Algebra 2 2011 – 2012
Final Review
Name: _______________ __ Period ____
Key things to review for the Algebra2 Final Exam.
FROM FIRST SEMESTER (refer to first semester study guide for more detailed information)
Chapter 1
Foundations For Functions
•
Number sets – “0.8 belongs to what sets”
•
Writing sets of numbers in interval or set builder notation
•
Simplify radical expression •
Simplify expressions with negative exponents or 0 as an exponent
Chapter 2
Linear Functions
•
Solve absolute value inequalities
•
Find x and y intercepts of a linear equation
•
Write a linear equation given two points on a line in slope-intercept form
•
Solve linear equations – 5(2x + 1) = 6(x – 4)
Chapter 3
Solving Linear Systems
•
Graph a system of linear inequalities – remember when a line is solid or dashed and how to shade
•
Solve 2x2 systems of equations using substitution or elimination
•
Solve 3x3 systems of equations
Chapter 5
Quadratic Functions
•
Find axis of symmetry of a quadratic equation (x=-b/2a))
•
Find vertex of a quadratic equation (x=-b/2a, plug in to find y)
•
Know how “a” affects whether a quadratic equation opens up or down
•
Find max or min of a quadratic equation
•
Find y-intercept of a quadratic equation
•
Know how a, h, and k transform an equation in the form
•
Find “zeros” of a quadratic equation using the quadratic formula
•
Add, Subtract, Multiply and Divide imaginary numbers like (2 – 3i)2
Chapter 6
Polynomial Functions
•
Use the binomial theorem to expand a binomial like (x-5)6
•
Multiply polynomials like (x+5)2(x2+3x+7)
•
Use long or synthetic division to divide polynomials or determine if a binomial is a factor of a polynomial
Chapter 9
Functions and Inverses
•
Perform operations to functions like (f+g)(x), (f-g)(x), (fg)(x), and
•
•
Find composition of functions 𝑓°𝑔(𝑥) and 𝑔°𝑓(𝑥)
Find the inverse of a function
𝑓
𝑔
(𝑥)
FROM SECOND SEMESTER
Chapter 7
Exponents and Logarithmic Functions
Sections
7.1
Review
Exponential Functions – Growth & Decay
f(x) =abx
f(x) = a· BaseExponent
Growth Function: a>0 and b>1
Decay Function: a>0 and 0<b<1
A(t) = a(1 ± r)t
Practice
1. Tell whether the graph of
f(x)=2(0.8)x is growth or decay
and graph it.
Write the equation and solve:
2. A bacterial culture contains 50
bacteria and doubles every hour.
How many bacteria will there be
after 12 hours
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2012
Algebra 2 2011 – 2012
7.2
7.3
Final Review
Name: _______________ __ Period ____
Inverses of Relations & Functions
Reflection across the line y =x
To find, swap x and y (f(x)) and then solve for y (ie f-1(x))
Find the inverse of f(x) and graph
both equations:
𝟑𝒙−𝟓
3. 𝒇(𝒙) =
Logarithmic Functions:
Exponential and logarithmic equations are inverses of each
other.
Complete the table:
4.
Exp. Eq.
53 = 125
bx = a

logba = x
𝟐
Log Eq.
log381 = x
𝟏
𝟕 =
𝟒𝟗
Simplify if possible:
5. log84 + log816
6. log4320 – log45
7. log4256
−𝟐
7.4
Properties of Logarithms:
Product Property
Quotient Property
Power Property
7.6
Chapter 8
Sections
8.2
log 𝑏
𝑚
= log 𝑏 𝑚 − log 𝑏 𝑛
𝑛
log 𝑏 𝑎𝑝 = 𝑝 ∙ log 𝑏 𝑎
Inverse Property of
logarithms
log 𝑏 𝑏 𝑥 = 𝑥
Inverse Property of
Exponents
𝑏 log𝑏 𝑥 = 𝑥
Change of Base
Formula
7.5
log 𝑏 𝑚𝑛 = log 𝑏 𝑚 + log 𝑏 𝑛
log 𝑏 𝑥 =
log 𝑎 𝑥
log 𝑎 𝑏
Exponential and Logarithmic Equations:
If bx = by, then x=y (b≠1, b≠0)
If a = b, then log(a) = log(b) (a>0, b >0)
The Natural Base e :
The value of e is approx. 2.718 (it is irrational like π)
The natural logarithm / exponent relationship:
ex = y then ln(y) = x
ln(ex) = x
Rate of Decay: N(t) = N0e-kt Compound Interest: A=Pert
N(t): Amount remaining
A: Final amount
k: decay constant
r: annual interest rate as a decimal
t: time
t: time in years
N0: Initial amount
P: Initial amount
Radical & Rational Functions
Review
Multiplying & Dividing Rational Expressions:
Factor the Numerator and denominator of both pieces first.
Multiply and divide just like fractions. (When dividing remember to
‘flip’ the 2nd fraction and then multiply).
Simplify by canceling any common factors.
Solve:
8. 8x = 2x + 6
9. log6(4x – 9) = log6(x)
Simplify:
10. ln(e2x + 10)
11. eln5x
12. How much money would you
have in 10 years if you invested
$1000 at 5% interest?
13. 2ex=16
Practice
Simplify:
14.
𝒙𝟐 −𝒙−𝟐
𝟑𝒙−𝟔
15.
𝒙𝟐 −𝟗 𝒙−𝟓
∙
𝟐𝒙+𝟏𝟎 𝒙−𝟑
16.
8.3
Adding & Subtracting Rational Expressions:
Factor the Numerator and denominator of both pieces first
Get common denominators
𝒙𝟐 −𝟏𝟔
𝒙−𝟒
÷ 𝒙+𝟏
𝒙𝟐 +𝟒𝒙+𝟑
Simplify:
𝒙
17.
−
𝒙+𝟏
𝟑
𝒙+𝟒
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2012
Algebra 2 2011 – 2012
8.4
8.5
Final Review
Name: _______________ __ Period ____
Add/subtract the numerators
Simplify as needed
Rational Functions:
Graph of equation below is a hyperbola
𝑎
𝑓(𝑥) =
+𝑘
𝑥−ℎ
|a|: vertical stretch or compression, a<0 reflection across x-axis
k: vertical translation (y = k is horiz. Asymptote)
h: horizontal translation (x = h is vertical asymptote)
Solving Rational Equations and Inequalities:
Use cross-products if possible to remove fractions. If more than 1
term on a side, multiply by Least Common Multiple to remove all
denominators.
Graph:
18. 𝒈(𝒙) =
Radical Expressions and Rational Exponents:
(𝑏)𝑚+𝑛
𝑚
𝑛
𝑏 =
8.7
𝑚
=𝑏 ∙𝑏
𝑚
𝑛
√𝑏 𝑚
(𝑏)𝑚−𝑛
(𝑏)−𝑚
=
=
𝑏𝑚
𝑏𝑛
1
𝑏𝑚
Radical Functions:I
Identify the Domain (valid x-values) and Range (valid y-values)
Graph the functions using 𝑓(𝑥) = √𝑥 as a guide
−𝟓
Solve:
19.
20.
8.6
𝟏
𝒙+𝟐
𝟔𝒙
𝒙+𝟓
=
𝟒
𝒙−𝟒
𝟐𝒙−𝟐𝟎
𝒙+𝟓
−𝒙
𝒙
= 𝒙−𝟒 + 𝟐
Simplify and write with Positive
Exponents:
𝟒 𝒙𝟖 𝒚𝟒
21. √
𝟑𝟐
Graph:
22. 𝒈(𝒙) = 𝟑√𝒙 + 𝟐 + 𝟒
Vertical Translation: f(x) + k
Horizontal Translation: f(x-h)
Vertical Stretch/Compress: a·f(x)
1
Horizontal Stretch/Compress: f(x) = x
b
Reflection: -(f(x)) –> across x-axis, f(-x) -> across y-axis
Chapter 10 Sections
10.2
10.6
(10.310.5
next
page…)
Conic Sections
Review
Circles:
(x-h)2 + (y-k)2 = r2
Center: (h,k)
Radius: r
Distance Formula from (x1,y1) to (x2,y2): 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Practice
Write the equation for the circle:
Identifying Conics:
Arrange the equation in standard form. Match form with shapes above.
Identify the conics:
25. X2 – 6y – 8x + 16 = 0
26. 10x2 + 15xy + 10y2 + 15x + 25y
+9=0
27. 6x2 = 14x + 12y2 – 16y + 20
28. X2 + y2 – 8y – 33 = 0
23. Center (2, -3), radius 5
24. Center (2, -3), point on
circle (8, -1)
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2012
Algebra 2 2011 – 2012
10.3,
10.4,
10.5
Final Review
Name: _______________ __ Period ____
Ellipses, Hyperbolas, Parabolas
Horizontal formula comparison: (Vertical formulas are similar, swap X &
Y axis.)
Formula
Ellipse
Hyperbola
(𝑥 − ℎ)2 (𝑦 − 𝑘)2
+
=1
𝑎2
𝑏2
(𝑥 − ℎ)2
𝑎2
−
Parabola
(𝑦 − 𝑘)2
𝑏2
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
=1
𝑥 = 𝑎(𝑦 − 𝑘)2 + ℎ
(𝑦 − 𝑘)2 (𝑥 − ℎ)2
−
=1
𝑏2
𝑎2
Vertices
On major axis
If x comes first, left
right a units from
center
If y comes first, up
down b units from
center
Co-Vertices
On minor axis
Not as important
Foci (and
directrix)
c2 = a2 – b2 OR
c2 = b2 – a2
c 2 = a2 + b 2
c units from center
(“pearl” inside the
hyperbola)
c units from center
on major axis
Chapter 11 Sections
11.1
Vertex is (h, k)
Write the equation of the
hyperbola in standard form and
graph it identifying all parts:
31. 9x2 – 4y2 – 54x – 40y – 55 = 0
32. Center (3, 1), Vertex (3,3), covertex (8,1)
Write the equation of the
parabola in standard form and
graph it identifying all parts:
1
33. vertex (2,3), focus (2,5)
p=
4𝑎
2
distance to the 34. 2x – 5x + y – 19 = 0
focus and
directrix from
vertex
Probability
Review
Permutations & Combinations:
Permutations – number of ways to select and arrange a subset of things
𝑛!
(order matters) nPr=
(𝑛−𝑟)!
Combinations – number of ways to select a subset of things
(order doesn’t matter) nCr=
𝑛!
𝑟!(𝑛−𝑟)!
Fundamental Counting Principle : If there are n items and m1 ways to
choose the 1st, m2 ways to choose the 2nd etc. , then the total number of
ways to choose is m1·m2· …·mn
11.2
Write the equation of the ellipse
in standard form and graph it
identifying all parts:
29. Center(-2, 5), vertex (7,5),
focus (5,5)
30. 6x2 + 4y2 + 84x – 24y + 306 = 0
Theoretical & Experimental Probability:
Probability of an event is how likely the event will occur. Experimental
probability uses the experimental results for the calculation.
# 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠
𝑃(𝑒𝑣𝑒𝑛𝑡) =
# 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒𝑠
The total of the favorable and unfavorable outcomes is always 1
P(not event) = 1 – P(event)
Practice
35. Jamie purchased 3 blouses, 3
jackets, 2 skirts. How many
different blouse/skirt/jacket
outfits are possible?
36. Nate is on a 7 day vacation.
He plans to spend 1 day
swimming and 1 day golfing.
How many ways can he
schedule the activities?
37. A teacher can send 4
students to the library daily.
There are 21 students in
class. How many ways can he
choose 4 students the first
day?
Color
Red
Green
Blue
Spins
5
8
7
38. What is the probability of
spinning Red?
39. What is the probability of
spinning red or blue?
4
2012
Algebra 2 2011 – 2012
Final Review
Name: _______________ __ Period ____
11.3
Independent & Dependent Events
Independent Events – 2 or more events where one event does not
impact the others.
Independent Probability (A and B) = P(A) · P(B)
Dependent Events – 2 or more events where one outcome impacts the
probability of the other outcomes.
Dependent Probability (A and B) = P(A) · P(B/A)
Where P(B/A) is the probability of event B given that A has occurred
11.4
Compound Events
A compound event is one made up of 2 or more simple events
Mutually Exclusive events: Events that cannot both occur in the same
trial.
𝑃(𝐴 ⋃ 𝐵) = 𝑃( 𝐴) + 𝑃(𝐵)
Inclusive events: Events that have one or more outcomes in common.
𝑃(𝐴 ⋃ 𝐵) = 𝑃( 𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ⋂ 𝐵)
(subtract the probability that A and B both happen)
11.5
Central Tendency & Variation
Mean: Average (add all elements and divide by # of elements)
Median: Middle number of elements arranged in order (mean of middle
two if even number of elements)
Mode: Element value repeated the most (may have more than one or
none)
Box & Whisker Plot: Display minimum value, maximum value, median, 1st
quartile and 3rd quartile (quartile is median of the given half of data)
11.6
Variance (σ2) – the average of the squared differences from the mean
Standard Deviation (σ) – the square root of the variance. Low values
indicate tightly clustered data, high values indicate widely spaced data.
To find Standard Deviation:
1) find the mean of the data
2) find the distance each point is from the mean
3) square the values from #2
4) variance: the average of the values from #3
5) standard deviation: squareroot of the variance
Binomial Distributions
Pascal’s Triangle or Combinations can be used to find the coefficients for
the binomial expansions.
Pascal's Triangle
Combinations (Binomial
Coefficients)
1
0C0
11
121
1331
1C0
2C0
3C0
Given {2, 4, 4, 6, 6, 6, 7, 8}
44. Find the mean, median and
mode.
Given {2, 4, 1, 4, 2, 2, 7, 4}
45. Make a box & whisker plot
and label all parts.
Given {3, 3, 4, 5, 5)
46. Find the variance
47. Find the standard deviation
Use the Binomial Theorem to
expand each binomial
48. (x - 3)6
Binomial Expansion
(x + y)0 = 1
1C1
2C1
3C1
A bag contains 20 checkers (10
black, 10 red)
40. What is the probability of
selecting 2 black checkers
when they are chosen at
random with replacement?
41. What is the probability of
selecting 2 black checkers
when they are chosen
without replacement?
The numbers 1-10 are written on
cards and placed in a bag. Find
each probability
42. Choosing a number greater
than 5 or an odd number.
43. Choosing an 8 or a number
less than 5.
3C2
(x + y)1 = x + y
2C2
3C3
(x + y)2 = x2 + 2xy + y2
(x+y)3= x3+ 3x2y+
3xy2+y3
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2012
Algebra 2 2011 – 2012
Chapter 12 -
Final Review
Name: _______________ __ Period ____
Sequences & Series
Sections
12.1
Review
Intro to Sequences:
Sequence is an ordered set of numbers. If it ends, it is finite. If it keeps
going, it is an infinite sequence.
A recursive formula is a rule used to find the nth term of a sequence and
is based on or more previous terms.
An explicit formula defines the nth term of a sequence as a function of n.
12.2
Series & Sum Notation:
A series is a sum of the terms of a sequence. Usually uses sum notation:
𝑏
∑ 𝐶𝑘
𝑘=𝑎
a = first value
b = last value
Ck = explicit formula for the sequence
Constant
Linear Series
Quadratic Series
Series
𝑛
𝑛
∑ 𝑐 = 𝑛𝑐
𝑘=1
12.3
12.4
12.5
𝑛(𝑛 + 1)
∑𝑘 =
2
𝑘=1
𝑛
∑ 𝑘2 =
𝑘=1
𝑛(𝑛 + 1)(2𝑛 + 1)
6
Arithmetic Sequences & Series:
Arithmetic Sequence – a seq. where the terms differ by the same number
(d = common difference)
Use slope method OR formula: an = a1 + (n-1)d
Arithmetic Series – the sum of an arithmetic sequence.
𝑛(𝑎1 + 𝑎𝑛 )
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑛 𝑡𝑒𝑟𝑚𝑠: 𝑆𝑛 =
2
Geometric Sequences & Series:
Geometric Sequence – a seq. where the terms are multiplied by the same
number (r = common ratio)
an = a1 · r(n-1)
Geometric Mean: If a & b are positive, their geom. Mean = √𝑎 ∙ 𝑏
Geometric Series – the sum of a geometric sequence.
𝑎1 (1 − 𝑟 𝑛 )
𝑠𝑢𝑚 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑛 𝑡𝑒𝑟𝑚𝑠: 𝑆𝑛 =
1−𝑟
Mathematical Induction & Infinite Geometric Series:
Geometric Series Rules:
 A series converges if |r| <1 (sum approaches a limit)
 A series diverges if |r| ≥ 1 (sum approaches infinity)

Sum of infinite geom.
Mathematical Induction Proof:
1. Show it works for n=1
2. Assume it works for n=k
3. Prove it works for n = k+1
𝑆=
𝑎1
1−𝑟
Practice
Find the first 5 terms of the
sequence
49. a1=1, an = 4an-1 – 1
50. an = -3n2
Write a possible explicit rule for
the nth term of each sequence
51. 6, 9, 12, 15, 18, ….
52. 9, 5, 1, -3, -7, …
Write each series in Sum
Notation
53. -1+4 – 9 + 16 – 25 + 36
Evaluate each series
54. ∑𝟑𝒌=𝟏 𝒌𝟑
55. ∑𝟕𝒌=𝟐(−𝟐)𝒌
𝟐
56. ∑𝟐𝟎
𝒌=𝟏 𝒌
Determine if the sequence could
be arithmetic and find the
difference:
57. 46, 39, 32, 25, 18, …
58. 28, 21, 15, 10, 6, ….
Find the 8th term of the arith. seq.
59. a4= 184 and a5=16.2
Determine if the sequence could
be geometric or arithmetic and
find the ratio or difference:
60. -10, -12, -14, -16, …
61. -320, -80, -20, -5, …
Find the 8th term of the geom.
seq.
62. 2, 6, 18, 54, 162, …
63. a4= -12 and a5= -4
64. a4 = -4 and a6 = -100
Determine whether the series
converges or diverges
65. 1 -5 + 25 – 125 + 625…
66. 27 + 18 + 12 + 8 + …
Find the Sum of the inf. Geom.
series
𝒌
67. ∑∞
𝒌=𝟏 𝟒(𝟎. 𝟐𝟓)
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2012