Advanced Math Senior Review:
Semester 2:
The following information is comprehensive for the second semester. Senior exam
will consist of up to 20 calculation/short answer questions, 5 points each and 5
constructed response questions worth 20 points each for a total of 200 points!!!
Graphing Advanced Rational Functions
Rational Expressions (PDF)
Rational Expressions Part II (PDF)
DOWNLOAD THIS DOCUMENT FOR TODAY'S NOTES: How to find
Asymptotes Notes
Given the following functions: y=1/(2x+6)
y = (6x+8)/(-3x+4)
+ 5x + 6)/(x+3) (don't forget there is a hole in this graph)
find:
Vertical Asymptote
Horizontal Asymptote
Oblique Asymptote
Hole
x- intercept
y-intercept
Domain
Range
Sketch a graph of each
y = (x^2
Polynomial Division
Here's a video to help with long division:
http://www.khanacademy.org/math/algebra2/polynomial_and_rational/dividing_polyn
omials/v/polynomial-division
Here's synthetic division:
http://www.khanacademy.org/math/algebra2/polynomial_and_rational/syntheticdivision/v/synthetic-division
Polynomial Functions Review
Determine the end behavior of each polynomial:
1) f(x) = 2x^5 - 2x + 8
2) g(x) = -4x^6 - 5
Simplify the following polynomials:
g(x) = -4x^6 - 5
f(x) = 2x^5 - 2x + 8
3) (f + h)(x)
4) (g - f)(x)
h(x) = x^2 - 3
5) h(x) times f(x)
Piecewise Functions
Given: a(x)=3x^2+2x b(x)=x+3
Find: 1. a/b of x
2. b(c(x))
5. (b-a)(x)
6. b/a of (-1)
c(x)= (-2x) + 4
3. a(x)c(x)
4. (b-c)(x)
7. (a+c)(2)
composition of functions #1-8 only
Composite Functions Practice
Properties of Exponents
Properties of Radicals
http://hotmath.com/hotmath_help/topics/properties-of-exponents.html
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut40_a
ddrad.htm
Watch this youtube video on how to find the domain of a function:
http://www.youtube.com/watch?v=pUAv94BH7y4
http://www.youtube.com/watch?v=_zy7Uro7iCg
Function Operations
Watch this youtube video on how to find the domain of a function:
http://www.youtube.com/watch?v=pUAv94BH7y4
http://www.youtube.com/watch?v=_zy7Uro7iCg
Properties of exponents
Properties of Radicals
Cramer's Rule Do numbers 1-6 only
Determinants 3x3 #1-5
Inverse Matrices Do #1-10 only
Matrix Equations 1-5
Answer the following questions:
1) What does the determinant tell you?
2) Any matrix times its inverse will equal what?
3) What is the identity matrix?
4) Can you find the solution to a system whose coefficient matrix has a determinant of
zero? Why or why not?
5) What is a coefficient matrix? Where is it used?
Videos to help:
How to multiply a matrix by a scalar:
http://virtualnerd.com/algebra2/matrices/scalar-multiply-example.php
How to find the determinant of a 3x3 matrix:
https://www.youtube.com/watch?v=V3e7m-qFDFU
How to multiply
matrices:
https://www.youtube.com/watch?v=kuixY2bCc_0
How to find the determinant of a 2x2 matrix: http://virtualnerd.com/algebra2/matrices/determinant-2-by-2.php
What is a coefficient matrix?
http://virtualnerd.com/algebra2/matrices/coefficient.php
Cramers Rule for a
2x2
https://www.youtube.com/watch?v=cidMyv6L7Gs
How to find the inverse of a
2x2 https://www.khanacademy.org/math/precalculus/precalcmatrices/inverting_matrices/v/inverse-of-a-2x2-matrix
Matrix Practice Problems (all)
Multi Operation Matrices #1-8 only
Cramer's Rule Do numbers 1-6 only
Answer the following questions:
1) What does the determinant tell you?
2) Any matrix times its inverse will equal what?
3) What is the identity matrix?
4) Can you find the solution to a system whose coefficient matrix has a determinant of
zero? Why or why not?
5) What is a coefficient matrix? Where is it used?
HOMEWORK 1/12/15
Complete the following scenario:
Matrix Scenario 2
http://www.virtualnerd.com/algebra-2/matrices/definition.php
http://www.virtualnerd.com/algebra-2/matrices/add-example.php
http://www.virtualnerd.com/algebra-2/matrices/add-example.php
http://www.virtualnerd.com/algebra-2/matrices/scalar-multiply-example.php
https://www.youtube.com/watch?v=kuixY2bCc_0
Complete the following over the weekend for basic operations practice.
Basic Matrix Operations numbers 1-6, 16-18, 23-24
Complete the following scenario:
Scenario matrix
Mathematics Success Tips: Asymptotes
*Warning* The information given here is not given in technical mathematical definitions and is not the complete
knowledge that you need. Instead it attempts to give a basic understanding of the concept using non-rigorous
terms.
An asymptote is a line that a graph gets closer and closer to, but never touches or crosses it. There are
three main types of asymptotes that functions may have: horizontal, vertical, or oblique. Using the
conditions that follow, you may be able to determine where the asymptotes exist.

Horizontal Asymptotes
o
If the denominator has a higher degree than the numerator, then y=0 is your horizontal
asymptote plus the constant.
1
6
4x  x  5
The highest degree of the numerator is to the power of 0.
f(x) 
2
The highest degree of the denominato r is to the power of 2.
Therefore, the horizontal asymptote is y  0  6  6
o
If the degrees of both the numerator and denominator are the same, then the ratio of
the lead coefficients of the two highest degrees will be the horizontal asymptote plus
the constant.
3x 2  x  4
x 2  6x  9
The highest degree of the numerator is to the power of 2. The coefficient is 3.
f(x) 
The highest degree of the denominato r is to the power of 2. The coefficient is 1.
Therefore, the horizontal asymptote is y 
o
3
or y  3.
1
If the numerator has a higher degree than the denominator, then there is an undefined
asymptote.
3x 2  2x  1
x7
The highest degree of the numerator is to the power of 3.
f(x) 
The highest degree of the denominato r is to the power of 1.
Therefore, the horizontal asymptote does not exist and therefore is undefined.

Vertical Asymptotes
o

Set the denominator of your function equal to zero and solve. The value(s) you get will
be vertical asymptote(s).
3x
f(x) 
(x - 2)(x  5)
If we set the denominato r to zero, then we would get :
( x - 2)(x  5)  0
If x - 2  0 then x  2
If x  5  0 then x  -5
Therefore, the vertical asymptotes are at x  2 and x  -5
Oblique or Slant Asymptotes
o
o
Slant asymptotes only exist when the numerator has a degree EXACTLY one greater than
the degree of the denominator. Use polynomial division (either long division or
synthetic division) or factor to find the quotient. The quotient will be your slant
asymptote (you can ignore the remainder).
Don’t forget that a special case exists when the numerator and denominator share a
common factor. In which case the remainder will be zero, the slant asymptote WILL be
the graph, and there will be a hole or holes (points of discontinuity) where the domain is
restricted.
3x 2  2x  1
f(x) 
x2
The highest degree of the numerator is to the power of 2.
The highest degree of the denominato r is to the power of 1.
Using synthetic division, find the quotient.
2 3 2 1
 6 16
3 8 15
The quotient is y  3x  8 with a remainder of 15.
Therefore, the slant asymptote is y  3x  8.
Mathematics Success Tips: Asymptotes
*Warning* The information given here is not given in technical mathematical definitions and is not the complete
knowledge that you need. Instead it attempts to give a basic understanding of the concept using non-rigorous terms.
An asymptote is a line that a graph gets closer and closer to, but never touches or crosses it. There are
three main types of asymptotes that functions may have: horizontal, vertical, or oblique. Using the
conditions that follow, you may be able to determine where the asymptotes exist.

Horizontal Asymptotes
o If the denominator has a higher degree than the numerator, then y=0 is your horizontal
asymptote plus the constant.
1
6
4x  x  5
The highest degree of the numerator is to the power of 0.
f(x) 
2
The highest degree of the denominato r is to the power of 2.
Therefore, the horizontal asymptote is y  0  6  6
o If the degrees of both the numerator and denominator are the same, then the ratio of the
lead coefficients of the two highest degrees will be the horizontal asymptote plus the
constant.
3x 2  x  4
f(x)  2
x  6x  9
The highest degree of the numerator is to the power of 2. The coefficient is 3.
The highest degree of the denominato r is to the power of 2. The coefficient is 1.
3
Therefore, the horizontal asymptote is y  or y  3.
1
o If the numerator has a higher degree than the denominator, then there is an undefined
asymptote.
3x 2  2x  1
x7
The highest degree of the numerator is to the power of 3.
f(x) 
The highest degree of the denominato r is to the power of 1.
Therefore, the horizontal asymptote does not exist and therefore is undefined.
Given the following functions: y=1/(2x+6)
forget there is a hole in this graph)
find:
Vertical Asymptote
Horizontal Asymptote
Oblique Asymptote
Hole
x- intercept
y-intercept
Domain
Range
Sketch a graph of each
y = (6x+8)/(-3x+4)
y = (x^2 + 5x + 6)/(x+3) (don't
What is the standard form of a polynomial function?
Define the following:
Local or relative maximum
Local or relative minimum
Global or absolute maximum
Global or absolute minimum
How do you know if a graph has a Point of inflection? What criteria must be met?
Determine if a curve is concave up or concave down
Be able to find the extrema and point of inflection given a graph
Know all four possible end behaviors
Determine the end behavior of each graph:
1) X2 – x -6
2. -9x5 – 4x + x
3. -0.00256x8 + 12.5524x – 0.11154
4. 1/3x7 – 12 x + 2
Polynomial operations:
Given f(x) = -2x4 – 2x2 + 6x -15
5. Find: (f +g)(x)
g(x) = 5x4+x3-6x2 – x +10
6) find (f – g)(x)
h(x)=(x + 1)
7)find h(x) times f(x)
Use the 4 graphs below to complete Part I and Part II:
Part I: Identify all relative extrema and points of inflection by circling each point and writing what it
is.
Part II: name the intervals of increasing and decreasing behavior for each of the following graphs.
Use synthetic substitution to evaluate the following polynomials:
Given f(x) = -2x4 – 2x2 + 6x -15
1. f(x) if x=2
g(x) = 5x4+x3-6x2 – x +10
2) g(x) if x = -1
h(x)= -3x4 +x2 + 2x -10
3) h(x) if x=-2
Answer the following questions below using the given polynomials:
Given: f(x) = -2x4 – 2x2 + 6x -15
g(x) = 5x4+x3-6x2 – x +10
h(x)= -3x4 +x2 + 2x -10
1) Evaluate (f+h)(2)
2) Determine the end behavior of (f – h)(x)
Polynomial behavior:
Determine the intervals in which the graph is decreasing:
How many points of inflection are there in the graph below and estimate their ordered pairs.
Determine the end behavior of each polynomial:
1) f(x) = 2x^5 - 2x + 8
2) g(x) = -4x^6 - 5
Simplify the following polynomials:
g(x) = -4x^6 - 5
f(x) = 2x^5 - 2x + 8
3) (f + h)(x)
Given: a(x)=3x^2+2x
Find: 1. a/b of x
5.
(b-a)(x)
4) (g - f)(x)
b(x)=x+3
2. b(c(x))
6.
h(x) = x^2 - 3
5) h(x) times f(x)
c(x)= (-2x) + 4
3. a(x)c(x)
b/a of (-1)
4.
7.
(b-c)(x)
(a+c)(2)
Answer the following questions:
1) What does the determinant tell you?
2) Any matrix times its inverse will equal what?
3) What is the identity matrix?
4) Can you find the solution to a system whose coefficient matrix has a determinant of zero? Why or why
not?
5) What is a coefficient matrix? Where is it used?
Videos to help:
How to multiply a matrix by a scalar:
http://virtualnerd.com/algebra-2/matrices/scalar-multiplyexample.php
How to find the determinant of a 3x3 matrix: https://www.youtube.com/watch?v=V3e7m-qFDFU
How to multiply matrices:
https://www.youtube.com/watch?v=kuixY2bCc_0
How to find the determinant of a 2x2 matrix: http://virtualnerd.com/algebra-2/matrices/determinant-2-by2.php
What is a coefficient matrix?
http://virtualnerd.com/algebra-2/matrices/coefficient.php
Cramers Rule for a 2x2
https://www.youtube.com/watch?v=cidMyv6L7Gs
How to find the inverse of a 2x2 https://www.khanacademy.org/math/precalculus/precalc matrices/inverting_matrices/v/inverse-of-a-2x2-matrix
Answer the following questions:
1) What does the determinant tell you?
2) Any matrix times its inverse will equal what?
3) What is the identity matrix?
4) Can you find the solution to a system whose coefficient matrix has a determinant of zero? Why or why
not?
5) What is a coefficient matrix? Where is it used?
Alec and Pete are the two top performing salesmen for Price LaBlanc Nissan. The highest
grossing salesman of the two will receive a 12% bonus! Given the specific automobile
sales prices, which salesman earned the bonus and how much was it? Use the
information below to figure it out!
Below are two charts that display the number of vehicles sold and on what day for Week 1.
Alex – Week 1
Frontier
Path Finder
300-Z
Altima
Monday
2
1
0
1
Wednesday
1
3
1
1
Friday
3
2
3
4
Saturday
5
1
2
3
Monday
1
2
2
2
Wednesday
2
1
0
0
Friday
4
3
3
4
Saturday
4
2
3
5
Pete – Week 1
Frontier
Path Finder
300-Z
Altima
1) What was the total number of vehicles sold during week 1 by Pete and Alex? Answer
must be in m x n matrix form.
2) Given the sales price of each vehicle: Frontier, $20,250: Path Finder, $34,255: 300-Z,
$56,275: Altima, $29,975
a. Write the vehicle pricing info in m x n matrix form.
3) What is the total of vehicles sold per day in week one by these two salesmen? Organize
data in a m x n matrix. Show how you calculated this figure.
4) What is the total gross sales amount (in dollars) in week 1 of each salesmen?
Show the matrix operations you used to calculate this amount.
5) In week one, what day experienced the highest grossing sales and what was its amount?
Show the matrix operations you used to calculate this amount.
6) Which of the two salesmen had the highest grossing sales at the end of week 1? How
much was this individual’s bonus?
The chart below displays the number of scoring options for four football players in a 10 year period:
Touchdown(TD)
Field Goal
Point after TD
Safety
Jim Thorpe
325
120
80
45
Guy Lombardi
120
15
25
10
Johnny Unitis
625
15
5
20
Knute Rockne
325
25
20
55
A) Rewrite the above information in the form of an m x n matrix and give the dimensions.
B) What does each column represent?
C) What does each row represent?
D) Given that a touchdown is worth 6 points, a field goal is worth 3 points, the point after TD is
worth 1 point and a safety is worth 2 points, answer the following:
a. Write the information from letter D in 4 x 1 matrix form.
E) Using the matrix from letter A and the matrix from letter D, determine the following:
a. How many total points did each player score in the given period of time? (Show your
work using matrix operations!) The answer must be in an m x n matrix form!
b. How many total points in each scoring option were scored? i.e. how many points in
touchdowns were scored, how many points in field goals were scored, etc… The answer
must be in an m x n matrix form!