Chapter 2 Trigonometry Diagram not drawn to scale.€¦ · Chapter 2 Trigonometry Part 1: ____ 1. What is the reference angle for 200q in standard position? A 100° B 70° C 20°

Mid-Term Review Math 2200 Name:________

Chapter 2 Trigonometry

Part 1: ____ 1. What is the reference angle for 200 in standard position?

A 100° B 70° C 20° D 110°

____ 2. If the point P(-3, 4) is on the terminal arm of Ѳ, find Ѳ.

A 53° B 127° C 233° D 307°

____ 3. The point (40, –9) is on the terminal arm of A. Which is the set of exact primary

trigonometric ratios for the angle?

A

, , B , ,

C , , D , ,

____ 4. The coordinates of a point P on the terminal arm of an angle are shown.

What are the exact trigonometric ratios for , , and ?

A

B

C

D

____ 5. Determine the length of x, to the nearest tenth of a centimetre.

A 26.6 C 11.2

B 36.5 D 17.1

____ 6. If , c = 10.3 cm, and b = 10.5 cm, and ABC is acute,

what is the measure of , to the nearest tenth of a degree?

A 57 C 30.5

(–8,6)

Diagram not drawn to scale.

x

y

A

aB C

bc


40º

70º

25 cm

x


Math 2200 Mid-Term Review 2

____7.Determine the measure of x, to the nearest tenth of a degree.

A 25.6 C 136.3

B 18.1 D 71.9

____ 8. What is the length of x, to the nearest tenth of a metre?

A 27.7 m C 26.1 m

B 21.8 m D 37.6 m

____ 9. If , r = 20 cm, and p = 23 cm, what is the length

of q, to the nearest centimetre?

A 21 cm C 12 cm

B 30 cm D 11 cm

____10. Solve the following triangle, rounding side lengths to the

nearest tenth of a unit and angle measures to the nearest degree.

A , , c = 5.0

B , , c = 5.0

C , , c = 28.7

D , , c = 28.2 , b = 19, a = 23.5

____ 11. If ,5.4,22 WYY and WXWY ,

what is the length of XY in WXY ?

A 2.4 C 4.5

B 8.3 D 11.1

Part 2: 12. The point ( – 3 , 4 ) lies on the terminal arm of an angle θ in standard position.

Determine the exact trigonometric ratios of sin θ, cos θ, tan θ.

40 cm

x

25 cm18 cm


x

18 m33 m


52º

q

pr


Q

P R

22 °

4.5

X

Y

W


13. Solve each of the following trigonometric equations for 0° ≤ θ < 360°.

(a) sin θ =2

1

(b) cos θ =

2

3

14. George (G) and Henry (H) are getting ready

to walk to the store (S) and George is 7.3 km

away from the store. If G = 62° and H = 68°

then how far apart are George and Henry before

traveling to the store?

15. In ΔABC, A = 36°, a = 20 cm and b = 27 cm. Solve the triangle by finding all the

missing sides and missing angles (to the nearest degree).

16. The point ( 2 , – 3 ) lies on the terminal arm of an angle θ in standard position.

Determine the exact trigonometric ratios of sin θ, cos θ, tan θ.

17. Solve the following trigonometric equation for 0° ≤ θ < 360°.

(a) sin θ =2

3

(b) cos θ =

2

1

18. Determine the value of x in the following diagram.

19. Two ships leave a drilling platform at the same time traveling in different directions.

Both ships are traveling at a constant speed of 2 km/h and 3 km/h, respectively.

If after 4 hours the ships are 16 km apart, determine the angle formed by the

courses of the ships, to the nearest degree.

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

12. sin θ =5

4 cos θ =5

3 tan θ =3

4 13. (a) θ = 30° , 150° (b) θ = 150° , 210°

14. 6.03 km 15. Acute Triangle: B = 53° , C = 89° , c = 34.02

Obtuse Triangle: B = 127° , C = 17° , c = 9.95

16 sin θ =13

3 cos θ =13

2 tan θ =2

3

17(a) θ = 240° , 300° (b) θ = 60° , 300°

18. x = 10.67 19. 104°

S

G H

A B

C

D

x

75° 30°

36°

9

11


Chapter 3: Quadratic Functions

____ 1. What is the axis of symmetry of ?

A x = 2 B x = –3 C x = –6 D x = 6

____ 2. What is the quadratic function in vertex form for the

parabola shown to the right?

A C

B D

____ 3. What is the vertex of ?

A (5, 4) B (–4, 5) C (–5, 4) D (7, –4)

____ 4. Which graph represents the quadratic function ?

A

B

C

D

1 2 3 4 5 6 7 8 9 10 11–1–2–3–4–5–6–7–8–9–10–11 x

1

2

3

4

5

6

7

8

9

10

11

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y


____ 5. What are the domain and range of ?

A Domain:

Range:

C Domain:

Range:

B Domain:

Range:

D Domain:

Range:

____ 6. The vertex of a parabola is located at . If the parabola has a y-intercept of 231, which

quadratic function represents the parabola?

A C

B D

____ 7. What information can be determined from the quadratic function ?

A the vertex is at (–2, –9) and the graph opens upward

B the vertex is at (–9, –2) and the graph opens downward

C the vertex is at (–2, –9) and the graph opens downward

D the vertex is at (–9, –2) and the graph opens upward

____ 8. Identify the characteristics of this graph.

A vertex: (–2, –5)

axis of symmetry:

y-intercept: 10.5

x-intercepts: –3 and –7

opens downward

C vertex: (–2, –5)

axis of symmetry:

y-intercept: 10.5


opens upward

B vertex: (–5, –2)

axis of symmetry:

y-intercept: 10.5


opens upward

D vertex: (–5, –2)

axis of symmetry:

y-intercept: 10.5

x-intercepts: 3 and 7

opens downward

2 4 6 8 10 12 14 16 18 20 22–2–4–6–8–10–12–14–16–18–20–22 x

2

4

6

8

10

12

14

16

18

20

22

–2

–4

–6

–8

–10

–12

–14

–16

–18

–20

–22

y


____ 9. What are the coordinates of the vertex of the quadratic function ?

A (–6, –1) B (8, –2) C (–1, –6) D (8, –6)

____10. What is the equation of the quadratic function in vertex form?

A C

B D

____11. State whether the function has a maximum or minimum value and identify the

coordinates of the vertex.

A maximum at C minimum at

B maximum at D minimum at

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

(a) y = 3( x + 2 )2 – 3 (b) y = 2

1 x2 + 4x – 6

Direction of Opening, Vertex, Equation of the Axis of Symmetry,

Maximum or Minimum Value, X–intercepts, Y–Intercept, Domain, and Range

13. A ball is kicked from an initial height of

1 foot and follows a parabolic path as shown.

After 4 seconds the ball reaches a maximum

height of 33 feet.

(a) Determine the quadratic function that

models the path traveled by the ball.

(b) Determine the height of the ball at 2 seconds.

14. A farmers has a rectangular field that is divided into three regions of equal area.

The farmer decides to enclose the entire field, and regions, using 160 m of fencing.

(a) Determine the function which gives

the area of the entire region as a function

of its width, x, and use this function

to calculate the maximum possible area.

(b) What is the length and width

of the entire rectangular field?

(c) State the domain and the range of the

variables in the function.

fencing

x x x x

x

y

1 ft


15. A theatre seats 200 people per show and is currently sold out with a ticket price

of $6. A survey shows that for every $1 per ticket price increase, 20 fewer people

will attend the show.

(a) Write a function to model the revenue generated from the show and use this

function to determine the maximum revenue the show can make.

(b) Determine the new ticket price that will result in the greatest revenue.

16. An orange grower has 400 crates of oranges ready for market and will have 20 more

crates available each day that shipment is delayed. The present price is $ 60 per

crate, however, for each day that shipment is delayed the price per crate

decreases by $ 2.

(a) Write a function to model the revenue generated from the sale of oranges

and determine the maximum revenue that can be generated.

(b) What will be the new price that will generate the maximum revenue?

Chapter 4 Quadratic Equations

____ 1. How many x-intercepts does the graph of the quadratic

function have?

A unknown C 1

B 2 D 0

____ 2. What are the x-intercepts of the quadratic function shown?

A 2 and –4 C 11.2

B –2 and 4 D 12.6

1 2 3 4 5 6–1–2–3–4–5–6 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

1 2 3 4 5 6–1–2–3–4–5–6 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

29.ANS: Answers 1. C 2. B 3. C 4. B 5. B 6. B 7. A 8. B 9. C 10.D 11. D

12. (a) Up, ( – 2 , – 3 ), x = – 2 , Min. of – 3, x-int: -3 and -1, y-int: 9, Dom: xR, Range: y ≥ -3

(b) Down, ( 4 , 2 ), x = 4 , Max. of 2, x-int: 2 and 6, y-int: -6, Dom: xR, Range: y ≤ 2

13. (a) h(t) = –2( t - 4 )2 + 33 (b) 25 feet

14. (a) A(x) = –2x2 + 80x , Max. Area = 800 m2 (b) L = 40 m , W = 20 m

(c) Domain: 0 < x ≤ 40 , Range: 0 < y ≤ 800

15. (a) R = –20x2 + 80x + 1200 Max Revenue of $ 1280 when x = 2 (b) $ 8

16. (a) R = –40x2 + 400x + 24 000 Max Revenue of $ 25 000 when x = 5 (b) $ 50


____ 3. What are the x-intercepts of the quadratic function

graphed here?

A 4.6 C –2.2 B there are none D 9.0

____ 4. What are the roots of the quadratic function ?

A –0.125 B 4 and 3 C –4 and –3 D 6

____ 5. Factor completely.

A C B D

____ 6. Factor completely.

A C B D

____ 7. Solve .

A x = 18 and x = –3 C x = and x =

B x = –18 and x = 3 D x = –144 and x = 24

____ 8. Determine the roots of the quadratic equation .

A x = –10 and x = –1 C x = 10 and x = 1

B x = –50 and x = –5 D x = and x =

____ 9. Solve .

A and C and

B and D and

____10. A rectangle has dimensions and , where x is in centimetres. If the area of the

rectangle is 72 cm2, what is the value of x, to the nearest tenth of a centimetre?

A x = 2.0 B x = –4.6 C x = 11.2 D x = -11.2

9

4

3

8

21

5

1 2 3 4 5 6–1–2–3–4–5–6 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y


____11. The vertex form of is

A C B D

____12. Which is the vertex form of ? Round coefficients to the nearest hundredth if

necessary.

A C B D

____13. Solve .

A 1 + and 1- C

B –1 + and –1 - D

____14. A rectangle with an area of is x centimetres wide and (x + 8) centimetres long. To

the nearest tenth of a centimetre, the width and length are

A 50.0 cm and 50.0 cm C 46.2 cm and 54.2 cm

B –46.2 cm and –54.2 cm D –14.0 cm and 114.0 cm

____15. When Alex rides his dirt bike off a ramp, his path can be modelled by

, where d is the horizontal distance from the ramp and h is the height, both in metres. How

far away from the ramp does he land, to the nearest tenth of a metre?

A 2.0m B 0.6m C 7.9m D 3.9m

____16. For a science experiment, a projectile is launched. Its path is given by

, where h is the height of the projectile above the ground and d is

the horizontal distance of the projectile from the launch pad, both in metres. How far away

from the launch pad is the projectile when it begins to fall, to the nearest tenth of a metre?

A 255.8 m C 0.3 m

B 7.7 m D 15.7 m

43 43 2 11

43 43 42


17. The trajectory of a rocket is represented by the function h(t) = –3t2 + 18t + 48,

where h is height in m and t is time in seconds.

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

(b) What is the height of the rocket after 2 seconds?

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

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

(e) When will the rocket land back on the ground?

18. A toy rocket is launched in the air from a launcher. The rocket’s path is described by

h(t) = -6t2 + 26t + 30 where h(t) is the height of the rocket above the ground t seconds

after its launch. Determine when the rocket reaches a height of 10 m.

19. A rectangular rug 8 feet by 6 feet is placed in a room

with floor area 120 ft2 such that a strip of bare floor

of uniform width surrounds the rug. Determine the width

of the strip of bare floor.

20. A rectangular garden, measuring 20 m by 12 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 128 m2, what is the width of the strip

that has been removed?

21. Use any method to determine the roots of the following quadratic equations.

(a) 4

1 x2 = 3x + 16 (b) 4( 3x + 1 )2 = 64

(c) 2( 4x + 8 )2 – 8( 4x + 8 ) = 0 (d) 3x2 – 6x – 6 = 0

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

13. B 14. C 15. D 16. B 17. (a) 48 m (b) 72 m (c) 3 sec (d) 75 m (e) 8 18. 5 sec

19. w = 2 m 20. w = 4 m 21(a) -4 , 16 (b)

1,3

5 (c) -2 , -1 (d) 31

Rug

x

x

x

x

20 m

12 m

x

x


Not to scale.

Not to scale.

Chapter 5 Radical Equations

____ 1. Find a simplified expression for the perimeter of this shape.

8

A C B D

____ 2. Simplify .

A 9 + C 9 +

B D 114

____ 3. Simplify .

A 4 + C –36

B 4 + D

____ 4. Express in simplest form.

A C B 252 D

____ 5. Find a simplified expression for the area of this shape.

A C 95

B D 31

____ 6. Express in simplest form.

A C 12

B D

____ 7. Solve .

A x = C x =

B x = D x =

4 5

7 5 2 6

238 2 2

2

2 15

3 7 4 2

3 7 4 2

2 3

25

49

5

7

25

7

5

49


____ 8. Solve .

A x = –6 C x = 6

B x = 24 D x = 12

____ 9. Solve .

A x = C x =

B x = D x =

____ 10. What is 3232 xxy written as an entire radical?

A 3 226 yx C 3 64

18 yx

B 3 4312 yx D 3 64

24 yx

11. Simplify: 2 1

12 4 200 48 6 503 2

12. Simplify: 2 3 6

3 6 2 3

13. A square shaped play ground has an area of 72m2. Determine an exact value for the perimeter

of the playground.

14. Solve for x: 3 13 3 2x x and identify any restrictions

15. The areas of congruent squares A and B are represented by 13 x square units and )1( x

square units, respectively. Algebraically determine the area of each square.

25 1

25

5 1

5

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

12.

13. 14. x=-3,1 ; 15. 25square units

A B

Area =

( x – 1 )

Area =

3 x + 1

Chapter 2 Trigonometry Diagram not drawn to scale.€¦ · Chapter 2 Trigonometry Part 1: ____ 1. What is the reference angle for 200q in standard position? A 100° B 70° C 20°

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