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Algebra II
Spring final Summary
Harding Charter Prep
2016-2017
Dr. Michael T. Lewchuk
Final Exam Info 1
• Covers the following: (but must know all material)
• THIS IS NOT A COMPLETE LIST!
• Your Semester notes are a complete list
– CH 8 – Exponents and Logarithms
– CH 9 – Rational Expressions
– CH 11 – Sequences and Series
– CH 12 – Probability and Statistics
– CH 4 - Matrices
– CH 13 – Trig functions
• You may use a scientific calculator but NOT a graphing
calculator
Final Exam Info 2
• Thursday May 11, 9:45
– Either Cafeteria or several other classrooms
– 40 multiple choice questions
• You may bring in 1 5.5” by 8.5” sheet with general equations
on it – must be handwritten!
– It may not have worked out problems or examples
– If I see them, the exam will be torn up and you will get a
zero!
– I will check notecards during the exam
– You may bring me your card outside of class time to
check it for acceptability but NOT accuracy
Final Exam Info 3
• If “I love math” appears near the top of the ZipScan sheet
you will get a bonus point on the Exam
• If your class section appears near your name on the
Zipscan you will get another bonus point
D :  , 
Parent Function of a Line
f ( x)  y  x
y
R :  ,  
x
Parent Function of an Absolute Value
f ( x)  y  x
y
D :  , 
R : 0, 
x
D :  , 
Parent Function of a Quadratic
f ( x)  y  x
R : 0, 
2
y
7
6
5
4
3
2
1
x
4
3
2
1
1
1
2
3
4
5
D :  , 
Parent Function of a Cubic
f ( x)  y  x
3
y
R :  ,  
x
D : 0,  
Parent Function of a Square Root
f ( x)  y  x
y
R : 0, 
x
D :  , 
Parent Function of a Cube Root
f ( x)  y  3 x
y
R :  ,  
x
Parent Function of a Rational Expression
1
f ( x)  y 
x
y
D :  ,0   0,  
R :  ,0   0,  
x
General Equations
Line
f ( x )  y  mx  b or ax  b
Absolute value
f ( x)  y  a | x  h | k
Quadratic
f ( x)  y  a  x  h   k
Square Root
f ( x)  y  a x  h  k
Cubic
f ( x)  y  a  x  h   k
Cube Root
f ( x)  y  a 3 x  h  k
Exponential
f ( x )  y  ab x h  k
Logarithm
f ( x )  y  a log b  x  h   k
Rational
Rat. Squared
2
3
a
f ( x)  y 
k
xh
a
f ( x)  y 
k
2
 x  h
Properties of Exponents Summary
a0
b0
a m a n  a mn
Product Rule
am
mn

a
an
Quotient Rule
a0  1
Zero Exponent
a m 
1
am
Negative-Exponent
a
 a mn
Power to Power Rule

m n
n n
ab

a
b
 
n
Power of a Product
n
n
a
a
 
   n
b b
Power of a Quotient
• General Equation for Exponential
Functions
Stretch or Shrink
-ve = Flip
Across asymptote
-ve = Flip
Across y
f ( x)  y  ab
Horizontal Shift
x h
k
Vertical Shift
Growth or Decay
Properties for Expanding and Condensing Logarithms
M , N and b are positive numbers and b  1
Product Rule for Logarithms
logb  MN   logb M  logb N
Quotient Rule for Logarithms
M 
log b    log b M  log b N
N
Power Rule for Logarithms
logb M p  p log b M
Simplifying Rational
Expressions
1. Factor the numerator and the denominator completely.
2. Divide both the numerator and the denominator by
any common factors.
Multiplying Rational
Expressions
1. Factor all numerators and denominators completely.
2. Divide numerators and denominators by common factors.
3. Multiply the remaining factors in the numerators and
multiply the remaining factors in the denominators.
Adding and Subtracting
Rational Expressions with the
Same Denominator
Add or subtract rational expressions with the same
denominator by:
(1) Adding or subtracting the numerators,
(2) Placing this result over the common denominator, and
(3) Simplifying, if possible.
Adding and Subtracting
Rational Expressions with
Different Denominators
Add or subtract rational expressions with different
denominators by:
(1) Finding the common denominator by listing factors
(2) Rewriting each expression to with the common
denominator
(3) Simplifying, if possible.
Adding and Subtracting Expressions
That Have Different Denominators
1. Find the LCD of the rational expressions.
2. Rewrite each rational expression as an equivalent
expression whose denominator is the LCD. To do so,
multiply the numerator and the denominator of each
rational expression by a factor(s) needed to convert
the denominator into the LCD.
3. Add or subtract numerators, placing the resulting
expression over the LCD.
4. If possible, simplify the resulting rational expression.
Finding the Least Common
Denominator
1. Factor each denominator completely.
2. List the factors of the first denominator.
3. Add to the list in step 2 any factors of the
second denominator that do not appear in the list.
4. Form the product of each different factor from
the list in step 3. This product is the least common
denominator.
Simplifying Complex Rational
Expressions
Complex fractions, those with more than one divisor, can
be simplified to an expression with a single numerator and
denominator by following the addition/subtraction and
multiplication/division rules for each part of the complex
fraction.
Solving Rational Equations
1. Factor all numerators and denominators completely
2. Find common denominators, if necessary
3. Simplify the equation as much as possible
4. Use cross multiplication to solve for the variable
5 Check that your solution(s) is/are consistent with the
original equation
Identifying The Horizontal Asymptote
If f ( x )  p( x ) / q( x ) is a rational function in general form, then the
degree of the leading terms in the numerator (n ) and denominator (m )
control the location of the horizontal asymptote.
1 If n  m, the x - axis, or y  0, is the horizontal asymptote.
2 If n  m, then the ratio of the leading coefficients determines the
y coordinate of the horizontal asymptote.
3 If n  m there is no horizontal asymptote, but there may be
a slant asymptote.
The graph of a rational function has a slant asymptote
if the degree of the numerator is one more than the
degree of denominator. The equation of the slant
asymptote can be found by division. It is the equation
of the dividend with the term containing the remainder
dropped.
Basic Variation equations
Direct Variation (both increase/decrease)
y  kx
Indirect Variation (one goes up the other down)
k
y
x
Joint Variation (varies with two or more variables)
y  kxz
General Variation equations
Direct Variation
y  kx or y  kx n
Indirect Variation
k
k
y
or y  n
x
x
Joint Variation
n
x
x
y  kxz or y  k
or y  kx n z n or y  k n
z
z
Sequences
A sequence is a set of numbers or objects arranged in a
particular pattern.
Sequences may be finite or infinite.
Sequences may be arithmetic, geometric or more complex.
Sequences may converge or diverge.
Each object in a sequence is called a term. Terms are
numbered a1, a2, a3 etc.
Example: The Fibonacci Sequence and its terms
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …….
a1
a5
a8
a13
Recursive Sequences
Sometimes a sequence requires knowledge of a previous
term in order to be solved.
This is called a recursive sequence.
Example: find the next 3 terms of an=3(an-1)+2 if a1=3
an=3(an-1)+2
a1=3
a2=3(a2-1)+2 =3(a1)+2=3(3)+2=11
a3=3(a2)+2=3(11)+2=35
a4=3(a3)+2=3(35)+2=107
Summation Notation
Sometimes we are interested in a series which is the sum
of several terms of a sequence. The sum of a portion of a
sequence is written using the sigma symbol ().
Example: find the sum of the first 5 terms of an=2n+4
This can also be written as
an=2n+4
a1=2(1)+4=6
a2=2(2)+4=8
a3=2(3)+4=10
a4=2(4)+4=12
a5=2(5)+4=14
5
 2n  4
n 1
5
 2n  4  6  8  10  12  14
n 1
5
 2n  4  50
n 1
Arithmetic Sequences
An arithmetic sequence is a sequence where the difference
between any two terms is a constant (d). We call this the
common difference of the sequence. For each successive
term of the sequence add d to the previous term.
The general term of an arithmetic sequence is written as
an  a1  (n  1)d
KNOW
THIS
The Sum of the First n Terms
of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence can
be determined by the equation below.
n
S n  a1  an 
2
or Sn  n
n is the number of terms to be summed
a1 is the first term
an is the nth term

a1  an
2

Geometric Sequences
A geometric sequence is a sequence where each successive
term is obtained by multiplying the previous term by a
nonzero constant (r). We call this ratio the common ratio of
the sequence.
The common ratio can be
obtained from the following:
an
r
an 1
KNOW
THIS
The general term of a geometric
sequence is written as
an  a1r
n 1
KNOW
THIS
The Sum of the First n Terms
of a Geometric Sequence
The series consisting of the sum of the first n terms of a
geometric sequence can be determined by the equation
below.
1 r n 

S n  a1 
 1 r 
n is the number of terms to be summed
a1 is the first term
r is the constant ratio, but r1
KNOW
THIS
The Steps in Word Problems
1 Figure out if the series is arithmetic or
geometric
2 Find the correct equation(s) (Recipe)
3 Extract data from statement (Groceries)
4 Plug the data in the equation(s) (Cook)
5 Write the correct answer with units (Eat dinner)
*you must show calculations and put the final
answer in sentence form, WITH UNITS!
Convergence, Divergence and the Sum
of an Infinite Geometric Sequence
When |r| <1 the sum (S) of an infinite geometric series may
converge to a constant value using the following formula:
a1
S
1 r
KNOW
THIS
a1 is the first term
r is the constant ratio such that 1  r  1
When |r| >1 of the infinite geometric does not have a sum
and the series diverges.
The 5 main Equations
Arithmetic
an  a1  (n  1)d
Geometric
an  a1r
n 1
n
n


1

r
Sn   a1  an  S  a
n
1

2
1 r

a1
S
1 r

KNOW
THESE
Independent Events
Two events are said to be independent if the occurrence of one
does not have an effect on the other.
The Fundamental Counting
Principle
The number of ways a series of successive independent events
can occur can be found by multiplying the number of ways each
individual event can occur.
Multiple Independent Events
Given two independent events, if one can occur m ways and the
other n ways then the number of ways both events can occur is
(m)(n).
If three or more independent events are involved then the number
of ways they can occur is the product of their individual
occurrences.
Factorial Notation
The factorial of a positive integer, n, is the product of all positive
integers from 1 to n.
It is identified by the symbol n!.
Example
7!   7  6  5  4  3 2 1  5040
Permutations
A permutation is an ordered arrangement of items that occurs when
1. No item is used more than once
2. The order of the items must be considered
n!
nP r 
(n  r )!
Combinations
A combination is an ordered arrangement of items that occurs when
1. No item is used more than once
2. The order of the items does not matter
n
n!
nC r 
 
 n  r ! r !  r 
Probability
Probability (P) provides a quantitative description of the chance or
likelihood of an event will occur. It ranges from 0 (never occurs) to
1 (always occurs).
Theoretical probability requires knowledge of all possible
outcomes. In this case the probability of an event, A is defined by:
P ( A) 
number of times A occurs
total number of outcomes
Experimental probability uses past knowledge to predict the
likelihood of future events.
Unions and Intersections
For two events, A and B, the occurrence of either is P(A) or
P(B) and the union of A or B is P(AB).
9 1
P ( A)  
18 2
9 1
P( B)  
18 2
P( A  B) 
18
1
18
For two events, A and B, the occurrence of both is the
intersection or (AB) of A and B.
P( A  B) 
6 1

18 3
Mutually Exclusive Events
If two events are mutually exclusive, when one event occurs,
the other cannot, and vice versa. There is no event where both
outcomes occur. The probability equation is a follows:
P( A or B )  P( A  B )  P( A)  P( B )
P ( A) 
9 1

18 2
P( B ) 
9 1

18 2
9 9 18
P( A or B )  P( A  B )   
1
18 18 18
Mutually Inclusive Events
If two events are not mutually exclusive, an outcome including
both possibilities (the intersection) must be considered. The
probability equation is a follows:
P( A or B)  P( A  B )  P( A)  P( B)  P( A  B)
9
P ( A) 
18
9
P( B) 
18
P( A or B)  P( A  B ) 
6
P( A  B) 
18
6 12 2
9 9
 


18 18 18 18 3
The Normal Distribution
The normal distribution can be described using the mean ( or 𝑥)
and a standard deviation ( or S).
In a standardized distribution the mean is equal to zero and the
standard deviation is equal to 1.
The Power of the Normal Distribution
Approximately
68.3% of results are within 1 SD
95.4% of results are within 2 SD
99.7% of results are within 3 SD
99.99% of results are within 4 SD
Matrix Notation
We can represent a matrix in two different ways.
1. A capital letter, such as A, B, or C , can denote a matrix.
2. A lowercase letter in brackets, such as that shown below can denote a matrix.
A   ai j 
A general element in matrix A is denoted by aij . This refers to the element in
the ith row and jth column. a32 is the element located in the 3rd row, 2nd column.
See below.
 a11 a12 a13 
a

a
a
22
23 
 21
 a31 a32 a33 
A matrix of order m  n has m rows and n columns. If m  n, a matrix has the same
number of rows as columns and is called a square matrix.
Equality of Matrices
Two matrices are said to be equal, if their dimensions are the same
and the entries in the corresponding positions are equal
Matrix Addition and Subtraction
To add or subtract matrices you simply add or subtract the
corresponding entries.
You can only add and subtract matrices with the same dimensions!
Matrices can be regrouped and added in any order
Commutative Property of Addition ie. A + B = B + A
Associative Property of Addition ie. (A + B) + C = A + (B + C)
Scalar Multiplication
Matrices may be multiplied by a scalar (any real number) by
multiplying each entry in the matrix by that scalar.
If more than one scalar multiplication is involved then the
multiplication may be done in any order.
  3 1  60 20 
 3 1
Example:  5 * 4  
 5* 4 




0
4
0
4
0
80


 

 
Scalar multiplication is Distributive ie. a(B+C)=aB+aC
  3 1  5 3  
  3 1   5 3 
Example: 5  

  5 
5




0
4
2

2
0
4
2

2
 

 



Summary of Matrix properties
Commutative Property of Addition ie. A + B = B + A
Associative Property of Addition ie. (A + B) + C = A + (B + C)
Distributive property of Addition/Subtraction ie. a(B + C)=aB + aC
Associative property of Scalar Multiplication ie. a(BC)=(aB)C=B(aC)
Associative property of Matrix Multiplication ie. (AB)C=A(BC)
Distributive property of Matrix Multiplication ie. A(B + C)=AB + AC
NOTE:
There is no commutative property of Matrix Multiplication
AB ≠ BA
Multiplying a Matrix by Another Matrix
Multiplying two matrices is not as easy as addition and subtraction.
The matrices do not have to be of the same order but the number of
columns in the first matrix must equal the number of rows in the
second matrix.
The order of the resultant matrix may be different than the original
matrices.
Example: if A is m x n matrix and B is a n x q matrix then the matrix
formed by A x B is an m x q matrix.
Right Triangle Definitions for the angle θ
of the Six Trigonometric Functions
Five Forms of Two Triangles
a 2
a
a
a
3
a
2
a
1
a
2
a 2
2
a 2
2
2a
a 3
a
a
3
2a
3
3
a
3
The Unit Circle in (x,y) Coordinate Space
 +y
2

2
3  1 3
3


(0,1)
1
3

,



3
 ,

2 2 
4
5
6



2 2

,


2
2



3 1
, 
 
2
2

-x
(-1,0)
7
6

3 1

,  

2
2

5
4

90°
120°
2 2 


 2 2
,


2
2
60°


45°
135°
 3 1
, 

2
2

30°
150°
180°
0°
330°
210°
225°

2
2
,
 

2
2


4   1 ,  3 
 2
2 

3
315°
240°
300°
270°
3
2
4
6
0
+x
(1,0)
 3 1
,  

2
2

 2
2
,


2
2


1

3 5
 , 

2
2

 3
-y (0,-1)

11
6
7
4