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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) y = (x^2 + 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 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/synthetic- division/v/synthetic-division Polynomial Functions Review Determine the end behavior of each polynomial:

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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) y = (x^2 + 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

Polynomial Division

Here's a video to help with long division:

http://www.khanacademy.org/math/algebra2/polynomial_and_rational/dividing_polynomials/v/polynomial-division

Here's synthetic division: http://www.khanacademy.org/math/algebra2/polynomial_and_rational/synthetic-

division/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 h(x) = x^2 - 3

3) (f + h)(x) 4) (g - f)(x) 5) h(x) times f(x)

Piecewise Functions

Given: a(x)=3x^2+2x b(x)=x+3 c(x)= (-2x) + 4

Find: 1. a/b of x 2. b(c(x)) 3. a(x)c(x) 4. (b-c)(x)

5. (b-a)(x) 6. b/a of (-1) 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/algebra-

2/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/algebra-2/matrices/determinant-2-by-2.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

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.

6 6 0y is asymptote horizontal theTherefore,

2. ofpower the toisr denominato theof degreehighest The

0. ofpower the toisnumerator theof degreehighest The

65x4x

1f(x)

2

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.

3.yor 1

3y is asymptote horizontal theTherefore,

1. ist coefficien The 2. ofpower the toisr denominato theof degreehighest The

3. ist coefficien The 2. ofpower the toisnumerator theof degreehighest The

96xx

4x3xf(x)

2

2

o If the numerator has a higher degree than the denominator, then there is an undefined asymptote.

undefined. is thereforeandexist not does asymptote horizontal theTherefore,

1. ofpower the toisr denominato theof degreehighest The

3. ofpower the toisnumerator theof degreehighest The

7x

12x3xf(x)

2

Vertical Asymptotes

o Set the denominator of your function equal to zero and solve. The value(s) you get will be vertical asymptote(s).

-5 xand 2at x are asymptotes vertical theTherefore,

-5 then x05 xIf

2 then x02- xIf

0 5)2)(x-x(

:get would then wezero, r todenominato set the weIf

5)2)(x-(x

3xf(x)

Oblique or Slant Asymptotes

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).

o 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.

8.3xy is asymptoteslant theTherefore,

15. ofremainder a with 83xy isquotient The

1583

166

1232

quotient. thefind division, synthetic Using

1. ofpower the toisr denominato theof degreehighest The

2. ofpower the toisnumerator theof degreehighest The

2x

12x3xf(x)

2

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.

6 6 0y is asymptote horizontal theTherefore,

2. ofpower the toisr denominato theof degreehighest The

0. ofpower the toisnumerator theof degreehighest The

65x4x

1f(x)

2

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.

3.yor 1

3y is asymptote horizontal theTherefore,

1. ist coefficien The 2. ofpower the toisr denominato theof degreehighest The

3. ist coefficien The 2. ofpower the toisnumerator theof degreehighest The

96xx

4x3xf(x)

2

2

o If the numerator has a higher degree than the denominator, then there is an undefined asymptote.

undefined. is thereforeandexist not does asymptote horizontal theTherefore,

1. ofpower the toisr denominato theof degreehighest The

3. ofpower the toisnumerator theof degreehighest The

7x

12x3xf(x)

2

Given the following functions: y=1/(2x+6) y = (6x+8)/(-3x+4) y = (x^2 + 5x + 6)/(x+3) (don 't

forget there is a hole in th is graph)

find:

Vertical As ym ptote

Horizontal As ym ptote

Oblique As ymptote

Hole

x- in tercept

y-in tercept

Domain

Range

Sketch a graph of each

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 g(x) = 5x4+x3-6x2 – x +10 h(x)=(x + 1)

5. Find: (f +g)(x) 6) find (f – g)(x) 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 g(x) = 5x4+x3-6x2 – x +10 h(x)= -3x4 +x2 + 2x -10

1. f(x) if x=2 2) g(x) if x = -1 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 o f each polynomia l:

1) f(x) = 2x^5 - 2x + 8

2) g(x) = -4x^6 - 5

Simpli fy the fo llowing polynomials :

g(x) = -4x^6 - 5 f(x) = 2x^5 - 2x + 8 h(x) = x^2 - 3

3) (f + h)(x) 4) (g - f)(x) 5) h(x) times f(x)

Given: a(x)=3x^2+2x b(x)=x+3 c(x)= (-2x) + 4

Find: 1 . a /b of x 2 . b(c(x)) 3 . a(x)c(x) 4 . (b-c)(x)

5 . (b-a)(x) 6 . b /a of (-1) 7 . (a+c)(2)

Answer the fo llowing questions:

1) What does the determinant tell you?

2) Any m atrix times i ts inverse will equal what?

3) What is the identi ty matrix?

4) Can you find the solution to a s ystem whose coefficient matrix has a determinant o f zero? Why or why

not?

5) What is a coefficient matrix? Where is i t used?

Videos to help:

How to mul tiply a matrix by a scalar: h ttp ://vi rtualnerd.com /algebra -2/matrices/scalar-mul tip ly-

example.php

How to find the determinant o f a 3x3 matrix: h ttps ://www.youtube.com/watch?v=V3e7m -qFDFU

How to mul tiply matrices : https ://www.youtube.com /watch?v=kuixY2bCc_0

How to find the determinant o f a 2x2 matrix: h ttp ://virtua lnerd.com /algebra-2/matrices /determinant-2-by-

2.php

What is a coefficient matrix? h ttp ://vi rtualnerd.com /algebra -2/matrices/coefficient.php

Cramers Rule for a 2x2 h ttps ://www.youtube.com/watch?v=cidMyv6L7Gs

How to find the inverse of a 2x2 h ttps://www.khanacadem y.org/math/pre calculus /precalc-

matrices/inverting_matrices/v/inverse -of-a-2x2-matrix

Answer the fo llowing questions:

1) What does the determinant tell you?

2) Any m atrix times i ts inverse will equal what?

3) What is the identi ty matrix?

4) Can you find the solution to a s ystem whose coefficient matrix has a determinant o f zero? Why or why

not?

5) What is a coefficient matrix? Where is i t 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

Monday Wednesday Friday Saturday

Frontier 2 1 3 5 Path Finder 1 3 2 1

300-Z 0 1 3 2 Altima 1 1 4 3

Pete – Week 1

Monday Wednesday Friday Saturday Frontier 1 2 4 4

Path Finder 2 1 3 2 300-Z 2 0 3 3

Altima 2 0 4 5

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!