2 4i

REWRITE QUOTIENTS When a quotient has an imaginary number in the denominator, rewrite the denominator as a real number so you can express the quotient in standard form.

7 1 5i 1 2 4i

7 1 5i 1 2 4i

1 1 4i 1 1 4i

}5}p}

1 28i 1 5i 1 20i 2 5 7} 2 1 1 4i 2 4i 2 16i

7 1 33i 1 20(21) 1 2 16(21)

Multiply using FOIL.

5}

Simplify and use i 2 5 1.

213 1 33i 5}

Simplify.

13 1 33 i 5 2} }

Write in standard form.

17

17

17

✓

Multiply numerator and denominator by 1 1 4i, the complex conjugate of 1 2 4i.

GUIDED PRACTICE

for Examples 3, 4, and 5

10. WHAT IF? In Example 3, what is the impedance of the circuit if the given

capacitor is replaced with one having a reactance of 7 ohms? Write the expression as a complex number in standard form. 11. i(9 2 i)

12. (3 1 i)(5 2 i)

5 13. } 11i

5 1 2i 14. } 3 2 2i

COMPLEX PLANE Just as every real number corresponds to a point on the real

number line, every complex number corresponds to a point in the complex plane. As shown in the next example, the complex plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis.

EXAMPLE 6

Plot complex numbers

Plot the complex numbers in the same complex plane. a. 3 2 2i

b. 22 1 4i

c. 3i

d. 24 2 3i

Solution a. To plot 3 2 2i, start at the origin, move 3 units to

the right, and then move 2 units down.

imaginary

22 1 4i 3i

b. To plot 22 1 4i, start at the origin, move 2 units

i

to the left, and then move 4 units up.

1

c. To plot 3i, start at the origin and move 3 units up. d. To plot 24 2 3i, start at the origin, move 4 units

to the left, and then move 3 units down.

278

n2pe-0406.indd 278

real

3 2 2i 24 2 3i

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:10:49 AM

For Your Notebook

KEY CONCEPT Absolute Value of a Complex Number

imaginary

The absolute value of a complex number z 5 a 1 bi, denoted z, is a nonnegative

uzu5

z 5 a 1 bi

a2 1 b 2 bi

}

real number defined as z 5 Ïa 2 1 b 2 .

a

This is the distance between z and the the origin in the complex plane.

EXAMPLE 7

real

Find absolute values of complex numbers

Find the absolute value of (a) 24 1 3i and (b) 23i. }

}

a. 24 1 3i 5 Ï (24)2 1 32 5 Ï 25 5 5 }

}

b. 23i 5 0 1 (23i) 5 Ï 02 1 (23)2 5 Ï 9 5 3 "MHFCSB

✓

GUIDED PRACTICE

at classzone.com

for Examples 6 and 7

Plot the complex numbers in the same complex plane. Then find the absolute value of each complex number. 15. 4 2 i

4.6

EXERCISES

16. 23 2 4i

17. 2 1 5i

HOMEWORK KEY

18. 24i

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 11, 29, and 67

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 21, 50, 60, 69, and 74

SKILL PRACTICE 1. VOCABULARY What is the complex conjugate of a 2 bi? 2. ★ WRITING Is every complex number an imaginary number? Explain. EXAMPLE 1 on p. 275 for Exs. 3–11

SOLVING QUADRATIC EQUATIONS Solve the equation.

3. x 2 5 228

4. r 2 5 2624

5. z 2 1 8 5 4

6. s 2 2 22 5 2112

7. 2x 2 1 31 5 9

8. 9 2 4y 2 5 57

9. 6t 2 1 5 5 2t 2 1 1

10. 3p2 1 7 5 29p2 1 4

11. 25(n 2 3)2 5 10

EXAMPLE 2

ADDING AND SUBTRACTING Write the expression as a complex number in

on p. 276 for Exs. 12–21

standard form. 12. (6 2 3i) 1 (5 1 4i)

13. (9 1 8i) 1 (8 2 9i)

14. (22 2 6i) 2 (4 2 6i)

15. (21 1 i) 2 (7 2 5i)

16. (8 1 20i) 2 (28 1 12i)

17. (8 2 5i) 2 (211 1 4i)

18. (10 2 2i) 1 (211 2 7i)

19. (14 1 3i) 1 (7 1 6i)

20. (21 1 4i) 1 (29 2 2i)

4.6 Perform Operations with Complex Numbers

n2pe-0406.indd 279

279

10/17/05 10:10:50 AM

21. ★ MULTIPLE CHOICE What is the standard form of the expression

(2 1 3i) 2 (7 1 4i)? A 24 EXAMPLES 4 and 5 on pp. 277–278 for Exs. 22–33

B 25 1 7i

C 25 2 i

D 51i

MULTIPLYING AND DIVIDING Write the expression as a complex number in

standard form. 22. 6i(3 1 2i)

23. 2i(4 2 8i)

24. (5 2 7i)(24 2 3i)

25. (22 1 5i)(21 1 4i)

26. (21 2 5i)(21 1 5i)

27. (8 2 3i)(8 1 3i)

7i 28. } 81i

6i 29. } 32i

22 2 5i 30. } 3i

4 1 9i 31. } 12i

7 1 4i 32. } 2 2 3i

21 2 6i 33. } 5 1 9i

EXAMPLE 6

PLOTTING COMPLEX NUMBERS Plot the numbers in the same complex plane.

on p. 278 for Exs. 34–41

34. 1 1 2i

35. 25 1 3i

36. 26i

37. 4i

38. 27 2 i

39. 5 2 5i

40. 7

41. 22

EXAMPLE 7

FINDING ABSOLUTE VALUE Find the absolute value of the complex number.

on p. 279 for Exs. 42–50

42. 4 1 3i

43. 23 1 10i

44. 10 2 7i

45. 21 2 6i

46. 28i

47. 4i

48. 24 1 i

49. 7 1 7i

50. ★ MULTIPLE CHOICE What is the absolute value of 9 1 12i?

A 7

B 15

C 108

D 225

STANDARD FORM Write the expression as a complex number in standard form.

51. 28 2 (3 1 2i) 2 (9 2 4i)

52. (3 1 2i) 1 (5 2 i) 1 6i

53. 5i(3 1 2i)(8 1 3i)

54. (1 2 9i)(1 2 4i)(4 2 3i)

(5 2 2i ) 1 (5 1 3i ) 55. } (1 1 i ) 2 (2 2 4i )

(10 1 4i ) 2 (3 2 2i ) 56. } (6 2 7i )(1 2 2i )

ERROR ANALYSIS Describe and correct the error in simplifying the expression.

57.

58.

(1 1 2i)(4 2 i)

}

2 2 3i 5 Ï22 2 32 }

5 4 2 i 1 8i 2 2i 2

5 Ï 25

5 22i 2 1 7i 1 4

5 iÏ5

}

59. ADDITIVE AND MULTIPLICATIVE INVERSES The additive inverse of a complex

number z is a complex number za such that z 1 za 5 0. The multiplicative inverse of z is a complex number zm such that z p zm 5 1. Find the additive and multiplicative inverses of each complex number. a. z 5 2 1 i

b. z 5 5 2 i

c. z 5 21 1 3i

60. ★ OPEN-ENDED MATH Find two imaginary numbers whose sum is a real

number. How are the imaginary numbers related? CHALLENGE Write the expression as a complex number in standard form.

a 1 bi 61. } c 1 di

280

n2pe-0406.indd 280

a 2 bi 62. } c 2 di

5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

a 1 bi 63. } c 2 di

★

a 2 bi 64. } c 1 di

5 STANDARDIZED TEST PRACTICE

10/17/05 10:10:52 AM

PROBLEM SOLVING EXAMPLE 3

CIRCUITS In Exercises 65–67, each component of the circuit has been labeled

on p. 277 for Exs. 65–67

with its resistance or reactance. Find the impedance of the circuit. 65.

66.

4Ω

67.

14 Ω

6Ω

6Ω

12 Ω 10 Ω

7Ω

9Ω

8Ω

8Ω GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

68. VISUAL THINKING The graph shows how you can geometrically

imaginary

6 1 6i

add two complex numbers (in this case, 4 1 i and 2 1 5i) to find their sum (in this case, 6 1 6i). Find each of the following sums by drawing a graph. a. (5 1 i) 1 (1 1 4i)

b. (27 1 3i) 1 (2 2 2i)

c. (3 2 2i) 1 (21 2 i)

5i 2

i

d. (4 1 2i) 1 (25 2 3i)

i 1 4

real

69. ★ SHORT RESPONSE Make a table that shows the powers of i from i 1 to

i 8 in the first row and the simplified forms of these powers in the second row. Describe the pattern you observe in the table. Verify that the pattern continues by evaluating the next four powers of i.

In Exercises 70–73, use the example below to determine whether the complex number c belongs to the Mandelbrot set. Justify your answer.

EXAMPLE

Investigate the Mandelbrot set

i

2

Consider the function f(z) 5 z 1 c and this infinite list of complex numbers: z 0 5 0, z1 5 f(z 0 ), z2 5 f(z1), z3 5 f(z2), . . . . If the absolute values of z 0, z1, z2, z3, . . . are all less than some fixed number N, then c belongs to the Mandelbrot set. If the absolute values become infinitely large, then c does not belong to the Mandelbrot set. Tell whether c 5 1 1 i belongs to the Mandelbrot set.

1

21 2i

The Mandelbrot set is the black region in the complex plane above.

Solution Let f (z) 5 z 2 1 (1 1 i). z0 5 0

z0 5 0 z1 5 f(0) 5 02 1 (1 1 i) 5 1 1 i

z1 ø 1.41

2

z2 ø 3.16

z2 5 f(1 1 i) 5 (1 1 i) 1 (1 1 i) 5 1 1 3i 2

z3 5 f(1 1 3i) 5 (1 1 3i) 1 (1 1 i) 5 27 1 7i

z3 ø 9.90

z 4 5 f(27 1 7i) 5 (27 1 7i)2 1 (1 1 i) 5 1 2 97i

z4 ø 97.0

c Because the absolute values are becoming infinitely large, c 5 1 1 i does not belong to the Mandelbrot set. 70. c 5 i

71. c 5 21 1 i

72. c 5 21

73. c 5 20.5i

4.6 Perform Operations with Complex Numbers

n2pe-0406.indd 281

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10/17/05 10:10:53 AM

}

}

}

74. ★ SHORT RESPONSE Evaluate Ï 24 p Ï 225 and Ï 100 . Does the rule }

}

}

Ïa p Ïb 5 Ïab on page 266 hold when a and b are negative numbers? 75. PARALLEL CIRCUITS In a parallel circuit, there is more than one pathway

through which current can flow. To find the impedance Z of a parallel circuit with two pathways, first calculate the impedances Z1 and Z2 of the pathways separately by treating each pathway as a series circuit. Then apply this formula: Z1Z2 Z5} Z1 1 Z2

What is the impedance of each parallel circuit shown below? a.

b.

7Ω

4Ω Z1

5Ω

3Ω

Z2

8Ω

c.

10 Ω

6Ω Z1

11 Ω

4Ω

3Ω

Z2

1Ω

Z1

6Ω

Z2

76. CHALLENGE Julia sets, like the Mandelbrot set shown on

page 281, are fractals defined on the complex plane. For every complex number c, there is an associated Julia set determined by the function f (z) 5 z2 1 c. For example, the Julia set corresponding to c 5 1 1 i is determined by the function f(z) 5 z 2 1 1 1 i. A number z 0 is a member of this Julia set if the absolute values of the numbers z1 5 f(z 0 ), z2 5 f(z1), z3 5 f(z2), . . . are all less than some fixed number N, and z 0 is not a member if these absolute values grow infinitely large.

A Julia set

Tell whether the given number z 0 belongs to the Julia set associated with the function f(z) 5 z 2 1 1 1 i. a. z 0 5 i

b. z 0 5 1

c. z 0 5 2i

d. z 0 5 2 1 3i

MIXED REVIEW Tell whether the relation is a function. Explain. (p. 72) 77. (5, 23), (0, 3), (22, 0), (2, 0), (25, 23)

78. (23, 1), (2, 22), (3, 21), (1, 21), (3, 0)

79. (0, 24), (1, 2), (21, 24), (2, 3), (1, 25)

80. (2, 6), (5, 6), (21, 4), (23, 5), (22, 3)

Perform the indicated operation. 81.

83.

F G F G F GF G 5 24 22 6

1

0 9 (p. 187) 1 28

2 21 0 4

0 4 3 25

82.

84.

(p. 195)

F G F G F GF G 6 3 25 21

2

1 0 4 21

23 2

9 23 (p. 187) 4 21

(p. 195)

PREVIEW

Solve the equation.

Prepare for Lesson 4.7 in Exs. 85–90.

85. 3x 2 2 3x 2 36 5 0 (p. 259)

86. 2x2 2 9x 1 4 5 0 (p. 259)

87. 6x 2 5 96 (p. 266)

88. 14x 2 5 91 (p. 266)

89. 2x2 2 8 5 42 (p. 266)

90. 3x 2 1 13 5 121 (p. 266)

282

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Lesson 4.6, p. 1013 Chapter 4 EXTRA QuadraticPRACTICE Functions andforFactoring

ONLINE QUIZ at classzone.com

10/17/05 10:10:54 AM

Use before Lesson 4.7

4.7 Using Algebra Tiles

to Complete the Square

M AT E R I A L S • algebra tiles

QUESTION

How can you use algebra tiles to complete the square for a quadratic expression?

If you are given an expression of the form x 2 1 bx, you can add a constant c to the expression so that the result x 2 1 bx 1 c is a perfect square trinomial. This process is called completing the square.

EXPLORE

Complete the square for the expression x 2 1 6x

STEP 1

STEP 2

STEP 3

Model the expression

Make a square

Complete the square

Use algebra tiles to model the expression x2 1 6x. You will need to use one x2-tile and six x-tiles for this expression.

Arrange the tiles in a square. You want the length and width of the square to be equal. Your arrangement will be incomplete in one of the corners.

Find the number of 1-tiles needed to complete the square. By adding nine 1-tiles, you can see that x2 1 6x 1 9 is equal to (x 1 3)2.

DR AW CONCLUSIONS

Use your observations to complete these exercises

1. Copy and complete the table at the right by

Completing the Square

following the steps above. 2. Look for patterns in the last column of

your table. Consider the general statement x2 1 bx 1 c 5 (x 1 d)2. a. How is d related to b in each case? b. How is c related to d in each case?

Number of 1-tiles needed to complete the square

Expression written as a square

x 2 1 2x 1 ?

?

?

x 2 1 4x 1 ?

?

Expression

? 2

c. How can you obtain the numbers in the

x 2 1 6x 1 ?

9

table’s second column directly from the coefficients of x in the expressions from the first column?

x 1 6x 1 9 5 (x 1 3) 2

x 2 1 8x 1 ?

?

?

?

?

2

x 1 10x 1 ?

4.7 Complete the Square

283

4.7 Before Now Why?

Key Vocabulary • completing the

Complete the Square You solved quadratic equations by finding square roots. You will solve quadratic equations by completing the square. So you can find a baseball’s maximum height, as in Example 7.

In Lesson 4.5, you solved equations of the form x2 5 k by finding square roots. This method also works if one side of an equation is a perfect square trinomial.

square

EXAMPLE 1 ANOTHER WAY You can also find the solutions by writing the given equation as x2 2 8x 2 9 5 0 and solving this equation by factoring.

Solve a quadratic equation by finding square roots

Solve x 2 2 8x 1 16 5 25. x2 2 8x 1 16 5 25

Write original equation.

(x 2 4)2 5 25

Write left side as a binomial squared.

x 2 4 5 65

Take square roots of each side.

x5465

Solve for x.

c The solutions are 4 1 5 5 9 and 4 2 5 5 21.

PERFECT SQUARES In Example 1, the trinomial x 2 2 8x 1 16 is a perfect square

because it equals (x 2 4)2. Sometimes you need to add a term to an expression x2 1 bx to make it a square. This process is called completing the square.

For Your Notebook

KEY CONCEPT Completing the Square

b 2. Words To complete the square for the expression x2 1 bx, add } 2

1 2

Diagrams In each diagram, the combined area of the shaded regions b 2 completes the square in the second diagram. is x 2 1 bx. Adding } 2

1 2

x

x

b

x2

bx

b Algebra x2 1 bx 1 } 2

1 2

284

n2pe-0407.indd 284

2

x

b 2

x

x2

sb2 dx

b 2

sb2 dx

sb2 d2

b x1 b 5 x1 b 5 x1} } }

1

2 21

22 1

22

2

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:11:50 AM

EXAMPLE 2

Make a perfect square trinomial

Find the value of c that makes x 2 1 16x 1 c a perfect square trinomial. Then write the expression as the square of a binomial. Solution

STEP 1

16 2

Find half the coefficient of x.

STEP 2 Square the result of Step 1.

82 5 64

STEP 3 Replace c with the result of Step 2.

x 2 1 16x 1 64

c The trinomial x 2 1 16x 1 c is a perfect square when c 5 64. Then x 2 1 16x 1 64 5 (x 1 8)(x 1 8) 5 (x 1 8)2.

✓

GUIDED PRACTICE

x

8

x

x2

8x

8

8x

64

}58

for Examples 1 and 2

Solve the equation by finding square roots. 1. x 2 1 6x 1 9 5 36

2. x 2 2 10x 1 25 5 1

3. x 2 2 24x 1 144 5 100

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 4. x 2 1 14x 1 c

5. x 2 1 22x 1 c

6. x 2 2 9x 1 c

SOLVING EQUATIONS The method of completing the square can be used to

solve any quadratic equation. When you complete a square as part of solving an equation, you must add the same number to both sides of the equation.

EXAMPLE 3

Solve ax 2 1 bx 1 c 5 0 when a 5 1

Solve x 2 2 12x 1 4 5 0 by completing the square. x 2 2 12x 1 4 5 0

Write original equation.

x 2 2 12x 5 24

Write left side in the form x 2 1 bx. 212 Add } 5 (26) 2 5 36 to each side.

(x 2 6)2 5 32

Write left side as a binomial squared. }

x 2 6 5 6Ï32

x 5 6 6 Ï32 For help with simplifying square roots, see p. 266.

1

2

2

Take square roots of each side.

}

REVIEW RADICALS

2

x 2 2 12x 1 36 5 24 1 36

}

x 5 6 6 4Ï 2

Solve for x. }

}

}

}

Simplify: Ï 32 5 Ï 16 p Ï 2 5 4Ï 2

}

}

c The solutions are 6 1 4Ï2 and 6 2 4Ï2 .

CHECK You can use algebra or a graph. Algebra Substitute each solution in the original

equation to verify that it is correct. Graph Use a graphing calculator to graph y 5 x 2 2 12x 1 4. The x-intercepts are } } about 0.34 ø 6 2 4Ï2 and 11.66 ø 6 1 4Ï2 .

Zero X=11.656854 Y=0

4.7 Complete the Square

n2pe-0407.indd 285

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EXAMPLE 4

Solve ax 2 1 bx 1 c 5 0 when a Þ 1

Solve 2x 2 1 8x 1 14 5 0 by completing the square. 2x 2 1 8x 1 14 5 0

Write original equation.

x 2 1 4x 1 7 5 0

Divide each side by the coefficient of x 2 .

x 2 1 4x 5 27

Write left side in the form x 2 1 bx. 2

4 Add } 5 22 5 4 to each side.

x 2 1 4x 1 4 5 27 1 4

122

(x 1 2)2 5 23

Write left side as a binomial squared. }

x 1 2 5 6Ï 23

Take square roots of each side. }

x 5 22 6 Ï23 }

x 5 22 6 i Ï 3

Solve for x. Write in terms of the imaginary unit i.

}

}

c The solutions are 22 1 i Ï 3 and 22 2 i Ï3 .

★

EXAMPLE 5

Standardized Test Practice

The area of the rectangle shown is 72 square units. What is the value of x?

x12 3x

ELIMINATE CHOICES

A 26

B 4

You can eliminate choices A and D because the side lengths are negative when x 5 26.

C 8.48

D 26 or 4

Solution Use the formula for the area of a rectangle to write an equation. 3x(x 1 2) 5 72

Length 3 Width 5 Area

2

3x 1 6x 5 72

Distributive property

x 2 1 2x 5 24

Divide each side by the coefficient of x 2 .

x 2 1 2x 1 1 5 24 1 1 (x 1 1)2 5 25

2 Add }

2

122

5 12 5 1 to each side.

Write left side as a binomial squared.

x 1 1 5 65

Take square roots of each side.

x 5 21 6 5

Solve for x.

So, x 5 21 1 5 5 4 or x 5 21 2 5 5 26. You can reject x 5 26 because the side lengths would be 218 and 24, and side lengths cannot be negative. c The value of x is 4. The correct answer is B. A B C D

✓

GUIDED PRACTICE

for Examples 3, 4, and 5

Solve the equation by completing the square. 7. x 2 1 6x 1 4 5 0 10. 3x 2 1 12x 2 18 5 0

286

n2pe-0407.indd 286

8. x 2 2 10x 1 8 5 0 11. 6x(x 1 8) 5 12

9. 2n2 2 4n 2 14 5 0 12. 4p(p 2 2) 5 100

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:11:52 AM

VERTEX FORM Recall from Lesson 4.2 that the vertex form of a quadratic function is y 5 a(x 2 h)2 1 k where (h, k) is the vertex of the function’s graph. To write a quadratic function in vertex form, use completing the square.

EXAMPLE 6

Write a quadratic function in vertex form

Write y 5 x 2 2 10x 1 22 in vertex form. Then identify the vertex. y 5 x 2 2 10x 1 22

Write original function.

2

y1䡵 ? 5 (x 2 10x 1 䡵 ? ) 1 22

Prepare to complete the square.

y 1 25 5 (x 2 2 10x 1 25) 1 22

210 Add } 5 (25) 2 5 25 to each side.

y 1 25 5 (x 2 5)2 1 22

Write x 2 2 10x 1 25 as a binomial squared.

y 5 (x 2 5)2 2 3

1

2

2

2

Solve for y.

c The vertex form of the function is y 5 (x 2 5)2 2 3. The vertex is (5, 23).

EXAMPLE 7

Find the maximum value of a quadratic function

BASEBALL The height y (in feet) of a baseball

t seconds after it is hit is given by this function: y 5 216t 2 1 96t 1 3 Find the maximum height of the baseball. Solution The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation. y 5 216t 2 1 96t 1 3

Write original function.

y 5 216(t 2 2 6t) 1 3

Factor 216 from first two terms.

y 1 (216)(䡵 ? ) 5 216(t 2 2 6t 1 䡵 ?)13 AVOID ERRORS When you complete the square, be sure to add (216)(9) 5 2144 to each side, not just 9.

y 1 (216)(9) 5 216(t 2 2 6t 1 9) 1 3 2

y 2 144 5 216(t 2 3) 1 3 y 5 216(t 2 3)2 1 147

Prepare to complete the square. Add (216)(9) to each side. Write t 2 2 6t 1 9 as a binomial squared. Solve for y.

c The vertex is (3, 147), so the maximum height of the baseball is 147 feet. "MHFCSB

✓

at classzone.com

GUIDED PRACTICE

for Examples 6 and 7

Write the quadratic function in vertex form. Then identify the vertex. 13. y 5 x 2 2 8x 1 17

14. y 5 x 2 1 6x 1 3

15. f(x) 5 x 2 2 4x 2 4

16. WHAT IF? In Example 7, suppose the height of the baseball is given by

y 5 216t 2 1 80t 1 2. Find the maximum height of the baseball. 4.7 Complete the Square

n2pe-0407.indd 287

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4.7

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 27, 45, and 65

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 12, 34, 58, 59, and 67 5 MULTIPLE REPRESENTATIONS Ex. 66

SKILL PRACTICE 1. VOCABULARY What is the difference between a binomial and a trinomial? 2. ★ WRITING Describe what completing the square means for an expression

of the form x 2 1 bx.

EXAMPLE 1 on p. 284 for Exs. 3–12

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

3. x 2 1 4x 1 4 5 9

4. x 2 1 10x 1 25 5 64

5. n2 1 16n 1 64 5 36

6. m2 2 2m 1 1 5 144

7. x 2 2 22x 1 121 5 13

8. x 2 2 18x 1 81 5 5

9. t 2 1 8t 1 16 5 45

10. 4u2 1 4u 1 1 5 75

11. 9x 2 2 12x 1 4 5 23

12. ★ MULTIPLE CHOICE What are the solutions of x 2 2 4x 1 4 5 21?

A 26i EXAMPLE 2 on p. 285 for Exs. 13–21

EXAMPLES 3 and 4 on pp. 285–286 for Exs. 22–34

B 22 6 i

C 23, 21

D 1, 3

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

13. x 2 1 6x 1 c

14. x 2 1 12x 1 c

15. x 2 2 24x 1 c

16. x 2 2 30x 1 c

17. x 2 2 2x 1 c

18. x 2 1 50x 1 c

19. x 2 1 7x 1 c

20. x 2 2 13x 1 c

21. x 2 2 x 1 c

COMPLETING THE SQUARE Solve the equation by completing the square.

22. x 2 1 4x 5 10

23. x 2 1 8x 5 21

24. x 2 1 6x 2 3 5 0

25. x 2 1 12x 1 18 5 0

26. x 2 2 18x 1 86 5 0

27. x 2 2 2x 1 25 5 0

28. 2k 2 1 16k 5 212

29. 3x 2 1 42x 5 224

30. 4x 2 2 40x 2 12 5 0

31. 3s 2 1 6s 1 9 5 0

32. 7t 2 1 28t 1 56 5 0

33. 6r 2 1 6r 1 12 5 0

34. ★ MULTIPLE CHOICE What are the solutions of x 2 1 10x 1 8 5 25? }

}

A 5 6 2Ï 3 EXAMPLE 5 on p. 286 for Exs. 35–38

B 5 6 4Ï 3

}

}

C 25 6 2Ï 3

D 25 6 4Ï3

GEOMETRY Find the value of x.

35. Area of rectangle 5 50 x

36. Area of parallelogram 5 48

x

x 1 10

x16

37. Area of triangle 5 40

38. Area of trapezoid 5 20 3x 2 1

x x14

288

n2pe-0407.indd 288

x x19

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:11:55 AM

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.

125 ft

39. At Buckingham Fountain in Chicago, the water’s height h (in feet)

above the main nozzle can be modeled by h 5 216t 2 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 y 5 0.0085x2 2 1.5x 1 120. Buckingham Fountain

EXAMPLES 6 and 7

WRITING IN VERTEX FORM Write the quadratic function in vertex

on p. 287 for Exs. 41–49

41. y 5 x 2 2 8x 1 19

42. y 5 x 2 2 4x 2 1

43. y 5 x 2 1 12x 1 37

44. y 5 x 2 1 20x 1 90

45. f(x) 5 x 2 2 3x 1 4

46. g(x) 5 x 2 1 7x 1 2

47. y 5 2x 2 1 24x 1 25

48. y 5 5x 2 1 10x 1 7

49. y 5 2x 2 2 28x 1 99

form. Then identify the vertex.

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

50.

51.

x2 1 10x 1 13 5 0

4x2 1 24x 2 11 5 0

x2 1 10x 5 213

4(x2 1 6x) 5 11

x2 1 10x 1 25 5 213 1 25

4(x2 1 6x 1 9) 5 11 1 9 4(x 1 3) 2 5 20

(x 1 5) 2 5 12 }

(x 1 3) 2 5 5

x 1 5 5 6Ï 12

}

}

x 1 3 5 6Ï 5

x 5 25 6 Ï 12

}

}

x 5 23 6 Ï5

x 5 25 6 4Ï 3

COMPLETING THE SQUARE Solve the equation by completing the square.

52. x 2 1 9x 1 20 5 0

53. x 2 1 3x 1 14 5 0

54. 7q2 1 10q 5 2q2 1 155

55. 3x 2 1 x 5 2x 2 6

56. 0.1x 2 2 x 1 9 5 0.2x

57. 0.4v 2 1 0.7v 5 0.3v 2 2

58. ★ OPEN-ENDED MATH Write a quadratic equation with real-number

solutions that can be solved by completing the square but not by factoring. 59. ★ SHORT RESPONSE In this exercise, you will investigate the graphical

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

y 5 x 2 1 2x

y 5 x 2 1 4x

y 5 x 2 2 6x

y 5 (x 1 1)2

y 5 (x 1 2)2

y 5 (x 2 3)2

b 2 . What happens to b. Compare the graphs of y 5 x 2 1 bx and y 5 x 1 } 2

1

2

the graph of y 5 x 2 1 bx when you complete the square?

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

1 2

2

5 k have

exactly 1 real solution? 2 real solutions? 2 imaginary solutions? 61. CHALLENGE Solve x 2 1 bx 1 c 5 0 by completing the square. Your answer

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

4.7 Complete the Square

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PROBLEM SOLVING EXAMPLE 7

62. DRUM MAJOR While marching, a drum major tosses a baton into the air and

catches it. The height h (in feet) of the baton after t seconds can be modeled by h 5 216t 2 1 32t 1 6. Find the maximum height of the baton.

on p. 287 for Exs. 62–65

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

63. VOLLEYBALL The height h (in feet) of a volleyball t seconds after it is hit can

be modeled by h 5 216t 2 1 48t 1 4. Find the volleyball’s maximum height. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

64. SKATEBOARD REVENUE A skateboard shop

sells about 50 skateboards per week for the price advertised. For each $1 decrease in price, about 1 more skateboard per week is sold. The shop’s revenue can be modeled by y 5 (70 2 x)(50 1 x). Use vertex form to find how the shop can maximize weekly revenue. 65. VIDEO GAME REVENUE A store sells about 40 video game systems each

month 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 modeled by y 5 (200 1 10x)(40 2 x). Use vertex form to find how the store can maximize monthly revenue. 66.

MULTIPLE REPRESENTATIONS The path of a ball thrown by a softball player can be modeled by the function

y 5 20.0110x2 1 1.23x 1 5.50 where x is the softball’s horizontal position (in feet) and y is the corresponding 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 of

x from 0 to 120 in increments of 10. c. Drawing a Graph Use your table to graph the function. What is the

maximum height of the softball? How far does it travel? 67. ★ EXTENDED RESPONSE Your school is adding a rectangular outdoor eating

section along part of a 70 foot side of the school. The eating section will be enclosed by a fence along its three open sides. The school has 120 feet of fencing and plans to use 1500 square feet of land 70 ft for the eating section. a. Write an equation for the area

of the eating section. b. Solve the equation. Explain

x x

why you must reject one of the solutions. c. What are the dimensions of

120 – 2x

the eating section?

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5 STANDARDIZED TEST PRACTICE

Eating section

5 MULTIPLE REPRESENTATIONS

10/17/05 10:11:59 AM

GEOMETRY REVIEW

68.

The volume of clay equals the difference of the volumes of two cylinders.

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

x cm

3 cm 3 cm x cm

9 cm x cm x cm

Top view

Side view

MIXED REVIEW PREVIEW

Evaluate b 2 2 4ac for the given values of a, b, and c. (p. 10)

Prepare for Lesson 4.8 in Exs. 69–74.

69. a 5 2, b 5 7, c 5 5

70. a 5 1, b 5 26, c 5 9

71. a 5 4, b 5 21, c 5 3

72. a 5 3, b 5 2, c 5 26

73. a 5 24, b 5 2, c 5 27

74. a 5 25, b 5 3, c 5 2

Write an equation of the line that passes through the given points. (p. 98) 75. (2, 5), (4, 9)

76. (3, 21), (6, 23)

77. (24, 24), (21, 2)

78. (22, 4), (1, 22)

79. (21, 25), (1, 1)

80. (6, 3), (8, 4)

Graph the system of inequalities. (p. 168) 81. x ≥ 2

y≤3

82. x ≥ 0

84. 4x 1 y ≥ 3

83. 3x 2 2y < 8

x1y<4

2x 1 y > 0

2x 2 3y < 6

QUIZ for Lessons 4.5–4.7 Solve the equation. 1. 4x 2 5 64 (p. 266)

2. 3(p 2 1)2 5 15 (p. 266)

3. 16(m 1 5)2 5 8 (p. 266)

4. 22z 2 5 424 (p. 275)

5. s 2 1 12 5 9 (p. 275)

6. 7x2 2 4 5 26 (p. 275)

Write the expression as a complex number in standard form. (p. 275) 7. (5 2 3i) 1 (22 1 5i) 10. (8 2 3i)(26 2 10i)

8. (22 1 9i) 2 (7 1 8i)

9. 3i(7 2 9i) 3 2 2i 12. } 28 1 5i

4i 11. } 26 2 11i

Write the quadratic function in vertex form. Then identify the vertex. (p. 284) 13. y 5 x 2 2 4x 1 9

14. y 5 x 2 1 14x 1 45

15. f(x) 5 x 2 2 10x 1 17

16. g(x) 5 x 2 2 2x 2 7

17. y 5 x 2 1 x 1 1

18. y 5 x 2 1 9x 1 19

19. FALLING OBJECT A student drops a ball from a school roof 45 feet above

ground. How long is the ball in the air? (p. 266)

EXTRA PRACTICE for Lesson 4.7, p. 1013

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4.8 Before Now Why?

Key Vocabulary • quadratic formula • discriminant

Use the Quadratic Formula and the Discriminant You solved quadratic equations by completing the square. You will solve quadratic equations using the quadratic formula. So you can model the heights of thrown objects, as in Example 5.

In Lesson 4.7, you solved quadratic equations by completing the square for each equation separately. By completing the square once for the general equation ax2 1 bx 1 c 5 0, you can develop a formula that gives the solutions of any quadratic equation. (See Exercise 67.) The formula for the solutions is called the quadratic formula.

For Your Notebook

KEY CONCEPT The Quadratic Formula

Let a, b, and c be real numbers such that a Þ 0. The solutions of the quadratic }

2b 6 Ï b2 2 4ac . equation ax2 1 bx 1 c 5 0 are x 5 }}}}}}}} 2a

EXAMPLE 1

Solve an equation with two real solutions

Solve x 2 1 3x 5 2. x2 1 3x 5 2 AVOID ERRORS Remember to write the quadratic equation in standard form before applying the quadratic formula.

Write original equation.

2

x 1 3x 2 2 5 0

Write in standard form. }

2b 6 Ïb2 2 4ac x 5 }}}}}}}} 2a

Quadratic formula

}}

23 6 Ï32 2 4(1)(22) x 5 }}}}}}}}}} 2(1)

a 5 1, b 5 3, c 5 22

}

23 6 Ï 17 x 5 }}}}}

Simplify.

2

}

}

23 1 Ï17 ø 0.56 and x 5 23 2 Ï 17 ø 23.56. c The solutions are x 5 }}}}} }}}}} 2

2

2

CHECK Graph y 5 x 1 3x 2 2 and note that the x-intercepts are about 0.56 and about 23.56. ✓

Zero X=.56155281 Y=0

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EXAMPLE 2

Solve an equation with one real solution

Solve 25x 2 2 18x 5 12x 2 9. 25x2 2 18x 5 12x 2 9 ANOTHER WAY You can also use factoring to solve this equation because the left side factors as (5x 2 3) 2.

Write original equation.

25x2 2 30x 1 9 5 0

Write in standard form. }}

30 6 (230)2 2 4(25)(9) 2(25)

Ï x 5 }}}}}}}}}}}

a 5 25, b 5 230, c 5 9

}

30 6 Ï0 x 5 }}}}

Simplify.

3 x5}

Simplify.

50

5

3. c The solution is } 5

CHECK

Graph y 5 25x2 2 30x 1 9 and note that 3. ✓ the only x-intercept is 0.6 5 } 5

EXAMPLE 3

Zero X=.6

Y=0

Solve an equation with imaginary solutions

Solve 2x 2 1 4x 5 5. 2x2 1 4x 5 5

Write original equation.

2

2x 1 4x 2 5 5 0

Write in standard form. }}

2

Ï x 5 }}}}}}}}}} 24 6 4 2 4(21)(25) 2(21)

a 5 21, b 5 4, c 5 25

}

24 6 Ï24 x 5 }}}}}

Simplify.

24 6 2i x 5 }}}}

Rewrite using the imaginary unit i.

x526i

Simplify.

22

22

c The solutions are 2 1 i and 2 2 i.

CHECK

Graph y 5 2x2 1 4x 2 5. There are no x-intercepts. So, the original equation has no real solutions. The algebraic check for the imaginary solution 2 1 i is shown. 2(2 1 i)2 1 4(2 1 i) 0 5 23 2 4i 1 8 1 4i 0 5 555✓

✓

GUIDED PRACTICE

for Examples 1, 2, and 3

Use the quadratic formula to solve the equation. 1. x2 5 6x 2 4

2. 4x 2 2 10x 5 2x 2 9

3. 7x 2 5x 2 2 4 5 2x 1 3

4.8 Use the Quadratic Formula and the Discriminant

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DISCRIMINANT In the quadratic formula, the expression b2 2 4ac is called

the discriminant of the associated equation ax2 1 bx 1 c 5 0. }

2b 6 Ïb2 2 4ac x 5 }}}}}}}}

discriminant

2a

You can use the discriminant of a quadratic equation to determine the equation’s number and type of solutions.

For Your Notebook

KEY CONCEPT Using the Discriminant of ax 2 1 bx 1 c 5 0 Value of discriminant Number and type of solutions Graph of y 5 ax 2 1 bx 1 c

b2 2 4ac > 0

b2 2 4ac 5 0

b2 2 4ac < 0

Two real solutions

One real solution

Two imaginary solutions

y

x

Two x-intercepts

EXAMPLE 4

y

y

x

x

One x-intercept

No x-intercept

Use the discriminant

Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. a. x2 2 8x 1 17 5 0

b. x 2 2 8x 1 16 5 0

c. x 2 2 8x 1 15 5 0

Solution Equation

Discriminant

Solution(s)

b2 2 4ac

6 Ï b2 2 4ac x 5 2b }}}}}}}}

(28)2 2 4(1)(17) 5 24

Two imaginary: 4 6 i

}

ax 2 1 bx 1 c 5 0 a. x2 2 8x 1 17 5 0 2

b. x 2 8x 1 16 5 0 c. x 2 8x 1 15 5 0

✓

GUIDED PRACTICE

2

One real: 4

2

Two real: 3, 5

(28) 2 4(1)(16) 5 0

2

2a

(28) 2 4(1)(15) 5 4

for Example 4

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

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4. 2x 2 1 4x 2 4 5 0

5. 3x 2 1 12x 1 12 5 0

6. 8x 2 5 9x 2 11

7. 7x 2 2 2x 5 5

8. 4x 2 1 3x 1 12 5 3 2 3x

9. 3x 2 5x 2 1 1 5 6 2 7x

Chapter 4 Quadratic Functions and Factoring

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MODELING LAUNCHED OBJECTS In Lesson 4.5, the function h 5 216t 2 1 h0

was used to model the height of a dropped object. For an object that is launched or thrown, an extra term v 0t must be added to the model to account for the object’s initial vertical velocity v 0 (in feet per second). Recall that h is the height (in feet), t is the time in motion (in seconds), and h0 is the initial height (in feet). h 5 216t 2 1 h0

Object is dropped.

2

h 5 216t 1 v0t 1 h0

Object is launched or thrown.

As shown below, the value of v0 can be positive, negative, or zero depending on whether the object is launched upward, downward, or parallel to the ground.

v0 > 0

v0 < 0

EXAMPLE 5

v0 5 0

Solve a vertical motion problem

JUGGLING A juggler tosses a ball into the air. The ball leaves the juggler’s hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?

Solution Because the ball is thrown, use the model h 5 216t 2 1 v0t 1 h0. To find how long the ball is in the air, solve for t when h 5 3. h 5 216t 2 1 v 0t 1 h0

Write height model.

3 5 216t 2 1 40t 1 4

Substitute 3 for h, 40 for v0, and 4 for h0.

2

0 5 216t 1 40t 1 1

Write in standard form.

}}

240 6 402 2 4(216)(1) 2(216)

Ï t 5 }}}}}}}}}}}

Quadratic formula

}

240 6 Ï 1664 t 5 }}}}}}

Simplify.

t ø 20.025 or t ø 2.5

Use a calculator.

232

c Reject the solution 20.025 because the ball’s time in the air cannot be negative. So, the ball is in the air for about 2.5 seconds.

✓

GUIDED PRACTICE

for Example 5

10. WHAT IF? In Example 5, suppose the ball leaves the juggler’s hand with an

initial vertical velocity of 50 feet per second. How long is the ball in the air?

4.8 Use the Quadratic Formula and the Discriminant

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4.8

EXERCISES

HOMEWORK KEY

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

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 12, 51, 55, 62, 69, 72, and 73

SKILL PRACTICE 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

h 5 216t 2 1 v0t 1 h0 but not the model h 5 216t 2 1 h0.

EXAMPLES 1, 2, and 3 on pp. 292–293 for Exs. 3–30

EQUATIONS IN STANDARD FORM Use the quadratic formula to solve the

equation. 3. x 2 2 4x 2 5 5 0

4. x 2 2 6x 1 7 5 0

5. t 2 1 8t 1 19 5 0

6. x 2 2 16x 1 7 5 0

7. 8w 2 2 8w 1 2 5 0

8. 5p2 2 10p 1 24 5 0

9. 4x 2 2 8x 1 1 5 0

10. 6u2 1 4u 1 11 5 0

11. 3r 2 2 8r 2 9 5 0

12. ★ MULTIPLE CHOICE What are the complex solutions of the equation

2x2 2 16x 1 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. 3w 2 2 12w 5 212

14. x 2 1 6x 5 215

15. s 2 5 214 2 3s

16. 23y 2 5 6y 2 10

17. 3 2 8v 2 5v 2 5 2v

18. 7x 2 5 1 12x2 5 23x

19. 4x 2 1 3 5 x2 2 7x

20. 6 2 2t 2 5 9t 1 15

21. 4 1 9n 2 3n2 5 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).

EXAMPLE 4 on p. 294 for Exs. 31–39

22. z2 1 15z 1 24 5 232

23. x 2 2 5x 1 10 5 4

24. m2 1 5m 2 99 5 3m

25. s 2 2 s 2 3 5 s

26. r 2 2 4r 1 8 5 5r

27. 3x 2 1 7x 2 24 5 13x

28. 45x 2 1 57x 1 1 5 5

29. 5p2 1 40p 1 100 5 25

30. 9n2 2 42n 2 162 5 21n

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

31. x 2 2 8x 1 16 5 0

32. s 2 1 7s 1 11 5 0

33. 8p2 1 8p 1 3 5 0

34. 24w 2 1 w 2 14 5 0

35. 5x 2 1 20x 1 21 5 0

36. 8z 2 10 5 z2 2 7z 1 3

37. 8n2 2 4n 1 2 5 5n 2 11

38. 5x 2 1 16x 5 11x 2 3x2

39. 7r 2 2 5 5 2r 1 9r 2

SOLVING QUADRATIC EQUATIONS Solve the equation using any method.

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40. 16t 2 2 7t 5 17t 2 9

41. 7x 2 3x 2 5 85 1 2x2 1 2x

42. 4(x 2 1)2 5 6x 1 2

43. 25 2 16v 2 5 12v(v 1 5)

3 y 2 2 6y 5 3 y 2 9 44. } } 4 2

9 x 2 4 5 5x 1 3 45. 3x 2 1 } } 4 2

46. 1.1(3.4x 2 2.3)2 5 15.5

47. 19.25 5 28.5(2r 2 1.75)2

48. 4.5 5 1.5(3.25 2 s)2

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:15:36 AM

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

49.

50.

3x 2 1 6x 1 15 5 0 }}

x 2 1 6x 1 8 5 2 }}

26 6 62 2 4(3)(15) 2(3)

26 6 Ï62 2 4(1)(8) x 5 }}}}}}}}} 2(1)

Ï x 5 }}}}}}}}}} }

}

26 6 Ï2144 5 }}}}}}

26 6 Ï4 5 }}}}}

26 6 12 5 }}}}

26 6 2 5 }}}}

5 1 or 23

5 22 or 24

6

2

6

2

51. ★ SHORT RESPONSE For a quadratic equation ax 2 1 bx 1 c 5 0 with two real b . How is this fact solutions, show that the mean of the solutions is 2}} 2a

related to the symmetry of the graph of y 5 ax2 1 bx 1 c?

VISUAL THINKING In Exercises 52–54, the graph of a quadratic function

y 5 ax 2 1 bx 1 c is shown. Tell whether the discriminant of ax 2 1 bx 1 c 5 0 is positive, negative, or zero. 52.

53.

y

54.

y

y x

x

x

55. ★ MULTIPLE CHOICE What is the value of c if the discriminant of

2x2 1 5x 1 c 5 0 is 223? A 223

B 26

C 6

D 14

THE CONSTANT TERM Use the discriminant to find all values of c for which the

equation has (a) two real solutions, (b) one real solution, and (c) two imaginary solutions. 56. x 2 2 4x 1 c 5 0

57. x 2 1 8x 1 c 5 0

58. 2x2 1 16x 1 c 5 0

59. 3x 2 1 24x 1 c 5 0

60. 24x2 2 10x 1 c 5 0

61. x 2 2 x 1 c 5 0

62. ★ OPEN-ENDED MATH Write a quadratic equation in standard form that has

a discriminant of 210. WRITING EQUATIONS Write a quadratic equation in the form ax 2 1 bx 1 c 5 0

such that c 5 4 and the equation has the given solutions. 63. 24 and 3

4 and 21 64. 2} 3

65. 21 1 i and 21 2 i

66. REASONING Show that there is no quadratic equation ax2 1 bx 1 c 5 0 such

that a, b, and c are real numbers and 3i and 22i are solutions. 67. CHALLENGE Derive the quadratic formula by completing the square to solve

the general quadratic equation ax2 1 bx 1 c 5 0.

4.8 Use the Quadratic Formula and the Discriminant

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PROBLEM SOLVING EXAMPLE 5 on p. 295 for Exs. 68–69

68. FOOTBALL In a football game, a defensive player jumps up to block a pass by

the opposing team’s quarterback. The player bats the ball downward with his hand at an initial vertical velocity of 250 feet per second when the ball is 7 feet above the ground. How long do the defensive player’s teammates have to intercept the ball before it hits the ground? GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

69. ★ MULTIPLE CHOICE For the period 1990–2002, the number S (in thousands)

of cellular telephone subscribers in the United States can be modeled by S 5 858t 2 1 1412t 1 4982 where t is the number of years since 1990. In what year did the number of subscribers reach 50 million? A 1991

B 1992

C 1996

D 2000

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

70. MULTI-STEP PROBLEM A stunt motorcyclist makes a jump from one ramp

20 feet off the ground to another ramp 20 feet off the ground. The jump 1 x 2 1 1 x 1 20 where x is the between the ramps can be modeled by y 5 2}} } 640

4

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 it

reaches 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 be

modeled by the function S 5 20.000013E2 1 0.042E 2 21 where E is the elevation (in meters). Predict the elevation(s) at which you would expect to find 10 species of ants. 72. ★ SHORT RESPONSE A city planner wants to create

adjacent sections for athletics and picnics in the yard of a youth center. The sections will be rectangular and will be surrounded by fencing as shown. There is 900 feet of fencing available. Each section should have an area of 12,000 square feet.

!THLETICS SECTION

0ICNIC SECTION

W

4 l. a. Show that w 5 300 2 } 3

b. Find the possible dimensions of each section.

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*

*

5 STANDARDIZED TEST PRACTICE

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73. ★ EXTENDED RESPONSE You can model the position (x, y) of a moving

object using a pair of parametric equations. Such equations give x and y in terms of a third variable t that represents time. For example, suppose that when a basketball player attempts a free throw, the path of the basketball can be modeled by the parametric equations x 5 20t y 5 216t 2 1 21t 1 6 where x and y are measured in feet, t is measured in seconds, and the player’s feet are at (0, 0). a. Evaluate Make a table of values giving the position (x, y) of the

basketball 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 the

backboard is (15, 12). Does the player make the free throw? Explain. 74. CHALLENGE The Stratosphere Tower in Las Vegas is 921 feet tall

Big Shot ride

and 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 the needle and then lets them fall back to the launching pad. a. The height h (in feet) of a rider on the Big Shot can be

modeled by h 5 216t 2 1 v 0t 1 921 where t is the elapsed time (in seconds) after launch and v0 is the initial vertical velocity (in feet per second). Find v 0 using the fact that the maximum value of h is 921 1 160 5 1081 feet. b. A brochure for the Big Shot states that the ride up the

needle takes two seconds. Compare this time with the time given by the model h 5 216t 2 1 v 0t 1 921 where v 0 is the value you found in part (a). Discuss the model’s accuracy.

MIXED REVIEW Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal, or is vertical. (p. 82) 75. (2, 27), (4, 9)

76. (28, 3), (4, 25)

77. (23, 22), (6, 22)

3, 2 , 1, 5 78. } } } 2 4 4

79. (21, 0), (21, 5)

1 , 7 , 4, 2 80. } } } 3 3 3

1

21

2

1

21

PREVIEW

Graph the inequality or equation in a coordinate plane.

Prepare for Lesson 4.9 in Exs. 81–86.

81. y ≤ 10 (p. 132)

82. 8x 2 4y < 216 (p. 132)

1 x 1 3y > 8 (p. 132) 83. } 2

4 x 2 7 (p. 132) 84. y ≥ 2} 9

85. y 5 3(x 1 1)(x 1 2) (p. 245)

86. y 5 22(x 2 3)(x 2 1) (p. 245)

2

87. HANG-GLIDING Suppose that t minutes after beginning a descent, a hang

glider has an altitude a (in feet) given by the model a 5 2000 2 250t. What is the height of the hang glider prior to the descent? How long does it take the hang glider to reach the ground? (p. 72)

EXTRA PRACTICE for Lesson 4.8, p. 1013 ONLINEFormula QUIZ atand classzone.com 4.8 Use the Quadratic the Discriminant

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4.9 Before Now Why?

Key Vocabulary

Graph and Solve Quadratic Inequalities You graphed and solved linear inequalities. You will graph and solve quadratic inequalities. So you can model the strength of a rope, as in Example 2.

A quadratic inequality in two variables can be written in one of the following forms:

• quadratic inequality

y < ax2 1 bx 1 c y ≤ ax2 1 bx 1 c y > ax2 1 bx 1 c y ≥ ax2 1 bx 1 c in two variables • quadratic inequality The graph of any such inequality consists of all solutions (x, y) of the inequality. in one variable

For Your Notebook

KEY CONCEPT

Graphing a Quadratic Inequality in Two Variables To graph a quadratic inequality in one of the forms above, follow these steps:

STEP 1

Graph the parabola with equation y 5 ax 2 1 bx 1 c. Make the parabola dashed for inequalities with < or > and solid for inequalities with ≤ or ≥.

STEP 2 Test a point (x, y) inside the parabola to determine whether the point is a solution of the inequality.

STEP 3 Shade the region inside the parabola if the point from Step 2 is a solution. Shade the region outside the parabola if it is not a solution.

EXAMPLE 1

Graph a quadratic inequality

Graph y > x 2 1 3x 2 4. Solution AVOID ERRORS Be sure to use a dashed parabola if the symbol is > or < and a solid parabola if the symbol is ≥ or ≤ .

STEP 1

Graph y 5 x2 1 3x 2 4. Because the inequality

symbol is >, make the parabola dashed.

STEP 2 Test a point inside the parabola, such as (0, 0).

1

(0, 0)

y 2x

y > x 2 1 3x 2 4 0? > 02 1 3(0) 2 4 0 > 24 ✓ So, (0, 0) is a solution of the inequality.

STEP 3 Shade the region inside the parabola. "MHFCSB

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Chapter 4 Quadratic Functions and Factoring

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EXAMPLE 2

Use a quadratic inequality in real life

RAPPELLING A manila rope used for rappelling down a cliff can safely support a weight W (in pounds) provided

W ≤ 1480d 2 where d is the rope’s diameter (in inches). Graph the inequality. Solution

W ≤ 1480d 2 2000 ? ≤ 1480(1)2

W 3000

Weight (lb)

Graph W 5 1480d 2 for nonnegative values of d. Because the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (1, 2000).

W ≤ 1480d 2

1000 0

2000 ≤ 1480 ✗

(1, 2000)

2000

0

0.5

1 1.5 2 Diameter (in.)

d

Because (1, 2000) is not a solution, shade the region below the parabola.

SYSTEMS OF QUADRATIC INEQUALITIES Graphing a system of quadratic inequalities is similar to graphing a system of linear inequalities. First graph each inequality in the system. Then identify the region in the coordinate plane common to all of the graphs. This region is called the graph of the system.

EXAMPLE 3

Graph a system of quadratic inequalities

Graph the system of quadratic inequalities. y ≤ 2x 2 1 4 y > x 2 2 2x 2 3

Inequality 1 Inequality 2

Solution

STEP 1

Graph y ≤ 2x 2 1 4. The graph is the

y

red region inside and including the parabola y 5 2x 2 1 4.

STEP 2 Graph y > x 2 2 2x 2 3. The graph is the blue region inside (but not including) the parabola y 5 x 2 2 2x 2 3.

STEP 3 Identify the purple region where the

1 1

x

y > x 2 2 2x 2 3 y ≤ 2x 2 1 4

two graphs overlap. This region is the graph of the system.

✓

GUIDED PRACTICE

for Examples 1, 2, and 3

Graph the inequality. 1. y > x 2 1 2x 2 8

2. y ≤ 2x 2 2 3x 1 1

3. y < 2x 2 1 4x 1 2

4. Graph the system of inequalities consisting of y ≥ x 2 and y < 2x 2 1 5.

4.9 Graph and Solve Quadratic Inequalities

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ONE-VARIABLE INEQUALITIES A quadratic inequality in one variable can be

written in one of the following forms: ax2 1 bx 1 c < 0

ax2 1 bx 1 c ≤ 0

ax2 1 bx 1 c > 0

ax2 1 bx 1 c ≥ 0

You can solve quadratic inequalities using tables, graphs, or algebraic methods.

EXAMPLE 4

Solve a quadratic inequality using a table

Solve x 2 1 x ≤ 6 using a table. Solution Rewrite the inequality as x 2 1 x 2 6 ≤ 0. Then make a table of values. MAKE A TABLE To give the exact solution, your table needs to include the x-values for which the value of the quadratic expression is 0.

x

25

24

23

22

21

0

1

2

3

4

x2 1 x 2 6

14

6

0

24

26

26

24

0

6

14

Notice that x2 1 x 2 6 ≤ 0 when the values of x are between 23 and 2, inclusive. c The solution of the inequality is 23 ≤ x ≤ 2.

GRAPHING TO SOLVE INEQUALITIES Another way to solve ax 2 1 bx 1 c < 0 is to

first graph the related function y 5 ax 2 1 bx 1 c. Then, because the inequality symbol is <, identify the x-values for which the graph lies below the x-axis. You can use a similar procedure to solve quadratic inequalities that involve ≤, >, or ≥.

EXAMPLE 5

Solve a quadratic inequality by graphing

Solve 2x 2 1 x 2 4 ≥ 0 by graphing. Solution The solution consists of the x-values for which the graph of y 5 2x2 1 x 2 4 lies on or above the x-axis. Find the graph’s x-intercepts by letting y 5 0 and using the quadratic formula to solve for x. 0 5 2x 2 1 x 2 4

1

}}

21 ± 12 2 4(2)(24) 2(2)

Ï x5}

25 21.69

y

1.19 x

}

21 ± Ï 33 x5} 4

y 5 2x 2 1 x 2 4

x ø 1.19 or x ø 21.69 Sketch a parabola that opens up and has 1.19 and 21.69 as x-intercepts. The graph lies on or above the x-axis to the left of (and including) x 5 21.69 and to the right of (and including) x 5 1.19. c The solution of the inequality is approximately x ≤ 21.69 or x ≥ 1.19.

✓

GUIDED PRACTICE

for Examples 4 and 5

5. Solve the inequality 2x 2 1 2x ≤ 3 using a table and using a graph.

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EXAMPLE 6

Use a quadratic inequality as a model

ROBOTICS The number T of teams that have

participated in a robot-building competition for high school students can be modeled by T(x) 5 7.51x2 2 16.4x 1 35.0, 0 ≤ x ≤ 9 where x is the number of years since 1992. For what years was the number of teams greater than 100? Solution You want to find the values of x for which: T(x) > 100 2

7.51x 2 16.4x 1 35.0 > 100 7.51x2 2 16.4x 2 65 > 0 Zero X=4.2299219 Y=0

Graph y 5 7.51x2 2 16.4x 2 65 on the domain 0 ≤ x ≤ 9. The graph’s x-intercept is about 4.2. The graph lies above the x-axis when 4.2 < x ≤ 9.

c There were more than 100 teams participating in the years 1997–2001.

EXAMPLE 7

Solve a quadratic inequality algebraically

Solve x 2 2 2x > 15 algebraically. Solution First, write and solve the equation obtained by replacing > with 5. x 2 2 2x 5 15 2

x 2 2x 2 15 5 0

Write equation that corresponds to original inequality. Write in standard form.

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

Factor.

x 5 23 or x 5 5

Zero product property

The numbers 23 and 5 are the critical x-values of the inequality x2 2 2x > 15. Plot 23 and 5 on a number line, using open dots because the values do not satisfy the inequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality. 25

24

23

22

21

Test x 5 24: (24)2 2 2(24) 5 24 > 15 ✓

0

1

2

3

Test x 5 1: 12 2 2(1) 5 21 >/ 15

4

5

6

7

Test x 5 6: 62 2 2(6) 5 24 > 15 ✓

c The solution is x < 23 or x > 5.

✓

GUIDED PRACTICE

for Examples 6 and 7

6. ROBOTICS Use the information in Example 6 to determine in what years at

least 200 teams participated in the robot-building competition. 7. Solve the inequality 2x 2 2 7x > 4 algebraically. 4.9 Graph and Solve Quadratic Inequalities

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4.9

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 17, 39, and 73

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 44, 45, 68, and 73 5 MULTIPLE REPRESENTATIONS Ex. 74

SKILL PRACTICE 1. VOCABULARY Give an example of a quadratic inequality in one variable

and an example of a quadratic inequality in two variables. 2. ★ WRITING Explain how to solve x 2 1 6x 2 8 < 0 using a table, by graphing,

and algebraically. EXAMPLE 1 on p. 300 for Exs. 3–19

MATCHING INEQUALITIES WITH GRAPHS Match the inequality with its graph.

3. y ≤ x 2 1 4x 1 3

4. y > 2x 2 1 4x 2 3

5. y < x 2 2 4x 1 3

A.

B.

C.

y

y

y

1

3 x

2

1 x

2

1

x

GRAPHING QUADRATIC INEQUALITIES Graph the inequality.

6. y < 2x 2

7. y ≥ 4x 2

9. y ≤ x 2 1 5x

8. y > x 2 2 9

10. y < x 2 1 4x 2 5

11. y > x 2 1 7x 1 12

12. y ≤ 2x 2 1 3x 1 10

13. y ≥ 2x 2 1 5x 2 7

14. y ≥ 22x 2 1 9x 2 4

15. y < 4x 2 2 3x 2 5

16. y > 0.1x 2 2 x 1 1.2

2 17. y ≤ 2}x 2 1 3x 1 1 3

ERROR ANALYSIS Describe and correct the error in graphing y ≥ x 2 1 2.

18.

19.

y

y

1

1 1

1

x

x

EXAMPLE 3

GRAPHING SYSTEMS Graph the system of inequalities.

on p. 301 for Exs. 20–25

20. y ≥ 2x 2 y < 2x 2 1 1

21. y > 25x 2 y > 3x 2 2 2

22. y ≥ x 2 2 4

23. y ≤ 2x 2 1 4x 2 4 y < 2x 2 1 x 2 8

24. y > 3x 2 1 3x 2 5 y < 2x 2 1 5x 1 10

25. y ≥ x 2 2 3x 2 6

y ≤ 22x 2 1 7x 1 4 y ≥ 2x 2 1 7x 1 6

EXAMPLE 4

SOLVING USING A TABLE Solve the inequality using a table.

on p. 302 for Exs. 26–34

26. x 2 2 5x < 0

27. x 2 1 2x 2 3 > 0

28. x 2 1 3x ≤ 10

29. x 2 2 2x ≥ 8

30. 2x 2 1 15x 2 50 > 0

31. x 2 2 10x < 216

32. x 2 2 4x > 12

33. 3x 2 2 6x 2 2 ≤ 7

34. 2x 2 2 6x 2 9 ≥ 11

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EXAMPLE 5

SOLVING BY GRAPHING Solve the inequality by graphing.

on p. 302 for Exs. 35–43

35. x 2 2 6x < 0

36. x 2 1 8x ≤ 27

37. x 2 2 4x 1 2 > 0

38. x 2 1 6x 1 3 > 0

39. 3x 2 1 2x 2 8 ≤ 0

40. 3x 2 1 5x 2 3 < 1

41. 26x 2 1 19x ≥ 10

1 x 2 1 4x ≥ 1 42. 2} 2

43. 4x 2 2 10x 2 7 < 10

44. ★ MULTIPLE CHOICE What is the solution of 3x 2 2 x 2 4 > 0? 4 A x < 21 or x > }

4 B 21 < x < }

4 or x > 1 C x < 2}

4 D 1

3

3

3

3

45. ★ MULTIPLE CHOICE What is the solution of 2x 2 1 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

EXAMPLE 7

SOLVING ALGEBRAICALLY Solve the inequality algebraically.

on p. 303 for Exs. 46–57

46. 4x 2 < 25

47. x 2 1 10x 1 9 < 0

48. x 2 2 11x ≥ 228

49. 3x 2 2 13x > 10

50. 2x 2 2 5x 2 3 ≤ 0

51. 4x 2 1 8x 2 21 ≥ 0

52. 24x 2 2 x 1 3 ≤ 0

53. 5x 2 2 6x 2 2 ≤ 0

54. 23x 2 1 10x > 22

55. 22x 2 2 7x ≥ 4

56. 3x 2 1 1 < 15x

57. 6x 2 2 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 y 5 7.51x2 2 16.4x 1 35.0 and y 5 100 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. 8x 2 2 3x 1 1 < 10

60. 4x 2 1 11x 1 3 ≥ 23

61. 2x 2 2 2x 2 1 > 2

62. 23x 2 1 4x 2 5 ≤ 2

63. x 2 2 7x 1 4 > 5x 2 2

64. 2x 2 1 9x 2 1 ≥ 23x 1 1

65. 3x 2 2 2x 1 1 ≤ 2x2 1 1

66. 5x 2 1 x 2 7 < 3x 2 2 4x

67. 6x 2 2 5x 1 2 < 23x2 1 x

68. ★ OPEN-ENDED MATH 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 2 bh parabola and a horizontal line is given by A 5 } 3

y

h

where b and h are as defined in the diagram. Find the area of the region determined by each pair of inequalities. a. y ≤ 2x2 1 4x

y≥0

b. y ≥ x2 2 4x 2 5

b

y≤3

x

4.9 Graph and Solve Quadratic Inequalities

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PROBLEM SOLVING EXAMPLE 2

70. ENGINEERING A wire rope can safely support a weight W (in pounds)

provided W ≤ 8000d 2 where d is the rope’s diameter (in inches). Graph the inequality.

on p. 301 for Exs. 70–71

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

71. WOODWORKING A hardwood shelf in a wooden bookcase can safely support

a weight W (in pounds) provided W ≤ 115x2 where x is the shelf’s thickness (in inches). Graph the inequality. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

EXAMPLE 6

72. ARCHITECTURE The arch of the Sydney Harbor Bridge in Sydney, Australia,

can be modeled by y 5 20.00211x 2 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 the water. For what distances x is the arch above the road?

on p. 303 for Exs. 72–74

pylon

y

52 m x 73. ★ SHORT RESPONSE The length L (in millimeters) of the larvae of the black

porgy fish can be modeled by L(x) 5 0.00170x 2 1 0.145x 1 2.35, 0 ≤ x ≤ 40 where x is the age (in days) of the larvae. Write and solve an inequality to find at what ages a larvae’s length tends to be greater than 10 millimeters. Explain how the given domain affects the solution. 74.

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

A(x) 5 0.0051x2 2 0.319x 1 15, 16 ≤ x ≤ 70 V(x) 5 0.005x2 2 0.23x 1 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 the

x-values for which A(x) is less than V(x). b. Making a Table Use a table to find the solution of the inequality

from part (a). Your table should contain x-values from 16 to 70 in increments of 6. c. Drawing a Graph Check the solution you found in part (b) by using

a graphing calculator to solve the inequality A(x) < V(x) graphically. Describe how you used the domain 16 ≤ x ≤ 70 to determine a reasonable solution. d. Interpret Based on your results from parts (b) and (c), do you think a

driver would react more quickly to a traffic light changing from green to yellow or to the siren of an approaching ambulance? Explain.

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5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

★

5 STANDARDIZED TEST PRACTICE

5 MULTIPLE REPRESENTATIONS

10/17/05 10:48:14 AM

75. SOCCER The path of a soccer ball kicked from the ground can be modeled by y 5 20.0540x 2 1 1.43x

where x is the horizontal distance (in feet) from where the ball was kicked and y is the corresponding height (in feet). a. A soccer goal is 8 feet high. Write and solve an inequality to find at what

values 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 feet

away. No one is blocking the goal. Will the player score a goal? Explain your reasoning. 76. MULTI-STEP PROBLEM A truck that is 11 feet tall and

7 feet wide is traveling under an arch. The arch can be modeled by

Y

%.42!.#%

y 5 20.0625x2 1 1.25x 1 5.75

where x and y are measured in feet. a. Will the truck fit under the arch? Explain your

reasoning. b. What is the maximum width that a truck 11 feet

tall can have and still make it under the arch?

X

c. What is the maximum height that a truck 7 feet

wide 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 0.1x2 2 0.5x 2 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 the

minimum x-value in the domain for which the function gives meaningful results.

MIXED REVIEW Graph the function. 78. y 5 3x 1 7 (p. 89)

79. f(x) 5 24x 1 5 (p. 89)

1 x (p. 123) 80. y 5 } ⏐⏐ 2

81. y 5 ⏐x 2 2⏐ (p. 123)

82. y 5 ⏐x 1 6⏐ 2 1 (p. 123)

83. g(x) 5 x 2 2 8 (p. 236)

84. f (x) 5 x 2 1 4x 1 3 (p. 236)

85. y 5 2x2 2 9x 1 4 (p. 236)

1 x 2 2 2x 1 1 (p. 236) 86. y 5 } 4

PREVIEW

Solve the system of equations. (p. 178)

Prepare for Lesson 4.10 in Exs. 87–92.

87. x 1 y 1 z 5 22

88. x 1 y 1 z 5 3

89. 4x 1 2y 1 z 5 26

90. x 1 y 1 z 5 8

91. x 1 y 1 z 5 5

92. x 1 y 1 z 5 1

4x 1 2y 1 z 5 3 z 5 23

9x 2 3y 1 z 5 0 4x 2 2y 1 z 5 21

2x 1 3y 2 z 5 28 z54

2x 2 3y 1 3z 5 9 2x 1 7y 2 z 5 11

EXTRA PRACTICE for Lesson 4.9, p. 1013

x 1 y 1 z 5 23 16x 1 4y 1 z 5 0

x2y1z51 3x 1 y 1 3z 5 3

ONLINE QUIZ at classzone.com

307

Use before Lesson 4.10

classzone.com Keystrokes

4.10 Modeling Data with a Quadratic Function M AT E R I A L S • compass • 50 pennies • graphing calculator

QUESTION

EXPLORE

How can you fit a quadratic function to a set of data?

Collect and model quadratic data

STEP 1 Collect data Draw five circles using a compass. Use diameters of 1 inch, 2 inches, 3 inches, 4 inches, and 5 inches. Place as many pennies as you can in each circle, making sure that each penny is completely within the circle.

STEP 2 Record data Record your results from Step 1 in a table like the one shown at the right. Also, record the number of pennies that would fit in a circle with a diameter of 0 inch.

STEP 3 Enter data

STEP 4 Display data

Enter the data you collected into two lists of a graphing calculator.

Display the data in a scatter plot. Notice that the points appear to lie on a parabola.

Diameter of circle (in.), x

Number of pennies, y

0

?

1

?

2

?

3

?

4

?

5

?

STEP 5 Find model Use the quadratic regression feature to find a quadratic model for the data.

L2 L1 0 ----1 2 3 4 L1(1)=0

DR AW CONCLUSIONS

EDIT CALC TESTS 1:1-Var Stats 2:2-Var Stats 3:Med-Med 4:LinReg(ax+b) 5:QuadReg 6:CubicReg

Use your observations to complete these exercises

1. Graph your model from Step 5 on the same screen as the scatter plot.

Describe how well the model fits the data. 2. Use your model from Step 5 to predict the number of pennies that will

fit in a circle with a diameter of 6 inches. Check your prediction by drawing a circle with a diameter of 6 inches and filling it with pennies. 3. Explain why you would expect the number of pennies that fit inside

a circle to be a quadratic function of the circle’s diameter. 4. The diameter of a penny is 0.75 inch. Use this fact to write a quadratic

function giving an upper limit L on the number of pennies that can fit inside a circle with diameter x inches.

308

Chapter 4 Quadratic Functions and Factoring

4.10 Before Now Why?

Key Vocabulary • best-fitting

Write Quadratic Functions and Models You wrote linear functions and models. You will write quadratic functions and models. So you can model the cross section of parabolic dishes, as in Ex. 46.

In Lessons 4.1 and 4.2, you learned how to graph quadratic functions. In this lesson, you will write quadratic functions given information about their graphs.

quadratic model

EXAMPLE 1

Write a quadratic function in vertex form

Write a quadratic function for the parabola shown. Solution

y 2

(3, 2)

Use vertex form because the vertex is given. x

1

y 5 a(x 2 h)2 1 k

Vertex form

y 5 a(x 2 1)2 2 2

Substitute 1 for h and 22 for k.

vertex (1, 22)

Use the other given point, (3, 2), to find a. 2 5 a(3 2 1)2 2 2

Substitute 3 for x and 2 for y.

2 5 4a 2 2

Simplify coefficient of a.

15a

Solve for a.

c A quadratic function for the parabola is y 5 (x 2 1)2 2 2.

EXAMPLE 2

Write a quadratic function in intercept form

Write a quadratic function for the parabola shown.

y 3

Solution Use intercept form because the x-intercepts are given.

(3, 2) 4

21 1

y 5 a(x 2 p)(x 2 q)

Intercept form

y 5 a(x 1 1)(x 2 4)

Substitute 21 for p and 4 for q.

x

Use the other given point, (3, 2), to find a. AVOID ERRORS Be sure to substitute the x-intercepts and the coordinates of the given point for the correct letters in y 5 a(x 2 p)(x 2 q).

2 5 a(3 1 1)(3 2 4)

Substitute 3 for x and 2 for y.

2 5 24a

Simplify coefficient of a.

1 5a 2} 2

Solve for a.

1 (x 1 1)(x 2 4). c A quadratic function for the parabola is y 5 2} 2

4.10 Write Quadratic Functions and Models

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EXAMPLE 3

Write a quadratic function in standard form

Write a quadratic function in standard form for the parabola that passes through the points (21, 23), (0, 24), and (2, 6). Solution

STEP 1

Substitute the coordinates of each point into y 5 ax2 1 bx 1 c to obtain

the system of three linear equations shown below. 23 5 a(21)2 1 b(21) 1 c

Substitute 21 for x and 23 for y.

23 5 a 2 b 1 c

Equation 1

24 5 a(0)2 1 b(0) 1 c

Substitute 0 for x and 24 for y.

24 5 c

Equation 2

6 5 a(2)2 1 b(2) 1 c

Substitute 2 for x and 6 for y.

6 5 4a 1 2b 1 c

Equation 3

STEP 2 Rewrite the system of three equations in Step 1 as a system of two

REVIEW SYSTEMS OF EQUATIONS

equations by substituting 24 for c in Equations 1 and 3.

For help with solving systems of linear equations in three variables, see p. 178.

a 2 b 1 c 5 23

Equation 1

a 2 b 2 4 5 23

Substitute 24 for c.

a2b51

Revised Equation 1

4a 1 2b 1 c 5 6

Equation 3

4a 1 2b 2 4 5 6

Substitute 24 for c.

4a 1 2b 5 10

Revised Equation 3

STEP 3 Solve the system consisting of revised Equations 1 and 3. Use the elimination method. a2b51 4a 1 2b 5 10

32

2a 2 2b 5 2 4a 1 2b 5 10 6a 5 12 a52

So 2 2 b 5 1, which means b 5 1. The solution is a 5 2, b 5 1, and c 5 24. c A quadratic function for the parabola is y 5 2x2 1 x 2 4.

✓

GUIDED PRACTICE

for Examples 1, 2, and 3

Write a quadratic function whose graph has the given characteristics. 1. vertex: (4, 25)

passes through: (2, 21)

2. vertex: (23, 1)

passes through: (0, 28)

3. x-intercepts: 22, 5

passes through: (6, 2)

Write a quadratic function in standard form for the parabola that passes through the given points. 4. (21, 5), (0, 21), (2, 11)

310

n2pe-0410.indd 310

5. (22, 21), (0, 3), (4, 1)

6. (21, 0), (1, 22), (2, 215)

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:49:35 AM

QUADRATIC REGRESSION In Chapter 2, you used a graphing calculator to perform

linear regression on a data set in order to find a linear model for the data. A graphing calculator can also be used to perform quadratic regression. The model given by quadratic regression is called the best-fitting quadratic model.

EXAMPLE 4

Solve a multi-step problem

PUMPKIN TOSSING A pumpkin tossing contest is held each

year in Morton, Illinois, where people compete to see whose catapult will send pumpkins the farthest. One catapult launches pumpkins from 25 feet above the ground at a speed of 125 feet per second. The table shows the horizontal distances (in feet) the pumpkins travel when launched at different angles. Use a graphing calculator to find the best-fitting quadratic model for the data. Angle (degrees)

20

30

40

50

60

70

Distance (feet)

372

462

509

501

437

323

Solution

STEP 1

Enter the data into two lists of a graphing calculator.

STEP 2 Make a scatter plot of the data. Note that the points show a parabolic trend.

L2 L1 30 462 40 509 50 501 60 437 70 323 L2(6)=323

STEP 3 Use the quadratic regression

STEP 4 Check how well the model

feature to find the bestfitting quadratic model for the data.

fits the data by graphing the model and the data in the same viewing window.

QuadReg y=ax2+bx+c a=-.2614285714 b=22.59142857 c=23.02857143

c The best-fitting quadratic model is y 5 20.261x2 1 22.6x 1 23.0.

✓

GUIDED PRACTICE

for Example 4

7. PUMPKIN TOSSING In Example 4, at what angle does the pumpkin travel the

farthest? Explain how you found your answer.

4.10 Write Quadratic Functions and Models

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4.10

EXERCISES

HOMEWORK KEY

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

★

5 STANDARDIZED TEST PRACTICE Exs. 2, 15, 16, 43, 44, and 51 5 MULTIPLE REPRESENTATIONS Ex. 50

SKILL PRACTICE 1. VOCABULARY Copy and complete: When you perform quadratic regression

on a set of data, the quadratic model obtained is called the ? . 2. ★ WRITING Describe how to write an equation of a parabola if you know

three points on the parabola that are not the vertex or x-intercepts. EXAMPLE 1

WRITING IN VERTEX FORM Write a quadratic function in vertex form for the

on p. 309 for Exs. 3–15

parabola shown. 3.

4.

y

vertex (22, 1)

(5, 6)

5.

y

y

1

1

2

x

(1, 21)

x

1

(21, 21) vertex (21, 23)

vertex (3, 2)

1

x

1

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

6. vertex: (24, 1)

7. vertex: (1, 6)

point: (22, 5)

8. vertex: (5, 24)

point: (21, 2)

9. vertex: (23, 3)

point: (1, 20)

10. vertex: (5, 0)

point: (1, 21)

11. vertex: (24, 22)

point: (2, 227)

12. vertex: (2, 1)

point: (0, 30)

13. vertex: (21, 24)

point: (4, 22)

14. vertex: (3, 5)

point: (2, 21)

point: (7, 23)

15. ★ MULTIPLE CHOICE The vertex of a parabola is (5, 23) and another point on

the parabola is (1, 5). Which point is also on the parabola? A (0, 3) EXAMPLE 2

B (21, 9)

C (21, 15)

D (7, 7)

16. ★ MULTIPLE CHOICE The x-intercepts of a parabola are 4 and 7 and another

point on the parabola is (2, 220). Which point is also on the parabola?

on p. 309 for Exs. 16–26

A (1, 21)

B (8, 24)

C (5, 240)

D (5, 4)

WRITING IN INTERCEPT FORM Write a quadratic function in intercept form for

the parabola shown. 17.

18.

y

y

(0, 6)

19. 23

1

y

3 x

1

(23, 3) 1

22

26

1 1

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n2pe-0410.indd 312

24 21

x

(1, 24)

3 x

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:49:36 AM

WRITING IN INTERCEPT FORM Write a quadratic function in intercept form

whose graph has the given x-intercepts and passes through the given point. 20. x-intercepts: 2, 5

21. x-intercepts: 23, 0

point: (4, 22)

22. x-intercepts: 21, 4

point: (2, 10)

point: (2, 4)

23. x-intercepts: 3, 7

24. x-intercepts: 25, 21

25. x-intercepts: 26, 3

point: (6, 29)

point: (27, 224)

point: (0, 29)

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

26. x-intercepts: 4, 23; point: (5, 25)

27. vertex: (2, 3); point: (1, 5)

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

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

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

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

23 5 29a

5 5 2a

1 3

5 2

1 3

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

5 2

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

EXAMPLE 3

WRITING IN STANDARD FORM Write a quadratic function in standard form for

on p. 310 for Exs. 28–39

the parabola shown. 28.

29.

y

y

(2, 6)

6

(2, 21)

2

(23, 4)

(4, 23)

23

30.

y

x

4

x

2 2

(26, 22)

(0, 22)

x

(24, 22) (24, 26)

(1, 26)

WRITING IN STANDARD FORM Write a quadratic function in standard form for the 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 has

the given characteristics. 40. passes through:

(20.5, 21), (2, 8), (11, 25)

41. x-intercepts: 211, 3

passes through: (1, 2192)

42. vertex: (4.5, 7.25)

passes through: (7, 23)

43. ★ OPEN-ENDED MATH Draw a parabola that passes through (22, 3). Write a

function for the parabola in standard form, intercept form, and vertex form. 44. ★ SHORT RESPONSE Suppose you are given a set of data pairs (x, y). Describe

how you can use ratios to determine whether the data can be modeled by a quadratic function of the form y 5 ax2. 45. CHALLENGE Find a function of the form y 5 ax2 1 bx 1 c whose graph passes

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

4.10 Write Quadratic Functions and Models

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PROBLEM SOLVING EXAMPLES 1 and 3

46. ANTENNA DISH Three points on the parabola formed by

y

the cross section of an antenna dish are (0, 4), (2, 3.25), and (5, 3.0625). Write a quadratic function that models the cross section.

on pp. 309–310 for Exs. 46–47

1

GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

x

1

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. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

48. MULTI-STEP PROBLEM The bar graph shows the

Yearly Time on the Internet

average number of hours per person per year spent on the Internet in the United States for the years 1997–2001.

on p. 311 for Exs. 48–50

150

Hours per person

EXAMPLE 4

a. Use a graphing calculator to create a scatter plot. b. Use the quadratic regression feature of the

calculator to find the best-fitting quadratic model for the data. c. Use your model from part (b) to predict the

134 106

100

50

0

average number of hours a person will spend on the Internet in 2010.

82 54 34

1997

1998

1999

2000

2001

49. RUNNING The table shows how wind affects a runner’s performance

in the 200 meter dash. Positive wind speeds correspond to tailwinds, and negative wind speeds correspond to headwinds. The change t in finishing time is the difference beween the runner’s time when the wind speed is s and the runner’s time when there is no wind. Wind speed (m/sec), s Change in finishing time (sec), t

26

24

22

0

2

4

6

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 with color TVs (millions)

21

47

63

78

90

94

101

a. Drawing a Graph Make a scatter plot of the data. Draw the parabola that

you think best fits the data. b. Writing a Function Estimate the coordinates of three points on the

parabola. 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 data

for the years listed in the original table above. Compare the numbers of households given by your function with the numbers in the original table.

314

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5 WORKED-OUT SOLUTIONS

Chapter 4 Quadratic on p. WS1Functions and Factoring

★

5 STANDARDIZED TEST PRACTICE

5 MULTIPLE REPRESENTATIONS

10/17/05 10:49:38 AM

51. ★ MULTIPLE CHOICE The Garabit Viaduct in France has a parabolic arch as

part 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. Which point 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 which

a circle can be divided using n chords. For example, the diagram shows that R 5 4 when n 5 2. Copy and complete the table. Then write a quadratic model giving R as a function of n. n

0

1

2

3

4

5

6

R

?

?

4

?

?

?

?

2 1

3 4

MIXED REVIEW PREVIEW

Evaluate the expression for the given value of the variable. (p. 10)

Prepare for Lesson 5.1 in Exs. 53–58.

53. x 2 2 3 when x 5 5

54. 3a5 2 10 when a 5 21

55. x4 when x 5 22

56. 4u3 2 15 when u 5 3

57. v 2 1 3v 2 5 when v 5 5

58. 2y 3 1 2y 1 5 when y 5 2

Solve the system of linear equations. (p. 160) 59. 4x 1 5y 5 18

2x 1 2y 5 15 62. 3x 1 y 5 10

2x 1 2y 5 20

60. 3x 1 7y 5 1

4x 1 5y 5 23 63. 4x 1 5y 5 2

23x 1 2y 5 33

61. 3x 1 4y 5 21

2x 1 6y 5 231 64. 2x 1 3y 5 21

10x 1 7y 5 21

QUIZ for Lessons 4.8–4.10 Use the quadratic formula to solve the equation. (p. 292) 1. x 2 2 4x 1 5 5 0

2. 2x2 2 8x 1 1 5 0

3. 3x 2 1 5x 1 4 5 0

5. y > 2x2 1 2x

6. y ≥ 2x 2 1 2x 1 3

8. 12 ≤ x 2 2 7x

9. 2x2 1 2 > 2 5x

Graph the inequality. (p. 300) 4. y < 23x2

Solve the inequality. (p. 300) 7. 0 ≥ x2 1 5

Write a quadratic function whose graph has the given characteristics. (p. 309) 10. vertex: (5, 7)

passes through: (3, 11)

11. x-intercepts: 23, 5

passes through: (7, 240)

12. passes through:

(21, 2), (4, 223), (2, 27)

13. SPORTS A person throws a baseball into the air with an initial vertical

velocity of 30 feet per second and then lets the ball hit the ground. The ball is released 5 feet above the ground. How long is the ball in the air? (p. 292)

EXTRA PRACTICE for Lesson 4.10, p. 1013

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QUIZ Functions at classzone.com 4.10 ONLINE Write Quadratic and Models

315

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MIXED REVIEW of Problem Solving

STATE TEST PRACTICE

classzone.com

Lessons 4.6–4.10 1. MULTI-STEP PROBLEM You are playing a lawn

version of tic-tac-toe in which you toss bean bags onto a large board. One of your tosses can be modeled by y 5 20.12x2 1 1.2x 1 2 where x is the bean bag’s horizontal position (in feet) and y is the corresponding height (in feet). a. Write the given function in vertex form. b. Graph the function. c. What is the bean bag’s maximum height? 2. MULTI-STEP PROBLEM A music store sells

about 50 of a new model of drum per month at a price of $120 each. For each $5 decrease in price, about 4 more drums per month are sold.

4. OPEN-ENDED Name three different complex

numbers with an absolute value of 25. Then plot your answers in the same complex plane. 5. GRIDDED ANSWER What is the product of

5 2 9i and its complex conjugate? 6. SHORT RESPONSE The diagram shows a design

for a hanging glass lamp. Write a quadratic function that models the parabolic cross section of the lamp. Explain how you can verify that your model is correct. Y

X

7. EXTENDED RESPONSE You throw a ball to your a. Write a function that models the store’s

revenue from sales of the new drum model. b. Write an inequality you can use to find the

prices that result in revenues over $6500. c. Solve the inequality from part (b)

algebraically and graphically. 3. EXTENDED RESPONSE The table shows the

average price of a VCR from 1998 through 2003. Years since 1998, t

Price (dollars), p

0

133

1

102

2

81

3

71

4

61

5

61

a. Use a graphing calculator to find the

best-fitting quadratic model for the data.

friend. The ball leaves your hand 5 feet above the ground and has an initial vertical velocity of 50 feet per second. Your friend catches the ball when it falls to a height of 3 feet. a. Write a function that gives the ball’s height h

(in feet) t seconds after you throw it. b. How long is the ball in the air? c. Describe three methods you could use

to find the maximum height of the ball. Then find the maximum height using each method. 8. SHORT RESPONSE You are

IN

designing notepaper with a solid stripe along the paper’s top and left sides as shown. The stripes IN will take up one third of the area of the paper. The paper measures 5 inches by 8 inches. What will the width x of the stripes be? X Explain why you must reject one of the solutions.

X

b. Graph the model and the data together. c. Do you think this model will give a good

estimate of the price of a VCR in 2010? Explain.

316

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9. GRIDDED ANSWER What is the discriminant

of the equation 3x2 1 5x 2 2 5 0?

Chapter 4 Quadratic Functions and Factoring

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4 Big Idea 1

CHAPTER SUMMARY

Algebra classzone.com Electronic Function Library

BIG IDEAS

For Your Notebook

Graphing and Writing Quadratic Functions in Several Forms You can graph or write a quadratic function in standard form, vertex form, or intercept form. Form Standard form

Equation

Information about quadratic function b • The x-coordinate of the vertex is 2} .

y 5 ax2 1 bx 1 c

2a

b • The axis of symmetry is x 5 2} . 2a

Vertex form

y 5 a(x 2 h) 2 1 k

• The vertex is (h, k). • The axis of symmetry is x 5 h.

Intercept form

• The x-intercepts are p and q.

y 5 a(x 2 p)(x 2 q)

p1q 2

• The axis of the symmetry is x 5 }.

Big Idea 2

Solving Quadratic Equations Using a Variety of Methods There are several different methods you can use to solve a quadratic equation. Equation contains: Binomial without x-term Factorable trinomial Unfactorable trinomial

Big Idea 3

Example

Method

5x2 2 45 5 0

Isolate the x2-term. Then take square roots of each side.

x2 2 5x 1 6 5 0

Factor the trinomial. Then use the zero product property.

x2 2 8x 1 35 5 0

Complete the square, or use the quadratic formula.

Performing Operations with Square Roots and Complex Numbers You can use the following properties to simplify expressions involving square roots or complex numbers. Square roots Complex numbers

}

Î

}

a b

Ï

}

a If a > 0 and b > 0, then Ï ab 5 Ïa p Ïb and } 5 } }. }

}

Ïb

}

• The imaginary unit i is defined as i 5 Ï21 , so that i 2 5 21. }

}

}

2

• If r is a positive real number, then Ï2r 5 i Ïr and (i Ïr ) 5 2r. • (a 1 bi) 1 (c 1 di) 5 (a 1 c) 1 (b 1 d)i • (a 1 bi) 2 (c 1 di) 5 (a 2 c) 1 (b 2 d)i }

• a 1 bi 5 Ïa2 1 b2

Chapter Summary

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4

CHAPTER REVIEW

classzone.com • Multi-Language Glossary • Vocabulary practice

REVIEW KEY VOCABULARY • quadratic function, p. 236

• standard form of a quadratic equation, p. 253

• imaginary number, p. 276

• root of an equation, p. 253

• complex conjugates, p. 278

• parabola, p. 236

• zero of a function, p. 254

• complex plane, p. 278

• vertex, p. 236

• square root, p. 266

• axis of symmetry, p. 236

• radical, radicand, p. 266

• absolute value of a complex number, p. 279

• minimum, maximum value,

• rationalizing the denominator,

• completing the square, p. 284

• standard form of a quadratic function, p. 236

p. 238

p. 267

• pure imaginary number, p. 276

• quadratic formula, p. 292

• vertex form, p. 245

• conjugates, p. 267

• discriminant, p. 294

• intercept form, p. 246

• imaginary unit i, p. 275

• monomial, binomial, trinomial,

• complex number, p. 276

• quadratic inequality in two variables, p. 300

• standard form of a complex number, p. 276

• quadratic inequality in one variable, p. 302

p. 252

• quadratic equation, p. 253

• best-fitting quadratic model, p. 311

VOCABULARY EXERCISES 1. WRITING Given a quadratic function in standard form, explain how to determine

whether the function has a maximum value or a minimum value. 2. Copy and complete: A(n) ? is a complex number a 1 bi where a 5 0 and b Þ 0. 3. Copy and complete: A function of the form y 5 a(x 2 h)2 1 k is written in ? . 4. Give an example of a quadratic equation that has a negative discriminant.

REVIEW EXAMPLES AND EXERCISES Use the review examples and exercises below to check your understanding of the concepts you have learned in each lesson of Chapter 4.

4.1

Graph Quadratic Functions in Standard Form

pp. 236–243

EXAMPLE Graph y 5 2x 2 2 4x 2 5.

1

Because a < 0, the parabola opens down. Find and plot the vertex (22, 21). Draw the axis of symmetry x 5 22. Plot the y-intercept at (0, 25), and plot its reflection (24, 25) in the axis of symmetry. Plot two other points: (21, 22) and its reflection (23, 22) in the axis of symmetry. Draw a parabola through the plotted points.

EXAMPLE 3 on p. 238 for Exs. 5–7

318

n2pe-0480.indd 318

(22, 21)

y 1x

EXERCISES Graph the function. Label the vertex and axis of symmetry. 5. y 5 x2 1 2x 2 3

6. y 5 23x2 1 12x 2 7

7. f(x) 5 2x 2 2 2x 2 6

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:50:52 AM

classzone.com Chapter Review Practice

4.2

Graph Quadratic Functions in Vertex or Intercept Form pp. 245–251 EXAMPLE Graph y 5 (x 2 4)(x 1 2). Identify the x-intercepts. The quadratic function is in intercept form y 5 a(x 2 p)(x 2 q) where a 5 1, p 5 4, and q 5 22. Plot the x-intercepts at (4, 0) and (22, 0).

y

(22, 0)

2

(4, 0) 6

x

Find the coordinates of the vertex. p1q 2

4 1 (22) 2

x5}5}51 y 5 (1 2 4)(1 1 2) 5 29

(1, 29)

Plot the vertex at (1, 29). Draw a parabola through the plotted points as shown.

EXERCISES EXAMPLES 1, 3, and 4 on pp. 245–247 for Exs. 8–14

Graph the function. Label the vertex and axis of symmetry. 8. y 5 (x 2 1)(x 1 5)

9. g(x) 5 (x 1 3)(x 2 2)

11. y 5 (x 2 2)2 1 3

12. f(x) 5 (x 1 6)2 1 8

10. y 5 23(x 1 1)(x 2 6) 13. y 5 22(x 1 8)2 2 3

14. BIOLOGY A flea’s jump can be modeled by the function y 5 20.073x(x 2 33)

where x is the horizontal distance (in centimeters) and y is the corresponding height (in centimeters). How far did the flea jump? What was the flea’s maximum height?

4.3

Solve x 2 1 bx 1 c 5 0 by Factoring

pp. 252–258

EXAMPLE Solve x 2 2 13x 2 48 5 0. Use factoring to solve for x. x2 2 13x 2 48 5 0

Write original equation.

(x 2 16)(x 1 3) 5 0 x 2 16 5 0

Factor.

or x 1 3 5 0

x 5 16 or

x 5 23

Zero product property Solve for x.

EXERCISES EXAMPLE 3

Solve the equation.

on p. 254 for Exs. 15–21

15. x 2 1 5x 5 0

16. z2 5 63z

17. s 2 2 6s 2 27 5 0

18. k 2 1 12k 2 45 5 0

19. x 2 1 18x 5 281

20. n2 1 5n 5 24

21. URBAN PLANNING A city wants to double the area of a rectangular

playground that is 72 feet by 48 feet by adding the same distance x to the length and the width. Write and solve an equation to find the value of x.

Chapter Review

319

4

CHAPTER REVIEW 4.4

Solve ax 2 1 bx 1 c 5 0 by Factoring

pp. 259–264

EXAMPLE Solve 230x 2 1 9x 1 12 5 0. 230x2 1 9x 1 12 5 0

Write original equation.

2

10x 2 3x 2 4 5 0

Divide each side by 23.

(5x 2 4)(2x 1 1) 5 0 5x 2 4 5 0

or

4 x5}

or

5

EXAMPLE 5 on p. 261 for Exs. 22–24

4.5

Factor.

2x 1 1 5 0

Zero product property

1 x 5 2} 2

Solve for x.

EXERCISES Solve the equation. 22. 16 5 38r 2 12r 2

23. 3x 2 2 24x 2 48 5 0

24. 20a2 2 13a 2 21 5 0

Solve Quadratic Equations by Finding Square Roots

pp. 266–271

EXAMPLE Solve 4(x 2 7)2 5 80. 4(x 2 7)2 5 80

Write original equation.

2

(x 2 7) 5 20

Divide each side by 4. }

x 2 7 5 6Ï20

Take square roots of each side. }

x 5 7 6 2Ï 5

Add 7 to each side and simplify.

EXERCISES EXAMPLES 3 and 4 on pp. 267–268 for Exs. 25–28

Solve the equation. 25. 3x 2 5 108

26. 5y 2 1 4 5 14

27. 3(p 1 1)2 5 81

28. GEOGRAPHY The total surface area of Earth is 510,000,000 square

kilometers. Use the formula S 5 4πr 2, which gives the surface area of a sphere with radius r, to find the radius of Earth.

4.6

Perform Operations with Complex Numbers

pp. 275–282

EXAMPLE Write (6 2 4i)(1 2 3i) as a complex number in standard form. (6 2 4i)(1 2 3i) 5 6 2 18i 2 4i 1 12i 2

320

n2pe-0480.indd 320

Multiply using FOIL.

5 6 2 22i 1 12(21)

Simplify and use i 2 5 21.

5 26 2 22i

Write in standard form.

Chapter 4 Quadratic Functions and Factoring

10/17/05 10:50:55 AM

classzone.com Chapter Review Practice

EXERCISES EXAMPLES 2, 4, and 5 on pp. 276–278 for Exs. 29–34

4.7

Write the expression as a complex number in standard form. 29. 29i(2 2 i)

30. (5 1 i)(4 2 2i)

31. (2 2 5i)(2 1 5i)

32. (8 2 6i) 1 (7 1 4i)

33. (2 2 3i) 2 (6 2 5i)

34. } 23 1 6i

4i

Complete the Square

pp. 284–291

EXAMPLE Solve x 2 2 8x 1 13 5 0 by completing the square. x2 2 8x 1 13 5 0

Write original equation.

x2 2 8x 5 213

Write left side in the form x2 1 bx.

x2 2 8x 1 16 5 213 1 16 (x 2 4)2 5 3

Write left side as a binomial squared. }

x 2 4 5 6Ï3

Take square roots of each side. }

x 5 4 6Ï 3 EXAMPLES 3 and 4 on pp. 285–286 for Exs. 35–37

4.8

2

1 228 2

Add } 5 (24) 2 5 16 to each side.

Solve for x.

EXERCISES Solve the equation by completing the square. 35. x 2 2 6x 2 15 5 0

36. 3x 2 2 12x 1 1 5 0

37. x2 1 3x 2 1 5 0

Use the Quadratic Formula and the Discriminant

pp. 292–299

EXAMPLE Solve 3x 2 1 6x 5 22. 3x2 1 6x 5 22

Write original equation.

3x2 1 6x 1 2 5 0

Write in standard form. }}

2

Ï x5 } 26 6 6 2 4(3)(2) 2(3)

Use a 5 3, b 5 6, and c 5 2 in quadratic formula.

}

23 6 Ï3 x5} 3

Simplify.

EXERCISES EXAMPLES 1, 2, 3, and 5 on pp. 292–295 for Exs. 38–41

Use the quadratic formula to solve the equation. 38. x2 1 4x 2 3 5 0

39. 9x2 5 26x 2 1

40. 6x 2 2 8x 5 23

41. VOLLEYBALL A person spikes a volleyball over a net when the ball is 9 feet

above the ground. The volleyball has an initial vertical velocity of 240 feet per second. The volleyball is allowed to fall to the ground. How long is the ball in the air after it is spiked? Chapter Review

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4

CHAPTER REVIEW 4.9

Graph and Solve Quadratic Inequalities

pp. 300–307

EXAMPLE Solve 22x 2 1 2x 1 5 ≤ 0. The solution consists of the x-values for which the graph of y 5 22x2 1 2x 1 5 lies on or below the x-axis. Find the graph’s x-intercepts by letting y 5 0 and using the quadratic formula to solve for x. }}

x5

22 6 Ï22 2 4(22)(5) } 2(22) }

}

5

22 6 Ï44 24

}

y

5

21 6 Ï 11 22

}

21.16 1

x ø 21.16 or x ø 2.16

2.16

Sketch a parabola that opens down and has 21.16 and 2.16 as x-intercepts. The solution of the inequality is approximately x ≤ 21.16 or x ≥ 2.16. EXAMPLE 5 on p. 302 for Exs. 42–44

4.10

x

1

EXERCISES Solve the inequality by graphing. 42. 2x2 2 11x 1 5 < 0

43. 2x2 1 4x 1 3 ≥ 0

1 x 2 1 3x 2 6 > 0 44. } 2

Write Quadratic Functions and Models

pp. 309–315

EXAMPLE Write a quadratic function for the parabola shown. Because you are given the x-intercepts p 5 23 and q 5 2, use the intercept form y 5 a(x 2 p)(x 2 q) 5 a(x 1 3)(x 2 2).

y

23

1

Use the other given point, (1, 22), to find a. 22 5 a(1 1 3)(1 2 2)

Substitute 1 for x and 22 for y.

22 5 24a

Simplify coefficient of a.

1 2

}5a

2 x

1

(1, 22)

Solve for a.

1 (x 1 3)(x 2 2). c A quadratic function for the parabola is y 5 } 2

EXERCISES EXAMPLES 1 and 2 on p. 309 for Exs. 45–48

Write a quadratic function whose graph has the given characteristics. 45. x-intercepts: 23, 2

passes through: (3, 12)

46. passes through:

(5, 2), (0, 2), (8, 26)

47. vertex: (2, 7)

passes through: (4, 2)

48. SOCCER The parabolic path of a soccer ball that is kicked from the ground

passes through the point (0, 0) and has vertex (12, 7) where the coordinates are in feet. Write a quadratic function that models the soccer ball’s path.

322

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Chapter 4 Quadratic Functions and Factoring

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4

CHAPTER TEST Graph the function. Label the vertex and axis of symmetry. 1. y 5 x2 2 8x 2 20

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

3. f(x) 5 2(x 1 4)(x 2 2)

4. x 2 2 11x 1 30

5. z2 1 2z 2 15

6. n2 2 64

7. 2s 2 1 7s 2 15

8. 9x2 1 30x 1 25

9. 6t 2 1 23t 1 20

Factor the expression.

Solve the equation. 10. x 2 2 3x 2 40 5 0

11. r 2 2 13r 1 42 5 0

12. 2w 2 1 13w 2 7 5 0

13. 10y 2 1 11y 2 6 5 0

14. 2(m 2 7)2 5 16

15. (x 1 2)2 2 12 5 36

Write the expression as a complex number in standard form. 16. (3 1 4i) 2 (2 2 5i)

31i 18. } 2 2 3i

17. (2 2 7i)(1 1 2i)

Solve the equation by completing the square. 19. x 2 1 4x 2 14 5 0

20. x 2 2 10x 2 7 5 0

21. 4x 2 1 8x 1 3 5 0

Use the quadratic formula to solve the equation. 22. 3x 2 1 10x 2 5 5 0

23. 2x2 2 x 1 6 5 0

24. 5x 2 1 2x 1 5 5 0

26. y < x2 1 4x 2 21

27. y > 2x2 1 5x 1 50

Graph the inequality. 25. y ≥ x2 2 8

Write a quadratic function whose graph has the given characteristics. 28. x-intercepts: 27, 23

29. vertex: (23, 22)

passes through: (21, 12)

30. passes through:

passes through: (1, 210)

(4, 8), (7, 24), (8, 0)

31. ASPECT RATIO The aspect ratio of a widescreen TV

is the ratio of the screen’s width to its height, or 16 : 9. What are the width and the height of a 32 inch widescreen TV? (Hint: Use the Pythagorean theorem and the fact that TV sizes such as 32 inches refer to the length of the screen’s diagonal.)

IN

X

X

32. WOOD STRENGTH The data show how the strength of Douglas fir wood is

related to the percent moisture in the wood. The strength value for wood with 2% moisture is defined to be 1. All other strength values are relative to this value. (For example, wood with 4% moisture is 97.9% as strong as wood with 2% moisture.) Use the quadratic regression feature of a graphing calculator to find the best-fitting quadratic model for the data. Percent moisture, m

2

4

6

8

10

Strength, s

1

0.979

0.850

0.774

0.714

Percent moisture, m

12

14

16

18

20

0.643

0.589

0.535

0.494

0.458

Strength, s

Chapter Test

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4

★ Standardized TEST PREPARATION CONTEXT-BASED MULTIPLE CHOICE QUESTIONS Some of the information you need to solve a context-based multiple choice question may appear in a table, a diagram, or a graph.

PROBLEM 1 The area of the shaded region is 56 square meters. What is the height of the trapezoid? A 3 meters

B 4 meters

C 6 meters

D 7.5 meters

x12 8m

2x 3x 1 2 13 m

Plan INTERPRET THE DIAGRAM You know the area of the shaded region. Use the

diagram to find the area of the rectangle, and write an expression for the area of the trapezoid. The difference of these two areas is the area of the shaded region.

STEP 1 Find expressions for the two areas.

Solution Area of rectangle:

Area of trapezoid: 1 (b 1 b )h A5} 1 2

A 5 lw

2

5 13(8)

1 [(x 1 2) 1 (3x 1 2)](2x) 5}

5 104 m 2

1 (2x)(4x 1 4) 5}

2 2

5 x(4x 1 4) 5 4x2 1 4x

STEP 2 Write an equation for the area of the shaded region and solve by factoring.

Area of shaded region 5 Area of rectangle 2 Area of trapezoid 56 5 104 2 (4x2 1 4x) 2

248 5 24x 2 4x 4x2 1 4x 2 48 5 0 2

x 1 x 2 12 5 0 (x 2 3)(x 1 4) 5 0 x 5 3 or x 5 24

STEP 3 Find the possible heights.

STEP 4 Reject the negative height.

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Substitute. Subtract 104 from each side. Write in standard form. Divide each side by 4. Factor. Zero product property

The height of the trapezoid is given by the expression 2x. Therefore, the possible heights are 2(3) 5 6 meters and 2(24) 5 28 meters. Because height cannot be negative, the height of the trapezoid is 6 meters. c The correct answer is C. A B C D

Chapter 4 Quadratic Functions and Factoring

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PROBLEM 2 The height h (in feet) of a lobbed tennis ball after t seconds is shown by the graph. What is the initial vertical velocity of the tennis ball? A 3 feet/second

B 16 feet/second

C 37.5 feet/second

D 47 feet/second

h

5

(3, 0)

(0, 3) 1

2

3

t

Plan INTERPRET THE GRAPH The graph is a parabola passing through the points (0, 3) and (3, 0). In order to find the initial vertical velocity of the tennis ball, you must write an equation of the parabola.

STEP 1 Write the model for an object that is launched.

STEP 2 Use the initial height in your model.

STEP 3 Find the initial vertical velocity by substituting a point on the parabola.

Solution Because the tennis ball is launched, the parabola has an equation of the form h 5 216t 2 1 v0t 1 h0 where v 0 is the initial vertical velocity and h0 is the initial height of the tennis ball. The graph passes through (0, 3), so the initial height of the tennis ball is 3 feet. When you substitute 3 for h0 in the model, you obtain h 5 216t 2 1 v 0t 1 3. Use the fact that the graph of h 5 216t 2 1 v 0t 1 3 passes through (3, 0) to find the initial vertical velocity v 0. 0 5 216(3)2 1 v 0 (3) 1 3

Substitute 0 for h and 3 for t.

0 5 2141 1 3v0

Simplify.

47 5 v 0

Solve for v0.

The initial vertical velocity is 47 feet per second. c The correct answer is D. A B C D

PRACTICE In Exercises 1 and 2, use the graph in Problem 2. 1. What is the maximum height of the tennis ball to the nearest tenth of a foot?

A 36.3 feet

B 36.8 feet

C 37.5 feet

D 38.0 feet

2. What does the x-coordinate of the vertex of the graph represent?

A The maximum height of the tennis ball B The number of seconds the ball is in the air C The number of seconds it takes the ball to reach its maximum height D The initial height of the tennis ball

Standardized Test Preparation

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4

★ Standardized TEST PRACTICE

CONTEXT-BASED MULTIPLE CHOICE In Exercises 1 and 2, use the parabola below.

5. The graph of which inequality is shown?

y

y

1

vertex

x

1 1 x

1

1. Which statement is not true about the

parabola? A The x-intercepts are 22 and 4.

A y ≥ 2x2 2 x 1 6

B The y-intercept is 22.

C y > 2x2 1 2x 2 12 D y ≥ 22x2 2 2x 1 12

C The maximum value is 3.

B y ≥ x2 1 x 2 6

In Exercises 6 and 7, use the information below.

D The axis of symmetry is x 5 1.

The graph shows the height h (in feet) after t seconds of a horseshoe tossed during a game of horseshoes. The initial vertical velocity of the horseshoe is 30 feet per second.

2. What is an equation of the parabola?

A y 5 (x 2 2)(x 1 4) 1 (x 1 2)(x 2 4) B y 5 2} 3

h

C y 5 2(x 1 2)(x 2 4) D y 5 23(x 1 2)(x 2 4)

4

(0, 2)

3. You are using glass tiles to make a picture

frame for a square photograph with sides 10 inches long. You want the frame to form a uniform border around the photograph. You have enough tiles to cover 300 square inches. What is the largest possible frame width x? A 3.6 inches

x

1

t

2

6. To the nearest tenth of a second, how long is

the horseshoe in the air? A 0.1 seconds

B 1.9 seconds

C 2.1 seconds

D 3.9 seconds

7. To the nearest tenth of a foot, what is the

B 5 inches

maximum height of the horseshoe? x

C 7.3 inches

10 in. x 10 in.

D 15 inches

x

A 15.9 feet

B 16.1 feet

C 29.2 feet

D 32.2 feet

8. The diagram shows a circle inscribed in 4. At a flea market held each weekend, an artist

sells handmade earrings. The table below shows the average number of pairs of earrings sold for several prices. Given the pattern in the table, how much should the artist charge to maximize revenue? Price

$15

$14

$13

$12

Pairs sold

50

60

70

80

A $5

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B $7.50

C $10

a square. The area of the shaded region is 21.5 square inches. To the nearest tenth of an inch, how long is a side of the square? A 4.6 inches B 8.7 inches

r

C 9.7 inches D 10.0 inches

D $15

Chapter 4 Quadratic Functions and Factoring

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STATE TEST PRACTICE

classzone.com

GRIDDED ANSWER

SHORT RESPONSE 17. Solve 3x 2 1 6x 1 14 5 0 by completing the

9. What is the value of k in the equation 2

6x 2 11x 2 10 5 (3x 1 2)(2x 2 k)?

square. Explain all of your steps.

10. What is the real part of the standard form of

18. The surface area y of a cube is given by the

function y 5 6x2 where x is an edge length. Graph the function. Compare this graph to the graph of y 5 x2.

the expression (5 1 i)(10 2 i)? 11. For what value of c is x 2 2 7x 1 c a perfect

square trinomial?

19. The height h (in feet) of an object after it is

12. What is the maximum value of the function

launched is given by the function

2

y 5 23(x 2 2) 1 6?

h 5 216t 2 1 v 0t 1 h0

13. What is the greatest zero of the function

y 5 x2 2 25x 1 66?

where v0 is the initial vertical velocity, h0 is the initial height of the object, and t is the time (in seconds) after the object is launched. Explain how this function is related to the general function for a dropped object.

14. What is the absolute value of 25 1 12i? 15. What is the x-coordinate of the vertex of the

parabola that passes through the points (0, 222), (2, 26), and (5, 212)?

20. At what two points do the graphs of 1 x 2 2 3x 1 4 y 5 2x2 2 5x 2 12 and y 5 } 2

16. What is the minimum value of the function

f(x) 5 4x2 1 24x 1 39?

intersect? Explain your reasoning.

EXTENDED RESPONSE 21. The table below shows the New York Yankees’ payroll (in millions of dollars)

from 1989 through 2004. Years since 1989

0

1

2

3

4

5

6

7

Payroll

21

21

28

36

41

45

47

52

Years since 1989

8

9

10

11

12

13

14

15

Payroll

59

63

88

93

112

126

153

184

a. Make a scatter plot of the data. b. Draw the parabola that you think best fits the data. c. Estimate the coordinates of three points on the parabola from part (b).

Then use a system of equations to write a quadratic model for the data. d. Use your model from part (c) to make a table of data for the years listed

in the original table. Compare the numbers given by your model with the actual data. Assess the accuracy of your model. 22. A volleyball is hit upward by a player in a game. The height h (in feet) of the

volleyball after t seconds is given by the function h 5 216t 2 1 30t 1 6. a. What is the maximum height of the volleyball? Explain your reasoning. b. After how many seconds does the volleyball reach its maximum height? c. After how many seconds does the volleyball hit the ground?

Standardized Test Practice

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