Name: _______________________________
Cumulative Exam Review Package
Sequences and Series Unit:
Arithmetic Sequence:

An arithmetic sequence is an ordered list of terms in which the difference between
consecutive terms is constant. In other words, the same value or variable is added
to each term to create the next term. This constant is called the common
difference.

The common difference is determined by subtracting two consecutive terms. For
example, 𝒕𝟑 − 𝒕𝟐 = 𝒅.
Example 1: For the arithmetic sequence -6, -1, 4, 9,… determine t12.
1
Name: _______________________________
Example 2: Determine the position of the term 170 in the arithmetic sequence -4, 2, 8,
…
Example 3: Two terms in an arithmetic sequence are t3 = 4 and t8 = 34. What is t1?
Arithmetic Series:

An arithmetic Series is a sum of terms that form an arithmetic sequence. For
example the arithmetic sequence 4, 7, 10, 13 the arithmetic series is represented by
4 + 7 + 10 + 13.
2
Name: _______________________________
Example 4: Determine the sum of the first fourteen terms of the arithmetic series 9 + 15
+ 21 + …
3
Name: _______________________________
Example 5: Find the sum of the terms in the sequence 17, 12, 7, …, -38.
Example 6: An arithmetic series has t1 = 5.5 and d = 2.5. Determine S40.
Geometric Sequence:

A geometric sequence is a sequence in which the ratio of consecutive terms is
constant. The Common Ratio can be found by taking any term, except the first,
and dividing that term by the preceding term.
𝑟=
𝑡𝑛
𝑡𝑛−1
4
Name: _______________________________

Note: The next term in an arithmetic sequence is determined by adding/subtracting
a constant to the previous term, which the next term in a geometric sequence is
determined by multiplying the previous term by a constant (common ratio).

Types of Variables:
Discrete Variables:
 Take on whole number values only.
 The line that connects the points in a discrete graph is a dotted line.
 Examples: price of movie ticket (can't purchase a ticket and a half)
 tn = arn - 1
Continuous Variables:
 Take on any values, including decimals.
 The line that connects the points in a continuous graph is a solid line.
 Examples: population question, bacteria growth
 y = abx
5
Name: _______________________________
Example 7: Determine the 12th term of the geometric sequence 512, -256, 128, -64, …
Geometric Series:
 A Geometric Series is the expression for the sum of the terms of a geometric
sequence.
For example,
6 + 18 + 54 +162
Example 8: Calculate the sum of this geometric series: −6 + 24 − 96 + ⋯ + 98 304.
6
Name: _______________________________
Infinite Geometric Series:

Infinite Geometric series is a geometric series that has an infinite number of terms,
and the series has no last term.

Types of Series:
 A Convergent Series is a series with an infinite number of terms, in which the
sequence of partial sums approaches a fixed value. For example, 1 + ½ + ¼ +
…
 A Divergent Series is a series with infinite number of terms, in which the
sequence of partial sums does not approach a fixed value. For example, 2 + 4 +
8+…

Note: As n gets larger and larger,
 rn gets closer and closer to zero, provided -1< r < 1 (convergent)
 rn gets larger and larger if r< -1 or r > 1 (divergent)
Example 9: What is the sum of the infinite geometric series below?
2
2 2
2 3
5 +5( ) + 5( ) + 5( ) + …
3
3
3
7
Name: _______________________________
Sequences and Series Practice Questions:
1) Determine the general term and the 50th term for each arithmetic sequence.
a) 6, 10, 14, …
b)3, 2 1 , 2, …
2
2) Determine the number of terms in each finite arithmetic sequence.
a) –6, –3, 0, …, 222
b)3 1 , 3 3 , 4 1 , … , 15 3
4
4
4
4
3) Determine the unknown terms in each arithmetic sequence.
a) 4, , , 16
b) , 8, , , 2
c) 20, , , , , –10
4) The 20th term of an arithmetic sequence is 107, and the common difference is 5. Determine the first term,
the general term, and the 40th term of this sequence.
5) Determine the sum of each arithmetic series.
a) 14 + 10 + 6 +  + (–86)
b) 5 + 6.5 + 8 +  + 26
3
13
49
c) + 2 +
++
4
4
2
6) For each arithmetic series, determine the number of terms.
a) t1 = 3, tn = 59, Sn = 465
b) t1 = –2, tn = –74, Sn = –950
c) t1 = 20, tn = –40, Sn = –210
7) A student is offered the opportunity to earn $6.00 for the first day, $11.00 for the second day, $16.00
for the third day, and so on, for 20 working days. Or, the student can accept $1000 for the whole job.
Which offer pays more?
8) Determine the number of terms in each geometric sequence.
a) 4, 12, 36, , 78 732
b) 5 2 , 10, 10 2 , , 640
1
5
1
c) t1 = 5, r =  , tn =
d)t1 = , r = 3, tn = 44 286.75
2
4
64
9) Determine the general term of each geometric sequence.
a) t1 = 2, r = 7
b)6, –18, 54, –164, …
1
1
c) t1 = 7, t5 = 1792
d)r = , t8 =
4
4
10) Determine the unknown terms in each geometric sequence.
a) 18, , , 6174
b) , 4, , , 108
8
Name: _______________________________
c) 5, , , , 80
11) An excavating company has a digger that was purchased for $240 000. It is depreciating at 12% per
year.
a) Determine the next three terms of this geometric sequence.
b) Determine the general term. Define your variables.
c) How much will the digger be worth in
7 years?
d) How long will it take before the equipment is worth less than
$120 000?
12) For each geometric series, state the
values of t1 and r. Then, determine each partial sum.
a) 0.43 + 0.0043 + 0.000 043 + …, (S6)
b) 5 – 5 + 5 – …, (S10)
c) –100 + 50 – 25 + …, (S7)
13) Determine the first term for each geometric series.
a) Sn = 3932.4, tn = 4915.2, r = –4
b)Sn = 292 968, n = 8, r = 5
14) Determine the number of terms in each geometric series.
a) 4 + 20 + 100 +
+ tn = 15 624
b)1792 – 896 + 448 –
– tn = 1197
15) A ball is dropped from the top of a 25-m ladder. In each bounce, the ball reaches a vertical height that
is 3 the previous vertical height. Determine the total vertical distance travelled by the ball when it
5
contacts the ground for the sixth time. Express your answer to the nearest tenth of a metre.
*16) State whether each geometric series is convergent or divergent.
5
40
80
a) 80 + 20 + 5 + + …
b)–30 + 20 –
+
–…
4
3
9
1
1
c) t1 = –5, r =
d)t1 = , r = –2
2
3
*17) Determine the sum of each geometric series, if it exists.
4
2
a) t1 = –4, r =
b)t1 = 10, r =
5
3
c) 10 + 10 3 + 30 + 30 3 + …
2
3
2
2
2
e) 8 + 8   + 8   + 8   + …
3
3
3
d)
5
5
5
5

+

+…
3
9
27
81
2
3
 3 
 3 
 3 
f)– 2 – 2   – 2   – 2   – …
 4 
 4 
 4 
*18) A ball is dropped from a height of 2.0 m onto a floor. On each bounce the ball rises to 75% of the
height from which it fell. Calculate the total distance the ball travels before coming to rest.
*19) A new oil well produces 12 000 m3/month of oil. Its production is known to be dropping by 2.5%
each month.
a) What is the total production in the first year?
9
Name: _______________________________
b) Determine the total production of the well.
Answers:
1) a) tn = 4n + 2; t50 = 202 b) tn = 7  1 n ; t50 = 21 1
2 2
2
2) a) 77 b) 26
3) a) 4, 8 , 12 , 16 b) 10 , 8, 6 , 4 , 2 c) 20, 14 , 8 , 2 , 4 , 10
4) t1 = 12, tn = 5n + 7, t40 = 207
1
5) a) –936 b) 232.5 c) 252.5 or 252
2
6) a) 15 b) 25 c) 21
7) 6 + 11 + 16 +
+ t20 = $1070. Therefore, the arithmetic series method pays more money.
8) a) 10 b) 14 c) 7 d) 12
n1
1
9) a) tn = 2(7)n – 1 b) tn = 6(–3) n – 1 c) tn = 7(4)n – 1 d) tn = 4096  
4
4
, 12, 36 c) 10, 20, 40
3
11) a) $211 200, $185 856, $163 553b) tn = 240 000(0.88) n – 1, tn = value of digger, in dollars, n – 1 = years since
purchase c) $98 082 d) 6 years
12. a) t1 = 0.43, r = 0.01, S6 = 43 b) t1 = 5, r = –1, S10 = 0 c) t1 = –100, r = –0.5, S7 = 1075
99
16
13. a) 1.2 b) 3
14. a) 6 b) 9
15. 94.2 m
16. a) convergent b) convergent c) convergent d) divergent
8
5
17. a) –20 b) 6 c) does not exist d)
e) 24 f) 
7
4
18. 14 m
19. a) 125 761 m3 b) 480 000 m3
10) a) 126, 882 b)
10
Name: _______________________________
Trigonometry Unit:

In geometry, an angle is formed by two rays with a common endpoint. The starting
position is called the initial arm and the final position is called the terminal arm of
an angle. If the angle rotation is counterclockwise, then the angle is positive. If the
angle rotation is clockwise, then the angle is negative.

Note: An angle is said to be an angle in standard position if its vertex is at the
origin of the coordinate grid and its initial arm coincides with the positive x-axis.

A reference angle is the acute angle (< 90o) formed between the terminal arm and
the x-axis. The reference angle is always positive and measures between 0 o and
90o. The right triangle that contains the reference angle and has one leg on the xaxis is known as the reference triangle.
11
Name: _______________________________
Example 1: Determine the measure of all the angles in standard position, 0 o ≤ θ ≤ 360o,
that have a reference angle of 35o.
Example 2: Determine the measure of the rotation angle given the reference angle and
the quadrant:
Reference Angle
Quadrant
25°
2
60°
4
8°
3
Rotation Angle
Special Triangles:
√2
12
Name: _______________________________
Angles in Standard Position:
 The CAST rule summarizes into,
 In Quadrant I: All were positive
 In Quadrant II: Sine was positive
 In Quadrant III: Tangent was positive
 In Quadrant IV: Cosine was positive
Notice the bold first letters. They form the acronym CAST, but you must start in the
4th quadrant.

Definition of Quadrantal Angle is an angle in standard position whose terminal
arm lies on one of the axes. Examples are 0o, 90 o, 180 o, 270 o, and 360 o.

Steps for Solving Angles Give their Sine, Cosine or Tangent
Step 1:
Determine which quadrants the solution will be in by looking at the sign (+
or -) of the given ratio. (CAST Rule)
Step 2:
Solve for the reference angle.
Step 3:
Sketch the reference angle in the appropriate quadrant. Use the diagram
to determine the measure of the related angle in standard position.
Example 3: Determine the exact values of sine, cosine and tangent ratios for 135o.
13
Name: _______________________________
Example 4: Suppose 𝜃 is an angle in standard position with terminal arm in quadrant IV
1
and 𝑡𝑎𝑛𝜃 = − 5 . Determine the exact values of 𝑆𝑖𝑛𝜃 and 𝐶𝑜𝑠𝜃.
Example 5: Solve for θ:
a) tan θ = -0.9004, 0o ≤ θ ≤ 360o
b) cos θ =
-
1
√2
, 0o ≤ θ ≤ 360o
14
Name: _______________________________
Sine Law:
Example 6: Determine the length of side b, to the nearest millimeter.
Example 7: Determine the measure of <B, to the nearest degree.
15
Name: _______________________________
The Cosine Law

The cosine law describes the relationship between the cosine of an angle and the
lengths of the three sides of any triangle.
B
a
C
c
b
A
For any ABC the cosine law states:
To solve for sides:
To solve for angles:
b2  c 2  a 2
2bc
2
a  c 2  b2
cos B 
2ac
2
a  b2  c 2
cos C 
2ab
cos A 
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b2  2ab cos C

To use the cosine law, you must have all sides and are looking for an angle
OR

You have a side-angle-side (SAS) situation.
Example 8: Determine the measure of <A, to the nearest degree.
16
Name: _______________________________
Example 9: A surveyor needs to find the length of a swampy area near Fishing Lake,
Manitoba. The surveyor sets up her transit at a point A. She measures the distance to
one end of the swamp as 468.2 meters, the distance to the opposite end of the swamp
as 692.6 meters and the angle of sight between the two as 78.6°. Determine the length
of the swampy area to the nearest tenth of a meter.
Trigonometry Practice Questions:
1) Determine the reference angle for each angle θ:
a. 355°
b. 135°
c. 260°
d. 70°
2) Determine the measures of the three other angles in standard position, 0    360 , that have
the given reference angle:
a. 40°
b. 89°
c. 27°
3) Determine the measure of each angle θ in standard position, 0    360 , given its reference
angle and which quadrant the terminal arms is in:
a. 37° in quad IV
b. 10° in quad II
c. 85° in quad IV
17
Name: _______________________________
4) Label the sides of the right triangles show with their exact lengths. The shortest side of each
triangle should be 1 unit. Complete the table with the exact values for sine, cosine, and tangent
for each angle.
5) Sketch an angle θ in standard position so that the terminal arm passes through (-4,-5). Then find
the exact values of sin  , cos , tan 
6) Solve for θ:
a. cos  0.8829 , 0    360
b. tan  1.9626 , 0    360
7)
8)
18
Name: _______________________________
9) Two ships are 1600 m apart. Each ship detects a wreck on the ocean floor. The wreck is
vertically below the line through the ships. From the ships, the angles of depression to the
wreck are 40° and 28°. To the nearest meter, how far is the wreck from each ship?
10) Afire spotter sees smoke on a bearing of 60°. At a point 20 km due east of the fire spotter, a
ranger see the same smoke on a bearing of 320°. How far is the smoke from each location?
11) * A cross country runner runs due east for 6 km, then changes course to E25°N and runs another
9 km. To the nearest tenth of a kilometer, how far is the runner from her starting point?
Answers:
1) a. 5° b. 45° c. 80° d. 70°
2) a. 140°,220°, 320° b. 91°, 269°, 271° c. 153°, 207°, 333°
3) a. 323° b. 170° c. 275°
4)
5) sin   
6)
7)
8)
9)
10)
11)
5
4
5
,cos  
, tan  
4
41
41
a. 28°, 332° b. 117°, 297°
a. 19°, 205°
a. 13.4 b. 27°
1109 m, 810m,
10 km, 16 km
14.7 km
Quadratic Functions:


For the quadratic function in vertex form, f(x) = a(x – p)2 + q, a ≠ 0, the graph
 Has the shape of a parabola
 Has a vertex at (p, q)
 Has an axis of symmetry with equation x = p
 Is congruent to f(x) = ax2 translated horizontally by p units and vertically by q
units.
 Domain: {x|x ϵ Ɍ}, Range: If the graph opens up  {y|y ≥ q, y ϵ Ɍ}
If the graph opens down  {y|y ≤ q, y ϵ Ɍ}
The parameter a gives the direction of opening and the vertical stretch factor.
 If a is positive, the graph opens upward and has a minimum y-value.
 If a is negative, the graph opens downward and has a maximum y-value.
 If -1 < a < 1, the parabola is compressed vertically.
 If a > 1 or a < -1, the parabola is stretched vertically.
19
Name: _______________________________

The general form of a quadratic function is f(x) = ax2 + bx + c or y = ax2 + bx + c,
where a, b, and c are real numbers with a ≠ 0.
 a determines the shape and whether the graph opens upward (positive a) or
downward (negative a)
 b influences the position of the graph
 c determines the y-intercept of the graph
Example 1: For each graph of a quadratic function identify the following:
f ( x)   x 2  2 x  8
a) The direction of opening: _________________________
b) The coordinates of the vertex: _____________________
c) The maximum or minimum value: __________________
d) The equation of the axis of symmetry: _______________
e) The x-intercepts and y-intercepts: __________________
f) The domain and range: __________________________
Example 2: Complete the table below.
Equation
Vertex
Transformations
Axis of
Symmetry
Domain and
Range
y = 2x2 + 1
y = 2(x + 1)2
20
Name: _______________________________
y = (x – 3)2 + 2

Changing From General {y = x2 + bx + c; a = 1} to Standard {y = a(x – p)2 + q} Form
Steps for completing the square:
𝑏 2
1. Divide the middle term by two, square this number (2)
𝑏 2
2. Add and subtract this number (2) from the equation.
3. Factor the new perfect square trinomial, and collect like terms.

Changing From General {y = ax2 + bx + c; a ≠ 1} to Standard {y = a(x – p)2 + q} Form
Completing the Square when there is a coefficient to x2:
Divide the first two terms by the coefficient of x2.
Take half of the new b and square it.
Add and subtract the squared number in step 2 INSIDE the bracket.
To pull the subtracted term out of the brackets, multiply by the coefficient that was
pulled out of the first two terms in step 1.
5. Factor the perfect square trinomial and collect like terms.
1.
2.
3.
4.
Example 3: Express y = x2 – 12x + 40 in standard form. Determine the vertex.
21
Name: _______________________________
Example 4: Convert y = 2x2 + 12x + 1 to vertex form.
Quadratic Functions Practice Questions:
Multiple Choice
For #1 to 5, select the best answer.
1. Which of the following does not represent a quadratic function?
A y = (x – 3)2 + 8
B y = (x + 6) – 2
C y = 3x2 – 7x + 2
D y + 3 = x2
2. Which of the following statements is true about the graph of the function y = x2 – 1?
A It has two x-intercepts.
B It opens downward.
C The axis of symmetry is x = 1.
D The vertex is at the origin.
3. What are the coordinates of the vertex for the quadratic function y = –4(x + 7)2 + 5?
A (7, 5)
B (7, –5)
C (–7, 5)
D (–7, –5)
22
Name: _______________________________
4. Which of the following statements is true about the quadratic function shown?
A The equation of the axis of symmetry could be 2x – 5 = 0.
B The range could be {y | y < 2, y  R}.
C The vertex could be (3, –5).
D The x-intercept could have a value of 5.
5.Suppose that the graph of the function
f (x) = 2x2 is reflected in the x-axis, translated 2 units to the left, and then translated 5 units upward.
What could the equation of the quadratic function of the resultant graph be?
A f (x) = –(x + 2)2 + 5
B f (x) = 2(–x – 2)2 + 5
C f (x) = –2(x – 2)2 + 5
D f (x) = –2(x + 2)2 + 5
Short Answer
6. Identify the coordinate, x or y, that is affected in every point (x, y) on the graph of y = x2 by the
following transformations.
a) horizontal translation
b) vertical translation
c) reflection in the x-axis
23
Name: _______________________________
7. Three graphs of the form f (x) = x2 are shown. Identify the graph(s) that fit each description.
a) a > 0
b) a < 0
8.Write a quadratic function in the form
f (x) = a(x – p)2 + q for each graph.
a)
b)
24
Name: _______________________________
9. Rewrite each quadratic function in vertex form by completing the square. Determine the coordinates
of the vertex.
a) y = x2 – 4x + 12
1
4
b) y = – x2 – 4x – 18
Extended Response
10. Determine the following characteristics of the quadratic function y =







1 2
x – 6x – 3.
2
vertex
axis of symmetry
direction of opening
domain
range
exact value for x-intercepts
y-intercept
11. Last year, a music theatre charged $60 admission, and at that price an average of 200 seats were sold
for each show. A survey predicts that for every $5 increase in ticket price, ten fewer people would be
expected to attend a show.
a) Write a quadratic function to model this situation.
b) Determine the admission price that would maximize revenue.
c) What is the maximum revenue?
d) How many seats would be empty when revenue is maximized compared to last year’s average?
Answers:
1. B 2. A 3. C 4. A 5. D
6. a) x b) y c) y
7. a) r and s b) t
8. a) f (x) = 1 (x + 4)2 + 2
3
b) f (x) = –
x2
3
+5
9. a) y = (x – 2)2 + 8; vertex (2, 8)
b) y = – 1 (x + 8)2 – 2; vertex(–8, –2)
4
10. vertex: (6, –21); axis of symmetry: x = 6; direction of opening: upward; domain: x  R;
range: y ≥ –21; x-intercepts: (6  42, 0);
y-intercept: (0, –3)
11. a) R = –50x2 + 400x + 12 000
b) $80
c) $12 800
d) 40 seats
25
Name: _______________________________
Quadratic Equations:

Solutions and roots to a quadratic equation could be determined using any of the
following methods,
1) Graphing
2) Factoring
3) Completing the square
4) Quadratic Formula, 𝑥 =

−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
The number of solutions (nature of the roots) of a quadratic equation can be
determined by the discriminant. The discriminant is the expression 𝑏 2 − 4𝑎𝑐
found under the radical sign in the quadratic formula. It is used to determine the
nature of the roots for a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0.
Example 1: Is x + 3 a factor of each trinomial?
a) 4x2 + 12x + 9
b) 2x2 + x – 15
26
Name: _______________________________
Example 2: Solve each equation,
a) x2 – 2x – 8 = 0
b) (2x – 3)(x + 1) = 3
Example 3: Solve the following quadratic equations by completing the square.
a) x2 – 10x + 3 = 0
b) -5x2 – 10x + 2 = 0
Example 4: Solve the following equations,
a) 2x = 3(x – 1)(x + 1)
b) (2x + 1)(x – 1) = 5x
27
Name: _______________________________
Example 5: Determine whether the equation 5x2 – 8x + 6 = 0 has one, two or no real
roots.
Example 6: Determine the values of k for which the equation 3x2 – 5x + k = 0 has two
real and distinct roots.
Example 7: A picture that measures 10 cm by 5 cm is to be surrounded by a mat
before being framed. The width of the mat is to be the same on all sides of the picture.
The area of the mat is to be twice the area of the picture. What is the width of the mat?
28
Name: _______________________________
Quadratic Equations Practice Questions:
1) Determine whether x  5 is a factor of:
a. 2 x 2  2 x  40
b. 3x 2  13x  10
2) Solve by factoring:
a. x 2  6 x  5  0
b. 3x 2  21x  54  0
c. x 2  6 x  27
d. 3x 2  6 x  105
3) A rectangular garden has dimensions 3 m by 4m. A path is built around the garden. The
area of the garden and path is 6 times as great as the area of the garden. What is the
width of the path?
4) Solve each equation by completing the square. Give your answers as exact values.
a. 3x 2  18 x  2  0
b. 5 x 2  20 x  8  0
c. 2 x 2  16 x  3  0
d. x 2  7 x  11  0
5) Solve each quadratic equation:
a. 3x 2  4 x  1
c. 2 x ( x  3)  4( x  3)  1
b. 4 x 2  1  7 x
d. (2 x  1)2  2  0
6) Josie’s rectangular garden measures 9m by 13m. She wants to double the area of her
garden by adding equal lengths to both dimensions. Determine this length to the bearest
centimeter.
7) Without solving each equation, determine whether it has one, two, or no real roots:
a. 2 x 2  9 x  4  0
b.  x 2  7 x  5  0
c. 2 x 2  16 x  32  0
d. 2.55 x 2  1.4 x  0.2  0
8) Determine the values for k for which:
a. kx 2  6 x  1  0 , has two real roots
Answers:
1) a. no b. yes
2) a.1,5 b. 9,-2 c. 9,-3 d. 7,-5
3) 2.5 m
4
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4) a. 3 
b. 2 
c. 4 
3
2
5
2 7
7  65
b.
3
8
6) 4.43 meters
7) a. 2 b. 2 c. 1 d. 2
3
8) a. k  9 b. k 
8
5) a.
c.
5 3
2
d.
b. 6 x 2  3x  k  0 has no real roots
7 5
2
d. no real roots
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