1

Math Analysis Semester Exam Review 2013-2014 LINEAR & QUADRATIC FUNCTIONS (Chapter 1 & Section 2-4) In problems 1 & 2, graph the equation of the line and identify the x & y intercepts. 1. 2x – y = 4 2. 3x + 2y = 6 In problems 3 & 4, write the equation of the line graphed. 3. 4. 5. The table shows the pay (p) Sue receives for working (h) hours. If the relation is linear, find the equation for her pay in terms of the hours she works.

Hours (h) 8 20 35 Pay (p) $100 $250 $437.50

Name: ___________________________ Period: _______ Date: ______________ 6. The table shows the number of hours studying for an exam (h) and the score on the exam (s). If the relation is linear, find the equation of the exam score in terms of hours studying. Number of hours studying (h) 4 6 9 Exam Score Percentage (s) 67 79 97 In problems 7 – 9, find the x & y intercepts for each quadratic equation. 7. ! = 2!! − 3! + 1 8. ! = 6!! + 5! − 6 9. ! =  −4!! + 8! + 5

  In problems 10 − 12, find the x value of the maxima or minima for each quadratic equation. 10. ! =  − ! − 1 ! + 5 11. ! =  −2!! − 3! + 4 12. ! =  !! + 2! + 5 13. Suppose you are using 200 feet of fencing to make a rectangular dog run area. One side of the run is a building and the other 3 sides will use the fencing. What length and width give the maximum area? 14. A rectangle has a perimeter of 100 ft. If the rectangle’s width is x, express the length and area in terms of x. What is the maximum area of the rectangle?

2

POLYNOMIAL EQUATIONS (Chapter 2) In problems 15 − 17, find the quotient and remainder. 15. 2!! − ! + 1 is divided by (! + 5). 16. (!! − 3!! + 2! − 8) is divided by ! − 2 . 17. (!! − 7!! + 14) is divided by (x − 5). 18. Give the real zeros of f(x) = . 19. Find all the roots (zeros) of 20. Find all the roots (zeros) round to the nearest tenth: 3 2( ) 6 46 88 42f x x x x= − + − 21. Graph: ( )2 4 ( 2)y x x= − + 22. Graph: 2 3( 3) ( 1)y x x= + − 23. Write an equation for the graph shown below.

24. Write an equation for the graph shown below. In problems 25 − 27, find the remainder. 25. ( 5 22 2 1x x− + ) is divided by (x + 1). 26. ( 4 213 32x x− + ) is divided by (x − 3). 27. 10 42 2 1x x− + is divided by (x + 1). 28. If a cubic polynomial has zeros of 6 and 2 + 3i, then what is the other zero? 29. If a quartic polynomial has zeros of 3+   2    !"#    2− 3! then what are the other two zeros. 30. Can a polynomial of degree 5 have zeros of ±3, (2 + i), and 3! . Explain your answer. In problems 31 – 33, find the complex zeros of the functions. 31. ! ! =  3!! + 9!! + 4! + 12 32. ! ! =  !! − 16 33. ! ! = !! + !! − 20

2 2( 4)( 9)x x+ −

4 3 2( ) 3 11 18 44 24f x x x x x= − + − +

(0, 12)

−4 −3 −2 −1 1 2 3 4 5

5

10

15

20

x

y

(2, 0)

(0, -4)

−4 −3 −2 −1 1 2 3 4 5

5

x

y

(-2, 0)

3

In problems 34 – 36 solve for x. 34. !! + 3!  + 1 = 0 35. 2!! + 7! = 4! 36. 3!! − 4!  –  7  –  0 RATIONAL FUNCTIONS (Chapter 19 & Supplement) In problems 37 – 44 perform the indicated operation and simplify the expression. 37. !

!!!!!!!!

∙ !!!

!!!!!!!"

38. !!

!!!!!!!!!

∙ !!!!"!!!!!!!"

39. !

!!!!!!"!!!!"

÷ !!!!!!!"

40. !!!

!!!!!!!!÷ !!!!"

!!!!!"!!!"

41. !

!!!+   !!!!

!!!!!!!

42. !

!!!!!!!"+ !!

!!!!

43. !!!!!!!!!

− !!!!!!!!!!!!

44. !!

!!!!!"#− !

!!!!"

In problems 45 – 48, give the vertical asymptotes and the horizontal asymptotes.

45. 4 31

xy

x!=!

46. y =

3x(x + 2)(x ! 5)(x + 7)(x ! 2)(x ! 5)

47. y =

(x + 2)(x !1)(x ! 2)(x + 3)

48. ! =   !!!

!!!!!

49. Which equation could be the graph shown?

A. 12

xyx+=−

B. 21

xyx−=+

C. ( 2)

( 1)( 3)xy

x x−=

− + D.

( 2)( 1)( 3)

xyx x

+=+ −

4

50. Which equation could be the graph shown?

A. 2

2

21xy

x−=

− B.

2

2

21

xyx

=−

C. 2

21xy

x−=

− D.

221xy

x−=−

51. Which equation could be the graph shown?

A. 3

( 1)( 4)xy

x x−=

+ − B.

31

xyx−=+

C. 13

xyx+=−

D. 3

( 1)( 4)xy

x x+=

− +

Functions (Chapter 4) In problems 52 – 57, let

! ! =3− !!! − 4 ,              ! ! =

! − 3! + 2  ,

ℎ ! =  ! + 2,                    ! ! =  6! − 7

Find each of the following: 52. !(ℎ −4 ) 53. !(ℎ 2 ) 54. !(ℎ ! ) 55. (! ∙ ℎ)(!) 56. (h – k)(x) 57. (f /g)(x) 58. A salesperson earns a 2% bonus on weekly sales over $4000. The following functions represent the above information: g(x) = 0.02x h(x) = x – 4000 Which composition h(g(x)) or g(h(x)) represents the weekly bonus? Write the function. What is the bonus if the sales for the week is $6000? 59. If a club’s yearly income is represented by I(x) and expenses are represented by E(x), write an expression for the balance at the end of the year. 60. If 2 2( ) and ( ) ( 2) 5f x x g x x= = + − then f(x) is shifted horizontally _____ units to the _________ and is shifted vertically ________ units ___________ to get g(x).

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

−8

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

7

8

x

y

5

61. If ( ) and ( ) 3f x x g x x= = − − then ( )f x is flipped over the _______ axis and is shifted horizontally ________ units to get ( ).g x 62. The graph of y = f (x) is shown. Graph y = f (x + 3) – 4 In problems 63 – 68, determine if the function is odd, even or neither. 63. ! ! =  6!! − 5! 64. ! ! = ! 65. ! ! = 5!! 66. ! ! = 3 ! 67. ! ! = !! 68. ! ! = 3!! − 2!! In problems 69 – 71, graph each piecewise function.

69. 2 4, 0

( )2, 0

x for xf x

x for x− ≥⎧

= ⎨ − <⎩

70. 2 4, 1

( ) 3, 12 , 1

x for xf x for x

x for x

− <⎧⎪= =⎨⎪− >⎩

71. f (x) =2x !1, for x " 0x2 + 3, for x < 0

#$%&

SEQUENCES – (Chapter 13) In problems 72 -75, decide whether the sequence is arithmetic or geometric. Write an explicit formula for the nth term of the sequence. 72. -7.8, -3.8, 0.2, … 73. 16, 24, 36, . . . 74. 2, 8, 14, 20, … 75. -6, -12, -24, … 76. What are the 2nd, 3rd and 4th terms of the recursive sequence represented by !! = 5 !! = −3!!!! + 7 77. What are the 3rd , 4th and 5th terms of the recursive sequence represented by !! = 4 !! = 5!!!! − 2

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

6

In problems 78 & 79, write a recursive definition for the sequence. 78. -6, -12, -24, … 79. 2, 8, 14, 20, … MATRICES – (Chapter 14) In problems 80 − 86, use the following matrices. If the problem is not possible, write Not Possible or NP.

1 23 2 6 1 3 1 1

5 14 5 3 2 2 0 4

0 3A B C D

! !" #! !" # " # " # $ %= = = = !$ % $ % $ % $ %! ! !& ' & ' & ' $ %& '

80. AB 81. BC 82. CB 83. CD 84. 2A − B 85. A-1 86. B-1

87. What are the dimensions of CD? 88. What is the multiplicative identity for a 3 X 3 matrix? 89. What is the additive identity for a 2 X 2 matrix? 90. What is the matrix equation for the following systems of equations? Solve the matrix equation. 4x – y = 10 −2x + 5y = 4 91. Solve the system of equations by using a matrix equation. 3x − 4y + z = 15 −2x − 6y + 3z = 4 2x + 2y – 2z = −1

In problems 92 & 93, write the communication matrix for the communication network given. 92.

93. COMBINATORICS –(Chapter 15) In problems 94 − 98, use the information below: In a survey at a shopping mall, 120 people were surveyed to determine their preferences for stores shopped. 62 - Dillard’s 71 - JCP 67 - Sears 37 - Dillard’s and JCP 32 - Dillard’s and Sears 40 - JCP and Sears 12 shopped at all three stores 94. Construct a Venn diagram of the information Determine the number of people who shopped in 95. Dillard’s only. 96. exactly two of the three stores. 97. Sears and JCP, but not Dillard’s. 98. Sears or JCP, but not Dillard’s.

A

B C

A B C D

7

In problems 99 & 100, find the value of the expression. 99. 7 P3 100. 7 C3 In problems 101 & 102, write the first 4 terms of the expansion. 101. ! − 3 ! 102. ! + 2! ! 103. You are packing to go on vacation and need to choose 4 shirts from 12 shirts in your closet. In how many ways can you choose the 4 shirts? Assume the order of the shirts does not matter. 104. How many different license plates can you make if you must use 3 letters followed by 4 digits? 105. If you have 12 students in a club, how many different 4-person committees can you form? 106. Suppose you are given a 10-question test. Each question has 4 choices. How many different ways can you answer the test? 107. How many 4 digit numbers contain no 2’s or 5’s? How many 4 digit numbers contain at least one 2 or 5? 108. How many numbers from 4000 to 6999 contain at least one 7? PROBABILITY – (Chapter 16) 109. Suppose a blue and red die are rolled. What is the probability that the sum of the numbers showing on the dice is 7 or 11? 110. Suppose a blue and red die are rolled. What is the probability that the sum is less than 12?

111. A coin is tossed 10 times. What the probability of exactly 2 “heads”? Use the binomial formula: n C k pk(1 − p)n − k 112. Consider the set of families with exactly 5 children. If P(child is a girl) = ½ , find the probability that one of these families picked at random has 3 girls. Use the binomial formula: n C k pk(1 − p)n − k 113. Three cards are drawn at random from a standard deck of cards without replacement. What is the probability that the three cards are an ace or a king? 114. Thirteen cards are draw at random from a standard deck of cards without replacement. What is the probability that the 13 cards contain exactly 4 kings and exactly 2 queens? 115. If P(X) = .7, P(Y) = .4 and X and Y are independent events, what is the P(X and Y)? In problems 116-121, use the tree diagram below.

116. Find the missing probabilities. 117. Find P(Y’|X). 118. Find P(Y|X’). 119. Find P(X and Y) 120. Find P(X’ and Y) 121. Find P(Y)

.4 ? ?

X X’ Y

.8 ? .3 ?

Y Y’ Y Y’

8

In problems 122 - 125, use the table below. Each student in a class of 25 takes one math class and one science class. Alg 2 Trig Calculus Total Chemistry 2 4 5 11 Biology 8 3 3 14 Total 10 7 8 25 122. Find the probability that a randomly chosen student studies Calculus. 123. Find the probability that a randomly chosen student studies Calculus given that the student studies Biology. 124. Find the probability that a randomly chosen student studies Trig given that the student studies Chemistry. 125. Find the probability that a randomly chosen student studies Alg 2 given that the student studies Biology.