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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 = โx4 + 4x3
(B) y = x4 โ 4x3
(C) y = x3 โ 4x2
(D) y = โx3 + 4x2
4. Which represents the value of k if the remainder is 5 for
(2x3 + 4x2 + kx โ 3) รท (x + 1) ?
(A) โ6 (B) โ2 (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 โ intercept is โ6
(A) P(x) = x4 โ 5x2 โ 6 (B) P(x) = โx4 + 3x3 + 6
(C) P(x) = โ(x + 2)(x + 3) (D) P(x) = โ x3+ x โ 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 โ 12x2 + 8x ?
(A) x = โ4, โ2, โ1 (B) x = โ2, โ1, 0
(C) x = 0, 1, 2 (D) x = 1, 2, 4
8. List all possible integral zeros for P(x) = x4 + 3x3 โ 2x2 โ 12x โ 8.
(A) ยฑ1, ยฑ8 (B) ยฑ1, ยฑ2, ยฑ3, ยฑ4, ยฑ6, ยฑ12
(C) ยฑ1, ยฑ2, ยฑ4, ยฑ8 (D) ยฑ2, ยฑ4
9. The volume of a rectangular prism is V= 2x3 โ 5x2 โ x + 6. If two of
the dimensions are x โ 2 and x + 1, what is an expression for the other
dimension?
(A) ๐ฅ โ 6 (B) ๐ฅ โ 6 (C) 2๐ฅ โ 3 (D) 2๐ฅ + 3
10. What are the x -intercepts of f (x) = x 2 (x + 3)(x โ 2)?
(A) โ3 and 2 (B) 3 and โ2
(C) 0, โ3, and 2 (D) 0, 3, and โ2
11. What is the quotient and remainder for (2 x 3 โ x 2 + 2 x + 4) รท ( x โ 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 ๐ฆ = ๐(๐ฅ) is stretched horizontally by a factor of 4 and
is reflected in the x-axis. What is the equation of the transformed function?
(A) โ๐ฆ = ๐ (1
4๐ฅ) (B) ๐ฆ = ๐ (โ
1
4๐ฅ)
(C) โy = f(4x) (D) y = f(โ4x)
15. The graph of ๐ฆ = ๐(๐ฅ) contains the point A(โ6, 4). What are
The coordinates of the image point, ๐ดโฒ, on the function 1
2(๐ฆ โ 2) = ๐(โ3(๐ฅ + 1))?
(A) (1, 10) (B) (3, 10) (C) (1, 0) (D) (3, 0)
16. The graph of ๐ฆ = ๐(๐ฅ) is transformed according to the mapping rule
(๐ฅ, ๐ฆ) โ (2๐ฅ โ 3, โ1
4๐ฆ). What is the equation of the resulting function?
(A) ๐ฆ = โ1
4๐(2๐ฅ โ 3) (B) ๐ฆ = โ
1
4๐ (
1
2๐ฅ + 3)
(C) ๐ฆ = โ1
4๐(2(๐ฅ โ 3)) (D) ๐ฆ = โ
1
4๐ (
1
2(๐ฅ + 3))
17. Given that ๐ฆ = ๐(๐ฅ) contains the point (๐, ๐), which of the following
points must lie on the graph of 1
3๐ฆ = ๐(๐ฅ + ๐) ?
(A) (2m, 3n) (B) (2๐,1
3๐)
(C) (0, 3n) (D) (0,1
3๐)
18. The function ๐ฆ = ๐(๐ฅ) is transformed to ๐ฆ = 3๐(4๐ฅ). Which describes how
๐ฆ = ๐(๐ฅ) 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 ๐ฆ = ๐(๐ฅ) is transformed to produce the graph of
๐ฆ = โ5๐(2๐ฅ โ 6) โ 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 ๐ฆ = ๐(๐ฅ)?
(A) y = f(x โ 4) (B) y = f(x) + 4
(C) y = 3f(x) (D) y = f(3x)
21. The graph of ๐ฆ = ๐(๐ฅ), shown below, is transformed to produce the
graph of โ2๐ฆ = ๐(๐ฅ + 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 ๐ฆ = ๐(๐ฅ), what is the vertical stretch factor
of โ2(๐ฆ + 1) = ๐(1
3๐ฅ)?
(A) โ2 (B) 2 (C) โ1
2 (D)
1
2
23. What transformations of the graph of ๐ฆ = 2๐(๐ฅ + 3) are required
to produce the graph of ๐ฆ = 6๐(๐ฅ + 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 ๐ฆ = ๐(๐ฅ) 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) ๐ฆ =1
3๐(๐ฅ + 5) โ 1 (B) ๐ฆ =
1
3๐(๐ฅ โ 5) + 1
(C) ๐ฆ = 3๐(๐ฅ + 5) โ 1 (D) ๐ฆ = 3๐(๐ฅ โ 5) โ 1
25. The mapping rule (๐ฅ, ๐ฆ) โ (5๐ฅ + 2, ๐ฆ โ 3) is applied to ๐ฆ = ๐(๐ฅ) to
produce a function of the form ๐ฆ = ๐๐(๐(๐ฅ โ โ)) + ๐. Which values
are correct for ๐, ๐, โ, and ๐?
(A) ๐ = 1, ๐ = 1
5, โ = 2, ๐ = โ3 (B) ๐ = 1, ๐ = 5, โ = โ2, ๐ = 3
(C) ๐ = 1, ๐ = 1
5, โ = โ2, ๐ = 3 (D) ๐ = 1, ๐ = 5, โ = 2, ๐ = โ3
26. Which combination of transformations is required to map ๐ฆ = ๐(๐ฅ)
onto ๐ฆ =1
2๐(โ๐ฅ)?
(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 ๐ฆ = ๐(๐ฅ) is {๐ฅ/โ12 โค ๐ฅ โค 6, ๐ฅ โ โ}. What is the
domain of โ(๐ฅ) = 3๐(2๐ฅ) ?
(A) {๐ฅ|โ24 โค ๐ฅ โค 12, ๐ฅ โ ๐ } (B) {๐ฅ|โ6 โค ๐ฅ โค 3, ๐ฅ โ ๐ }
(C) {๐ฅ|โ4 โค ๐ฅ โค 2, ๐ฅ โ ๐ } (D) {๐ฅ|โ36 โค ๐ฅ โค 18, ๐ฅ โ ๐ }
28. The function ๐ฆ = ๐(๐ฅ) has zeroes ๐ฅ = โ4 and ๐ฅ = 2. What are the
zeroes of the function ๐(๐ฅ) = 3๐ (โ1
2๐ฅ)?
(A) x = 2 and x = โ1 (B) x = 8 and x = โ4
(C) x = 6 and x = โ3 (D) x = 6 and x = โ12
29. The graphs of ๐ฆ = ๐(๐ฅ) and ๐ฆ = ๐(๐ฅ) are shown below. Which
mapping rule would map ๐ฆ = ๐(๐ฅ) onto ๐ฆ = ๐(๐ฅ)?
(A) (x, y) โ (x + 2, โ2y โ 1)
(B) (x, y) โ (x โ 2, โ2y โ 1)
(C) (x, y) โ (x + 2, โ2y + 2)
(D) (x, y) โ (x โ 2, โ2y + 2)
30. A function is defined as f(x) = x2 โ 5. What is the minimum value of the
function y = โ๐(๐ฅ) ?
(A) โ5 (B) 0 (C) 1 (D) 5
31. The equation of a function is given as ๐(๐ฅ) = ๐ฅ2 + 1. What is the domain
of ๐ฆ = โ๐(๐ฅ)?
(A) ๐ฅ โ (โโ, โ) (B) ๐ฅ โ [1, โ)
(C) ๐ฅ โ (1, โ) (D) ๐ฅ โ (โโ, โ1]โ[1, โ)
32. Which set of transformations would map ๐ฆ = โ๐ฅ
onto ๐ฆ โ 3 = โโ4(๐ฅ + 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) ๐ฆ = โโ(๐ฅ โ 2)
(B) ๐ฆ = โโ๐ฅ โ 2
(C) ๐ฆ = โโ๐ฅ โ 2
(D) ๐ฆ = โโ๐ฅ โ 2
34. The graph of a function ๐ฆ = ๐(๐ฅ) is shown. Which statement about the
function ๐ฆ = โ๐(๐ฅ) is true?
(A) It has two invariant points and has range ๐ฆ โค 1
(B) It has three invariant points and has range ๐ฆ โค 1
(C) It has two invariant points and has range ๐ฆ ๐ [0, 1]
(D) It has three invariant points and has range ๐ฆ ๐ [0, 1]
35. Which function has domain {๐ฅ/๐ฅ โฅ โ2, ๐ฅ โ ๐ } and {๐ฆ/๐ฆ โค 3, ๐ฆ โ ๐ }?
(A) ๐ฆ โ 3 = โโ๐ฅ + 2 (B) ๐ฆ + 3 = โโ๐ฅ โ 2
(C) ๐ฆ โ 3 = โ๐ฅ โ 2 (D) ๐ฆ + 3 = โ๐ฅ + 2
36. The function ๐ฆ = ๐(๐ฅ) contains the point (9, 4). Which point must lie on
the graph of ๐ฆ = โ๐(๐ฅ)?
(A) (3, 2) (B) (9, 2) (C) (3, 4) (D) (2, 3)
37. The point (๐, โ2) lies on the graph of the function ๐ฆ = 2โ๐ฅ โ 1 โ 6 ?
Which value is correct for ๐?
(A) 2 (B) 3 (C) 5 (D) 50
38. Which mapping rule would map ๐ฆ = โ๐ฅ onto โ1
3(๐ฆ + 2) = โ2๐ฅ + 6 ?
(A) (๐ฅ, ๐ฆ) โ (1
2๐ฅ โ 6, โ3๐ฆ โ 2) (B) (๐ฅ, ๐ฆ) โ (
1
2๐ฅ โ 3, โ3๐ฆ โ 2)
(C) (๐ฅ, ๐ฆ) โ (2๐ฅ โ 6, โ1
3๐ฆ + 2) (D) (๐ฅ, ๐ฆ) โ (2๐ฅ โ 3, โ
1
3๐ฆ + 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 ๐ฆ = โ๐ฅ 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: ๐ฅ โค โ4 Range: ๐ฆ โค 2
(B) Domain: ๐ฅ โค โ4 Range: ๐ฆ โฅ 2
(C) Domain: ๐ฅ โฅ โ4 Range: ๐ฆ โค 2
(D) Domain: ๐ฅ โฅ โ4 Range: ๐ฆ โฅ 2
40. Convert 160ยฐ to radians.
(A) 9ฯ
16 (B)
9ฯ
8 (C)
8ฯ
9 (D)
5ฯ
6
41. Which best approximates the value of cot(200ยฐ) + csc(3)?
(A) 0.3273 (B) 1.7374 (C) 9.8336 (D) 21.8548
42. If ๐๐๐ก๐ < 0 and ๐ ๐๐๐ > 0, in which quadrant would the terminal arm of
๐ lie?
(A) I (B) II (C) III (D) IV
43. What is the length of the arc cut by a 120ยฐ sector in a circle
having radius 6 cm?
(A) 4ฯ (B) ฯ
9 (C)
ฯ
3 (D) 5ฯ
44. Which expression represents all angles co-terminal with 2๐
3 ?
(A) 2ฯ
3+ kฯ, k โ R (B)
2ฯ
3+ kฯ, k โ I
(C) 2ฯ
3+ 2kฯ, k โ R (D)
2ฯ
3+ 2kฯ, k โ I
45. Which represents the missing coordinate if the point ๐ (๐ฅ,3
4)
lies on the terminal arm of the unit circle in the second quadrant?
(A) โ1
4 (B) โ
โ7
4 (C) โ
5
4 (D) โ
โ2
2
46. Which represents a circle centered at the origin with a radius of 5โ3 ?
(A) ๐ฅ2 + ๐ฆ2 = 15 (B) ๐ฅ2 + ๐ฆ2 = 5โ3
(C) ๐ฅ2 + ๐ฆ2 = 75 (D) ๐ฅ2 + ๐ฆ2 = 30
47. What is the measure of the central angle if P(ฮธ) = (โ2
2, โ
โ2
2)?
(A) ฯ
4 (B)
5ฯ
4 (C)
5ฯ
3 (D)
7ฯ
4
48. What is the exact value of: csc4ฯ
3 ?
(A) โโ3
2 (B) โ
1
2 (C) โ2 (D) โ
2โ3
3
49. If the point P(5, โ12) lies on the terminal arm of ำจ, then which
represents the ratio for sec ำจ ?
(A) โ13
12 (B)
13
5 (C) โ
12
13 (D)
5
13
50. If ๐๐๐ก ๐ =7
24, ๐ < ๐ <
3๐
2, then which ratio is true?
(A) ๐๐๐ ๐ =7
25 (B) ๐๐ ๐๐ =
25
24
(C) ๐ ๐๐๐ = โ25
24 (D) ๐ ๐๐๐ = โ
24
25
51. Solve for x: cos ำจ = 1
(A) kฯ, k โ I (B) 2ฯk, k โ I
(C) ฯ
2+ kฯ, k โ I (D)
ฯ
2+ 2ฯk, k โ I
52. If ๐ ๐๐ ๐ = โ2.5, 0ยฐ โค ๐ โค 180ยฐ, what is a possible value of ๐?
(A) 23.6ยฐ (B) 66.4ยฐ (C) 113.6ยฐ (D) 156.4ยฐ
53. Which represents the exact value of ๐ก๐๐2 5๐
6+ ๐๐๐ก2 5๐
6 ?
(A) 3
10 (B)
10
3 (C)
4
3 (D) 1
54. Solve for ฮธ: 2sin2ฮธ = sin ฮธ where 0 โค ฮธ โค 2ฯ.
(A) {ฯ
6,
5ฯ
6} (B) {
ฯ
3,
2ฯ
3}
(C) {0,ฯ
6,
5ฯ
6, ฯ, 2ฯ} (D) {0,
ฯ
3,
2ฯ
3, ฯ, 2ฯ}
55. What is the period of: y = 3cos [1
2x]?
(A) ๐
2 (B) (C) 4 (D) 8
56. Which represents the sine function with amplitude 3 and a period of 5๐
6 ?
(A) y = 3sin (5ฯ
6ฮธ) (B) y = 3sin (
6
5ฯฮธ)
(C) y = 3sin (5
12ฮธ) (D) y = 3sin (
12
5ฮธ)
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 ๐(๐ฅ) = 3๐ฅ2 + 6๐ฅ โ 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 ๐๐ ๐๐
3+ ๐๐๐ก
11๐
4
69. Solve for x:
(a) 2๐๐๐ ๐ฅ + โ3 = 0, ๐ฅ โ [0,180ยฐ)
(b) ๐๐ ๐๐ฅ + 12 = 2 โ 4๐๐ ๐๐ฅ, ๐ฅ โ [๐
2,
5๐
2)
(c) 3 csc2x โ 4 = 0 where 0 โค x โค 2ฯ
(d) 2๐ ๐๐2๐ฅ + 3๐ ๐๐๐ฅ = โ1, ๐ฅ โ [โ180ยฐ, 540ยฐ)
(๐) 3๐ก๐๐2๐ฅ = 14 tan ๐ฅ + 5, ๐ฅ โ [0,3๐)
(๐) 2๐ ๐๐2(๐ฅ) = sec(๐ฅ) + 15, where ๐ฅ โ [โ2๐, 2๐)
70. Determine the length of ๐ด๏ฟฝฬ๏ฟฝ (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 = โ34 58. โ1, โ1, 2,3
2 59. ๐ฆ = โ
1
2๐ฅ(๐ฅ + 2)2(๐ฅ โ 3)
60. 0 < x < 6, a 2m x 2m square must be removed from each corner
61. 62. y = โ2f(x + 3)
63.
64. Restriction: x โฅ โ1 ๐ฆ = โ๐ฅ+4
3โ 1
65. Domain: โ1 โค x โค 3 Range: 0 โค y โค โ8
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) (๐ฅ, ๐ฆ) โ (โ1
2๐ฅ + 4, 3๐ฆ + 1)
(ii) Domain: x โค 4 Range: y โฅ 1
(iii)
67.
68. 2โ3โ3
3
69.(a) 150แต (b) 7ฯ
6,
11ฯ
6 (c)
ฯ
3,
2ฯ
3,
4ฯ
3,
5ฯ
3
(d) โ30ยฐ, โ90ยฐ, โ150ยฐ, 210ยฐ, 270ยฐ, 330ยฐ
(e) 1.4 ๐๐๐, 2.8 ๐๐๐, 4.5 ๐๐๐, 5.98 ๐๐๐, 7.7 ๐๐๐
(f)
โ5.05 ๐๐๐, โ4.3 ๐๐๐, โ1.98 ๐๐๐, โ1.23 ๐๐๐, 1.23 ๐๐๐, 1.98 ๐๐๐, 4.3 ๐๐๐, 5.05 ๐๐๐
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:
{๐ฅ = 0 ๐๐๐ ๐ฅ = 2}