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Study Guide: ReviewCHAPTER

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Organizer Vocabularyquadratic model 376

quadratic regression 376

real part 351root of an equation 334

........ 324

........ 336

. 318

. 318

.. 333

imaginary number 350

imaginary part 351

imaginary unit 350maximum value 326

absolute value of a complexnumber 382

axis of symmetry 323binomial 336

completing the sqi

complex conjugate

complex number .

complex plane ..discriminant ....

Objective: Help studentsorganize and review key conceptsand skills presented in Chapter 5.

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EXERCISESEXAMPLES•.~ Lesson Tutorial Videosv4iIJI1 CD-ROM

• Using the graph of f(x) = X2 as a guide,describe the transformations, and then graphg(x) = tx2 + 3.

Graph each function by using a table.@f(X) = _X2 - 2x 0f(X) = tx2 + 3x - 4

Using the graph off(x) = X2 as a guide, describe thetransformations, and then graph each function.

S. g(x) = 4(x - 2)2 9. g(x) = -2(x + 1)2

10. g(x) = tx2 - 3 11. g(x) = -(x + 2)2 +6

Answers1. imaginary number; complex

number

2. zero of a function

3. vertex of a parabola"

4. discriminant

5. minimum value

g(x) = iX2 + 3 isfvertically compressedby a factor of i andtranslated 3 units up.

Use the description to write each quadratic functionin vertex form.

12. f(x) = X2 is reflected across the x-axis andtranslated 3 units down to create g.

13. f(x) = X2 is vertically stretched by a factor of 2 andtranslated 4 units right to create g.

14. f(x) = X2 is vertically compressed by a factor of iand translated 1 unit left to create g.

• Use the description to write a quadraticfunction in vertex form. The functionf(x) = X2 is translated 1 unit right to create g.

translation 1 unit right: h = 1g(x) = a(x - h)2 + k -t g(x) = (x - 1)2

6.

Chapter 5 Quadratic Functions

9. 9 is f reflected across the x-axis,vertically stretched by a factor of 2,and translated 1 unit left.

8. 9 is f vertically stretched by a factor of4 and translated 2 units right.

5-2 Properties of Quadratic Functions in Standard Form (pp. 323-330)

EXAMPLE EXERCISESFor f(x) = _X2 + 2x + 3, (a) determinewhether the graph opens upward ordownward, (b) find the axis of symmetry,(c) find the vertex, (d) find the y-Intercept,and (e) graph the function.

a. Because a < 0, the e.parabola opensdownward.

b. axis of symmetry:x = _l!.... = __ 2_ = 1

2a 2(-1)c. f(I) = _12 + 2(1) + 3 = 4

The vertex is (1, 4).d. Because c = 3, the j-intercept is 3.

For each function, (a) determine whether the graphopens upward or downward, (b) find the axis ofsymmetry, (c) fmd the vertex, (d) find the j-intercept,and (e) graph the function.15. f(x) = X2 - 4x + 3 1? g(x) = X2 + 2x + 3

17. hex) = X2 - 3x 18; j(x) = ±x2 - 2x + 4

Find the minimum or maximum value of eachfunction.?19. f(x) = X2 + 2x + 6

@f(x) = X2 - 5x + 1

~ f(x) = _X2 - 4x + 8'-" .

@g(X) = 6x - 2X2

ffi: g(x) = _2X2 - 8x + 10

24~ g(x) = 3x2 + 7

-3 Solving Quadratic Equations by Graphing and Factoring (pp. 333-340)

EXERCISESFind the roots of X2 + x = 30 by factoring.

X2 + x - 30 = 0 Rewrite in standard form.

(x - 5)(x + 6) = 0 Factor.

x - 5 = 0 or x + 6 = 0 Zero Product Property.x = 5or x = -6 Solve each equation.

Write a quadratic function with zeros8 and-8.

x = 8or x = -8 Write zeros as solutions.

x - 8 = 0 or x + 8 = 0 Set equations equal to O.

(x - 8)(x + 8) = 0 Converse Zero ProductProperty

f(x) = X2 - 64 Replace 0 with f(x).

-4 Completing the Square (pp. 342-349)

Find the roots of each equation by factoring.h5. X2 - 7x - 8 = 0 . (26. X2 - 5x + 6 = 0

27: X2 = 144 28. X2 - 2Ix = 0

29. 4x2 - 16x + 16 = 0

31. X2 + 14x = 32•.. 30:' 2X2 + 8x + 6 = 0

32. 9x2 + 6x + 1 = 0

Write a quadratic function in standard form for eachgiven set of zeros.33. 2 and -3 34. 1 and-l

35. 4 and 5

37. -5 and-5

36. -2 and-3

38)9 and 0

XAM PLE EXERCISESSolve X2 - 8x = 12 by completing the square.

X2 - 8x + • = 12 + • Set up equation.

X2 - 8x + 16 = 12 + 16 Add (~r.(x - 4)2 = 28 Factor.

x - 4 = ±Y2B Takesquare roots.

X = 4 ± 2V7 Solve for x.

~plve each equation by completing the square.39;1 X2 - 16x + 48 = 0 40. X2 + 20x + 84 = 0

l!.V X2 - 6x = 16 42 X2 - 14x = 13

Write each function in vertex form, and identifyits vertex.43. f(x) = X2 - 4x + 9 44. g(x) = X2 + 2x - 7

Answers19. min.: 520. max.: 4.521. min.: -5.2522. max.: 18

23. max.: 1224. min.: 725. x = - 1 or x = 8

26.x=2orx=3

27. x = - 12 or x = 12

~:::::_QOLX=21

29.x= 230. x = - 1 or x = -3

31. x = 2 or x = - 16----~-

Study Guide: Review 393

32.x = _1.3

33. Possible answer: f(x) = X2 + X - 634. Possible answer: f(x) = X2 - 135. Possible answer: f(x) = X2 - 9x + 2036. Possible answer: f(x) = X2 + 5x + 637. Possible answer: f(x) = X2 + lOx + 2538. Possible answer: f(x) = X2 - 9x

39.x = 4 or x = 1240. x = -14 or x = -6

41. x = -2 or x = 8

42.x.= 7 ±-J6243. f(x) = (x - 2)2 + 5; (2, 5)44.g(x) = (x + 1?-8; (-1, -8)

Answers10. 9 is f vertically compressed by a

factor of t and translated 3 unitsdown.

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21\I x

0 1Y,1

11. 9 is f reflected across the x-axisand translated 2 units left and6 units up.

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12. Possible answer: g(x) = _X2 - 313. Possible answer:

g(x) = 2(x - 4)2

14. Possible answer:

g(x) = ~ (x + 1?15. opens upward; x = 2; (2, -1); 3

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1\ Ii x0 !

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16. opens upward; x = -1;(-1,2);3

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17. opens upward; x = 1.5;(1.5, -2.25); 0

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0\ 1/;-

18. opens upward; x = 2; (2, 2); .

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Study Guide: Review

Answers45.x = ±9i46.x = ±5i~=-3 ±i"'-_A = -6 ± 3i49.x = 7 ± i..f2650.x = 11 ± 2iV351. -5i-452.3 -iVS53. 3 +2..[41

54.5 ± 2iV3

552. + i-JT1

. 2 - 2

56. -; ± i v:57' 2. + . V1s

. 2 -' 2

58. 1 distinct real solution59. 2 real solutions60.2 nonreal complex solutions61. 2 real solutions62. 2 nonreal complex solutions63. 2 nonreal complex solutions64. -" 11

\ ,'-\ 6 ,- ....•--\ 4- -t. ,I-- ~-"

I

X

0 !

I

66.x s -3 or x ~ 1

67. -4 < x <-168. x < 1 or x > 569. -3 s; x s; 370. -V3 < x < V371.-2 < x < 1.- - 3

5-5 Complex Numbers and Roots (pp. 350-355)

EXAMPLE EXERCISES• Solve X2 - 22x+ 133 = O. Solve each equation.

45. X2 = -81

47. X2 + 6x + 10 = 0

49. X2 - 14x + 75 = 0

X2 - 22x + = -133 +

X2 - 22x + 121 = -133 + 121

(x - 1l)2 = -12x - II = ± v' -12

Rewrite.

Factor.

Take squareroots.

Solve.

46. 6x2 + 150 = 0

, 48. X2 + 12x + 45 = 0

50. X2 - 22x + 133 = 0

Find each complex conjugate.(51'.5i-4 .G 3+iVS

5-6 The Quadratic Formula (pp. 356-363)

EXERCISES

x=1l±2iV3

EXAMPLES• Find the zeros off(x) = 3x2 - 5x + 3 by using

the Quadratic Formula.-b ± Vb2 - 4acx = ---=-=-"':'2~a-~~ Quadratic

Formula

Find the zeros of each function by using theQuadratic'Formula.53. f(x) = X2 - 3x - 8

54. hex) = (x - 5)2 + 12

55. f(x) = 2X2 - lOx + 18

56. g(x) = X2 + 3x + 3

57. hex) = X2 - 5x + 10

Find the type and number of solutions for eache uation.58. 2X2 - 16x + 32 = 0 59 X2 - 6x = -5

60. x2 + 3x + 8 = 0

@2+5X=-12

61. X2 - 246x = -144

63. 3x2 - 5x + 3 = 0

-( -5) ± V (_5)2 - 4(3)(3)x = 2(3) Substitute.

_ 5±v"=ll_~± ,m- 6 - 6 I 6 Simplify.

• Find the type and number of solutions forX2 + 9x + 20 = O.

b2 - 4ac = 92 - 4(1)(20)= 81 - 80 = 1

There are two distinct real roots because thediscriminant is positive.

5-7 Solving Quadratic Inequalities (pp.366-373)

EXAMPLE EXERCISES

• Solve X2 - 4x - 9 ;:: 3 by using algebra.

Write and solve the related equation.X2 _ 4x - 12 = 0 Write in standard form.

(x + 2)(x - 6) = 0 Factor.

x = -2 or x = 6 Solve.

The critical values are -2 and 6, These valuesdivide the number line into three intervals:x ~ -2, -2 ~ x ~ 6, and x;:: 6.

Testing an x-value in each interval gives thesolution of x ~ -2 or x ;::6.

Graph each inequality.

64. y > X2 + 3x + 4 65. Y s 2X2 - X - 5

Solve each inequality by ~tables or graphs.

@X2+2X-4::::-1 LV-x2-5X>4 •

Solve each inequality by using algebra.

8. X2 + 6x < 5 69. 3x2 - 25 ~ 2

70. X2 - 3 < 0 B 3x2 + 4x - 3 ~ 1

394 Chapter 5 Quadratic Functions

---~------

Chapter 5

5-8 Curve Fitting with Quadratic Models (pp.374-381)

EXAMPLE EXERCISESa quadratic model for the wattage of

\,~.escent bulbs Fgiven the comparableincandescent bulb wattage 1.Use the modelto estimate the wattage of a fluorescent bulbthat produces the same amount of light as a120-watt incandescent bulb.

Incandescent (watts)

Fluorescent (watts)

nter the data into two lists in a graphingcalculator. Use the quadratic regression feature.

QuadReg":j=ax~+bx+ca=.0016259823b=.0481326935c=6.480005564R~=.9898561939

he model is F(I) "'> 0.0016[2 + 0.0481I + 6.48.36-watt fluorescent bulb produces about

the same amount of light as a 120-wattincandescent bulb.

Write a quadratic function that fits each set of points.72. (-1,8), (0,6), and (1, 2)

73. (0,0), (1, -1), and (2, -6)

100

Construction For Exercises 74-77, use the table ofcopper wire gauges.

28

Common U.S. Copper Wire Gauges

Resistance perGauge Diameter (in.) 1000 ft (ohms)

24 0.0201 25.67

22 0.0254 16.14

20 0.0320 10.15

18 0.0403 6.385

74. Find a quadratic regression equation to model thediameter given the wire gauge.

75. Use your model to predict the diameter for a12-gauge copper wire.

76. Find a quadratic regression equation to model theresistance given the wire gauge.

77. Use your model to predict the resistance for a26-gauge copper wire.

Operations with Complex Numbers (pp.382-389)

erform each indicated operation, and write thesult in the form a + hi.

1-2+4il

V (_2)2 + 42 = V4+16 = V20 = 2 V5

(3 + 2i)(4 - 5i)

12 - 15i + 8i - 10i2

12 -7i -10(-1) = 22 -7i

-5 + 3i (1 + 2i) = -5 - 7i + 6f1 - 2i 1+ 2i 1- 4i2

-11-7i 11 7.= =----1

1 + 4 5 5

EXERCISESPerform each indicated operation, and write theresult in the form a + hi.

78. 1-3il80. 112- 16il

82. (1 + 5i) + (6 - i)84. (5 - i) - (n - i)

86:(5 - 2i)(6 + 8i)

88. (4 + i)(1 - 5i)

79. 14 - z.]81. 17il

83. (9 + 4i) - (3 + 2i)

85. -5i(3 - 4i)

87. (3 + 2i)(3 - 2i)

(-7 + 4i)(3 + 9i)90. i32

92. 2 + 9i-21

94. 8 - ~i1 + 1

91 _5i21

93. 5 + 2~3 - 41

95. -12 + 26i2 + 4i

Study Guide: Review 395

Answers72. Y = _X2 - 3x + 673. Y = _2x2 + x

74.y:::::0.000188x2 - 0.0112x + 0.182

75. :::::0.074 in.76. y::::: 0.360X2 - 11.9x + 10577.:::::37.8 ohms

78.379.2V580.2081.782.7 + 4i83.6 + 2i84. -685. -20 - 15i

86.46 + 28i87.1388.9- 19i

89. -57 - 51i

90.191. -5i

92. -; + i

93 7 26·'25' + 25'194.2 - 6i

95.4+ 5i