1

Advanced Math Midterm Exam Review Name:___________________ January 2016

Use the following schedule to complete the semester exam review.

You must complete all of the highlighted problems. Any other problems are for extra practice.

Homework will be checked in every day.

Homework answers will be provided at the beginning of each class period for you to check your work from the previous night.

SEMESTER EXAM SCHEDULE: The first exam each day begins promptly at 8:00 a.m.

Tuesday, January 26th 1st & 2nd hour exams [90 min each]

Wednesday, January 27th 4th & 5th hour exams [90 min each]

Thursday, January 28th 6th & 7th hour exams [90 min each]

Friday, January 29th 3rd hour exam [90 min]

Date Assigned Assignment Due Date

Fri., Jan. 15, 2016 Chapter P: Page 2-5 All odds

Tues., Jan. 19, 2016

Tues., Jan. 19, 2016 Chapter 1 Day 1: Page 6-9 #1-55 odds SKIP #37, 45

Wed., Jan. 20, 2016

Wed., Jan. 20, 2016

Chapter 1 Day 2: Page 9-10 #57-65 odds, 69 Chapter 2 Day 1: Page 11-13 #1-37 odds SKIP #5, 11

Thurs., Jan. 21, 2016

Thurs., Jan. 21, 2016

Chapter 2 Day 2: Page 13-19 #39, 41, 45-53 odds, 57, 59, 65, 67, 71, 73, 79, 81, 82, 84, 86 Chapter 4: Page 19 #1 , 2

Fri., Jan. 22, 2016

Fri., Jan. 22, 2016 Chapter 3: Page 20-22 All odds

Mon., Jan. 25, 2016

Mon., Jan. 25, 2016 Review & Notecard

**NO LATE WORK WILL BE ACCEPTED**

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Chapter P Simplify each expression. 1. −2𝑥 − 4 [ 7 − 3 (5𝑥 + 3) − 4𝑥 ] 2. 6 − 4(2𝑥 + 3) Evaluate each expression.

3a. Let 𝑥 = −2 and 𝑦 = 3 3b. Let 𝑥 = −1, 𝑦 = −2 and 𝑧 = 3 −2𝑥3 + 4𝑥𝑦 − 3𝑦2 2𝑥 − 4𝑦(6𝑧 + 𝑦3) Evaluate or simplify each exponential expression. When raising a numeric value to a fractional exponent, use radicals to show your work.

4. − 2 6 5. − 2 2 (−3)2 6. 3 − 4 7. (2

3)

− 3

8. 3

𝑥− 3 9.

𝑥− 2

𝑦− 3 10. 25

1

2 11. − 27 2

3

12. 49 − 1

2 13. 3

81−

14

14. (−3𝑥 − 3 𝑦 3)− 3 15. (𝑥 − 12) (𝑥

56)

3

Evaluate or simplify each exponential expression. When raising a numeric value to a fractional exponent, use radicals to show your work.

16. (2𝑎 1

4 𝑏− 1

2)4

(−3𝑎 1

3 𝑏− 1

6)3

17. 𝑎

23 𝑏

− 34

𝑎 43 𝑏

− 74

18. (−3𝑥 − 3 𝑦 3)− 3

(4𝑥 − 2 𝑦 3)− 1 19. (8𝑥

54

𝑥 12

)

2

3

Simplify each radical expression. Assume variables represent real numbers.

20. √48𝑎3𝑏8 21. √−128𝑎3𝑏73

22. 𝑏√8𝑎3𝑏3 + 2 √18𝑎3𝑏5 23. (3 + √6)(7 − 2√6)

24. (2 + √3)2

25. 5

√8

26. 4

2 + √7 27.

6

4 − √3

4

Perform the indicated operation and express the result in standard form. 28. (6𝑥 − 1)(3𝑥 + 4) 29. (3𝑥3 + 4𝑥 − 6) − (−2𝑥3 + 5𝑥 − 10) 30. (2𝑥 − 3)(2𝑥 + 3) 31. (2𝑥 + 1)(𝑥2 − 3𝑥 − 4) Factor the following completely. 32. 𝑥2 + 7𝑥 − 18 33. 12𝑥3𝑦4 + 24𝑥2𝑦2 − 6𝑥2𝑦 34. 4𝑥2 − 25 35. 3𝑥2 − 4𝑥 − 15

5

Factor the following completely. 36. 𝑥4 + 2𝑥2 − 3 37. 27𝑥3 − 64 38. 64𝑥2𝑦 − 36𝑦3 39. 𝑥4 − 81 Simplify each rational expression.

40. 3𝑥 − 15

2𝑥2− 50 ∙

2𝑥2+ 16𝑥 + 30

6𝑥 + 9 41.

𝑥2− 81

𝑥2− 16 ÷

𝑥2− 𝑥 − 20

𝑥2+ 5𝑥 − 36

42. 6𝑥2− 19𝑥 + 10

2𝑥2+ 3𝑥 − 20 43.

𝑥

𝑥 − 5 +

7𝑥

𝑥 + 3

6

Chapter 1 Solve the following equations Check your solutions.

1. 6[3𝑦 − 2(𝑦 − 1)] − 2 + 7𝑦 = 0 2. 6(𝑔 + 1.5) = 12𝑔 3. 3

4𝑥 +

1

2=

2

3

4. 4

9𝑥 −

1

2=

2

3 5. |𝑥 − 6| = 9 6. |

𝑥−4

2| = 8 7. 3|𝑥 + 2| − 16 = 2

Perform the indicated operation and write the answer in simplest form. 8. 𝑖 2 9. 𝑖 3 10. 𝑖 4 11. 𝑖 402

12. 𝑖 207 13. √− 36 14. √− 98 15. (−2 + 8𝑖) − (6 − 3𝑖)

16. ( 3 − 2𝑖) + ( 2 + 5𝑖) 17. (2 − 𝑖)2 18. 3𝑖 (6 − 4𝑖) 19. ( 2 − 3𝑖)( 4 − 5𝑖)

20. 2 − 4𝑖

1 + 4𝑖 21.

3 − 𝑖

2 − 5𝑖 22.

2

𝑖 23.

3

8𝑖

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Given the graph of the quadratic equation, estimate the solutions [zeros/roots]. 24. 25. 26. Estimate the solutions [zeros/roots] for the following equations. Then use the zeros/roots to graph. 27. 𝑦 = −(𝑥 + 2)(𝑥 + 8) 28. 𝑦 = 𝑥2 + 3𝑥 + 2 29. 𝑦 = 5𝑥2 − 35𝑥 + 50 Solve the following quadratics by factoring. 30. 4𝑥2 − 2𝑥 = 0 31. 2𝑥2 − 8 = 0 32. 𝑥2 − 144 = 0 33. 𝑥2 + 𝑥 = 30 34. 𝑥2 + 16 = 8𝑥 35. 12𝑥2 − 25𝑥 + 12 = 0 36. 36𝑥2 − 12𝑥 = −1 37. 2𝑥2 + 4𝑥 = 6 38. 12𝑥2 − 22𝑥 − 14 = 0

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Solve the following equations that are in quadratic form. 39. 𝑥4 − 29𝑥2 + 100 = 0 40. 𝑥4 − 17𝑥2 + 16 = 0 Solve the following quadratics using square roots. 41. (𝑥 + 2)2 = −25 42. (𝑥 − 6)2 = 81 43. (𝑥 − 5)2 = − 32 Solve the following quadratics by completing the square. 44. 𝑥2 − 8𝑥 = −14 45. 𝑥2 + 6𝑥 − 10 = 0 46. 𝑥2 − 3𝑥 − 5 = 0 47. 𝑥2 + 12𝑥 − 4 = 0 48. 𝑥2 + 16𝑥 + 68 = 0 49. 𝑥2 − 2𝑥 + 8 = 0 Find the value of the discriminant. Describe the type of roots for the following quadratics (how many and what type). 50. 𝑥2 − 10𝑥 + 16 = 0 51. 𝑥2 + 6𝑥 + 12 = 0 52. 12𝑥2 + 15𝑥 = −7

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Solve the following quadratics using the quadratic formula. 53. 2𝑥2 − 5𝑥 + 1 = 0 54. 𝑥2 = −6𝑥 − 9 55. 3𝑥2 + 6𝑥 + 7 = 0 Solve each equation. Be sure to check your solution(s).

56. 𝑥1

2 = 10 57. 𝑥5

6 = 32 58. 3𝑥2

3 − 22 = 53

59. 7𝑥3

5 = 56 60. 2𝑥3

4 + 6 = 60 61. 𝑥4 − 9𝑥2 + 14 = 0

62. 4𝑥4

3 − 54 = 270 63. (𝑥 − 2)1

3 − 5 = −2 64. 3 + (𝑥 + 4)1

3 = 5

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State the excluded value for each equation, then solve the equation. Make sure to check your answers! If the excluded value is one of your solutions, you cannot use it!

65. 2

2𝑥+

1

16=

4

4𝑥 66.

12

𝑥+

−3

𝑥 + 5=

−1

𝑥

67. 3𝑥 + 2

𝑥 − 2 =

4𝑥 − 1

𝑥 − 2 68.

9

𝑥 − 3 =

3𝑥

𝑥 − 3− 2

69. 5

𝑥 − 1 + 2 =

3𝑥 + 1

𝑥 − 1 70.

4

𝑥 − 1 =

5

𝑥 − 1 −

7

𝑥 + 7

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Chapter 2 Find the distance between the set of points. 1. (6, 4) and (−8, 11) 2. (40, 32) and (36, 20) Find the midpoint between the set of points. 3. (6, 4) and (−8, 11) 4. (40, 32) and (36, 20) 5. (12, −2) and (7, 2) 6. (5, 3) and (−6, 9) State whether the equation defines 𝒚 as a function of 𝒙. 7. 2𝑥 + 4𝑦 = 8 8. 𝑦 = 𝑥2 9. 𝑦 = 3 ± 𝑥 10. 𝑦 = |𝑥 + 3| State whether the set of ordered pairs defines 𝒚 as a function of 𝒙. 11. {(5, 2), (3, 7), (4, −3), (5, 6)} 12. {(6, 3), (−7, 3), (8, 3), (9, 3)} Evaluate the function.

13. Given 𝑓(𝑥) = √𝑥2 + 5 a. 𝑓(−2) b. 𝑓(0) c. 𝑓(3) 14. Given 𝑓(𝑥) = 3𝑥2 + 1 a. 𝑓(−2) b. 𝑓(0) c. 𝑓(3)

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Write each interval as an inequality and graph on a number line. 15. [−2, 5) 16. (3, ∞) 17. (−∞, 6] 18. [−1, 1] Write each inequality in interval notation. 19. −4 < 𝑥 ≤ 1 20. 𝑥 < 3 ∪ 𝑥 ≥ 6 21. −8 < 𝑥 < −2 22. −3 ≤ 𝑥 ≤ 2 23. 𝑥 < 6 24. 𝑥 > −5 Determine the domain of the function represented by the given equation. Write your answer in INTERVAL NOTATION. 25. 𝑓(𝑥) = 2𝑥 + 5 26. 𝑓(𝑥) = 𝑥2 − 5𝑥

27. 𝑓(𝑥) = √6 + 𝑥 28. 𝑓(𝑥) = √𝑥 − 3

29. 𝑓(𝑥) =4

𝑥−6 30. 𝑓(𝑥) =

7

𝑥+8

Find the real value or values of 𝒂 in the domain of 𝒇 for which 𝒇(𝒂) equals the given number. 31. 𝑓(𝑥) = 3𝑥 − 2; 𝑓(𝑎) = 8 32. 𝑓(𝑥) = 2 − 5𝑥; 𝑓(𝑎) = 7 33. 𝑓(𝑥) = 𝑥2 + 2𝑥 − 2; 𝑓(𝑎) = 1 34. 𝑓(𝑥) = |𝑥 + 2|; 𝑓(𝑎) = 6

13

Find the zeros, or 𝒙-intercepts, of 𝒇(𝒙). 35. 𝑓(𝑥) = 3𝑥 + 2 36. 𝑓(𝑥) = 6 + 2𝑥 37. 𝑓(𝑥) = 𝑥2 − 5𝑥 − 24 38. 𝑓(𝑥) = 𝑥2 + 4𝑥 − 21 Find the slope of the line through the given points. 39. (8, −2) and (−4, 10) 40. (−2, −7) and (4, −9) Graph 𝒚 as a function of 𝒙 by finding the slope and 𝒚-intercept. 41. 𝑦 = −3𝑥 + 5 42. 𝑦 = 4𝑥 − 2 slope:_________ 𝑦-intercept:_________ slope:_________ 𝑦-intercept:_________ 43. 2𝑥 − 3𝑦 = 12 44. −3𝑥 + 5𝑦 = −15 slope:_________ 𝑦-intercept:_________ slope:_________ 𝑦-intercept:_________

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Find the equation of the indicated line in slope-intercept form. 45. Slope: −17 𝑦-intercept: (0, −5) 46. Slope: 6 𝑦-intercept: (0, 2) 47. Slope: 12 Through (4, −6) 48. Slope: −3 Through (−5, −1) 49. Through (5, −6) and (2, −8) 50. Through (−5, 6) and (−3, −4) Find the equation of the line, in slope-intercept form, that satisfies the given condition.

51. The graph is parallel to the graph of 𝑦 = −3

4𝑥 + 3 and passes through the point whose coordinates are

(−4, 2). 52. The graph is parallel to the graph of 𝑦 = 3𝑥 − 1 and passes through the point whose coordinates are

(−3, −5). 53. The graph is perpendicular to the graph of 3𝑥 − 2𝑦 = 5 and passes through the point whose coordinates

are (−3, 4). 54. The graph is perpendicular to the graph of 𝑦 = −𝑥 + 3 and passes through the point whose coordinates

are (−5, 2).

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Determine the center and radius of the circle with the given conditions. 55. (𝑥 − 3)2 + (𝑦 + 7)2 = 64 56. (𝑥 + 8)2 + 𝑦2 = 81 Center:_________ Radius:_________ Center:_________ Radius:_________ 57. (𝑥 − 8)2 + (𝑦 − 9)2 = 13 58. (𝑥 − 2)2 + (𝑦 − 7)2 = 17 Center:_________ Radius:_________ Center:_________ Radius:_________ Find an equation of a circle that satisfies the given conditions. Write your answer in standard form. 59. Center: (−5, 2); Radius: 7 60. Center: (4, −3); Radius: 12 61. Center: (1, 3); Passing Through: (4, −1) 62. Center: (−2, 5); Passing Through: (1, 7) 63. The circle has a diameter with endpoints (2, 3) and (−4, 11). 64. The circle has a diameter with endpoints (−2, 3) and (4, 3). 65. Use the Vertical Line Test to determine which of the following graphs are functions. a. b. c. d.

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Use the indicated graph to identify the intervals over which the function is increasing, decreasing, or constant. Write your answers in INTERVAL NOTATION. 66. 67.

Increasing:____________________________ Increasing:____________________________ Decreasing:___________________________ Decreasing:___________________________ Constant:_____________________________ Constant:_____________________________ 68. 69.

Increasing:____________________________ Increasing:____________________________ Decreasing:___________________________ Decreasing:___________________________ Constant:_____________________________ Constant:_____________________________

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Use the equation of the quadratic function to determine the following: (a) the vertex, (b) the max or min value of the vertex, (c) if the vertex is a max or min (circle either max or min), and (d) the equation for the axis of symmetry. 70. 𝑦 = (𝑥 − 4)2 + 6 71. 𝑦 = −3(𝑥 + 7)2 + 12

a. vertex:________ a. vertex:________

b. max or min value:_________ b. max or min value:_________

c. max / min (circle one) c. max / min (circle one)

d. axis of symmetry:_________ d. axis of symmetry:_________

Use the method of completing the square to find the standard form of the quadratic function. 72. 𝑦 = 𝑥2 − 14𝑥 + 12 73. 𝑦 = 𝑥2 − 8𝑥 + 12

Use the vertex formula 𝒙 = −𝒃

𝟐𝒂 to find the vertex of the given quadratic. Then use the value for 𝒂 to write the

equation in vertex form. 74. 𝑦 = 5𝑥2 + 20𝑥 − 7 75. 𝑦 = 2𝑥2 + 8𝑥 − 17 76. 𝑦 = 3𝑥2 + 12𝑥 + 8 77. 𝑦 = 4𝑥2 + 32𝑥 − 10

Use the vertex formula 𝒙 = −𝒃

𝟐𝒂 to find the vertex of the given quadratic.

78. 𝑦 = 𝑥2 − 10𝑥 79. 𝑦 = −𝑥2 + 4𝑥 + 1

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Find the maximum or minimum value of the function. State whether it is a maximum or minimum. Find the range of the function and leave your answer for the range in INTERVAL NOTATION. 80. 𝑦 = 𝑥2 + 10𝑥 − 3 81. 𝑦 = −𝑥2 + 6𝑥 + 2

a. max or min value:_________ a. max or min value:_________

b. max / min (circle one) b. max / min (circle one)

c. range: __________________ c. range: __________________

82. Match each equation with its graph. 𝑦 = (𝑥 + 2)2 + 1 _______ 𝑦 = (𝑥 − 2)2 + 1 _______ 𝑦 = −𝑥2 − 3_______ 𝑦 = 𝑥2 − 3 _______

A

B

C

D

83. Let 𝑓(𝑥) = 3𝑥 + 2 and 𝑔(𝑥) = 2𝑥 − 4. Find the following: a. (𝑓 + 𝑔)(𝑥) b. (𝑓𝑔)(𝑥) c. (𝑓 − 𝑔)(−4) 84. Let 𝑓(𝑥) = 3𝑥3 − 4𝑥2 + 3 and 𝑔(𝑥) = 2𝑥2 − 6. Find the following: a. (𝑓 − 𝑔)(𝑥) b. (𝑓 + 𝑔)(𝑥) c. (𝑓 + 𝑔)(7)

19

85. Let 𝑓(𝑥) = 3𝑥2 and 𝑔(𝑥) = 6𝑥 − 10. Find the following: a. 𝑓(𝑔(𝑥)) b. 𝑔(𝑓(𝑥)) c. 𝑔(𝑓(4)) 86. Let 𝑓(𝑥) = 4𝑥 − 3 and 𝑔(𝑥) = 𝑥2 + 7. Find the following: a. 𝑓(𝑔(𝑥)) b. 𝑔(𝑓(𝑥)) c. 𝑓(𝑔(3)) 87. Determine whether or not 𝑓(𝑥) and 𝑔(𝑥) are inverses of each other. Show work!

𝑓(𝑥) = 2𝑥 + 3; 𝑔(𝑥) =𝑥 − 3

2

Chapter 4 Find 𝒇−𝟏(𝒙). State any restrictions on the domain of 𝒇−𝟏(𝒙). 1. 𝑓(𝑥) = 4𝑥 − 8 2. 𝑓(𝑥) = −3𝑥 − 7

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Chapter 3 Use Synthetic Division to divide the first polynomial by the second.

1. 4𝑥3 + 5𝑥2 + 6𝑥 − 9, 𝑥 + 2 2. 8𝑥4 − 12𝑥3 + 10𝑥2 + 5𝑥 + 1, 𝑥 −1

2

3. 𝑥4 − 5𝑥2 − 8𝑥 + 3, 𝑥 − 3 4. 𝑥3 + 64, 𝑥 + 4 Use Synthetic Division and the Remainder Theorem to find 𝑷(𝒄). 5. 𝑃(𝑥) = 2𝑥3 − 𝑥2 + 3𝑥 − 1, 𝑐 = 3 6. 𝑃(𝑥) = 3𝑥4 − 6𝑥2 + 7, 𝑐 = −2 Synthetic Division: Synthetic Division: Remainder Theorem: Remainder Theorem: Use Synthetic Division and the Factor Theorem to determine whether the given binomial is a factor of 𝑷(𝒙). 7. 𝑃(𝑥) = 3𝑥3 + 4𝑥2 − 27𝑥 − 36, 𝑥 + 4 8. 𝑃(𝑥) = 𝑥4 − 25𝑥2 + 144, 𝑥 − 3 Use Synthetic Division to show that 𝐜 is a zero of 𝑷.

9. 𝑃(𝑥) = 2𝑥3 + 5𝑥2 − 4𝑥 − 3, 𝑐 = −1

2 10. 𝑃(𝑥) = 2𝑥4 − 34𝑥3 + 70𝑥2 − 153𝑥 + 45, 𝑐 = 15

Examine the leading term and the degree of the polynomial to determine the far-left and far-right behavior (end behavior) of the graph of the polynomial function.

11. 𝑃(𝑥) = −6𝑥4 + 3𝑥2 − 𝑥 − 2 12. 𝑃(𝑥) = 2

3𝑥3 − 4𝑥2 + 7𝑥 − 1

13. 𝑃(𝑥) = −3𝑥3 + 4𝑥2 + 5𝑥 − 6 14. 𝑃(𝑥) = 𝑥2 − 4𝑥 + 4

21

Find the real zeros of the polynomial function by factoring. Identify the multiplicity of each zero. 15. 𝑃(𝑥) = 𝑥4 − 6𝑥3 + 8𝑥2 16. 𝑃(𝑥) = 𝑥4 − 13𝑥2 + 36

17. 𝑃(𝑥) = −𝑥3 + 2𝑥2 + 15𝑥 18. 𝑃(𝑥) = 𝑥5 − 5𝑥3 + 4𝑥 Use the Intermediate Value Theorem to verify that 𝑷 has a zero between 𝐚 𝐚𝐧𝐝 𝐛. 19. 𝑃(𝑥) = 3𝑥3 + 7𝑥2 + 3𝑥 + 7, 𝑎 = −3, 𝑏 = −2 20. 𝑃(𝑥) = 𝑥4 − 𝑥2 − 𝑥 − 4, 𝑎 = 1.7, 𝑏 = 1.8 Determine the x-intercepts of the graph of 𝑷(𝒙). For each x-intercept, use the Even and Odd Powers of (𝒙 − 𝒄) Theorem to determine whether the graph of 𝑷(𝒙) crosses the x-axis (passes through) or intersects but does not cross the 𝒙-axis (hits and bounces). 21. 𝑃(𝑥) = (2𝑥 + 1)(𝑥 − 2)(𝑥 + 5) 22. 𝑃(𝑥) = −𝑥4 + 4𝑥3 − 3𝑥2 Using x-intercepts and end behavior, sketch the graph of #21 and #22. Do not use a graphing calculator. 23. 24.

22

Use the Rational Zero Theorem to list the possible rational zeros for each polynomial function. Then find ALL the zeros (real and/or imaginary). Don’t forget to check for multiplicities! 25. 𝑃(𝑥) = 𝑥3 − 19𝑥 − 30 26. 𝑃(𝑥) = 2𝑥3 + 𝑥2 − 25𝑥 + 12 27. 𝑃(𝑥) = 𝑥4 − 4𝑥3 + 8𝑥2 − 16𝑥 + 16 28. 𝑃(𝑥) = 𝑥4 − 2𝑥3 − 2𝑥2 + 2𝑥 + 1 Find a polynomial function of lowest degree with integer coefficients that has the given zeros. 29. 5, −2, 4 30. −1, 1, −2, 2 31. 5, 2𝑖, −2𝑖 32. 𝑖, −𝑖, 5𝑖, −5𝑖