MHF4U – Exam Review Practice Questions
Unit 1 and 2: Polynomial Functions
1. Determine the zeros and their order of the function 𝑓(𝑥) = 3(𝑥 2 − 25)(4𝑥 2 + 4𝑥 + 1).
2. Determine the equation of:
a) A cubic function with zeros 1, −2, −5 and y-intercept 10
b) A quartic function with roots at 2, −1, 3 (𝑜𝑟𝑑 2) and passes through the point (4, −10)
3. Determine the equation of the polynomial function represented by the following table of values. |
𝑥
𝑦
−2
1
−1
0
0
1
1
16
2
81
3
256
4
625
4. Find the remainder when:
a) 2𝑥 3 − 4𝑥 + 7 is divided by 2𝑥 − 1.
b) 𝑥 3 − 2𝑥 2 + 3𝑥 − 4 is divided by 𝑥 2 + 𝑥 − 6
5. Find the value of 𝑘 if (𝑥 − 5) if a factor of 4𝑥 3 − 5𝑥 + 𝑘𝑥 − 2.
6. When 𝑥 3 + 𝑚𝑥 2 + 𝑔𝑥 − 4 is divided by 𝑥 + 2, the remainder is −2. When it is divided by 𝑥 + 1, the
remainder is −3. Find 𝑚 and 𝑔.
7. Divide 3𝑦 3 − 2𝑦 2 + 12𝑦 − 9) ÷ (𝑦 2 + 2).
8. Factor fully:
a) 3𝑥 3 + 8𝑥 2 + 3𝑥 − 2
c) 3𝑥 3 − 81
b) 𝑥 4 − 2𝑥 3 + 2𝑥 − 1
d) 𝑥 3 + 5𝑥 2 + 3𝑥 + 15
9. Solve for 𝑥, 𝑥 ∈ 𝐶
a) 𝑥 3 − 7𝑥 + 6 = 0
d) 3𝑥 3 − 7𝑥 2 = 2 − 6𝑥
b) 2𝑥 3 + 𝑥 2 − 8𝑥 − 4 = 0
e) 𝑥 2 (2𝑥 − 1) = 2𝑥 − 1
10. Solve the following polynomial inequalities.
a) 𝑥 3 − 2𝑥 2 − 3𝑥 > 0
b) 𝑥 3 + 4𝑥 2 + 𝑥 − 6 < 0
c) 9𝑥 3 − 4𝑥 = 0
c) 𝑥(𝑥 − 2)(𝑥 + 1)(𝑥 + 5) ≥ 0
11. For the function 𝑔(𝑥) = −2𝑥 3 (𝑥 − 2)2 :
a) State the degree of the function and describe its end behaviours
b) State the zeros and their order, then sketch the function
c) State the interval where 𝑔(𝑥) < 0
12. Three consecutive integers have a product of −504. Determine the three integers.
13. Determine algebraically whether the following functions are even, odd or neither.
a) 𝑓(𝑥) = −2𝑥 3 + 7𝑥 − 1
b) 𝑔(𝑥) =
4𝑥 2 −2
2𝑥 1 +1
c) ℎ(𝑥) =
−5𝑥 3 +𝑥
3𝑥−5
14. Find the average rate of change of 𝑓(𝑥) = 𝑥 3 + √𝑥 on the interval:
a) 𝑥 ∈ [0, 1]
b) 𝑥 ∈ [1, 9]
Unit 3: Rational Functions
𝑥+2
15. Find the equation of all of the asymptotes of the function 𝑓(𝑥) = 3𝑥−2.
16. Sketch the graph of 𝑓(𝑥) =
𝑥 2 +4𝑥+3
𝑥−2
𝑥−2
17. Given 𝑔(𝑥) = 𝑥 2 +5𝑥+6
a)
b)
c)
d)
Determine the x- and y- intercepts.
State the domain.
Determine the equation of any asymptotes.
Determine the coordinate of any holes.
𝑥−3
18. Given ℎ(𝑥) = 𝑥 2 −9
a)
b)
c)
d)
Determine the x- and y- intercepts.
State the domain.
Determine the equation of any asymptotes.
Determine the coordinate of any holes.
19. Create a function that has a graph with the given features:
a) Vertical asymptote at 𝑥 = −2 and horizontal asymptote at 𝑦 = 1
b) Two vertical asymptotes at 𝑥 = 3, 𝑥 = −4 and a hole at 𝑥 = 1
c) Vertical asymptote at 𝑥 = −3, horizontal asymptote at 𝑦 = 0, no x-intercepts and a y-intercept
at −2
20. Crossover points are points in which the function intersects with its horizontal or oblique asymptote.
Determine the coordinates of the crossover points for each function.
a) 𝑓(𝑥) =
(2−𝑥)(3𝑥+2)
𝑥2
b) 𝑓(𝑥) =
2𝑥 3 −3𝑥 2 +𝑥−3
𝑥 2 +1
21. Solve and state any restrictions.
a)
𝑥−1
𝑥+3
= 𝑥+4
𝑥−3
𝑥
1
2
b) 𝑥−1 + 𝑥+1 = 𝑥 2 −1
22. Sketch 𝑓(𝑥) = −𝑥 2 + 9 and its reciprocal on the same grid.
23. Solve.
a)
2(𝑥−2)
>0
𝑥+1
b)
3𝑥
>6
𝑥−2
10
c) 0 = 2 − 𝑥 2
24. Graph 𝑓(𝑥) =
𝑥 2 −16
. State its domain, intercepts, the equation of any asymptotes and end
𝑥
behaviours.
25. For each function, determine its intercepts, equations of any asymptotes and coordinates of any
holes. (For extra practice, sketch!)
3𝑥 2
𝑥+5
a) 𝑦 = 𝑥 2 −25
b) 𝑦 = 𝑥 2 −1
𝑥 2 −25
d) 𝑦 = 𝑥+5
e) 𝑦 =
𝑥 2 +3
2𝑥
c) 𝑦 =
𝑥 2 +4
𝑥
𝑥2
f) 𝑦 = 1−𝑥2
Unit 4 and 5: Trigonometry and Trigonometric Functions
26. Sketch the following graphs in the interval −2𝜋 ≤ 𝜃 ≤ 2𝜋.
1
2
a) 𝑦 = 2𝜃 − 1
1
b) 𝑦 = −4 cos ( 𝜃 − 𝜋) + 3
𝜋
b) 𝑦 = 2 sin [2 (𝜃 + 4 )]
d) 𝑦 = −3 cos(4𝜃 + 𝜋) − 2
27. Find the exact value of each.
a) csc 𝜃, given that sin 𝜃 =
3
√5
b) cot 𝜃, given that csc 𝜃 =
7
3
28. Solve, 𝑥 ∈ [0, 2𝜋]
1
𝑥
a) sin 2𝑥 = 2
3
d) cos2 𝑥 = 4
g) 4 cos2 2𝑥 − 1 = 0
j) 2 cos 𝑥 = 1 − sin2 𝑥
1
𝜋
1
b) cos (2) = − 2
c) cos (𝑥 + 3 ) = 2
e) 2 sin2 𝑥 + sin 𝑥 − 1 = 0
f) 10 cos 2 2𝑥 + 7 cos 2𝑥 = 6
h) 2 tan2 𝑥 + tan 𝑥 − 3 = 0
k) 2 cos 𝑥 + sin 2𝑥 = 0
i) 6𝑐𝑜𝑠 2 𝑥 − sin 𝑥 − 4 = 0
l) cot 𝑥 cos2 𝑥 = 2 cot 𝑥
29. Evaluate the following without a calculator.
5𝜋
11𝜋
a) cos 6
b) sec 3
𝜋
11𝜋
c) sin 12
d) tan 12
𝜋
𝜋
𝜋
30. Write as a single trig function.
a) sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
d)
2𝜃
1 − 2 sin2 ( 3 )
2 tan 𝑥
b) 1−tan2 𝑥
c) 10 sin 𝑥 cos 𝑥
𝑥
𝑥
e) −4 sin 2 cos 2
f) 1−tan 3𝑥 tan 4𝑥
tan 3𝑥−tan 4𝑥
31. Develop a formula for 𝑠𝑖𝑛3𝜃 in terms of sin 𝜃
4
12
𝜋
𝜋
32. If sin 𝑥 = 5 and cos 𝑦 = − 13, where 0 ≤ 𝑥 ≤ 2 and 2 ≤ 𝑦 ≤ 𝜋, determine the exact value of:
a) sin (𝑥 − 𝑦)
b) tan(𝑥 + 𝑦)
2 𝜋
33. If cot 𝑥 = − 3 , 2 < 𝑥 < 𝜋, determine the exact value of:
a) tan 2𝑥
b) csc 2𝑥
𝜋
e) cos 4 cos 12 − sin 4 sin 12
34. Prove the following identities.
a)
d)
1+tan2 𝑥
= csc 2 𝑥
tan2 𝑥
cos(𝑥−𝑦)
= 1 + tan 𝑥 tan 𝑦
cos 𝑥 cos 𝑦
g) cot 𝑥 + tan 𝑥 = 2 csc 2𝑥
cos4 𝑥−sin4 𝑥
cos3 𝑥−sin3 𝑥
4
=
cos
𝑥
c)
= 1 + sin 𝑥 cos 𝑥
4
1−tan 𝑥
cos 𝑥−sin 𝑥
1−cos 2𝑥
e)
= sin 2𝑥
f) 2 cos 𝑥 − 2𝑐𝑜𝑠 3 𝑥 = sin 𝑥 sin 2𝑥
tan 𝑥
sin 2𝑥
cos 2𝑥
h) sec 𝑥 = sin 𝑥 − cos 𝑥
b)
35. A Ferris wheel with a diameter of 40m makes a full rotation in 3 minutes. Passengers board at the
bottom of the Ferris wheel, which is 2m above ground.
a) Find the equation that best models the height of a passenger after boarding the wheel.
b) Determine the height of a passenger who has been riding the Ferris wheel for 150 seconds.
c) Determine the times at which the passenger is 25m above the ground within the first rotation.
36. Find the equations of the following graphs in terms of both a sine and a cosine function.
Unit 6 and 7: Exponential and Logarithmic Functions
37. Sketch the following functions.
a) 𝑦 = − log 𝑥 + 2
b) 𝑦 = log(𝑥 + 4)
c) 𝑦 = − log(−2𝑥) + 2
38. Evaluate the following without a calculator.
3
a) log 7 49
b) log 2 16
c) log 2 √32
d) 6−2 log6 8
f) log12 576 − log12 4
g) log 6 18 + log 6 12
39. Solve.
a) log 𝑥 = log 5 + 2 log 3
b) log 6 (𝑥 + 3) + log 6(𝑥 − 2) = 1
3
e) log 4 (64 × √16 )
b) log 3 𝑥 − log 3 4 = log 3 12
d) log(𝑥 + 4) − log 𝑥 = log (𝑥 + 2)
40. Express as a single log:
a)
1
1
1
log 5 𝑥 + 3 log 5 𝑦 − 4 log 5 𝑧
2
1
b) 2 [log 4 𝑥 + 3 log 4 𝑦] − 2[log 4 𝑎 + log 4 𝑏]
41. Solve.
a) 32𝑥−1 = 5
b) 22𝑥 + 3(2𝑥 ) − 10 = 0
d) d) 6(102𝑥 ) + 10𝑥 − 2 = 0
1
c) (64)
e) 42𝑥+1 − 5 × 4𝑥 + 1 = 0
𝑥+2
= 82𝑥
Unit 8: Combining Functions
1
42. Given 𝑓(𝑥) = 𝑥 2 − 4 and 𝑔(𝑥) = 𝑥, find:
a) 𝑓(𝑔(2))
b) 𝑔 ∘ 𝑓(2)
c) 𝑓(𝑔(𝑥))
d) 𝑔 ∘ 𝑓(𝑥)
Answers
1
1. 5 (𝑜𝑟𝑑 1), −5 (𝑜𝑟𝑑 1), − 2 (𝑜𝑟𝑑 2)
2. a) 𝑓(𝑥) = −𝑥 3 − 6𝑥 2 − 3𝑥 + 10
3. 𝑦 = 𝑥 4 + 4𝑥 3 + 6𝑥 2 + 4𝑥 + 1
21
b) 𝑓(𝑥) = −𝑥 4 + 7𝑥 3 − 13𝑥 2 − 3𝑥 + 18
b) 12𝑥 − 22
4. a) 4
5. −94.6
6. 𝑚 = 3, 𝑔 = 1
7.
3𝑦 3 −2𝑦 2 +12𝑦−9
6𝑦−5
= 3𝑦 − 2 + 𝑦2 +2
𝑦 2 +2
8. a) (𝑥 + 1)(3𝑥 − 1)(𝑥 + 2)
c) 3(𝑥 − 3)(𝑥 2 + 3𝑥 + 9)
1
9. a) 𝑥 = 1, 2, −3
d) 𝑥 = 1,
b) (𝑥 − 1)3 (𝑥 + 1)
d) (𝑥 + 5)(𝑥 2 + 3)
b) 𝑥 = ±2, − 2
2±√2𝑖
3
2
c) 𝑥 = 0, ± 3
1
e) 𝑥 = ±1, 2
10. a) 𝑥 ∈ (−1, 0) ∪ (3, ∞)
b) 𝑥 ∈ (−∞, −3) ∪ (−2, 1)
c) 𝑥 ∈ (−∞, −5] ∪ [−1, 0] ∪ [2, ∞)
11. a) Degree is 5, End Behaviours: As 𝑥 → ∞, 𝑦 → −∞ and As 𝑥 → −∞, 𝑦−> ∞
a) 0 (ord 2), 2 (ord 2)
b) 𝑔(𝑥) < 0 when 𝑥 ∈ (0, 2) ∪ (2, ∞)
12. The integers are −9, −8, −7.
13. a) Neither
b) Even
c) Neither
365
14. a) 2
2
b) 4
1
15. VA: 𝑥 = 3 HA: 𝑦 = 3
16.
17. a) x-int: 2
1
y-int: − 3
b) 𝐷 = {𝑥|𝑥 ∈ 𝑅, 𝑥 ≠ −3, −2}
c) VA: 𝑥 = −3, 𝑥 = −2
18. a) x-int: None
b) VA: 𝑥 = −3
d) There are no holes
1
y-int: 3
b) 𝐷 = {𝑥|𝑥 ∈ 𝑅, 𝑥 ≠ ±3}
HA: 𝑦 = 0
d) Hole: (3, 6)
19. Answers may vary. Sample answers: a) 𝑦 =
20. a) (−1, −3)
1
𝑥
𝑥+2
b) 𝑦 =
𝑥−1
(𝑥−1)(𝑥−3)(𝑥+4)
c) 𝑦 =
b) (0, −3)
5
21. a) 𝑥 = − 3 , 𝑥 ≠ 3, −4
b) 𝑥 = −3, 1, 𝑥 ≠ ±1
c) 𝑥 = ±√5, 𝑥 ≠ 0
22.
23. a) 𝑥 ∈ (−∞, −1)
b) 𝑥 ∈ (2, 4)
24. 𝐷 = {𝑥|𝑥 ∈ 𝑅, 𝑥 ≠ 0}, x-int: ±4, no y-int, VA: 𝑥 = 0, OA: 𝑦 = 𝑥,
EB: As 𝑥 → ∞, 𝑦 → ∞, As 𝑥 → −∞, 𝑦 → −∞, As 𝑥 → 0+ , 𝑦 → −∞, As 𝑥 → 0− , 𝑦 → ∞
25.
a)
x-int
None
b)
c)
0
None
y-int
1
−
5
0
None
Asymptotes
VA: 𝑦 = 5 HA: 𝑦 = 0
VA: 𝑥 = 1, 𝑥 = −1 HA: 𝑦 = 3
1
VA: 𝑥 = 0 OA: 𝑦 = 2 𝑥
Holes
1
)
10
None
None
(−5,
−6
𝑥+3
d)
e)
f)
5
None
0
−5
None
0
(−5, 0)
None
None
None
VA: 𝑥 = 0 OA: 𝑦 = 𝑥
VA: 𝑥 = 1, 𝑥 = −1 HA: 𝑦 = −1
26.
√5
27. a) 3
b)
𝜋 5𝜋
2√10
3
4𝜋
28. a) 12 , 12
b) 3
f)
g) ,
𝜋 5𝜋 7𝜋 11𝜋
, , ,
6 6 6
6
𝜋 3𝜋
j) 2 , 2
√3
29. a) − 2
4𝜋
𝜋 5𝜋 7𝜋 11𝜋
c) 0, 3
𝜋 7𝜋 𝜋 4𝜋 2𝜋 5𝜋 5𝜋 11𝜋
, , , , , ,
6 6 3 3 3 3 6
6
𝜋 3𝜋
𝜋 3𝜋
k) 2 , 2
l) 2 , 2
√6−√2
b) 2
c) 4
30. a) sin(𝐴 − 𝐵)
b) tan 2𝑥
𝜋 5𝜋 3𝜋
d) 6 , 6 , 6 , 6
e) 6 , 6 , 2
h) ,
i) ,
𝜋 5𝜋
, 2.16, 5.30
4 4
1
d) 2 + √3
e) 2
4𝜃
3
c) 5 sin 2𝑥
d) cos
b) 39.32𝑚
c) 1.64, 1.36 min
3
𝜋 5𝜋
, 3.87, 5.55
6 6
e) −2 sin 𝑥
f) − tan 𝑥
31. sin 3𝜃 = 3 sin 𝜃 − 4 sin 𝜃
63
33
32. a) −
b)
33. a)
12
5
65
56
b) −
13
12
34. Solutions may vary.
𝜋
35. a) ℎ(𝑡) = −20 cos 𝑡 + 22
3
1
𝜋
1
36. a) 𝑦 = 2 sin [3 (𝑥 − 2 )] + 2 or 𝑦 = 2 cos 3𝑥 + 2
7𝜋
𝜋
b) 𝑦 = 2 sin (𝑥 − 4 ) or 𝑦 = 2 cos (𝑥 − 4 )
37.
38. a) 2
b) 4
39. a) 𝑥 = 45
b) 𝑥 = 48
3
√𝑥 ∙ √𝑦
)
4
𝑧
√
40. a) log (
41. a) 𝑥 = 1.23
15
42. a) −
16
5
11
d) 64
c) 𝑥 = 3
e) 3
d) 𝑥 =
f) 2
−1+√17
2
√𝑥𝑦 3
b) log 4 ( 𝑎2 𝑏2 )
b) 𝑥 = 1
b) 𝐷𝑁𝐸
1
c) 3
c) 𝑥 = −1
1
c) 2 − 4
𝑥
1
d) 2
𝑥 −4
d) 𝑥 = −0.3
e) 𝑥 = 0, −1
g) 3