Mathematics 3200

Midyear Exam Review

1. If )2( x is a factor of )3(47 23 kxxx , what is the value of k?

(A) -31 (B) -28 (C) 28 (D) 31

2. Which graph below represents the graph of an even degree function?

(A) (B)

(C) (D)

3. Which of the equations below is best represented by the given graph?

(A) y = –x4 + 4x

3

(B) y = x4 – 4x

3

(C) y = x3 – 4x

2

(D) y = –x3 + 4x

2

4. Which represents the value of k if the remainder is 5 for

(2x3 + 4x

2 + kx – 3) ÷ (x + 1) ?

(A) –6 (B) –2 (C) 2 (D) 6

x-4 -2 2 4

y

-6

-4

-2

2

4

6

x-4 -2 2 4

y

-10

-8

-6

-4

-2

2

4

6

x-4 -2 2 4

y

-10

-8

-6

-4

-2

2

4

6

x-4 -2 2 4

y

-6

-4

-2

2

4

6

x- 2 2 4 6

y

- 8- 6- 4- 2

2468

10121416182022242628

5. Which function has each of the characteristics:

an even function

end behavior in the third and fourth quadrants

y – intercept is –6

(A) P(x) = x4 – 5x

2 – 6 (B) P(x) = –x

4 + 3x

3 + 6

(C) P(x) = –(x + 2)(x + 3) (D) P(x) = – x3+ x – 6

6. Which statement is true for a polynomial function?

(A) All even degree polynomial functions have at least one x-intercept.

(B) Some odd degree polynomial functions have no x-intercepts.

(C) Even degree polynomial functions always have an even number

of x-intercepts.

(D) All odd degree polynomials have at least one x-intercept.

7. What are the x-intercepts of P(x) = 4x3 – 12x

2 + 8x ?

(A) x = –4, –2, –1 (B) x = –2, –1, 0

(C) x = 0, 1, 2 (D) x = 1, 2, 4

8. List all possible integral zeros for P(x) = x4 + 3x

3 – 2x

2 – 12x – 8.

(A) ±1, ±8 (B) ±1, ±2, ±3, ±4, ±6, ±12

(C) ±1, ±2, ±4, ±8 (D) ±2, ±4

9. The volume of a rectangular prism is V= 2x3 – 5x

2 – x + 6. If two of

the dimensions are x – 2 and x + 1, what is an expression for the other

dimension?

(A) (B) (C) (D)

10. What are the x -intercepts of f (x) = x 2 (x + 3)(x − 2)?

(A) −3 and 2 (B) 3 and −2

(C) 0, −3, and 2 (D) 0, 3, and −2

11. What is the quotient and remainder for (2 x 3 − x

2 + 2 x + 4) ÷ ( x − 3)?

(A) The quotient is 2x 2 + x + 5, and the remainder is 19.

(B) The quotient is 2x 2 + 5x + 7, and the remainder is 29.

(C) The quotient is 2x 2 + 5x + 17, and the remainder is 55.

(D) The quotient is 2x 2 + x + 3, and the remainder is 7.

12. Which sketch best represents the graph of edxcxbxaxy 234

if 0a and 0e ?

(A) (B)

(C) (D)

13. How many x-intercepts are possible for the polynomial

function P(x) = ax5 + bx

4 + cx

3 ?

(A) 1 (B) 3 (C) 4 (D) 5

14. The function is stretched horizontally by a factor of 4 and

is reflected in the x-axis. What is the equation of the transformed function?

(A)

(B)

(C) –y = f(4x) (D) y = f(–4x)

15. The graph of contains the point A(–6, 4). What are

The coordinates of the image point, , on the function

?

(A) (1, 10) (B) (3, 10) (C) (1, 0) (D) (3, 0)

16. The graph of is transformed according to the mapping rule

. What is the equation of the resulting function?

(A)

(B)

(C)

(D)

17. Given that contains the point , which of the following

points must lie on the graph of

?

(A) (2m, 3n) (B)

(C) (0, 3n) (D)

18. The function is transformed to . Which describes how

is transformed?

Horizontal Stretch Vertical Stretch

A

B

C

D

19. The graph of is transformed to produce the graph of

. Which describes the horizontal translation?

(A) 6 units left (B) 6 units right

(C) 3 units left (D) 3 units right

20. Which function would produce a graph with the

same y-intercept as the graph of ?

(A) y = f(x – 4) (B) y = f(x) + 4

(C) y = 3f(x) (D) y = f(3x)

21. The graph of , shown below, is transformed to produce the

graph of . Which of the labelled points (A, B, and C)

would be invariant?

(A) A and B only

(B) C only

(C) A, B, and C

(D) None of them would be invariant

x

y

A B

C

22. When compared to , what is the vertical stretch factor

of

?

(A) –2 (B) 2 (C)

(D)

23. What transformations of the graph of are required

to produce the graph of ?

(A) Vertical Stretch of 4 and a horizontal translation of 2 units left

(B) Vertical Stretch of 3 and a horizontal translation of 2 units left

(C) Vertical Stretch of 4 and a horizontal translation of 2 units right

(D) Vertical Stretch of 3 and a horizontal translation of 2 units right

24. The function is stretched vertically by a factor of 3, and

translated 5 units right and 1 unit down. What is the equation of the

resulting function?

(A)

(B)

(C) (D)

25. The mapping rule is applied to to

produce a function of the form . Which values

are correct for ?

(A)

(B)

(C)

(D)

26. Which combination of transformations is required to map

onto

?

(A) Reflection in the x-axis, Stretched vertically by a factor of 2

(B) Reflection in the x-axis, Stretched vertically by a factor of

(C) Reflection in the y-axis, Stretched vertically by a factor of 2

(D) Reflection in the y-axis, Stretched vertically by a factor of

27. The domain of is . What is the

domain of ?

(A) (B)

(C) (D)

28. The function has zeroes . What are the

zeroes of the function

?

(A) x = 2 and x = –1 (B) x = 8 and x = –4

(C) x = 6 and x = –3 (D) x = 6 and x = –12

29. The graphs of are shown below. Which

mapping rule would map onto ?

(A) (x, y) → (x + 2, –2y – 1)

(B) (x, y) → (x – 2, –2y – 1)

(C) (x, y) → (x + 2, –2y + 2)

(D) (x, y) → (x – 2, –2y + 2)

30. A function is defined as f(x) = x2 – 5. What is the minimum value of the

function y = ?

(A) –5 (B) 0 (C) 1 (D) 5

31. The equation of a function is given as . What is the domain

of ?

(A) (B)

(C) (D)

32. Which set of transformations would map

onto ?

(A) Reflection in the x-axis, horizontal stretch by a factor of 4,

translation of 2 units right and 3 units down.

(B) Reflection in the x-axis, horizontal stretch by a factor of

,

translation of 2 units left and 3 units up.

(C) Reflection in the y-axis, horizontal stretch by a factor of 4,

translation of 2 units right and 3 units down.

(D) Reflection in the y-axis, horizontal stretch by a factor of

,

translation of 2 units left and 3 units up.

x- 6 - 4 - 2 2 4 6

y

- 8

- 6

- 4

- 2

2

4

6

8

y = f(x)

y = g(x)

33. Which function best represents the graph shown below?

(A)

(B)

(C)

(D)

34. The graph of a function is shown. Which statement about the

function is true?

(A) It has two invariant points and has range

(B) It has three invariant points and has range

(C) It has two invariant points and has range

(D) It has three invariant points and has range

35. Which function has domain and ?

(A) (B)

(C) (D)

36. The function contains the point (9, 4). Which point must lie on

the graph of ?

(A) (3, 2) (B) (9, 2) (C) (3, 4) (D) (2, 3)

37. The point lies on the graph of the function ?

Which value is correct for ?

(A) 2 (B) 3 (C) 5 (D) 50

38. Which mapping rule would map onto

?

(A)

(B)

(C)

(D)

x

y

x- 4 - 3 - 2 - 1 1 2 3 4 5 6

y

- 1

1

2

3

39. The graph of the function is reflected in the x-axis and is translated

4 units left and 2 units up. Which describes the domain and range of the

resulting function?

(A) Domain: Range:

(B) Domain: Range:

(C) Domain: Range:

(D) Domain: Range:

40. Convert 160° to radians.

(A)

(B)

(C)

(D)

41. Which best approximates the value of cot(200°) + csc(3)?

(A) 0.3273 (B) 1.7374 (C) 9.8336 (D) 21.8548

42. If and , in which quadrant would the terminal arm of

lie?

(A) I (B) II (C) III (D) IV

43. What is the length of the arc cut by a 120° sector in a circle

having radius 6 cm?

(A) (B)

(C)

(D)

44. Which expression represents all angles co-terminal with

?

(A)

(B)

(C)

(D)

45. Which represents the missing coordinate if the point

lies on the terminal arm of the unit circle in the second quadrant?

(A)

(B)

(C)

(D)

46. Which represents a circle centered at the origin with a radius of ?

(A) (B)

(C) (D)

47. What is the measure of the central angle if

?

(A)

(B)

(C)

(D)

48. What is the exact value of

?

(A)

(B)

(C) (D)

49. If the point P(5, –12) lies on the terminal arm of Ө, then which

represents the ratio for sec Ө ?

(A)

(B)

(C)

(D)

50. If

, then which ratio is true?

(A)

(B)

(C)

(D)

51. Solve for x: cos Ө = 1

(A) (B)

(C)

(D)

52. If , , what is a possible value of ?

(A) 23.6 (B) 66.4 (C) (D)

53. Which represents the exact value of

?

(A)

(B)

(C)

(D)

54. For what value of k will the polynomial 3 2( ) 4 3 6P x x x kx have the

same remainder when it is divided by both 1 and 3x x ?

55. Give the function, 65632)( 234 xxxxxP , determine the other roots.

56. Given the graph, determine the equation of the polynomial in factored form.

57. An open top box is made from a 16 m by 12 m rectangular piece of sheet

metal by cutting congruent squares of length x from each corner and folding

up the sides. Identify any restrictions on x and algebraically determine what

size squares must be removed to produce a box with a volume of 192 m3.

58. Sketch the graph of the function 3 22 5 4 3y x x x and clearly label the

x-intercept(s) and the y-intercept.

59. The graph of the function y g(x) represents a transformation

of the graph of y f (x). Determine the equation of g(x) in the

form y af (b(x h)) k.

x

y

x- 10 - 5 5 10

y

- 15

- 10

- 5

5

10

y = f (x)

y = g(x)

60. The graph of )(xfy is shown below. Sketch the graph of the transformed

function ))1(2(63 xfy . Identify the specific values of a, b, h and k

required and state the mapping rule.

61. Algebraically determine the inverse function for ,

stating any necessary restictions.

62. Determine the domain and range of 8)1(2 2 xy .

Show algebraic workings.

63. (i) Write the mapping rule that maps xy onto

the function 1)4(23 xy .

(ii) State the domain and range of the transformed function.

(iii) Sketch the graph on the grid provided, showing the image points for

those shown on the graph of xy .

Mapping Rule: ______________

Domain: ___________________

Range: ___________________

64. Determine the approximate solution to each equation graphically

xx 392 2 . Verify your answer algebraically.

65. Determine the exact value of

66. Solve for x, where 0 ≤ x ≤ 2π 9 csc2x – 4 = 0

67. Determine the length of (in cm), to the nearest tenth of a unit.

B (-15,8)

A (8,15)

ANSWERS:

1.D 2. B 3. A 4. A 5. C 6. D 7. C 8. C 9. C 10. C

11. C 12. D 13. B 14. A 15. A 16. D 17. C

18. B 19. D 20. D 21. A 22. D 23. D 24. D

25. A 26. D 27. B 28. B 29. A 30. B 31. A

32. B 33. A 34. D 35. A 36. B 37. C 38. B

39. A 40. C 41. C 42. D 43. A 44. D 45. B

46. C 47. D 48. D 49. B 50. D 51. B 52. C

53. B

54. k = –34 55.

56. y = –(x + 2)

2(x – 3)

57. x > 0, a 2x2 square must be removed from each corner

58. 59. y = –2f(x + 3) – 2

60.

61. restriction: x ≥ –1

62. Domain: –1 ≤ x ≤ 3 Range: 0 ≤ y ≤

63. (i)

65.

(ii) Domain: x ≤ 4 Range: y ≥ 1

(iii) 66.

67. 26.7 cm

64.

x- 6 - 4 - 2 2 4 6

y

- 8

- 6

- 4

- 2

2

4

6

x

y

A'

B' C' D'

x-6 -4 -2 2 4 6

y

-2

2

4

6

8

10