WESTLANE SECONDARY SCHOOL
MHF 4U Practice FINAL EXAM
Part A: Answers Only – 1 Mark Each
Write the simplified answer in the box provided. Include units where appropriate.
1.
Is the function f ( x) = x 3 + x − 1 even, odd, or neither?
2.
What is the degree of the function g ( x) = −2 x(2 x − 1) x 2 + 1 ?
3.
State the remainder when f ( x) = 2 x 6 − x 3 + 1 is divided by ( x + 1) .
4.
For the function g ( x) =
(
)
2
2x − 3
, state a) the vertical asymptote
x+2
b) the horizontal asymptote
c) the x-intercept
d) the y-intercept
5.
What is the domain of the function h( x) =
6.
Given the following graph of f (x) , state
x
?
x −1
2
a) the domain
b) the range
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
−2
1
2
3
4
5
6
−3
−4
5π
into degrees.
3
7.
Convert the angle
8.
Convert the angle 315° into radians.
9.
State the exact value of cos
π
6
.
10. State the exact value of tan
7π
.
6
11. State the exact value of csc
7π
.
4
1

12. For the function y = 2 sin  (x + 3π ) + 5
4

state a) the amplitude
b) the period
c) the phase shift
d) the maximum value
MHF 4U
Practice Final Exam
Page 2 of 6
13. If sin x = k and cos x = n , state the value of a) cos(π + x )
 3π

b) sin 
+ x
 2

14. Evaluate the following logarithms
a) log 4 64
1
)
b) log(100
c) log 27 9
d) log 2 16 500
Part B: Full Solutions
All work must be shown using proper form to receive full marks.
1.
Consider the function f ( x) = −2 x 4 + 8 x 3 + 6 x 2 − 36 x
a) Factor f (x) and state the zeros.
_
4
b) Use the zeros to sketch a graph of f (x) . (You do not need to find min/max points)
_
2
2.
Determine the equation of the following polynomial function:
10
−5
−4
−3
−2
−1
1
2
3
4
5
8
6
_
3
4
2
−4
−3
−2
−1
1
−2
−4
−6
−8
2
3
4
MHF 4U
3.
Practice Final Exam
Page 3 of 6
Use the TI-83+ to solve the following inequality. x 3 − 5 x 2 + 10 > 0
(Include a sketch of the graph and round to 2 decimal places.)
_
3
4.
Consider the function g ( x) =
2x − 6
x+3
6
a) State the x and y intercepts.
4
_
1
2
b) State the equations of all asymptotes.
−6
_
1
−4
−2
2
−2
c) Sketch the function.
−4
−6
_
2
5.
_
3
State the equation of the oblique asymptote of the function g ( x) =
4 x 2 + 5x + 2
.
x+2
4
6
MHF 4U
6.
Practice Final Exam
Page 4 of 6
Determine the equation of the rational function g (x) shown below.
6
4
_
4
2
−6
−4
−2
2
4
6
−2
−4
−6
7.
After a race, a runner’s heart rate (in bpm) is modelled by the function R(t ) = 120(0.85) + 70 ,
where t represents the time, in minutes.
t
a) What is the average rate of change of heart rate over the first 5 minutes?
_
2
b) What is the instantaneous rate of change after 10 minutes?
_
2
c) How many minutes does it take for the heart rate to lower to 90 bpm?
_
2
8. Solution A has a pH of 5.6. Solution B has a pH of 10. How much of solution B must be added
to 2.5 litres of solution A to neutralize it (pH = 7)?
_
4
MHF 4U
9.
Practice Final Exam
Page 5 of 6
 5π 
Evaluate exactly: cos 
 12 
_
4
10. Prove the following trig identity: cot(2 x) = csc(2 x) − tan x
_
4
11. For the following trig function, write the transformation using mapping notation, determine the
key image points, and sketch 2 complete cycles.
 
π 
y = −0.5 sin 3 x +  + 1.5
6 
 
Key Points
x
y
Image Points
x
y
Mapping Notation
x
y
_
5
2
1
−π
−2π/3
−π/3
π/3
−1
2π/3
π
MHF 4U
Practice Final Exam
Page 6 of 6
12. The following graph shows the velocity of a person’s breath while at rest.
Velocity (litres/sec)
1.0
0.8
0.6
0.4
0.2
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
Time (s)
Express the graph in terms of both sine and cosine functions.
_
5
13. The height of a bungee jumper during a jump is given by the function
t
h(t ) = 20(0.98) cos(1.047t ) + 25 , where h is the height above the ground in metres, and t is the
time in seconds.
a) Use the TI-83+ to graph the height of the jumper over the first 45 seconds.
(Remember to use radian mode). Sketch your graph below.
_
1
b) What is the initial height of the jumper?
_
1
c) Determine the average rate of change of height of the jumper between t = 2 and t = 4
seconds. Explain the meaning of the result.
_
2
__
80 Marks Total
End of Exam