Math 30-1 Name: _________________

Review

1. Given the graph of :

Sketch the graph of the given transformation on the same grid

Describe how the transformed graph relates to the graph of

Write the equation of the image of after the transformation

a.

b.

2. Using replacement notation, state the replacement(s) that would be necessary to make the following transformation(s): A horizontal translation of 7 units to the left and a vertical translation of 3 units down

3. The function is transformed to . If the point lies on the graph of ,

which of the following points must lie on the graph of to ?

a. b. c. d.

4. Given a point on the function , which of the following maps the image of the point after a

replacement of ?

a. (

) b. (

) c. d.

5. The function has been transformed into the function defined by the equation

(

) . One of the transformation applied to the function is a horizontal

stretch about the y-axis by a factor of:

a.

b.

c. d.

6. The function has been transformed into the function defined by the equation

. The correct order to perform these transformations to the graph of

is:

a. A horizontal reflection in the y-axis, a vertical stretch by a factor of 3 about the x-axis, a vertical

translation of 7 units down, and a horizontal translation of 4 units to the left.

b. A vertical stretch by a factor of 3 about the x-axis, a horizontal reflection in the y-axis, a vertical

translation of 7 units down, and a horizontal translation of 4 units to the left.

c. A horizontal reflection in the y-axis, a vertical stretch by a factor of

about the x-axis, a vertical

translation of 7 units down, and a horizontal translation of 4 units to the left.

d. A vertical stretch by a factor of

about the x-axis, a horizontal reflection in the y-axis, a vertical

translation of 7 units down, and a horizontal translation of 4 units to the left.

7. The function √ has been transformed into the function ( ) .

The correct equation for is:

a.

√

b. √

c.

√

d. √

8. Algebraically factor and determine the zeros of the following polynomials:

a. b.

9. Algebraically divide the following polynomials. State your answers in the form .

a.

b.

10. Use the Factor Theorem to show that is a factor of the polynomial .

11. Write a binomial factor of the polynomial function with integral coefficients if (

) .

12. Find the integral coefficients and in if the remainder is when divided by

and the remainder is when divided by .

13. If , a second degree factor, with integral coefficients, of the polynomial could be:

a.

b.

c. d.

14. If the possible integral zeros of are , the smallest positive value

is:

e. 1 f. 2 g. 10 h. 100

15. is a polynomial function, which when divided by has a remainder of 4. Therefore, when the

polynomial is divided by , the remainder is:

i. 4 j. 8 k. 2 l. 0

16. Which of the following is a factor of the polynomial ?

m. n. o. p.

17. A potential zero of the polynomial function is:

q.

r.

s. -1 t. 0

18. The graph of a third degree polynomial function, , with a leading coefficient of 1 is shown to the right.

Write the equation of in factored form.

19. A cubic polynomial function with equation has a zero at -2 with multiplicity one, and

a zero at 5 with multiplicity 2. Sketch the graph of the function . Determine the values of .

20. If the graph of a polynomial function has a right arm falling and a left arm rising, which of the following

statements are true?

a. The degree of the polynomial is even and the leading coefficient is positive.

b. The degree of the polynomial is even and the leading coefficient is negative.

c. The degree of the polynomial is odd and the leading coefficient is positive.

d. The degree of the polynomial is odd and the leading coefficient is negative.

21. If a polynomial function has a zero at with even multiplicity, then the graph of the function at :

a. is going straight through the x-axis

b. is tangent to the x-axis

c. changes concavity

d. is non-real

22. The polynomial function has:

a. Four real roots

b. Two real and two non-real roots

c. Two real roots and one non-real root

d. Three real and one non-real root

23. is an integral polynomial function with zeros at At which of the

following points does cross the y-axis?

a.

b.

c.

d.

24. Given the graph of the function √ :

a. Describe the series of transformations required to transform the graph of √ to the graph

of the given function.

b. Using replacement notation, state the replacements

for the variables and .

c. Sketch the graph on the grid provided.

d. State the domain and range of the function.

25. What is the domain and range of the function √ if ?

26. If , the domain and range of the function √

a. { } , { }

b. { } , { }

c. { } , { } d. { } , { }

27. If is a continuous function with range { }, determine the range of √ .

e. { }

f. { }

g. { } h. { }

28. Consider the graph of the function . When sketching the graph of √ there will be an

invariant point at:

i. j. k. l.

29. When , the graph of √ lies _________ the graph of .

m. on n. above o. below p. parallel to

30. is a continuous function with zeros -5, -1, and 4, such that in the intervals and .

Which of the following values is NOT on the graph of √ ?

q. r. s. t.

31. The domain of the function √ is:

u. { } v. { } w. { } x. { }

32. Given the graph of shown below, the domain of the function √ would be:

a. { }

b. { }

c. { }

d. { }

33. Determine the angles in the domain that are coterminal to .

a. State the principal angle.

34. Calculate the arc length of a sector of a circle with diameter 12 if the sector angle is .

35. A circle with centre C, minor arc AB, and diameter measuring is shown. If

find the length of minor arc AB to the nearest whole centimetre.

36. Draw

in standard position.

a. Write an expression that represents all angles in the domain that are coterminal with the given angle.

37. Rewrite

as the same trigonometric function of a positive acute angle.

38. The point √ lies on the terminal arm of a rotation angle in standard position. Determine the value

of

39. Algebraically determine the value of to the nearest degree, where in the interval

.

40. If √

, , determine the value of .

41. Convert to radians. Answer as an exact value.

42. Convert

to degrees. Answer to the nearest tenth of a degree.

43. Is the point (

) on the unit circle? Why or why not?

A

B

C

44. The point (

) is on the unit circle in quadrant 4. Determine the value of .

45. Determine the exact roots of √ for .

46. A beach ball is riding the waves in the ocean. The ball goes up and down with the waves according to the

formula (

) , where is the height, in metres, above sea level, and is the time, in seconds.

a. In the first 10 s, when is the ball at sea level?

b. When does the ball reach its greatest height above sea level? (First time and an expression for every time

after).

c. What is the most the ball goes below sea level?

47. The average number of air conditioners sold in western Canada varies seasonally and depends on the month of

the year. The formula (

) gives the expected sales, , in thousands, according to the

month, , where represents January, is February, etc.

a. In what month are sales of 8300 air conditioners expected?

b. In what month are sales expected to be least?

c. In what month are sales expected to be greatest?

d. State the domain and range.

48. (

) is a point on the terminal arm of an angle intersecting the unit circle. Determine the

coordinates of point .

49. Determine the general solution to √ √ .

50. Determine the period, amplitude, minimum phase shift, and vertical displacement of the following functions and

sketch the graph.

e.

f. (

)

51. Write the equation of the graph of after the following transformations:

g. Vertical stretch by a factor of

about the x-axis

h. Horizontal stretch by a factor of 3 about the y-axis

i. Reflected in the x-axis

j. Translated

units to the right and 1 unit down

52. Determine the equation of the sine function with amplitude 2, maximum (

) and nearest maximum to the

right at (

)

53. Graphically solve (

) , .

54. A Ferris wheel with a radius of 10 m rotates once every 60 s. Passengers get on board at a point 2 m above the

ground at the bottom of the Ferris wheel.

k. Write an equation to model the path of a passenger on the Ferris wheel, where the height is a function of

time.

l. If a passenger is at the bottom of the Ferris wheel when it begins to move, determine her height above the

ground, to the nearest tenth of a metre, when the wheel has been in motion 2 minutes and 18 seconds.

55. The exact value of

can be found by:

a. √

b.

√

c.

√

d.

√

56. An object rotates counter clockwise from point ( √

) to point (

√

) on the unit circle. Through what

angle has it rotated? Answer as an exact value in radians.

a.

b.

c.

d.

57. When the equation of the following graph is written in the form ( ) , the value of is:

a.

b.

c.

d.

58. The range of the function (

) is:

a. { }

b. { }

c. { } d. { }

59. Consider the following equations:

a. Solve for as an exact value where .

b. State the general solution.

i.

ii.

60. Solve for .

61. State the inverse of the logarithmic function .

62. If , determine the value of .

63. As increases, and if , the exponential function increases / decreases.

64. Solve for .

65. Consider . Determine the domain, range, y-intercept, and the equation(s) of any asymptotes of

the graph of the function in terms of and .

66. If , describe the transformations that must be applied to the graph of to obtain the transformed

function (

) .

67. The point (

) is on the graph of the logarithmic function , and the point is on the

graph of the inverse, . Determine the value of .

68. Use the change of base identity to algebraically evaluate using a base of 4.

69. If , determine the value of to the nearest whole number.

70. Evaluate .

71. Express √ as a single logarithm in simplest form. State any restrictions on the

variable.

72. Determine whether is true where and . Provide justification and explain the

restrictions placed on the c-value.

73. Simplify

. Describe, in order, a series of transformations that could be applied to the

graph of to obtain the graph of (

) .

74. Solve for to two decimal places.

75. Determine whether is true where and . Provide justification.

76. If , then √ can be represented as an algebraic expression, in terms of , as:

a.

b. c.

d.

77. Express

as a single logarithm in simplest form.

78. Express

as a single logarithm in simplest form. State any restrictions on

the variable.

79. Solve the following equations for .

a.

b.

c.

80. Copper has a half-life of 1.25 hours.

a. Write an exponential function to model this situation.

b. What fraction of a sample of Copper will remain after 10 hours?

c. How long will it take for a sample of Copper to decay to

of its original mass?

81. In how many different ways can you arrange all of the letters in the word SUMMER?

82. In how many ways can four girls and two boys be arranged in a row if the boys must be together?

83. Simplify

.

84. Solve .

85. From a standard deck of 52 playing cards, five cards are selected. In how many ways can you:

a. Select five cards if at most one of them is red?

b. Select five cards if at least two of them are black?

86. How many terms are in the expansion of ?

e. What is the sum of the exponents on the variables for each term?

87. Use the binomial theorem to expand and simplify .

88. Determine the sum of the numbers in the first 9 rows of Pascal’s triangle.

89. Prove that .

90. Determine the middle term in the expansion of .

91. Determine the constant term in the expansion of (

)

.

92. Determine the value of in the expansion of , if one term in the expansion is .

93. How many pathways are possible to go from A to B?

A

B

94. For which of the following functions is a horizontal stretch about the y-axis by a factor of 2 equivalent to a

vertical translation of one unit down?

A) x2log B) x5.0log C)

x5.0 D) x2

95. On Oct. 2, 2010, the tide at New Westminster reached a maximum height of 10.8 feet at midnight. At 9 am the tide reached the next minimum height of 5.8 feet. Assuming the relationship is sinusoidal, what was the height of the tide at 7 am?

A. 6.1 feet B. 6.4 feet C. 8.7 feet D. 9.9 feet

96. Determine the 4th term in the expansion of (

)

A. −7168x5 y3 B. −128x5y3 C. 16x4y4 D. 1120x4y4

Use the following information to answer questions 4 and 5.

The function y = f(x) has domain and range .

97. f(x) could be which type of function?

A. Exponential B. Rational C. Sinusoidal D. Radical

98. If then the domain of g(x) is:

A. B. C. D.

99. Suppose

. Then the expression is equivalent to:

A. 2x + 1 B. 2x + 2 C. 4x + 1 D. 4x + 2

100. A sample of a radioactive substance decreases from a mass of 80g to a mass of 5g in 56 days. What is the half-life of the substance?

A. 14 days B. 16 days C. 18 days D. 28 days

101. If 3

3cos , 0 , determine the exact value of

6sin

.

A. √

B.

√

C.

√ √

D.

√ √

102. The graph of the function

has a discontinuity at (a,b) and a horizontal asymptote y = c.

The value of a + b + c is:

A. 2.5 B. 5.5 C. 8.5 D. 10.5

103. The solution to 2)23(log1log xx xx is also a solution to which of the following equations?

A. 3x2 + x – 4 = 0 B. 3x2 + x = 0 C. 2x2 + x – 2 = 0 D. 3x2 – x – 2 = 0

104. A family of 6 (2 parents and 4 children) sit in a row at a theatre. A parent must sit at either end with the 4 children between them. In how many ways can the family be seated?

A. 24 B. 48 C. 120 D. 720

105. If the value of a stock increases by 8% every 3 years, how many years will it take until the stock is worth three times its initial value?

A. 38 B. 43 C. 49 D. 56 106. How many permutations are there of all of the letters of the word RETREAT if they must begin and end

with the same letter?

107. I have 3 quarters, 2 dimes, and 2 pennies in my pocket. If I select at least one coin from my pocket, how many different amounts of money can I make?

108. How many positive integers less than 700 can be formed from the digits 2, 4, 6, 8, 9 if:

a) Repetition is allowed? _____________

b) Repetition is not allowed? _____________

109. 3 ,12 223 nPP nn . Find ‘n’.

110. In how many ways can the digits 1,2,3,4,5 and 6 be placed in the boxes below (one digit per box) so that the sum of each column is 7 ?

111. Celine always has lunch with one of 3 friends, and never with the same friend on consecutive days. From Monday to Friday, how many different lunch schedules could Celine have?

112. How many permutations of all of the letters of the word PATHFINDER:

a) have no two vowels together

b) have all the vowels together