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Unit 1-Introduction to Quadratic Function (26 marks)

1. State domain, range and if the following relation is a function. (12 marks)

a) 22, xRxD

33, yRyR

Not a function

b) 11,5,3,2D

7,3,2,1R

Not a function

c)

5,4,3,2,1D or 51, xNxD

29,18,10,6,4R

Function

d) RxD

RyR

Function

2. A function h is defined by )7()( xxxh . Evaluate the following. (2 marks)

a) h (2) b) h (-3)

=2(7-2)=10 =-3(7+3)=-30

3. Describe the transformations that have been applied to the graph of 2xy , in order to obtain

the graph 6)2(2 2xy . (4 marks)

VR about x-axis. VS by factor of 2. HT left 2 units. VT down 6 units.


4. Graph and state domain and range. (8 marks)

Unit 2-The Algebra of Quadratic Expressions (18 marks)

1. Simplify. (3 marks)

a) 3x(2x – 1) – 4x(x + 3) b) (3y – 1)(2y – 3) c) (3x – 2)2

=6x2 – 3x -4x

2 – 12x =6y

2 – 11y + 3 =9x

2 – 12x + 4

=2x2 – 15x

2. Factor Completely. (15 marks)

a) 20x-15 b) xyxx 2614 23 c) 12x2y + 3xy – 21xy

2

=5(4x – 3) =2x(7x2 + 3xy – 1) =3xy(4x + 1 – 7y)

d) x2 + 2x –24= e) 42132 mm f) 3652 cc

=(x + 6)(x - 4) =(m - 6)(m – 7) =(c + 9)(c – 4)

g) 21172 2 aa h) 5x 6172 x i) 21236 2 xx

=(2a - 3)(a - 7) =(5x - 2)(x – 3) =(3x - 7 )(2x - 3)

j) 4129 2 xx k) 81364 2 xx l) 366025 2 xx


=(3x – 2)2 =(2x + 9)

2 =(5x – 6)

2

m) 16x2-25 n) 22 1449 yx o) 4x

2-1

=(4x + 5)(4x – 5) =9(x2 – 16) =(2x + 1)(2x – 1)

=9(x + 4)(x – 4)

Unit 3-Working with Quadratic Functions: Standard and Factored Forms (32 marks)

1. Solve the following quadratic equations by factoring. (15 marks)

a) 02452 xx b) 03522 xx c) 01272 xx

(x-8)(x+3)=0 (x+7)(x-5)=0 (x+4)(x+3)=0

x=8, -3 x=-7, 5 x=-4, -3

d) 0352 2 mm e) 010116 2 aa f) 025204 2 aa

(2m-1)(m+3)=0 (3a+2)(2a-5)=0 (2a-5)2=0

m= ½ , -3 m=-2/3, 5/2 a=5/2

g) 0204 2 xx h) 0169 2x

4x(x-5)=0 (3x+4)(3x-4)=0

x=0,5 x=-4/3, 4/3

2. Find the zeroes, the axis of symmetry, the y-intercept and the vertex of the following

quadratic functions. Use these points to sketch the function. (12 marks)

a) )4)(2()( xxxf b) )3)(52()( xxxf

0=(x+2)(x-4) 0=(2x-5)(x+3)

x=-2, 4 x=5/2, -3

x=(-2+4)/2=1 x=(5/2-3)/2=-1/4 (-0.25)

y=(1+2)(1-4)=(3)(-3)=-9 y=(2(-1/4)-5)(-1/4+3)=-121/8 (-15.125)

V(1, -9) V(-1/4,-121/8) (-0.25, -15.125)


3. At a school supplies store, the weekly revenue function for binders sold can be modelled

with 2400213)( 2 ccxR , where both the revenue R(x), and the cost, c, of a binder are in

dollars. (5 marks)

a) Write the function in factored form.

b) When will the revenue be zero?

c) What cost will give the maximum revenue?

d) What is the maximum revenue?

a) R(c)=-3(c-32)(c+25)

b) 0=-3(c-32)(c+25)

C=32, -25 (inad.) Zero revenue if binders cost $32.00.

c) C=(32-25)/2=3.5 Max revenue if binders cost $3.50.

d) R(3.50)=$2436.75 Max revenue is $2436.75.

Unit 4-Working with Quadratic Models: Standard and Vertex Forms (44 marks)

1. Provide an example of a quadratic function in each of the following forms. What

information is known about the parabola in each form? (6 marks)

a) Standard Form b) Vertex Form c) Factored Form

f(x)=2x2-3x+1 f(x)=2(x-5)

2+1 f(x)=-2(x-3)(x+1)

y-intercept Vertex x-intercepts

VR/VS/VC VR/VS/VC VR/VS/VC


2. Change the following to qpxay 2)( form and state the vertex. (12 marks)

a) 462 xxy b) 182 xxy c) 342 2 xxy

d) 2164 2 xxy e) 2303 2 xxy f) 2605 2 xxy

a) y=(x-3)2-5 V(3,5) b) y=(x-4)

2-15 V(4,-15) c) y=2(x-1)

2+1 V(1,1)

d) y=4(x+2)2-18 V(-2,-18) e) y=-3(x+5)

2+73 V(-5,73) f) y=-5(x-6)

2+182 V(6,182)

3. At a clothing store, the weekly revenue function for t-shirts sold can be modelled with

7500903)( 2 ccxR , where both the revenue R(x), and the cost, c, of a t-shirt are in

dollars. (5 marks)

a) Write the revenue function in vertex form.

b) What should the clothing store charge for a t-shirt in order to maximum revenue?

c) What is the maximum revenue?

a) R(x)=-3(c-15)2+8175

b) Maximum revenue is $8175.

c)

d) Charge $15 to maximize revenue. 4. At a large sporting goods store, the weekly profits for soccer balls can be modelled with

8625150030)( 2 xxxP , where both the profit P(x) and the price, x, of a soccer ball are in

dollars. (5 marks)

a) Write the profit function in vertex form.

b) What should the sporting goods store charge for a soccer ball in order to maximize

profit?

c) What is the maximum profit?

a) P(x)=-30(x-25)2+7000

b) Charge $25 to maximize profit.

c) Maximum profit is $7000.


5. Solve the following quadratic equations by using the Quadratic Formula (accurate to 2

decimal places) (6 marks)

a) 0172 2 xx b) 0623 2 mm c) 4.2 4.23.12 kk

4

417x

6

762x

4.8

01.423.1x

=3.35, 0.15 =1.12, 1.79 0.62, -0.93

6. A stone is thrown into the air from a bridge over a river. The stone falls into the river. The

height of the stone, h meters, above the river t seconds after the stone is thrown is given by

7105 2 tth . (2 marks)

When does the stone hit the water to the nearest tenth of a second?

10

24010t

-0.5 (inad.), 2.5 Therefore, stone hits water after 2.5s.

7. The height of a ball at a given time can be modelled with the quadratic function

1205)( 2 ttth , where h(t), is in metres and time, t, is in seconds. How long is the ball in

the air to the nearest hundredth of a second? (2 marks)

10

42020t

-0.05 (inad.), 3.05 Therefore, ball is in the air for 3.05s.

8. Find the value of the discriminate and state the number of real roots. (6 marks)

a) 0332 2 xx b) 0122 xx c) 0253 2 xx

D=-15 No real roots D=0 1 Real root D=49 2 Real Roots

Unit 5: Trigonometry and Acute Triangles (23 marks)

1. Find x using the primary trigonometric ratios (not Sine or Cosine Law). (8 marks)

a) b) 13 c) d)

32

75

x

x 4m

7cm

x

2 7m

10 m

x


x

o 432sin

13

2cos x

x

o 775tan

7

10tan x

x=7.5m <X=56o x=1.9cm <X=55

o

2. Find the value of x using the Sine or Cosine Law. (6 marks)

a) b) c)

40cos)8)(10(2810 222x )13)(15(2

121315cos

222

x 4.10

75sin

3.8

sin x

x=6.4 <X=50o <X=50

o

3. The shadow of a building that is 9 m tall measures 7 m in length. Determine the angle of

elevation of the sun. (2 marks)

7

9tan x <x=52

o The angle of elevation of the sun is 52

0.

4. A triangular backyard is enclosed by a fence. Two sides of the fence are 13 m and 17 m in

length separated by an angle of 70 . What is the distance between the two ends of the fence?

(2 marks)

70cos)17)(13(21713 222x

x=17.5 m The distance between the two ends of the fence is 17.5 m.

A

B C

L

M N

P

Q R

10.4 cm 8.3 cm

75 40

8

10 12

13 15 x

x

x


5. The theatre at point T is 8 km from Arial’s house at point A. From Bob’s house at point B,

the theatre and Arial’s house are separated by an angle of 25 . From Arial’s house, the

theatre and Bob’s house are separated by an angle of 65 . How far is Bob from the theatre?

Draw a diagram. (2 marks)

25sin

8

65sin

x

X=17.2 km Bob is 17.2 km from the theatre.

6. Given triangle ABC, if a = 57 m, c = 105 m and °, what is the area of triangle ABC

to the nearest square metre? Draw a diagram. (3 marks)

105

35sinho h=60.22552582 427486.1716)22552582.60)(57(

2

1A

The area of triangle is 1716 m2.

Unit 6: Sinusoidal Functions (26 marks)

1. State the amplitude, equation of the axis and range for the following trigonometric functions.

(6 marks)

a) 260sin2y b) 190sin3y

amp=2 amp=3

equ of axis: y=-2 equ of axis: y=1 04, yRyR 42, yRyR

2. Graph the following trigonometric functions. (4 marks)

a) 190sin3y b) 130sin2y


3. Given 260sin5)(xf , explain how the graph of f(x) differs from the graph of sin x.

(4 marks)

VR about x-axis, VS by factor of 5, HT right 60o, VT up 2 units

4. A ferris wheel has a radius of 15 m and rotates once every 1 minute. At its lowest point, a

rider is 1 m above the ground. (6 marks)

a) Graph the height of a passenger above the ground for 2 complete revolutions of the

wheel.

b) What are the period, range, amplitude, and the equation of the axis?

Period=1 minute


311, yRyR

Amp=15 m

Equation of Axis: h=16

5. A bicycle has wheels with a diameter of 20 cm. It takes 1.2 s to complete one cycle.

(6 marks)

a) The outside of the tire has a speck of paint on it. Sketch the graph of h against t for 2

cycles. Assume the speck is at its highest point when t = 0.

b) What are the period, range, amplitude, and the equation of the axis?

Period=1.2s 200, yRyR

Amp=10 cm

Equation of Axis: h=10

Unit 7: Exponential Functions (24 marks)

1. Evaluate. (12 marks)

a) 42 66 b) 4

2

5

5 c)

42

1 d)

3

3

2 e) 00

)3(22

36

16 2 2552 1624

8

27

2

33

1+2=3

f) 2

1

16 g) 3

2

8 h) 5

3

)32( i) 3

2

125 j) 3

2

8

27

4 4 -8 25

1

4

9

k) 2

1

25

9 l) 21 22

3

5

4

3


2. Simplify. (2 marks)

a) 3

24

5

55 b)

23

25

2

2

125 16

1

3. Use transformations to graph the following. State the domain and range for each. (6 marks)

a) 12)( 2xxf b) )3(2

1)( xxf

4. A small country that had 2 million inhabitants in 1990 has experienced an average growth in

population of 4% per year since then. Assume that the growth rate remains stable to answer

the following. (2 marks)

a) Write an equation that models the population, P, of this country as a function of the

number of years since then.

b) Use your equation to determine the population in 2005.

a) P(n)=2 000 000(1.04)n

b) P(15)=2 000 000(1/04)15

=3 601887 Population will be 3 601 887 after 15 years.

5. A computer loses 20% of its value every 3 months after it is purchased. (2 marks)

a) Write an equation that models the value of the computer, V, as a function of the number

of months since it was purchased, given a purchase price of $2500.

b) Determine the value of the computer after 2 years.

a) 3)80.0(2500)(

n

nV

b) 43.419$)80.0(2500)( 3

24

nV Value of computer after 2 years is $419.43.


Unit 8: Financial Applications (13 marks)

1. What regular deposit must Connie make at the end of each year into an account that pays

2.3%/a compounded annually to have an amount of $15 000 after 10 years? (2 marks)

22.1351$1023.1

)023.0(1500010

R

2. Jane invests $1500 for 6 years at 6% per annum, compounded quarterly. How much will she

have saved? (2 marks)

25.2144$)015.1(1500 24A

3. Sam plans to deposit $50 every month for the next 4 years. What amount will he have after

4 years if it is invested at 3%/a compounded monthly? (2 marks)

56.2546$0025.0

)10025.1(50 48

A

4. John wants to have $2000 saved up for school in 2 years. If money is invested at 4% per

year, compounded semi-annually, how much must he invest now? How much interest will

he make? (3 marks)

69.1847$02.1

20004

P I=2000-1847.69=$152.31

5. How much does Sarah have to deposit now in order to receive $1000 every 6 months for the

next 3 years if interest is 2%/a compounded semi-annually? (2 marks)

48.5795$01.0

)01.11(1000 6

P

6. Don just received a bonus of $5000 and wants to spend it in equal amounts every month for

the next 2 years. How much can Don spend every month if interest is 2.4% per year,

compounded monthly? (2 marks)

58.213$002.11

)002.0(500024

R

unit 1-introduction to quadratic function · unit 1-introduction to quadratic function ... mcf3m1...

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