MCF 3MIEXAM REVIEW

WATERLOO-OXFORD DISTRICT SECONDARY SCHOOLFINAL EXAMINATION

Department: Mathematics Date: Thursday June 20 – Section 2Course: MCF 3MI Friday June 21 – Section 3Sections: 02, 03 Length: 90 minutesTeachers: Mr. G. Albrecht Page: 1 of 8

Student Name:______________________________ Mark: ______ / 130

PERMITTED AIDS

1. A scientific calculator.No iPods, cell phones or other electronic storage devices may be used as a calculator.

2. Show clear and thoughtful solutions.

3. All work is to be done on the exam paper.

FORMULAS:

Quadratics

x=−b±√b2−4 ac2 a y = a (x – h)2 + k

y=a(x−r )(x−s) y=a x2+bx+c

Trigonometry

SOH-CAH-TOA

asin A

= bsin B

= csin C

a2=b2+c2−2bc coscos A

coscos A=b2+c2−a2

2bc

Sinusoidal Functions

y = a sin (x – d) + c

Exponential Growth/Decay

y=a(b)x

Finance

I = Prt A=P (1+i )n P= A(1+i )n

=A (1+ i)−n

2

A=R [(1+i )n−1 ]

iPV =

R[1−(1+i)−n ]i

Unit 1: Functions

Definition: A function is a relation where every x-value maps to only one y-value

Example 1: Is the relation a function? State the domain and range.

a) {(1, 2), (3, 2), (-1, 0), (1, 4)}

b) c)

Example 2: Graph the quadratic function then state the following

vertex:

axis of symmetry:

max or min value:

domain:

range:

x-intercepts:

y-intercept:

Example 3: For the function f(x) = 2x2 - 1, determine

3

a) f(-3) b) f(x + 3)

Unit 1: Functions – Practice Problems

1. State the domain and range for the following functions. (a) (b)

2. Determine whether the following relations are functions. State the domain and range.

(a) (b) (c)

3. If , determine

(a) (b)

4. A relation is given by . Evaluate.

(a) (b) (c)

5. In words, describe the transformations to the graph to get ,

if g ( x )=12(x+4 )2−3.

6. Graph each of the following and then state domain and range.

(a) (b) (c) 7. Create a first- and second-difference table for the following data.

x y-1 10 21 -32 -143 -31

(a) What conclusion can be made from the first difference?

4

(b) What conclusion can be made from the second difference?

EXTRA QUESTIONS – Chapter 1 p. 186 # 1 – 8.Unit 2: The Algebra of Quadratics

1) Expand and simplifya) (3x - 1)(2x + 5) b) 2(a + 1) - 3(a + 4)2

2) Factora) 4a2b3 - 6a3b + 8a4b2 b) x2 - 3x - 28

c) x2 + 8xy + 16y2 d) 9x2 - 16y2

e) 8m2 + 6m – 5 f) -30x3 - 62x2y - 28xy2

Unit 2: The Algebra of Quadratics – Practice Problems

1. Expand and simplify.

(a) (b) 2. Common factor each of the following polynomials.

(a) (b) 3. Factor fully.

(a) (b) (c)

(d) (e) (f)

4. What are ALL possible integer values, , such that can be factored?5. Factor fully.

(a) (b)

5

6. Name an integer, , such that the quadratic can be factored.

EXTRA QUESTIONS – Unit 2 p. 122 # 1b, 5, 6, 8, 9. p. 186 # 9 – 11.

Unit 3: Standard and Factored Forms of the Quadratic Function

Vertex Form

Standard Form

Factored FormExample 1: Solve by factoring

a) 0 = 2y2 - 8y b) 5 = 12x2 + 28

Example 2: Consider the quadratic Function f(x) = 2x2 - 8x – 42. Complete the chart

Factored form

x-intercepts

y-intercept

Axis of symmetry

Vertex

Max or min and its value

Domain

Range

6

Example 3: You throw your geography textbook off a cliff. Its flight is modelled by h(t)= -5t2 + 10t + 40

where h is the height in metres above the water below and t is the time in seconds.a) How high is the cliff?

b) When does the book hit the water below?

Example 4: Determine the equation in factored form of the graph.

Unit 3: Standard & Factored Forms of the Quadratic Function – Practice Problems

1. Write each of the following in standard form.

(a) (b) 2. Write each of the following in factored form.

(a) (b) (c) 3. Determine the zeros, the axis of symmetry, and the maximum and minimum value for each of the

following quadratic equations. Show your work.

(a) (b) 4. Find the equation of the following parabolas. Leave your answer in factored form.

(a) (b) The function has zeros at x = 2 and x = 7 and passes through the point (0, – 4).

5. Can all quadratic equations be solved by factoring? Explain.6. Solve for by factoring. Show your work.

7

(a) (b) 7. A firecracker is fired from the ground. The height of the firecracker at a given time is modelled by

the function , where is the height in metres and is time in seconds. (a) When will the firecracker hit the ground?(b) What is the maximum height of the firecracker?(c) When does the firecracker reach a maximum height?

(d) When will the firecracker reach a height of ?

8. The population of a city is modeled by the function , where is the population in thousands and is time in years. NOTE: represents the year 2000. According to the model,(a) in what year will the population reach 312 000?(b) will the population reach over 2 million people by the year 2050? Show your work.

EXTRA QUESTIONS – Chapter 3p. 188 #12-18Unit 4: Standard and Vertex Form

Example 1: Consider the function a) Describe the transformations

b) Graph the function using transformations

c) State the features

vertex: _________________

axis of symmetry: ______________

max or min value: _______________

x-intercepts: _________________

y-intercept: __________________

domain: ___________________

range: ____________________

Example 2: Convert to vertex form by completing the square: f(x) = –2x2 – 20x + 4

8

Example 3: Solve using the quadratic formula

0 = 3x2 – 4x – 5 Remember the discriminant?

Remember for word problems, there are three types...1) Given x, find y

2) Given y, find x

3) What is the max or min?

Unit 4: Quadratic Models: Standard & Vertex Forms - Practice Problems

1. Write the function in standard form.

2. For the function , complete the table:VertexAxis of SymmetryMax/Min ValueDomainRange

3. Use the vertex form to determine the equation of the parabola. Leave your answer in vertex form.

4. Write each function in vertex form and state the vertex.

(a) (b)

5. The cost, , of operating a cement-mixing truck is modeled by the function

9

, where is the number of minutes the truck is running. What is the minimum cost of operating the truck? Show your work.6. Solve using the quadratic formula. State your answers correct to 2 decimal places.

(a) (b)

7. A theatre company’s profit can be modeled by the function where is the price of a ticket in dollars. What is the break-even price of the tickets?

8. A model rocket is launched into the air. Its height, , in metres after seconds is

.(a) When is the rocket at a height of 62 m (correct to 2 decimal places)?(b) What is the height of the rocket after 6 seconds?(c) What is the maximum height of the rocket?

9. Without solving, determine the number of solutions of each equation. Show your work for full marks.

(a) (b) (c) EXTRA QUESTIONS – Chapter 4 p. 382 # 1-5, 16-24

Unit 5: Trigonometry

In what situation is each formula used?

Pythagorean Theorem a2+b2=c2

Primary Trig Ratios SOH CAH TOA

Sine Law

asin A

= bsin B

= csin C

sin Aa

=sin Bb

=sin Cc

Cosine Law a2=b2+c2−2 bc cos Acos A=b2+c2−a2

2bc

Example 1: Find x Example 2: Find angle A

Example 3: Solve the triangle

10

Unit 5: Trigonometry - Practice Problems

1. Use a calculator to evaluate to four decimal places.(a) (b) (c)

2. Use a calculator to find to the nearest degree.(a) (b)

3. Determine all the interior angles in correct to the nearest degree.

4. Solve where and . Include a diagram.

5. A ladder can be used safely only at an angle of with the horizontal. How high, to the nearest metre, can the ladder reach? Include a diagram.

6. A surveyor wants to calculate the distance across a river. He selects a position, , so that

is , and he measures and as and , respectively. Calculate the distance to the nearest tenth of a metre.

7. Two sides of a parallelogram measure and . The longer diagonal is long. How long, to the nearest centimeter, is the other diagonal? (Include a diagram).

8. A temporary support cable for a radio antenna is long and has an angle of elevation of . Two other support cables are already attached, each at an angle of elevation of . How long, to the nearest centimetre, is each of the shorter cables?

EXTRA QUESTIONS – Chapter 5 p. 382 #6-9,27-29

11

Unit 6: Sinusoidal Functions

What makes a graph periodic? sinusoidal? What are the five key points of the sine function?

Graph y = sinθ and list the important key features:

Period:

Amplitude:

Equation of Axis:

Domain:

Range:

Transformations of y = sin θ

How does each letter affect the graph of y = sin θ? Key Features?a

c

d

Example 1: Write the equation of a sine function that has an amplitude of 0.25, a phase shift of 30o to the right and an equation of the axis at y = -3.

Example 2: Graph y = 3 sin (x + 45o) + 1Amplitude:

Period:

Equation of Axis:

Domain:

Range:

12

Example 3:Based on graph below showing the height of a person on a swing, determine the following:

How high off the ground is the swing?

What is the period of the graph?

What does the period represent?

Assuming the person first starts going fowards on the swing, at what height will they be after 15 seconds? Will they be on their way up/down and going foward/backward? Explain.

Unit 6: Sinusoidal Functions – Practice Problems

1. Information about the movement of a Ferris wheel is shown below. Time is on the x-axis and height is on the y-axis.

(a) How long does it take for the Ferris wheel to make five complete rotations?(b) What is the height of the axle supporting the Ferris wheel?

2. Given the following graph, complete the given analysis.

Amplitude: __________

Period: _________

Range: ___________________

Number of cycles from -540 to 540: _____

Axis: ______________

3. Describe the transformation and then sketch it.

13

4. What is the range for each of the following sinusoidal functions?

(a) (b)

5. The function has been translated to the right , vertically stretched by a factor of 3 and reflected in the x-axis. Write the new equation.

6. Write the equation for the sinusoidal function.(a) (b)

7. Complete the chart below. Sinusoidal Function Maximum Minimum

(a)

(b)

(c)

8. The height of a Ferris wheel is modeled by the function , where is in metres and is the number of degrees the wheel has rotated from the boarding position of a rider. (a) Sketch the curve (on graph paper).(b) When the rider has rotated from the boarding position, how high above the ground is the

rider?9. Sketch each sinusoidal function.

(a)

(b)

EXTRA QUESTIONS – Chapter 6 p. 383 #10 – 15, 25, 26

Unit 7 - Exponential Functions

1. Use exponent laws to simplify. Express answers with positive exponents. Do not evaluate.

a) 52 ×(5−3)2÷ 5−3 b)(a8)(a−3)

(a2)5c) 34× 9

34 ×√❑

14

2. Compute the first and second difference for each table of values and identify the relation as linear, quadratic or exponential. If it is exponential, state the value of the growth/decay rate.

x y-2 15-1 11

0 91 82 7.5

x y

-2 4-1 30 41 72 12

3. Consider the exponential function f ( x )=3x

a) Is this exponential growth or decay? Explain.

b) State the y-intercept:

Equation of the asymptote:

Domain:

Range:

c) Graph the function on the grid provided.

4. There are 3400 bacteria in a culture that is growing at a rate of 5% per hour. a) Write an equation to model this growth. b) Determine the population after 1 day.

15

c) What was the population 3 hours ago? d) When will the population reach 20, 000?

5. How would the equation from # 4 change if the culture was decaying by 5% per hour?

Unit 7: Exponential Functions – Practice Problems

1. Write as a single power. Express answers with positive exponents. DO NOT EVALUATE.

(a) (b) (c)

(d) (e) (f) 2. Evaluate WITHOUT using a calculator.

(a) (b) (c)

(d) (e) (f)

3. Complete the table.

Exponential Form Radical Form Evaluationof Expression

3√274

4. Use your calculator to evaluate each expression. Express answers to two decimals.

(a) (b) (c) (d)

5. Complete the table.Function Exponential Growth

or Decay?Initial Value (y-intercept)

Growth/Decay rate

16

6. Calculate first and/or second differences to classify each function as linear, quadratic, exponential or none of those.(a) (b)

7. An ant colony triples in number every month. Currently, there are 24000 in the nest.(a) What is the monthly growth rate of the colony? What is the initial population?(b) Write an equation that models the number of ants in the colony, given the number of months.(c) Use your equation to predict the size of the colony in three months.(d) Use your equation to predict the size of the colony five months ago.

8. A police diver is searching a harbour for stolen goods. The equation that models the intensity of

light per metre of depth is .(a) At what rate does the light diminish per metre?(b) Determine the amount of sunlight the diver will have at a depth of 18 m, relative to the

intensity at the surface.9. Ryan purchases a used vehicle for $11, 899. If the vehicle depreciates at a rate of 13% yearly, what

will the car be worth, to the nearest dollar, in ten years? Show your work.10. After being filled, a basketball loses 3.2% of its air every day. The initial amount of air in the ball

was 840 (a) Write an equation to model this situation.(b) Determine the volume after 4 days.(c) Will this model be valid after 6 weeks? Explain.

11. List four characteristics of an exponential function.

EXTRA QUESTIONS: Chapter 7 p. 526 # 1 – 8

Unit 8: Financial Applications - (all $ formulas will be given to you)

1. Describe the difference between simple and compound interest.

2. Greg invests $750 in a bond that pays 4.3% per year.(a) Calculate, to the nearest penny, what Greg’s total amount will be after 4 years.

(Show your work)

17

(b) How much money did $750 earn in four years?

(c) If Greg is planning to enter University in 2025, would his money have doubled by then? (Show your work).

3. You invested $4000 at 6%/a compounded monthly. What is your investment worth after 5 years?

4. You invested $400 every 3 months at 8%/a compounded quarterly. What is your investment worth after 5 years?

CHAPTER 8: Financial Problems Involving Exponential Functions – Practice Problems

1. Complete the table (to the nearest penny).Prinicpal ($) Annual Interest

Rate (%)Time Simple Interest

Paid ($)Amount

400 7.25 5 years13 months 328.99

5.5 180.00 940.60

2. Kurtis earned $279.40 in simple interest by investing a principal of $400 in a Treasury bill. If the interest rate was 3.35%/a, for how many years did he have his investment?

3. Complete the table (correct to 2 decimal places).Principal

($)Annual

Interest Rate (%)

YearsInvested

CompoundingPeriod

Amount ($) Interest Earned ($)

350 2.75 10 monthly2500 8.5 2 semi-annually

18

7 annually 315.50

12 000 7 weekly 15 053.88

4. Calculate the amount you would end up with if you invested $2500 at /a compounded semi-annually for 8 years?

5. Johnny borrowed money from a friend. The interest rate was 5.75%/a compounded monthly. If Johnny will repay $5667 over the next 6 years. How much money did Johnny borrow?

6. Kay wants to travel to Australia in 20 months. The trip will cost $5600. How much should she deposit at the end of each month in an account that pays 9%/a compounded monthly to save the amount needed for the trip?

7. Since the birth of their daughter, the Tranters have deposited $450 every three months in an education savings plan. The interest rate is 7.5% compounded quarterly. What is the plan’s value when their daughter turns 17?

EXTRA QUESTIONS – Chapter 8 p. 526 #9, 10, 11, 14

19