MHF4U
Name: ______________________
Final Exam Information and Review
*
The MHF4U exam will take place on Wednesday January 29 from 1-3 p.m. in the cafeteria.
It will be worth 30% of the final course mark.
*
Possible Exam Format:
*
Study Tips
1. Start with the review questions provided. They give you an idea of how the focus is narrowed for each BIG IDEA
studied in the course (but note that the review is NOT an exact replica of the exam).
2. Go through your unit quizzes and tests – these questions were important enough to be on tests, so they are
likely important enough to be on the exam ( in some form)
3. Try other questions – from lessons, homework test reviews, or new ones you choose from the textbook but
realize that not everything will be covered on the exam (so focus on the same types of questions as those in
your exam review).
4. Create a one-page summary for each unit. This will help prevent “blanking out” when you write the exam.
5. Form a small study group and spend time going through specific sections of the course during the next month.
6. Create a study schedule – use a calendar and block off days you cannot commit to studying due to outside
commitments, assignment due dates and tests for other courses – whatever days are left, commit an hour or so
to reviewing
SHORT ANSWER (Including Multiple Choice and Communication)
FULL SOLUTION (Including Graphing, Application and Problem Solving)
EXAM TOPICS
Polynomials
 Polynomial functions and their properties:
Domain, range, intervals of increase and
decrease, turning points (maxima, minima), end
behaviour, zeros, y-intercept, even vs. odd
functions
 Graphing polynomial functions
 Transformations
Polynomial Equations and Inequalities
 Remainder Theorem and Factor Theorem
 Division of polynomials (long division, synthetic
division)
 Factoring Polynomial functions
 Solving Polynomial equations and inequalities
 Application of Polynomial functions including
Average and Instantaneous rates of change
Trigonometry
 Trigonometric ratios of any angle in radian
measure and exact values using trigonometric
formulas
 Graphs of sinusoidal functions and their
properties
 Graphs of reciprocal functions and their
properties
 Transformations of sinusoidal functions
 Rates of change of trigonometric functions
 Trigonometric Identities
 Trigonometric Equations
 Applications of trigonometric functions
Exponential and Logarithmic Functions
 Exponential and Logarithmic Functions
 Exponent Laws and Laws of Logarithms
 Solving exponential and logarithmic
equations
 Graphs of exponential and logarithmic
functions and their characteristics
 Transformations of exponential and
logarithmic functions
 Applications of exponential and
logarithmic functions
 Rates of change of exponential and
Logarithmic functions
Page 1 of 6
Rational Functions
 Rational Functions and their Properties
 Domain restrictions
 Asymptotes: Vertical, Horizontal, Oblique & end
behaviour
 Graphing Rational Functions (all types)
 Application of Rational Functions including rates
of change
Composition of Functions
 Operations (+ , − , ×, ÷ ) of functions
 Composition of functions
 Intersection of functions graphically
Expressions
1. Factor fully.
a) x 3  3x 2  25x  75
b) x 3  x 2  14x  24
c) 64x 3  27y 3
2. Find the exact value of the following.
a)
7log7
5
b) log 64 6 8
3. Write as a single logarithm.
2
log 5 w  2
3
a)
a log 5 (x  7) 
c)
1
log6 12  log6 8  log6 2
2
d)
e)
f)
30x 3  17x 2  8x  4
x 6  124x 3  125
8 x 6  125 y 6
c)
log 8 6  log 8 3  log 8 4
d)
log 9 37  5 81
b)
x  1  log( x  7)
d)
1
log 2 10  log 2 9
3
b)
56 radians
b)
-24˚


4. Convert to degrees.
a)
11
radians
15
5. Convert to radians.
a) 420˚
6. Find the exact value of the following.
3
4
11
b) tan
6
a)
cos
7. Given: sin A 
c)
d)
 3 
csc 
 2 
7
sin
12
6 3
2
3
 A  2 , and tan B  ,   B 
,
7
2
3
2
Find the exact value of the following.
a) sec A
b) cos 2B
c)
d)
sin(A  B)
tan(A  B)
Page 2 of 6
8.
Given:
f (x) 
1
2
and g(x)  x  8
x5
Find:
a)
 f  g (x)
e)
g og(x)
i)
g  f (3)
b)
 g
 f  (x)
f)
f 1 (x)
j)
 fg (1)
g)
g 1 (x)
k)
c)
d)
 f og (x)
g o f (x)
h)
f o f (x)
1
 f og (5)
g o f (5)
l)
Equations and Identities
9.
Solve. Exact answers are required, where possible. Otherwise, express answers correct
to one decimal place.
x 3  3x 2  4x  12
3
2
b) x  5x  5x  1
3
2
c) x  4x  9x  10  0
1
0
d) x 
x4
2x
1
2

 2
e)
x  1 x  3 x  4x  3
x 2
f) 4(7 )  8
g) log 4 (x  3)  2
h) log 9 (x  5)  log 9 (x  3)  1
a)
i)
32 x  2(3x )  15  0
j)
2cos2x  1
k)
sin2 x  2sin x  3  0 (0  x  2 )
l)
cos2x  cos x (0  x  2 )
m)
(0  x  2 )
2 tan x cos x  tan x (0  x  2 )
1
n) (42 )(22𝑥−3 ) = (16𝑥−2 ) ( )
√2
p) 𝑙𝑜𝑔7 (𝑥 + 2) = 1 − 𝑙𝑜𝑔7 (𝑥 − 4)
o) 𝑙𝑜𝑔4 (𝑥 + 3) = 2
10. Solve.
a)
x(x  1)(x  2)(x  4)  0
d)
b)
(x  7)2 (x  3)3  0
e)
c)
2x 3  3x 2  11x  6
x2
0
x2  9
5
3

0
x  3 x 1
11. Prove.
1  tan 
a) cos   sin  
sec 
1
1

 2 cot 2 
b)
1  sec  1  sec 
𝜋
csc 2   2
c)
 cos 2
csc 2 
d) cos 2  cos
2
2
  sin 2   sin 2 2
𝜋
e) sin(𝜋 + 𝑥) + cos ( − 𝑥) + 𝑡𝑎𝑛 ( + 𝑥) = −𝑐𝑜𝑡𝑥
2
2
f) cos(𝑥 + 𝑦) cos(𝑥 − 𝑦) = 𝑐𝑜𝑠 2 𝑥 + 𝑐𝑜𝑠 2 𝑦 − 1
12. If 𝑙𝑜𝑔𝑏 𝑎 =
1
𝑥
and 𝑙𝑜𝑔𝑎 √𝑏 = 3𝑥 2 , show that 𝑥 =
1
6
13. If ℎ2 + 𝑘 2 = 23ℎ𝑘 where ℎ > 0, 𝑘 > 0, show that 𝑙𝑜𝑔 (
ℎ+𝑘
5
1
) = (𝑙𝑜𝑔ℎ + 𝑙𝑜𝑔𝑘)
2
Page 3 of 6
Graphs
14. Determine whether each of the following functions is even, odd, or neither.
a)
𝑓(𝑥) =
1
𝑥 3 +1
b) 𝑔(ℎ) = 2ℎ4 + 3ℎ2
c) 𝑝(𝑥) = 2𝑥(𝑥 − 4)2 (𝑥 + 4)2
15. Write an equation to represent each function.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m) Consider the above graphs. Which functions:
i) are even?
ii) are odd?
iii) have inverses that are functions?
16. Graph the following functions. Label all key features.
y  x(x  3)(x  4)
2
3
b) y  (x  2) (x  3)
2
c) y 
x 1
5x  3
d) y 
2x  1
a)
e)
f)
g)
h)
y  3x 2  1
y  log 2 (8x 2 )


y  2 sin  x   , (2  x  2 )

3

1
y  cos  x    1 , (2  x  2 )
2
4
Page 4 of 6
17. When is the function 𝑓(𝑥) =
4
𝑥−1
−3+
−3𝑥 2
5−4𝑥−𝑥 2
, below the horizontal asymptote?
18. State the characteristics (properties) of each function:
a)
𝜋
𝜋
𝑓(𝑥) = −2 − 3𝑐𝑜𝑠 (3𝑥 − )
b) 𝑦 = cot (𝑥 − )
4
6
Given the graphs of y=f(x) and y=g(x),
graph:
a)  f  g (x)
Page 4 of 4
b)  fg (x)
19. y
 f
(x)
g 
c) 

x
d)
f 1 (x)
e)
g 1 (x)
Which functions are even? Odd?
Applications
20. When x  4x  ax  bx  1 is divided by (x-1), the remainder is 7. When it is divided by (x+1), the remainder is 3.
Determine the values of a and b.
4
3
2
21. An open box, no more than 5 cm in height, is to be formed by cutting four identical squares from the corners of a sheet of metal
25 cm by 32 cm, and folding up the metal to form sides. The capacity of the box must be 1575 cm 3. What is the side length of
the squares removed?
22. Consider all rectangles with an area of 200 m2. Let x be the length of one side of the rectangle.
a) Express the perimeter as a function of x.
b) Find the dimensions of a rectangle whose perimeter is 70 m.
23. Determine the intercepts and the equations of all asymptotes of
y
x 3  2x 2  x  2
then sketch.
x2  x  6
24. Pollution affects the clarity of water. The intensity of light below a particular polluted river’s
surface is reduced by 5% for each metre below the surface of the water.
a) Write an exponential function to model this relationship.
b) What percent of the original intensity of light penetrates to 4 m below the surface of the water?
c) At what depth does 40% of the original light intensity remain?
25. Energy is needed to transport a substance from outside a living cell to inside the cell. This energy is measured in kilocalories per


gram molecule, and is given by: E  1.4 log C1  log C2 , where C1 represents the concentration of the substance outside
the cell and C2 represents the concentration of the substance inside the cell.
a) Rewrite the formula as a single logarithm.
b) Find the energy needed to transport the exterior substance into the cell if the concentration
of the substance outside the cell is double the concentration inside the cell.
c) What is the sign of E if C1 < C2? Explain what this means in terms of the cell.
26. Strontium-87 is used in the study of bones. After one hour, 78.1 mg remains of a 100 mg substance.
What is the half-life of Strontium-87?
Page 5 of 6
27. A circular arc has length 3 cm, and the radius of the circle is 2 cm. What is the measure of the angle subtended by the arc, in both
radians and in degrees?
28. Tidal forces are greatest when the Earth, the sun and the moon are in line. When this occurs at the Annapolis Tidal Generating
Station, the water has a maximum depth of 9.6 m at 4:30 pm and a minimum depth of 0.4 m 6.2 h later.
a) Write an equation for the depth of the water at any time t.
b) Calculate the depth of water at 9:30 am and at 6:45 pm.
29.
The function, S (t )  3 cos

12
t  22 , represents the surface temperature S of a pond, in degrees Celsius, and t represents
the number of hours since sunrise at 06:00. At what time is the surface temperature of the water 20C?
30. The meat department manager discovers that he could sell m(x) kg of ground beef in a week, where
if he sold it at $x/kg. He pays the supplier $3.21/kg for the beef.
a) Determine an algebraic expression for P(x), where P represents the total profit from selling
ground beef for one week.
m(x)  14700  3040x ,
1
b) Find P (x) .
c) What selling price will earn $1900 in profit for the week?
31. Estimate IRC of each function at the given 𝑥 value using a centered interval of 0.001.
a) 𝑓(𝑥) = 𝑥 3 + 𝑥 2 at 𝑥 = 2
b) 𝑓(𝑥) = −𝑥 4 + 1 at 𝑥 = 3
32. A Ferris wheel with radius 10 m makes 2 rotations in 4 minutes. What is the speed of the Ferris wheel in metres per second.
Page 6 of 6