Midyear Exam Review - WordPress.comΒ Β· Midyear Exam Review 1. If (x 2) is a factor of 3 7 2 4x (3...
Transcript of Midyear Exam Review - WordPress.comΒ Β· Midyear Exam Review 1. If (x 2) is a factor of 3 7 2 4x (3...
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 = βx4 + 4x3
(B) y = x4 β 4x3
(C) y = x3 β 4x2
(D) y = βx3 + 4x2
4. Which represents the value of k if the remainder is 5 for
(2x3 + 4x2 + kx β 3) Γ· (x + 1) ?
(A) β6 (B) β2 (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 β intercept is β6
(A) P(x) = x4 β 5x2 β 6 (B) P(x) = βx4 + 3x3 + 6
(C) P(x) = β(x + 2)(x + 3) (D) P(x) = β x3+ x β 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 β 12x2 + 8x ?
(A) x = β4, β2, β1 (B) x = β2, β1, 0
(C) x = 0, 1, 2 (D) x = 1, 2, 4
8. List all possible integral zeros for P(x) = x4 + 3x3 β 2x2 β 12x β 8.
(A) Β±1, Β±8 (B) Β±1, Β±2, Β±3, Β±4, Β±6, Β±12
(C) Β±1, Β±2, Β±4, Β±8 (D) Β±2, Β±4
9. The volume of a rectangular prism is V= 2x3 β 5x2 β x + 6. If two of
the dimensions are x β 2 and x + 1, what is an expression for the other
dimension?
(A) π₯ β 6 (B) π₯ β 6 (C) 2π₯ β 3 (D) 2π₯ + 3
10. What are the x -intercepts of f (x) = x 2 (x + 3)(x β 2)?
(A) β3 and 2 (B) 3 and β2
(C) 0, β3, and 2 (D) 0, 3, and β2
11. What is the quotient and remainder for (2 x 3 β x 2 + 2 x + 4) Γ· ( x β 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 π¦ = π(π₯) is stretched horizontally by a factor of 4 and
is reflected in the x-axis. What is the equation of the transformed function?
(A) βπ¦ = π (1
4π₯) (B) π¦ = π (β
1
4π₯)
(C) βy = f(4x) (D) y = f(β4x)
15. The graph of π¦ = π(π₯) contains the point A(β6, 4). What are
The coordinates of the image point, π΄β², on the function 1
2(π¦ β 2) = π(β3(π₯ + 1))?
(A) (1, 10) (B) (3, 10) (C) (1, 0) (D) (3, 0)
16. The graph of π¦ = π(π₯) is transformed according to the mapping rule
(π₯, π¦) β (2π₯ β 3, β1
4π¦). What is the equation of the resulting function?
(A) π¦ = β1
4π(2π₯ β 3) (B) π¦ = β
1
4π (
1
2π₯ + 3)
(C) π¦ = β1
4π(2(π₯ β 3)) (D) π¦ = β
1
4π (
1
2(π₯ + 3))
17. Given that π¦ = π(π₯) contains the point (π, π), which of the following
points must lie on the graph of 1
3π¦ = π(π₯ + π) ?
(A) (2m, 3n) (B) (2π,1
3π)
(C) (0, 3n) (D) (0,1
3π)
18. The function π¦ = π(π₯) is transformed to π¦ = 3π(4π₯). Which describes how
π¦ = π(π₯) 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 π¦ = π(π₯) is transformed to produce the graph of
π¦ = β5π(2π₯ β 6) β 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 π¦ = π(π₯)?
(A) y = f(x β 4) (B) y = f(x) + 4
(C) y = 3f(x) (D) y = f(3x)
21. The graph of π¦ = π(π₯), shown below, is transformed to produce the
graph of β2π¦ = π(π₯ + 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 π¦ = π(π₯), what is the vertical stretch factor
of β2(π¦ + 1) = π(1
3π₯)?
(A) β2 (B) 2 (C) β1
2 (D)
1
2
23. What transformations of the graph of π¦ = 2π(π₯ + 3) are required
to produce the graph of π¦ = 6π(π₯ + 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 π¦ = π(π₯) 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) π¦ =1
3π(π₯ + 5) β 1 (B) π¦ =
1
3π(π₯ β 5) + 1
(C) π¦ = 3π(π₯ + 5) β 1 (D) π¦ = 3π(π₯ β 5) β 1
25. The mapping rule (π₯, π¦) β (5π₯ + 2, π¦ β 3) is applied to π¦ = π(π₯) to
produce a function of the form π¦ = ππ(π(π₯ β β)) + π. Which values
are correct for π, π, β, and π?
(A) π = 1, π = 1
5, β = 2, π = β3 (B) π = 1, π = 5, β = β2, π = 3
(C) π = 1, π = 1
5, β = β2, π = 3 (D) π = 1, π = 5, β = 2, π = β3
26. Which combination of transformations is required to map π¦ = π(π₯)
onto π¦ =1
2π(βπ₯)?
(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 π¦ = π(π₯) is {π₯/β12 β€ π₯ β€ 6, π₯ β β}. What is the
domain of β(π₯) = 3π(2π₯) ?
(A) {π₯|β24 β€ π₯ β€ 12, π₯ β π } (B) {π₯|β6 β€ π₯ β€ 3, π₯ β π }
(C) {π₯|β4 β€ π₯ β€ 2, π₯ β π } (D) {π₯|β36 β€ π₯ β€ 18, π₯ β π }
28. The function π¦ = π(π₯) has zeroes π₯ = β4 and π₯ = 2. What are the
zeroes of the function π(π₯) = 3π (β1
2π₯)?
(A) x = 2 and x = β1 (B) x = 8 and x = β4
(C) x = 6 and x = β3 (D) x = 6 and x = β12
29. The graphs of π¦ = π(π₯) and π¦ = π(π₯) are shown below. Which
mapping rule would map π¦ = π(π₯) onto π¦ = π(π₯)?
(A) (x, y) β (x + 2, β2y β 1)
(B) (x, y) β (x β 2, β2y β 1)
(C) (x, y) β (x + 2, β2y + 2)
(D) (x, y) β (x β 2, β2y + 2)
30. A function is defined as f(x) = x2 β 5. What is the minimum value of the
function y = βπ(π₯) ?
(A) β5 (B) 0 (C) 1 (D) 5
31. The equation of a function is given as π(π₯) = π₯2 + 1. What is the domain
of π¦ = βπ(π₯)?
(A) π₯ β (ββ, β) (B) π₯ β [1, β)
(C) π₯ β (1, β) (D) π₯ β (ββ, β1]β[1, β)
32. Which set of transformations would map π¦ = βπ₯
onto π¦ β 3 = ββ4(π₯ + 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) π¦ = ββ(π₯ β 2)
(B) π¦ = ββπ₯ β 2
(C) π¦ = ββπ₯ β 2
(D) π¦ = ββπ₯ β 2
34. The graph of a function π¦ = π(π₯) is shown. Which statement about the
function π¦ = βπ(π₯) is true?
(A) It has two invariant points and has range π¦ β€ 1
(B) It has three invariant points and has range π¦ β€ 1
(C) It has two invariant points and has range π¦ π [0, 1]
(D) It has three invariant points and has range π¦ π [0, 1]
35. Which function has domain {π₯/π₯ β₯ β2, π₯ β π } and {π¦/π¦ β€ 3, π¦ β π }?
(A) π¦ β 3 = ββπ₯ + 2 (B) π¦ + 3 = ββπ₯ β 2
(C) π¦ β 3 = βπ₯ β 2 (D) π¦ + 3 = βπ₯ + 2
36. The function π¦ = π(π₯) contains the point (9, 4). Which point must lie on
the graph of π¦ = βπ(π₯)?
(A) (3, 2) (B) (9, 2) (C) (3, 4) (D) (2, 3)
37. The point (π, β2) lies on the graph of the function π¦ = 2βπ₯ β 1 β 6 ?
Which value is correct for π?
(A) 2 (B) 3 (C) 5 (D) 50
38. Which mapping rule would map π¦ = βπ₯ onto β1
3(π¦ + 2) = β2π₯ + 6 ?
(A) (π₯, π¦) β (1
2π₯ β 6, β3π¦ β 2) (B) (π₯, π¦) β (
1
2π₯ β 3, β3π¦ β 2)
(C) (π₯, π¦) β (2π₯ β 6, β1
3π¦ + 2) (D) (π₯, π¦) β (2π₯ β 3, β
1
3π¦ + 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 π¦ = βπ₯ 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: π₯ β€ β4 Range: π¦ β€ 2
(B) Domain: π₯ β€ β4 Range: π¦ β₯ 2
(C) Domain: π₯ β₯ β4 Range: π¦ β€ 2
(D) Domain: π₯ β₯ β4 Range: π¦ β₯ 2
40. Convert 160Β° to radians.
(A) 9Ο
16 (B)
9Ο
8 (C)
8Ο
9 (D)
5Ο
6
41. Which best approximates the value of cot(200Β°) + csc(3)?
(A) 0.3273 (B) 1.7374 (C) 9.8336 (D) 21.8548
42. If πππ‘π < 0 and π πππ > 0, in which quadrant would the terminal arm of
π lie?
(A) I (B) II (C) III (D) IV
43. What is the length of the arc cut by a 120Β° sector in a circle
having radius 6 cm?
(A) 4Ο (B) Ο
9 (C)
Ο
3 (D) 5Ο
44. Which expression represents all angles co-terminal with 2π
3 ?
(A) 2Ο
3+ kΟ, k β R (B)
2Ο
3+ kΟ, k β I
(C) 2Ο
3+ 2kΟ, k β R (D)
2Ο
3+ 2kΟ, k β I
45. Which represents the missing coordinate if the point π (π₯,3
4)
lies on the terminal arm of the unit circle in the second quadrant?
(A) β1
4 (B) β
β7
4 (C) β
5
4 (D) β
β2
2
46. Which represents a circle centered at the origin with a radius of 5β3 ?
(A) π₯2 + π¦2 = 15 (B) π₯2 + π¦2 = 5β3
(C) π₯2 + π¦2 = 75 (D) π₯2 + π¦2 = 30
47. What is the measure of the central angle if P(ΞΈ) = (β2
2, β
β2
2)?
(A) Ο
4 (B)
5Ο
4 (C)
5Ο
3 (D)
7Ο
4
48. What is the exact value of: csc4Ο
3 ?
(A) ββ3
2 (B) β
1
2 (C) β2 (D) β
2β3
3
49. If the point P(5, β12) lies on the terminal arm of Σ¨, then which
represents the ratio for sec Σ¨ ?
(A) β13
12 (B)
13
5 (C) β
12
13 (D)
5
13
50. If πππ‘ π =7
24, π < π <
3π
2, then which ratio is true?
(A) πππ π =7
25 (B) ππ ππ =
25
24
(C) π πππ = β25
24 (D) π πππ = β
24
25
51. Solve for x: cos Σ¨ = 1
(A) kΟ, k β I (B) 2Οk, k β I
(C) Ο
2+ kΟ, k β I (D)
Ο
2+ 2Οk, k β I
52. If π ππ π = β2.5, 0Β° β€ π β€ 180Β°, what is a possible value of π?
(A) 23.6Β° (B) 66.4Β° (C) 113.6Β° (D) 156.4Β°
53. Which represents the exact value of π‘ππ2 5π
6+ πππ‘2 5π
6 ?
(A) 3
10 (B)
10
3 (C)
4
3 (D) 1
54. Solve for ΞΈ: 2sin2ΞΈ = sin ΞΈ where 0 β€ ΞΈ β€ 2Ο.
(A) {Ο
6,
5Ο
6} (B) {
Ο
3,
2Ο
3}
(C) {0,Ο
6,
5Ο
6, Ο, 2Ο} (D) {0,
Ο
3,
2Ο
3, Ο, 2Ο}
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 (5Ο
6ΞΈ) (B) y = 3sin (
6
5ΟΞΈ)
(C) y = 3sin (5
12ΞΈ) (D) y = 3sin (
12
5ΞΈ)
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 π(π₯) = 3π₯2 + 6π₯ β 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 ππ ππ
3+ πππ‘
11π
4
69. Solve for x:
(a) 2πππ π₯ + β3 = 0, π₯ β [0,180Β°)
(b) ππ ππ₯ + 12 = 2 β 4ππ ππ₯, π₯ β [π
2,
5π
2)
(c) 3 csc2x β 4 = 0 where 0 β€ x β€ 2Ο
(d) 2π ππ2π₯ + 3π πππ₯ = β1, π₯ β [β180Β°, 540Β°)
(π) 3π‘ππ2π₯ = 14 tan π₯ + 5, π₯ β [0,3π)
(π) 2π ππ2(π₯) = sec(π₯) + 15, where π₯ β [β2π, 2π)
70. Determine the length of π΄οΏ½ΜοΏ½ (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 = β34 58. β1, β1, 2,3
2 59. π¦ = β
1
2π₯(π₯ + 2)2(π₯ β 3)
60. 0 < x < 6, a 2m x 2m square must be removed from each corner
61. 62. y = β2f(x + 3)
63.
64. Restriction: x β₯ β1 π¦ = βπ₯+4
3β 1
65. Domain: β1 β€ x β€ 3 Range: 0 β€ y β€ β8
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) (π₯, π¦) β (β1
2π₯ + 4, 3π¦ + 1)
(ii) Domain: x β€ 4 Range: y β₯ 1
(iii)
67.
68. 2β3β3
3
69.(a) 150α΅ (b) 7Ο
6,
11Ο
6 (c)
Ο
3,
2Ο
3,
4Ο
3,
5Ο
3
(d) β30Β°, β90Β°, β150Β°, 210Β°, 270Β°, 330Β°
(e) 1.4 πππ, 2.8 πππ, 4.5 πππ, 5.98 πππ, 7.7 πππ
(f)
β5.05 πππ, β4.3 πππ, β1.98 πππ, β1.23 πππ, 1.23 πππ, 1.98 πππ, 4.3 πππ, 5.05 πππ
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:
{π₯ = 0 πππ π₯ = 2}