MHF4U Final Exam Review
Exam Time Limit: 2.0 hours
Name _________________
Exam Instructions:
1. Answer all questions in the space provided.
2. If you need additional paper or graph paper ask for it, otherwise all answers should be
written on the exam paper.
Part I: Place your answers for Part I in the space provided.
1. The instantaneous rate of change of a function is _____________.
(____)
A. the domain divided by the range
B. the change in the dependent variable divided by the change in the independent
variable from one point to another
C. the range divided by the domain
D. how quickly the dependent variable is changing with respect to a unit change in the
independent variable at an instant in time
2. The domain of a function is _______________.
(____)
A. the entire set of function values (of the dependent variable)
B. x  
C. y  
D. the entire set of values (of the independent variable) for which the function is
defined
3. The range of a function is _______________.
A. the entire set of function values (of the dependent variable)
B. x  
C. y  
D. the entire set of values (of the independent variable) for which the function is
defined
(____)
4. Which one of the following is a polynomial function?
(____)

2 
A. n  x   3cos 6  x 
4
3 

B. f  x  
3 3
x  6x 2  x  11
4
C. q  x   3log2 4x
 
D. f x 
3
5x  4
5. The ____ row of differences for quartic functions are all the same.
(____)
A. 1st
B. 2nd
C. 3rd
D. 4th
6. Even functions have which of the following characteristics?
A. f   x   f  x 
B. f   x   f  x 
(____)
1  1
C. f  x   f  x 
a  a
7. The Remainder Theorem states ___________.
A. When a polynomial P  x  is divided by x  b the remainder is P b 
(____)
B. If P b   0 , then x  b is a factor of P  x 
C. The remainder in the division P  x   b is P b 
D. If x  b divides P  x  with a remainder of zero, then x  b is a factor of P b 
8. The Factor Theorem states ___________.
A. When a polynomial P  x  is divided by x  b the remainder is P b 
(____)
B. If P b   0 , then x  b is a factor of P  x 
C. The remainder in the division P  x   b is P b 
D. If x  b divides P  x  with a remainder of zero, then x  b is a factor of P b 
 
9. What is the domain of h x  
A. x   x  2
B. y   y  
9
7
C. x   x  2
D. x  
9x  3
?
7 x  14
(____)
 
10. What is the horizontal asymptote for f x 
A. y  5
B. x 
10x
?
2x  3
(____)
3
2
C. y  10
D. y  2
 
11. What is the vertical asymptote for f x 
A. x  2
B. x 
3
2
C. x 
2
3
10x
?
2x  3
(____)
D. y  2
12. In a right angled triangle sin 
opposite
opposite
A.
B.
hypotenuse
adjacent
13. In a right angled triangle cos 
opposite
opposite
A.
B.
hypotenuse
adjacent
14. In a right angled triangle csc 
opposite
opposite
A.
B.
hypotenuse
adjacent
15. In a right angled triangle tan 
opposite
opposite
A.
B.
hypotenuse
adjacent
16. In a right angled triangle sec 
opposite
opposite
A.
B.
hypotenuse
adjacent
17. In a right angled triangle cot 
opposite
opposite
A.
B.
hypotenuse
adjacent
C.
C.
C.
C.
C.
C.
adjacent
hypotenuse
D.
opposite
hypotenuse
adjacent
hypotenuse
D.
opposite
hypotenuse
adjacent
hypotenuse
D.
opposite
hypotenuse
hypotenuse
hypotenuse
D.
opposite
adjacent
E.
E.
E.
E.
adjacent
opposite
adjacent
opposite
adjacent
opposite
adjacent
opposite
adjacent
opposite
hypotenuse
hypotenuse
D.
opposite
adjacent
E.
hypotenuse
hypotenuse
D.
opposite
adjacent
adjacent
E.
opposite
(____)
(____)
(____)
(____)
(____)
(____)
7

6
1
A.
3
18. tan
(____)
B. 1
4

3
2
A.
3
C.
3
D. 
1
E.  3
3
19. csc
(____)
B. 1
3

2
A. -1
C.
3
D. 2
2
3
E. 
20. cot
(____)
B. 1
C.
3
D. 0
E. undefined
21. To convert degrees to radians, multiply by _________
2


A.
B.
C. 
D.
180
180
360
22. csc 
1
A.
tan 
B.
23. sec 
1
A.
tan 
B.
24. cot 
1
A.
tan 
B.
1
cos 
1
cos 
1
cos 
C.
C.
C.
1
cot 
D.
1
cot 
D.
1
cot 
D.
 5
25. Which of the following is equivalent to sin  
 6
 3 
 7 
 5 
A. sin  
B. tan 
C. sin 



 4 
 2 
 6 
26. Using the diagram at the right, sec  
A.
146
5
B.
146
11
C.
11
5
D.
5
11
1
sin 
180
E.

1
csc 
E.
1
sin 
1
csc 
E.
1
sin 
1
csc 
E.

?

(____)
(____)
(____)
(____)
D. cos
15
4
5
E.
(____)
11
146
 4 
E. cos 

 3 
β
11
1
(____)


27. sin   x  
2


A. sin  x  
2

(____)
B. cosx
C. sinx
D. cosx

E. csc   x 
2



28. tan   x  
2

(____)
B. cosx
A.  cotx
C. cotx

E. csc   x 
2

D. cosx
29. Determine the equation of the function pictured at the right?
1
A. y  4 cos x
2
(____)
B. y  4 sin2x
1
D. y  4 sin x
2
C. y  4 cos2x
1
E. y  3cos x
2
30. Determine the equation of the function pictured at the right?




A. y  5cos2  x    1
B. y  5 sin2  x    1
6
6


1
1

C. y  5cos  x    1
D. y  4 sin x
2
2
6
(____)
E. y  5cos 4x  1
31.  sinx 
A. sinx

B. cos   x 
2

C. cosx

D. cos   x 
2


E. sec  x  
2

(____)
32. The graph at the right is ______.
A. y  cot x
(____)
B. y  csc x
C. y  tan x
D. y  sin x
E. y  sec x

2 
33. What is the amplitude of y  3cos2  x 
 5?
3 

2
right
A.
B. 3
C. 5 down
D. 2
3
(____)
E. -2


34. What is the amplitude of y   sin  x    6 ?
4

A.

left
4
B. 2
C. 6 up
D. 1

35. What is the period of y  4 sin5  x    2 ?
4

2

A. 4
B.
C.
D. 2 down
right
5
4
36. The graph shown is ________.
A. y  2x
B. y  3x
C. y  4x
1
D. y   
4
x
x
1
E. y   
3
(____)
E. -1
(____)
E. 5
(____)
37. The graph shown is ________.
(____)
A. y  2
x
B. y  3x
x
1
C. y   
2
x
1
D. y   
3
1
E. y   
4
x
p
38. Write h  3 in its equivalent logarithmic form.
A. p  log3 h
B. h  log3 p
C. 3  log p h
(____)
D. log3 p  h
39. Write loga b  c in its equivalent exponential form.
A. b c  a
40. Simplify
B. a c  b
D. c b  a
1
log2 x 2  2log2 y  8 log2 z into one logarithm.
3
A. log2  3 x 2 z 8y 2 

C. c a  b
(____)

 3 x 2z 8 

 y2 


B. log2 
C. log2
F
GGH
x 2y 2
z8
(____)
I
JJK
D. log2
F
GH
2
x 3z 8
y2
I
JK
Part 2: Show all work you do in the space provided for each question in part 2.
1. Use the graph at the right for
each of the following questions.
a) Determine Albert’s
average velocity from 2 to
6 seconds.
b) Determine Albert’s
instantaneous velocity at 4
seconds.
2. Use differences to determine the degree of the polynomial functions that the following
tables represent.
a)
x
y
-2
-1
0
1
2
3
27
-1
-7
-3
-1
-13
b)
x
y
-2
-1
0
1
2
3
43
0
-7
-2
15
68
3. Solve completely by factoring.
a) 2x 3  5x 2  x  6  0
b) 3x 4  8x 3  4x 2  16x
4. Solve without using technology.
a) x 2  4x  32  0
b) 2x 2  11x  6  0
y
5. Sketch the function
f x  
5
. Be sure to
2x  8
10
indicate all asymptotes and
intercepts (if any). Show how to
get each behavior near any
5
asymptotes.
x
-10
-5
5
-5
-10
10
6. Find the asymptotes for each of the following functions.
a) f x 
6x  3
4x  12
b) f  x  
7x  1
x x 2
 
2
7. Solve each of the following rational equations.
1
7
4


a)
3
5m
m
b)
x
3


x 4
x 2
18
 x  2  x  4 
8. Use a numerical method to approximate the instantaneous rate of change for the
2x  7
function f  x  
where x = 2.8.
x 3
9. The radius of the earth is 6380 km. A space shuttle 300 km above the earth travels
11
around the planet through an angle of
radians. How far does the shuttle move?
6
10. Use a compound angle identity to rewrite sin

sin
3
trigonometric ratio and then find its exact value.
3

3
 cos cos
as one single
4
3
4
11. Determine the average rate of
change for the function at the right
from where x = 0.2 to x = 1.1.
y






















12. Solve on the interval 0    2 .
a) 2sin2   7 sin   4  0
b) 2cos2   3cos   2  0
x






13. Prove each of the following.
a) sin   tan  cos 
b) sin   cos  cot  
sec 
tan 
c) 2csc2x tan x  sec x
2
14. The sunset time varies sinusoidally with time. The sunset time (S) in the town of Parry
 2

Sound can be modelled by the equation S  1.75cos 
d  172    18.4 , where d is the

 365

day of the year. How long a period of time (in days) is the sun setting earlier than 7:00
p.m.? You may use technology to solve this problem. If you do use technology, include a
neat labelled sketch with your solution.
15. Evaluate log6 93 . Explain how you can check this answer.
16. Solve each of the following.
a) 4x  50
b) 95x  221
17. Evaluate without the use of the log function on a calculator. (Hint: use the log laws)
a) log9 3  log9 27
b) 3log5 25
18. Simplify each of the following using the three laws of logarithms.
a) log3 144  log3 6
c)
b) 4log7 2x   log7 8xy 3 
19. Solve each of the following.
a) log9 (x  5)  1  log9 (x  3)
1
log z  5log x  2log u 3 
3
b) log4 (x  2)  log4 10  log4 (x 2  5x  14)
c) 93x 1  27 x [Write the answer in exact form]
d) 52x  4 5x   5  0
20. The magnitude, M, of an earthquake is measured using the Richter scale, which is defined
I
as M  log . I is the intensity of the earthquake, and Io is the intensity of a standard
Io
low level earthquake.
a) What is the Richter Scale reading for an earthquake that is 45000 times more
powerful than the standard low level earthquake?
b) How much more powerful is a 6.8 Richter Scale earthquake than a 4.0 scale quake?
21. A 60 mg sample of a radioactive isotope decays to 50 mg after 4.7 days.
a) Determine the half-life.
b) How long will it take for the amount
of the isotope to decrease to 20 mg?
 
a) Graph h  x   f  x   g  x 
 
22. Given f x  4x 2  2x  1 and g x  3x .
y
10
5
b) Solve the inequality
4x 2  2x  1  3x
x
-10
-5
5
-5
-10
  
c) Find and simplify g f x
  
and f g x
  
d) Explain why you would expect g f x
  
and f g x
to be different.
10