Mathematics 2201

Midterm Exam Review

Chapter 4: Radicals Chapter 6: Quadratic Functions Chapter 7: Quadratic Equations

1. Evaluate: 3 816

(A) – 2 (B) – 4 (C) 2 (D) 4

2. Express 3 32 as an entire radical.

(A) 12

(B) 3 12

(C) 3 24

(D) 24

3. What is the perimeter of the given diagram in simplest radical form?

(A) 710

(B)

714

(C)

718

(D)

78

4. Simplify: 223

(A) 5 – 62

(B) 1 – 62

(C) 5 + 62

(D) 1 + 62

5. Rationalize the denominator: 3

6

(A) 33 (B) 36 (C) 32 (D) 3

6. Determine the width, w, of the given rectangle.

(A) 103 (B) 106

(C) 1012 (D)

1018

7. Simplify: xxx 42 5

(A) xx 82 6 (B) 23 82 xx (C) xx 82 3 (D) 26 82 xx

8. Simplify: 2

6

3

27

x

x

(A) 29x (B) 49x (C) 23x (D) 43x

9. Solve for x:

243 x

(A) x = – 2 (B) x = 2 (C) x = – 4 (D) x = 4

7

28 28

63

Area = 8027

l = 29

w

10. Solve for x:

3633 x

(A) x = 1 (B) x = 2 (C) x = 3 (D) x = 7

11. Solve for x: 682 x

(A) x = 14 (B) x = 22 (C) x = – 14 (D) x = – 22

12. Solve for x:

20104 x

(A) x = 20 (B) x = 10 (C) x = 35 (D) x = 15

13. Perform the operations indicated and express the answer in simplest radical form.

(a) 75284482

1632

(b) 122332233

(c) 24

453202

14. Solve for x: 11593 x

15. Which is the vertex for the quadratic function y = 2(x + 4)2 – 8?

(A) (4, –8) (B) (–4, 8) (C) (–4, –8) (D) (4, 8)

16. Which is the vertex for the quadratic function y = x2 – 2x + 3?

(A) (2, 3) (B) (–2, 3) (C) (–1, 6) (D) (1, 2)

17. Which is a quadratic function?

(A) ( ) 3 4f x x (B) ( ) 2 ( 5)f x x x

(C) 2( ) 0 2 5f x x x D) 3 2( ) 3 7 1f x x x x

18. Which is the y–intercept for the quadratic function f(x) = –(x + 2)(4x – 3)?

(A) –6 (B) –2 (C) 0 (D) 6

19. Which is the y–intercept for the quadratic function y = –3x2 + 12x ?

(A) –2 (B) 2 (C) 0 (D) 12

20. What is the range for the function 3)7(2 2 xy ?

(A) 3y (B) 3y (C) 3y (D) 3y

21. Which represents the range for the graph?

(A) 𝑦 ≤ 9

(B) 𝑦 ≤ 12

(C) Ryy ,62

(D) Ry

x-5 -4 -3 -2 -1 1 2 3 4 5 6 7

y

-5

5

10

15

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

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

22. Which statement about the quadratic function y = –2x2 – 12x – 23 with vertex (–3 , –5) is correct?

(A) There is a minimum value of –23. (B) There is a minimum value of –5.

(C) There is a maximum value of –23. (D) There is a maximum value of –5.

23. A parabola opens down and has x–intercepts at –4 and 6. Which represents the

function in factored form?

(A) y = –(x – 4)(x + 6) (B) y = –(x + 4)(x – 6)

(C) y = (x – 4)(x + 6) (D) y = (x + 4)(x – 6)

24. Which is the function in Vertex Form represented by the graph?

(A) y = –2(x – 3)2 + 1

(B) y = –2(x + 3)2 + 1

(C) y = – 1

2(x – 3)

2 + 1

(D) y = – 1

2(x + 3)

2 + 1

25. Which is the equation of the axis of symmetry for the function y = 2x2 – 12x + 19 ?

(A) x = – 6 (B) x = – 3 (C) x = 3 (D) x = 6

26. Which is the equation of the axis of symmetry for the function y = 4x2 – 7 ?

(A) x = –7 (B) x = 0 (C) x = 4 (D) x= 7

27. Which is the equation of the axis of symmetry for the function y = (x + 4)(x – 6) ?

(A) x = – 5 (B) x = – 1 (C) x = 1 (D) x= 5

28. The height of an object in meters, thrown upward from a window, after t seconds

is given by the equation 24.9 19.6 4h t t When is the maximum height reached?

(A) 4 sec (B) 2 sec (C) 19.6 sec (D) 4.9 sec

29. The area of a rectangular enclosure is given by the function A(x) = – 5x2 + 150x,

where x is the width, in meters. What is the width that will produce a maximum area?

(A) 15 m (B) 5 m (C) 150 m (D) 30 m

30. Which represents the number of x-intercepts for the function . y = – 1

2(x + 4)

2 + 1

(A) 1 (B) 2 (C) 3 (D) none

31. Which quadratic function doesn’t have x – intercepts?

(A) y = 2x2 + 1 (B) y = –3x

2 + 1

(C) y = (x – 2)(x + 3) (D) y = –3(x – 2)2

32. Which represents the quadratic function y = –3(x – 1)(x + 3) in standard form?

(A) y = –3x2 + 2x – 3 (B) y = –3x

2 + 9

(C) y = –3x2 – 6x + 9 (D) y = –3x

2 + 6x – 9

33. The Beatles Fan Club has 6000000 members and charges $5.00 per month. If the Club

raises membership fees by $1.00 per month they expect 10000 fewer members per

month. Which represents the revenue function?

(A) )100005)(16000000( xxR (B) )100005)(16000000( xxR

(C) )15)(100006000000( xxR (D) )15)(100006000000( xxR

34. A farmer constructs a rectangular enclosed fence in an open field using 100m of fencing.

Which quadratic function models the maximum area of the enclosed region?

(A) xxxA )100()( (B) xxxA )2100()(

(C) xxxA )50()( (D) xxxA )250()(

35. Determine the following information from the graph.

Equation of Axis of Symmetry:

Vertex:

Maximum or Minimum Value:

Y – intercept: X – intercepts: ______

Domain: Range:

36. Determine the quadratic function, of the parabola graphed below, in factored form.

x- 2 2 4 6

y

- 10

- 8

- 6

- 4

- 2

2

x- 4 - 2 2 4 6

y

- 6

- 4

- 2

2

4

6

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8

y

- 13- 12- 11- 10

- 9- 8- 7- 6- 5- 4- 3- 2- 1

12345678

t1 2 3 4 5

h(t)

2

4

6

8

10

12

14

16

18

37. The flight of a golf ball is represented by the function h(t) = –3t2 + 12t, where height, h,

is given in meters and time, t, is given in seconds. The path of the ball can be seen in the

graph below.

(a) What is the height of the ball at 3 seconds?

(b) What is the maximum height of the ball?

(c) When does the ball reach its maximum height?

(d) How long is the ball in the air?

(e) What is the domain and the range?

38. Given the function y = 2x2 – 8x + 4 determine the following information and

sketch the graph.

Equation of Axis of Symmetry: ___________

Vertex: __________

Maximum or Minimum value is __________

Number of x – intercepts:

Y–intercept:

Domain:

Range:

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

y

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

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

y

-6

-5

-4

-3

-2

-1

1

2

39. Determine the following information and sketch the graph of the given function.

y = –2( x + 2 )2 – 1

Direction of Opening:

Vertex:

Equation of the Axis of Symmetry:

Maximum or Minimum Value:

Number of x–intercepts:

Y–Intercept:

Domain:

Range:

40. Determine the quadratic function, in vertex form, for the given graph.

41. The trajectory of a rocket is represented by the function h(t) = – 5t2 + 20t + 2, where h is

height in meters and t is time in seconds.

(a) What is the initial height of the rocket before it takes flight?

(b) What is the height of the rocket at 1.5 seconds?

(c) At what time does the rocket reach its maximum height?

(d) What is the maximum height reached by the rocket?

42. A storage space is to be constructed using 400 m of wire mesh fencing. If the warehouse is to

be used as one side of the storage space, what dimensions will produce a maximum area?

What is the maximum area of the storage space?

43. A dairy farmer has a rectangular field for his cows to graze. The farmer decides to surround the

field with 160 m of fencing.

(a) Determine the quadratic function that models the area

as a function of its width, x.

(b) Determine the maximum area of the rectangular field.

(c) What is the length and the width of the rectangular field?

(d) State the domain and the range of the variables in the function.

44. Global Gym charges it members $ 40 for a monthly membership. The club has 600 members.

Global Gym estimates that for each $ 4 increase in the monthly fee will cause them to lose

20 members.

(a) Determine the revenue function.

(b) Determine the maximum revenue that the gym can generate.

(c) What will be the new gym membership fee that will create a maximum revenue?

WAREHOUSE

x x

45. A travel agency offers a group rate of $1800 per person for a week in Ireland if 12 people

sign up for the tour. For each additional person who signs up, the price per person is

reduced by $100.

(a) Determine the revenue function.

(b) Determine the maximum revenue that can be generated.

(c) What will be the new price per person be to generate the maximum revenue?

46. Which represents the roots for: 2x2 –72x = 0?

(A) 36 (B) 0 and 36 (C) 6 and –6 (D) 72

47. Which are the zeros of the quadratic function: y = (3x – 5)( x + 2)?

(A) x = 2 and x = 3

5 (B) x = 2 and x =

3

5

(C) x = –2 and x =3

5 (D) x = –2 and x =

3

5

48. A diver springs off a diving board into a quadratic trajectory that is

modeled by the function h(t) = –t2 + t + 6 where h(t) represents height

in meters and t is time in seconds. Determine the time it takes for the

diver to hit the water.

(A) 0.5 sec (B) 2 sec (C) 3 sec (D) 6 sec

49. Based on the roots of x2 – 2x = 0, which graph represents the quadratic function y = x

2 – 2x?

(A) (B)

(C) (D)

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 4

- 3

- 2

- 1

1

2

3

4

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 4

- 3

- 2

- 1

1

2

3

4

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 4

- 3

- 2

- 1

1

2

3

4

x- 4 - 3 - 2 - 1 1 2 3 4

y

- 4

- 3

- 2

- 1

1

2

3

4

50. Which quadratic function has zeros of 2 and 4

3?

(A) f(x) = ( x + 2 )( 4x + 3 ) (B) f(x) = ( x + 2 )( 4x – 3 )

(C) f(x) = ( x – 2 )( 4x + 3 ) (D) f(x) = ( x – 2 )( 4x – 3 )

51. Which quadratic equation has roots of –3 and 1?

(A) x2 – 2x + 3 = 0 (B) x

2 + 2x + 3 = 0 (C) x

2 – 2x – 3 = 0 (D) x

2 + 2x – 3 = 0

52. What are the zeros of the quadratic equation 25x2 – 9 = 0?

(A) x = 3

5

(B) x =

5

3

(C) x =

9

25

(D) x =

25

9

53. An eagle swoops down to catch a rodent on the ground. If the path traveled by the eagle is

represented by h(t) = t2 – 2t – 8, where h(t) represents height in meters and t is time in

seconds, at what time does the eagle catch the rodent?

(A) 1 sec (B) 2 sec (C) 4 sec (D) 8 sec

54. Which graph represents a quadratic function with two equal, real zeros?

(A) (B)

(C) (D)

55. When will the graph of a quadratic function have two real roots?

(A) When b2 – 4ac > 0 (B) When b

2 – 4ac = 0

(C) When b2 – 4ac < 0 (D) Never

56. Mike used the quadratic formula, as shown below, to solve the quadratic equation x2 – 8x + 9 = 0

He made an error in his calculations. In which step did Mike first make his mistake?

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1: )1(2

)9)(1(4648

Step 2: 2

288

Step 3: 2

748

Step 4: 724

x

yy

x

y

x

y

x

57. A rectangular rug 8 feet by 6 feet is placed in a room such that a strip of bare floor of uniform

width surrounds the rug. If the total area of the bare floor and rug combined is 120 ft2, which

quadratic equation represents this situation?

(A) ( 8 – x )( 6 – x ) = 120

(B) ( 8 + x )( 6 + x ) = 120

(C) ( 8 – 2x )( 6 – 2x ) = 120

(D) ( 8 + 2x )( 6 + 2x ) = 120

58. The sum of the squares of two consecutive even integers is 100. Which equation models

this situation?

(A) x2 + ( x + 2 )

2 = 100 (B) x

2 + ( x + 1 )

2 = 100

(C) x2 + ( x + 2 )

2 = 100

2 (D) x

2 + ( x + 1 )

2 = 100

2

59. Solve the following quadratic equations using factoring:

a) 4𝑥2 = 8𝑥 b) 25𝑥2 − 4 = 0 c) 3𝑥2 = 𝑥 + 10 d) 2𝑥 𝑥 + 2 = 6

e) 𝑥 𝑥 + 10 = 4𝑥 + 27 f) −2.5𝑥2 = 80 − 30𝑥 g) 1

4𝑥2 + 2𝑥 − 5 = 0

60. Solve the following using the quadratic formula:

a) 𝑥2 − 4𝑥 + 2 = 0 b) 𝑥2 + 10𝑥 + 7 = 0

61. Determine the EXACT roots of the following:

a) 𝑥2 − 2𝑥 − 4 = 0 b) 𝑥 𝑥 − 4 = −3 c) 2𝑥2 − 12𝑥 + 14 = 0

d) 3𝑥2 + 11𝑥 = 4 b) e) 2.1𝑥2 = 18.9

62. Determine the quadratic equation, 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, that has the following roots.

a) 𝑥 = 9 𝑥 = 2 b) 𝑥 = −3

4 𝑥 = 1 c) 𝑥 = −

1

4 𝑥 =

3

5 d) 𝑥 = ± 5

63. A javelin is thrown into the air and its height is modeled by the function

h(t) = –4t2 + 22t + 1 where h(t) represents height, in meters, t seconds after

being released.

(a) What is the initial height of the javelin just prior to being released?

(b) What is the height of the javelin at 2 seconds?

(c) At what times does the javelin attain a height of 25 meters?

64. A rocket is fired into the air according to the equation ℎ 𝑡 = −2𝑡2 + 4𝑡 + 48

where 𝑡 is the time in minutes and ℎ is the height in meters.

(a) Determine the time(s) the rocket is at a height of 32 meters.

(b) Determine when the rocket hits the ground.

Rug

x

x

x

x

65. A missile’s path when fired from a ship is given by h(t) = –3t2 + 2t + 8, where h(t) is the

height of the missile in meters and t is the time in seconds.

(a) When does the missile hit the water?

(b) Approximately when does the missile reach a height of 4m?

66. The revenue made by a drama theater is represented by R = –20x2 + 80x + 3200,

where x is the number of shows the drama group performs. How many shows

should the drama group have to make a profit of $ 2000?

67. A rectangular swimming pool has length 30 m and width 20 m. There is a deck of

uniform width surrounding the pool. The area of the pool is the same as the area of the

deck. Write a quadratic equation that models this situation and use it to determine the

width of the deck.

68. Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m. If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x?

Pool

69. A rectangular garden, measuring 20 m by 15 m, has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway. If the area of the remaining garden is 204 m2, what will be the

width of the concrete walkway?

70. (a) The product of two odd consecutive integers is 63. Determine the integers.

(b) The sum of the squares of two even consecutive integers is 52. Determine the integers.

x

x

x

x

x

x

x

x

30 m

20 m

x x

x

x

20 m

10 m

20 m

15 m

x

x

x-2 -1 1 2 3 4 5

y

-5

-4

-3

-2

-1

1

2

3

4

5

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

y

-8

-7

-6

-5

-4

-3

-2

-1

1

2

ANSWERS

1. C 2. C 3. D 4. A 5. C 6. B 7. C 8. C 9. A 10. D 11. B 12. D

13.(a) 3372

(b) 629

(c) 8

105

14. 15x

15. C 16. D 17. B 18. D 19. C 20. C 21. B 22. D 23. B 24. B 25. C 26. B

27. C 28. B 29. A 30. B 31. A 32. C 33. D 34. C

35. Axis of symmetry: x = 3 Vertex: ( 3 , – 9 ) Minimum value of –9 y–intercept = 0

x–intercepts = 0 and 6 Domain: xR Range: y ≥ – 9

36. y = 2

1( x + 2 )( x – 4 )

37(a) 12 m (b) 16 m (c) 2 sec (d) 4 sec (e) domain: 0 ≤ t ≤ 4 range: 0 ≤ h ≤ 16

38. Axis of Symmetry: x = 2

Vertex: ( 2 , – 4 )

Minimum value of –4

Number of x–intercepts: 2

y–intercept = 4

Domain: xR

Range: y ≥ – 4

39. Direction: downwards

Vertex: ( – 2 , – 1 )

Axis of symmetry: x = –2

Maximum value of –1

Number of x–intercepts: 0

y–intercept = – 9

Domain: xR

Range: y ≤ –1

40. y = 2( x + 3 )2 – 4

41(a) 2 m (b) 20.75 m (c) 2 sec (d) 22 m

42. Dimensions: 200 m x 100 m Maximum Area = 20 000 m2

43(a) A(x) = –x2 + 80x (b) 1600 m

2 (c) l = 40 m , w = 40 m (d) Domain: 0 ≤ x ≤ 80 Range: 0 ≤ A ≤ 1600

44(a) R = –80x2 + 1600x + 24 000 (b) $32 000 (c) $ 80

45(a) R = –100x2 + 600x + 21 600 (b) $22 500 (c) $1500

46. C 47. C 48. C 49. A 50. C 51. D 52. B 53. C 54. C 55. A 56. C 57. D 58. A

59(a) x = 0 , 2 (b) x = 5

2

(c) x =

3

5 , 2 (d) x = – 3 , 1 (e) x = – 9 , 3 (f) x = 4 , 8 (g) x = – 10 , 2

60(a) x = 22 (b) x = 235

61(a) x = 51 (b) x = 1 , 3 (c) x = 23 (d) x = – 4 , 3

1 (e) x = – 3 , 3

62(a) x2 – 11x + 18 = 0 (b) 4x

2 – x –3 = 0 (c) 20x

2 – 7x – 3 = 0 (d) x

2 – 5 = 0

63(a) 1 m (b) 29 m (c) 1.5 sec , 4 sec

64(a) 4 seconds (b) 6 seconds

65(a) 2 seconds (b) 1.54 seconds

66. 10 shows

67. 4x2 + 100x – 600 = 0 , width = 5 m

68. 4x2 + 60x – 100 = 0 , width = 1.51 m

69. width = 3 m

70(a) { – 9 , – 7 } , { 7 , 9 } (b) { – 6 , – 4 } , { 4 , 6 }