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 + 4x3

(B) y = x4 – 4x3

(C) y = x3 – 4x2

(D) y = –x3 + 4x2

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

(2x3 + 4x2 + 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 – 5x2 – 6 (B) P(x) = –x4 + 3x3 + 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 – 12x2 + 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 + 3x3 – 2x2 – 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 – 5x2 – x + 6. If two of

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

dimension?

(A) 𝑥 − 6 (B) 𝑥 − 6 (C) 2𝑥 − 3 (D) 2𝑥 + 3

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 + bx4 + cx3 ?

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

14. The function 𝑦 = 𝑓(𝑥) is stretched horizontally by a factor of 4 and

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

(A) −𝑦 = 𝑓 (1

4𝑥) (B) 𝑦 = 𝑓 (−

1

4𝑥)

(C) –y = f(4x) (D) y = f(–4x)

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

The coordinates of the image point, 𝐴′, on the function 1

2(𝑦 − 2) = 𝑓(−3(𝑥 + 1))?

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

16. The graph of 𝑦 = 𝑓(𝑥) is transformed according to the mapping rule

(𝑥, 𝑦) → (2𝑥 − 3, −1

4𝑦). What is the equation of the resulting function?

(A) 𝑦 = −1

4𝑓(2𝑥 − 3) (B) 𝑦 = −

1

4𝑓 (

1

2𝑥 + 3)

(C) 𝑦 = −1

4𝑓(2(𝑥 − 3)) (D) 𝑦 = −

1

4𝑓 (

1

2(𝑥 + 3))

17. Given that 𝑦 = 𝑓(𝑥) contains the point (𝑚, 𝑛), which of the following

points must lie on the graph of 1

3𝑦 = 𝑓(𝑥 + 𝑚) ?

(A) (2m, 3n) (B) (2𝑚,1

3𝑛)

(C) (0, 3n) (D) (0,1

3𝑛)

18. The function 𝑦 = 𝑓(𝑥) is transformed to 𝑦 = 3𝑓(4𝑥). Which describes how

𝑦 = 𝑓(𝑥) is transformed?

Horizontal Stretch Vertical Stretch

A 1

4

1

3

B 1

4 3

C 4

1

3

D 4 3

19. The graph of 𝑦 = 𝑓(𝑥) is transformed to produce the graph of

𝑦 = −5𝑓(2𝑥 − 6) − 1 . 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 𝑦 = 𝑓(𝑥)?

(A) y = f(x – 4) (B) y = f(x) + 4

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

21. The graph of 𝑦 = 𝑓(𝑥), shown below, is transformed to produce the

graph of −2𝑦 = 𝑓(𝑥 + 3). 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 𝑦 = 𝑓(𝑥), what is the vertical stretch factor

of −2(𝑦 + 1) = 𝑓(1

3𝑥)?

(A) –2 (B) 2 (C) −1

2 (D)

1

2

23. What transformations of the graph of 𝑦 = 2𝑓(𝑥 + 3) are required

to produce the graph of 𝑦 = 6𝑓(𝑥 + 1)?

(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 𝑦 = 𝑓(𝑥) 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) 𝑦 =1

3𝑓(𝑥 + 5) − 1 (B) 𝑦 =

1

3𝑓(𝑥 − 5) + 1

(C) 𝑦 = 3𝑓(𝑥 + 5) − 1 (D) 𝑦 = 3𝑓(𝑥 − 5) − 1

25. The mapping rule (𝑥, 𝑦) → (5𝑥 + 2, 𝑦 − 3) is applied to 𝑦 = 𝑓(𝑥) to

produce a function of the form 𝑦 = 𝑎𝑓(𝑏(𝑥 − ℎ)) + 𝑘. Which values

are correct for 𝑎, 𝑏, ℎ, and 𝑘?

(A) 𝑎 = 1, 𝑏 = 1

5, ℎ = 2, 𝑘 = −3 (B) 𝑎 = 1, 𝑏 = 5, ℎ = −2, 𝑘 = 3

(C) 𝑎 = 1, 𝑏 = 1

5, ℎ = −2, 𝑘 = 3 (D) 𝑎 = 1, 𝑏 = 5, ℎ = 2, 𝑘 = −3

26. Which combination of transformations is required to map 𝑦 = 𝑓(𝑥)

onto 𝑦 =1

2𝑓(−𝑥)?

(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 1

2

(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 1

2

27. The domain of 𝑦 = 𝑓(𝑥) is {𝑥/−12 ≤ 𝑥 ≤ 6, 𝑥 ∈ ℝ}. What is the

domain of ℎ(𝑥) = 3𝑓(2𝑥) ?

(A) {𝑥|−24 ≤ 𝑥 ≤ 12, 𝑥 ∈ 𝑅} (B) {𝑥|−6 ≤ 𝑥 ≤ 3, 𝑥 ∈ 𝑅}

(C) {𝑥|−4 ≤ 𝑥 ≤ 2, 𝑥 ∈ 𝑅} (D) {𝑥|−36 ≤ 𝑥 ≤ 18, 𝑥 ∈ 𝑅}

28. The function 𝑦 = 𝑓(𝑥) has zeroes 𝑥 = −4 and 𝑥 = 2. What are the

zeroes of the function 𝑔(𝑥) = 3𝑓 (−1

2𝑥)?

(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 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) are shown below. Which

mapping rule would map 𝑦 = 𝑓(𝑥) onto 𝑦 = 𝑔(𝑥)?

(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 = √𝑓(𝑥) ?

(A) –5 (B) 0 (C) 1 (D) 5

31. The equation of a function is given as 𝑓(𝑥) = 𝑥2 + 1. What is the domain

of 𝑦 = √𝑓(𝑥)?

(A) 𝑥 ∈ (−∞, ∞) (B) 𝑥 ∈ [1, ∞)

(C) 𝑥 ∈ (1, ∞) (D) 𝑥 ∈ (−∞, −1]⋃[1, ∞)

32. Which set of transformations would map 𝑦 = √𝑥

onto 𝑦 − 3 = −√4(𝑥 + 2)?

(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 1

4,

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 1

4,

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) 𝑦 = √−(𝑥 − 2)

(B) 𝑦 = √−𝑥 − 2

(C) 𝑦 = −√𝑥 − 2

(D) 𝑦 = −√𝑥 − 2

34. The graph of a function 𝑦 = 𝑓(𝑥) is shown. Which statement about the

function 𝑦 = √𝑓(𝑥) is true?

(A) It has two invariant points and has range 𝑦 ≤ 1

(B) It has three invariant points and has range 𝑦 ≤ 1

(C) It has two invariant points and has range 𝑦 𝜖 [0, 1]

(D) It has three invariant points and has range 𝑦 𝜖 [0, 1]

35. Which function has domain {𝑥/𝑥 ≥ −2, 𝑥 ∈ 𝑅} and {𝑦/𝑦 ≤ 3, 𝑦 ∈ 𝑅}?

(A) 𝑦 − 3 = −√𝑥 + 2 (B) 𝑦 + 3 = −√𝑥 − 2

(C) 𝑦 − 3 = √𝑥 − 2 (D) 𝑦 + 3 = √𝑥 + 2

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

the graph of 𝑦 = √𝑓(𝑥)?

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

37. The point (𝑛, −2) lies on the graph of the function 𝑦 = 2√𝑥 − 1 − 6 ?

Which value is correct for 𝑛?

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

38. Which mapping rule would map 𝑦 = √𝑥 onto −1

3(𝑦 + 2) = √2𝑥 + 6 ?

(A) (𝑥, 𝑦) → (1

2𝑥 − 6, −3𝑦 − 2) (B) (𝑥, 𝑦) → (

1

2𝑥 − 3, −3𝑦 − 2)

(C) (𝑥, 𝑦) → (2𝑥 − 6, −1

3𝑦 + 2) (D) (𝑥, 𝑦) → (2𝑥 − 3, −

1

3𝑦 + 2)

x

y

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

y

- 1

1

2

3

39. The graph of the function 𝑦 = √𝑥 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: 𝑥 ≤ −4 Range: 𝑦 ≤ 2

(B) Domain: 𝑥 ≤ −4 Range: 𝑦 ≥ 2

(C) Domain: 𝑥 ≥ −4 Range: 𝑦 ≤ 2

(D) Domain: 𝑥 ≥ −4 Range: 𝑦 ≥ 2

40. Convert 160° to radians.

(A) 9π

16 (B)

9π

8 (C)

8π

9 (D)

5π

6

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 𝑐𝑜𝑡𝜃 < 0 and 𝑠𝑒𝑐𝜃 > 0, in which quadrant would the terminal arm of

𝜃 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) 4π (B) π

9 (C)

π

3 (D) 5π

44. Which expression represents all angles co-terminal with 2𝜋

3 ?

(A) 2π

3+ kπ, k ∈ R (B)

2π

3+ kπ, k ∈ I

(C) 2π

3+ 2kπ, k ∈ R (D)

2π

3+ 2kπ, k ∈ I

45. Which represents the missing coordinate if the point 𝑃 (𝑥,3

4)

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

(A) −1

4 (B) −

√7

4 (C) −

5

4 (D) −

√2

2

46. Which represents a circle centered at the origin with a radius of 5√3 ?

(A) 𝑥2 + 𝑦2 = 15 (B) 𝑥2 + 𝑦2 = 5√3

(C) 𝑥2 + 𝑦2 = 75 (D) 𝑥2 + 𝑦2 = 30

47. What is the measure of the central angle if P(θ) = (√2

2, −

√2

2)?

(A) π

4 (B)

5π

4 (C)

5π

3 (D)

7π

4

48. What is the exact value of: csc4π

3 ?

(A) −√3

2 (B) −

1

2 (C) −2 (D) −

2√3

3

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

represents the ratio for sec Ө ?

(A) −13

12 (B)

13

5 (C) −

12

13 (D)

5

13

50. If 𝑐𝑜𝑡 𝜃 =7

24, 𝜋 < 𝜃 <

3𝜋

2, then which ratio is true?

(A) 𝑐𝑜𝑠𝜃 =7

25 (B) 𝑐𝑠𝑐𝜃 =

25

24

(C) 𝑠𝑒𝑐𝜃 = −25

24 (D) 𝑠𝑖𝑛𝜃 = −

24

25

51. Solve for x: cos Ө = 1

(A) kπ, k ∈ I (B) 2πk, k ∈ I

(C) π

2+ kπ, k ∈ I (D)

π

2+ 2πk, k ∈ I

52. If 𝑠𝑒𝑐 𝜃 = −2.5, 0° ≤ 𝜃 ≤ 180°, what is a possible value of 𝜃?

(A) 23.6° (B) 66.4° (C) 113.6° (D) 156.4°

53. Which represents the exact value of 𝑡𝑎𝑛2 5𝜋

6+ 𝑐𝑜𝑡2 5𝜋

6 ?

(A) 3

10 (B)

10

3 (C)

4

3 (D) 1

54. Solve for θ: 2sin2θ = sin θ where 0 ≤ θ ≤ 2π.

(A) {π

6,

5π

6} (B) {

π

3,

2π

3}

(C) {0,π

6,

5π

6, π, 2π} (D) {0,

π

3,

2π

3, π, 2π}

55. What is the period of: y = 3cos [1

2x]?

(A) 𝜋

2 (B) (C) 4 (D) 8

56. Which represents the sine function with amplitude 3 and a period of 5𝜋

6 ?

(A) y = 3sin (5π

6θ) (B) y = 3sin (

6

5πθ)

(C) y = 3sin (5

12θ) (D) y = 3sin (

12

5θ)

57. 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 ?

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

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

60. 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.

61. 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.

62. 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.

63. 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.

64. Algebraically determine the inverse function for 𝑓(𝑥) = 3𝑥2 + 6𝑥 − 1,

stating any necessary restictions.

x- 10 - 5 5 10

y

- 15

- 10

- 5

5

10

y = f (x)

y = g(x)

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

Show algebraic workings.

66. (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: ___________________

67. Determine the approximate solution to each equation graphically

xx 392 2 . Verify your answer algebraically.

68. Determine the exact value of 𝑐𝑠𝑐𝜋

3+ 𝑐𝑜𝑡

11𝜋

4

69. Solve for x:

(a) 2𝑐𝑜𝑠𝑥 + √3 = 0, 𝑥 ∈ [0,180°)

(b) 𝑐𝑠𝑐𝑥 + 12 = 2 − 4𝑐𝑠𝑐𝑥, 𝑥 ∈ [𝜋

2,

5𝜋

2)

(c) 3 csc2x – 4 = 0 where 0 ≤ x ≤ 2π

(d) 2𝑠𝑖𝑛2𝑥 + 3𝑠𝑖𝑛𝑥 = −1, 𝑥 ∈ [−180°, 540°)

(𝑒) 3𝑡𝑎𝑛2𝑥 = 14 tan 𝑥 + 5, 𝑥 ∈ [0,3𝜋)

(𝑓) 2𝑠𝑒𝑐2(𝑥) = sec(𝑥) + 15, where 𝑥 ∈ [−2𝜋, 2𝜋)

70. Determine the length of 𝐴�̂� (in cm), to the nearest tenth of a unit.

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.D

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. C 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. C 55. C 56.D

57. k = –34 58. −1, −1, 2,3

2 59. 𝑦 = −

1

2𝑥(𝑥 + 2)2(𝑥 − 3)

60. 0 < x < 6, a 2m x 2m square must be removed from each corner

61. 62. y = –2f(x + 3)

63.

64. Restriction: x ≥ –1 𝑦 = √𝑥+4

3− 1

65. Domain: –1 ≤ x ≤ 3 Range: 0 ≤ y ≤ √8

B (-15,8)

A (8,15)

x- 6 - 4 - 2 2 4 6

y

- 8

- 6

- 4

- 2

2

4

6

x

y

A'

B' C' D'

66. (i) (𝑥, 𝑦) → (−1

2𝑥 + 4, 3𝑦 + 1)

(ii) Domain: x ≤ 4 Range: y ≥ 1

(iii)

67.

68. 2√3−3

3

69.(a) 150ᵒ (b) 7π

6,

11π

6 (c)

π

3,

2π

3,

4π

3,

5π

3

(d) −30°, −90°, −150°, 210°, 270°, 330°

(e) 1.4 𝑟𝑎𝑑, 2.8 𝑟𝑎𝑑, 4.5 𝑟𝑎𝑑, 5.98 𝑟𝑎𝑑, 7.7 𝑟𝑎𝑑

(f)

−5.05 𝑟𝑎𝑑, −4.3 𝑟𝑎𝑑, −1.98 𝑟𝑎𝑑, −1.23 𝑟𝑎𝑑, 1.23 𝑟𝑎𝑑, 1.98 𝑟𝑎𝑑, 4.3 𝑟𝑎𝑑, 5.05 𝑟𝑎𝑑

70. 26.7 cm

x-6 -4 -2 2 4 6

y

-2

2

4

6

8

10

x-4 -2 2 4

y

-2

2

4

Solutions:

{𝑥 = 0 𝑎𝑛𝑑 𝑥 = 2}