MCR 3U Culminating- Review Package
How to study:
1. BE ORGANIZED *make sure you have all of the notes
2. Use past review sheets from each unit as a guideline of learning goals in each unit
3. Complete exam review package *attempt all questions & check answers
4. Review all your notes *including past tests and quizzes
Unit 1: Determining Equivalent Algebraic Expressions
Expand and simplify polynomials, factor, simplify rational expressions, radical expressions and state
restrictions.
Unit 2 – Representing Functions
Know how to identify a function using different methods, function notation, transformations of parent
functions, and finding inverse functions.
Unit 3: Solving Problems Using Quadratic Equations
Finding and identifying key points on a parabola, finding the max and min using various methods, finding
the family of a quadratic function, solving linear-quadratic systems and word problems.
Unit 4: Exponential Functions
Simplifying exponential expressions and equations, graphing transformations of exponential functions
and exponential growth/decay problems
Unit 5: Trigonometry
Solving for exact angels (special triangles), CAST rule, SOH CAH TOA, sine law, cosine law, word
problems and solving trig identities
Unit 6: Sinusoidal Functions
Graphing and stating transformations of the Sine and Cosine function, identifying key features.
Unit 1: Determining Equivalent Algebraic Expressions
1. Expand and simplify.
2
a) 6x  3  2 x  5

b) 3x  1 x 2  6 x  4


2
c)  3z  1 2z  3z  4

2. Simplify and state restrictions.
3x 2  3x
a)
4x2  4x
x2  9
b) b) 2
x  x  12
3xy 4 12 x 5 y

c) c)
4 x 3 y 15x 4 y 2
x 2  16
8 x  40
 2
d) 2
x  x  20 3x  10 x  8
a2  3a  10 a2  2a  15

e) 2
a  a  12 a2  7a  12
4
5

3y  1 1  3y
x
x 1
g) 2
 2
x  5 x  24 x  x  6
y2
10 y 2 2 y 2  8 y


h) 2
y
y  3 y  28 y  7
f)
Unit 2: Representing Functions
3. Explain whether each relation represents a function.
a) y  5 x
b) y  2x  12
c) x 2  y 2  4
d)
e)
x
-1
0
2
2
4
y
0
3
-3
9
7
4. Given 𝑓(𝑥) = 4(𝑥 – 1)2 – 11
a) Find 𝑓(−2).
b) b) Find x when 𝑓(𝑥) = 53.
5. State the domain and range of the following functions:
a)
b)
4
c)
y
3
3
y
y
2
2
2
1
1
1
x
-3 -2 -1-1
1
-2
2
3
4
-3
-2
-1
-1
-2
6. State the domain and range of each of the following:
a) f  x   4 x  3
b) y  2  x  3   5
2
x
x
1
2
3
4
-2
-1
-1
-2
1
2
3
4
7. Find the equation of the inverse f-1(x) for each function:
a) f  x   4 x  3
b) g  x  
4x  8
c) y  2  x  3   5
2
8. Graph the inverse f-1(x) for the function shown to the right.
9. Graph each function along with the parent function.
a)
f x   x 2 , y  2 f x  1  3
g x  
1
( x  2) 2  1
2
c) hx   x , y  h x  1  2
b)
d) ix   3 2 x  2
e)
j x  
1
1

, y   j  ( x  3) 
x
2

f)
k x  
1
2x  2
Unit 3: Solving Problems Using Quadratic Equations
10. Simplify:
a)
54
b)
3



6  21
c) 2 3  3 2

54 3

11. For each parabola, determine the maximum or minimum value (state which).
a) 𝑦 = 𝑥 2 + 2𝑥 + 2
b) b) 𝑦 = 2𝑥 2 + 12𝑥 − 5
c) c) 𝑓(𝑥) = 2𝑥 2 − 24𝑥 + 5
d) 𝑓(𝑥) = 5𝑥 2 + 20𝑥 + 23
e) 𝑓(𝑥) = −3𝑥 2 + 21𝑥 − 35
f) 𝑦 = 𝑥 2 − 18𝑥 + 80
12. Solve each quadratic equation using the quadratic formula
a) 2𝑥 2 − 𝑥 − 3 = 0
b) −2𝑥 2 + 8𝑥 − 3 = 0
13. Solve each quadratic equation by factoring
a) 𝑓(𝑥) = 10𝑥 2 + 13𝑥 − 3
b) 𝑓(𝑥) = 2𝑥 2 + 5𝑥 − 12
c) 𝑓(𝑥) = 40𝑥 2 − 4000𝑥
14. Determine how many roots without solving the equation.
x 2  5x  7  0
2
b) 3 x  5 x  2  0
2
c) 4 x  20 x  25  0
a)
15. Find the values of k so that the graph of each function has one x-intercept
a) 𝑓(𝑥) = 𝑘𝑥 2 − 8𝑥 + 2
b) 𝑓(𝑥) = 2𝑥 2 + 𝑘𝑥 + 50
c) 𝑓(𝑥) = 𝑥 2 + 𝑘𝑥 + 𝑥 + 1
16. Determine the points of intersection algebraically
a) 𝑓(𝑥) = −2𝑥 2 − 4𝑥 − 3 and 𝑔(𝑥) = 2𝑥 − 3
b) 𝑓(𝑥) = −2𝑥 2 + 12𝑥 − 19 and 𝑔(𝑥) = 3𝑥 − 4
17. A walkway of uniform width is to be built inside the edge of a garden measuring 8 metres by 12 metres. The
new area of the garden must be 75% of its original area. Determine the exact width of the walkway.
18. During a game, Rachel hits a tennis ball into the air. The path of the ball is modeled by the function
h(t) = -2t2 + 20t + 1.5 where h(t) is the height of the ball in metres t seconds after it is hit.
a) From what height is the ball initially hit?
b) What is the maximum height reached by the ball?
c) When does the ball reach this maximum height?
d) When does the ball hit the ground?
19. Last year, H & M sold 200 basic tees at $15 each. This year due to the recession, they are planning to decrease
their prices. For every $1 price decrease, there will be 25 more shirts sold. What selling price will maximize
their t-shirt revenue?
Unit 4: Exponential Functions
20. Simplify & evaluate where possible using exponent laws.
4
a)  16
1
b) b)
1 3
(8)
c) c) 27

1
3

2
 27  3
d) d) 

 125 
e) (2 x 3 y 5 z 2 ) 3 (3x 5 y 9 z ) 2
 24x9 y 7  15x3 y11 


4 
3 
  8 x y  18xy 
f) 
10
12
1
2
9
15
g) (49 x y ) (27 x y )
h)
( 4 x 3 y 2 ) 2 (3x 5 y 4 ) 3
(6 x 7 y 2 ) 2


3 5 3
i)  4 x y z
2
 6a  3b c 

k) 
 12a 2b 1 


2
l)
1
3
3
2
27( xy) 9
8 x12 y 6
24. Population in a small town is predicted to grow at a rate of 9% per year. Currently there are 13 800 people in
town. Calculate the population in 2030.
25. A scientist has discovered a new strain of bacteria. The bacteria culture initially contained 630
bacteria and the bacteria are doubling every hour. How many bacteria will there be after 6 hours?
26. Kiaan inherited land from his grandparents that was purchased for $40 000 in 1960. The value of the land
increased by approximately 4.5% per year.
a) Write an exponential function to model the situation.
b) What is the approximate value of the land that Kiaan will inherit in 2016?
27. Cs-137 is used in medical radiation therapy devices for treating cancer. The half life of Cs-137 is 30 years.
a) If a science lab holds a sample of 8 kg, how much will remain after 150 years?
b) b) How long will it take for 0.0625 kg of the original sample to remain?
Unit 5: Trigonometry
28. Identify the ratio for csc, sec , and cot for the diagram to the right:
13 cm
5 cm
29. State the exact value of the following.
a) sin 45
b)
csc 60cot 30

12 cm
cos 45
30. Use special triangles and CAST to determine the exact value for the following. Include a fully labelled diagram
as part of your solution.
a) cos 210
b) sin 330
c) cot 315 d) sec120
31. Determine the possible value(s) for  over the domain 0    360 . Include a fully labelled CAST diagram
with rotations clearly labelled.
a.
sin   0.3420
b) sin  
3
2
c) cos   
1
3
d)
sec  2
32. Find the possible measures of each of the following angles. Include a sketch as part of your solution.
a. Triangle PQR has R = 400, r = 22 cm and q = 27 cm, find angle Q.
b. Triangle JKL has J = 300, j = 7.3 m and k = 14.6 m, find angle L.
c. Triangle EFG has F = 330, f = 1.3 cm and g = 2.8 cm, find angle G.
33. From the top of a 20 m lighthouse, the angles of depression to two ships, the Acadian and the Bounty, are 52◦
and 63◦ respectively. If the angle between the ships is 120◦ , how far apart are they?
34. A soccer player takes a shot on a standard net that is 7.3 m wide. If the player is 10 m from one goalpost and 14
from the other, through what angle can a goal be made?
35. Prove the following trigonometric identities.
tan x
sin x
cos  1
1  cos 
b) 1  sec 
c)

 csc  cot 
2
cos x 1  sin x
cos
sin 
sin 2 x
1
1
 1  cos x
d) tan x 
e)

tan x sin x cos x
1  cos x
a)
Unit 6: Sinusoidal Functions
36. Explain what each item means for a periodic curve. Show each item on a labelled diagram.
a) cycle
b) period
c) amplitude
d) axis of curve
e) maximum or minimum
37. Graph the following functions from 0    360 . State the domain and range of each.
a) f  x   2 cosx  45  1
b) g  x    sin 3 x  30 
c) y  3 sin[ 2( x  45)]
d) y  
1
cos2 x   3
2
38. Graph a cosine function with the following properties over the domain 𝟎 ≤ 𝒙 ≤ 𝟑𝟔𝟎°:
i. Amplitude of 4, Period of 120°, Axis of the curve at 𝑦 = −1
39. John took a ride on a Ferris wheel. The Ferris wheel has a radius of 12 m and its maximum height is 28 m above the
ground. The wheel completes 1 revolution in 20 seconds. John gets on the Ferris wheel at the bottom and the ride lasts
2 revolutions.
a) Graph his height, in metres, above the ground versus time, in seconds, for the ride.

y














x























b) Write an equation of a sine function that models John’s height above the ground as a function of time in the
h(t )  a sin[ k (t  d )]  c
c) Use the equation to determine the height of John at 7 seconds. (Round to the nearest metre).