# Instructor Notes for Chapters 3 & 4

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 Algebra for Calculus Fall 0 Graph quadratic functions Similar to obtaining the quadratic formula by completing the square on a quadratic equation, we obtain a shortcut for finding the vertex by completing the square on a quadratic function. Be sure students understand that the maximum or minimum value of a function refers to its output, not the input. Spend today doing the procedure of completing the square and graphing. The next class will be devoted to applications. Suggested HW: MLP HW 3.3A Section 3.3, Day : Analyzing Graphs of Quadratic Function and Modeling Find the maximum or minimum value of a given quadratic function in applications Pick up from where you left off yesterday, then emphasize finding the max or min in context. Students often have difficulty knowing which coordinate for which the problem is asking. You might encourage students to write the two changing quantities as an ordered pair. For example, for a falling object problem they might write (time, height). You might then work through example 6 in the text. Modeling is another issue. As you probably know, modeling is difficult for students. Certainly emphasize the area type, like example 4 on p. 6 (back in section.). The Instructor website has a worksheet with extra modeling problems. You might let students work in groups and try to figure it out. Another option is to guide them through it, then let them work on the rest of the problems on the sheet, offering help as needed. The goal of the sheet is writing the models. Suggested HW: MLP HW 3.3B 4. Polynomial Functions and Models Determine the end behavior of the graph of a polynomial function using the leading term test Find the zeros and their multiplicities of polynomial functions by factoring After introducing basic terminology, you might do # 0 odd on p. 306 together in class. Discuss the characteristics of the graphs of polynomial functions. The graphing calculator is a great tool to illustrate the Leading-Term Test. You might try this example: (also available on the website as an investigative activity for students):

3 Algebra for Calculus Fall In your graphing calculator, let Y x 7x 8x 6 and let Y x. Use the viewing window [ 5,5] by [ 5,5]. Compare the two graphs: How are they the same? How are they different? Try changing the viewing window to [ 40,40] by [ 5000,5000]. This is a bird s eye view. How do the graphs compare now? Hopefully this visual approach will convince students that for large values of x, the leading term is sufficient for determining the behavior of the graph. Review factoring and the connection between factors, zeros, and x-intercepts. Example 5 is good. For even and odd multiplicities, you might try this: In your graphing calculator, let Y ( x ) ( x ) What happens at the zeros of the function? Does the graph cross or touch? Why do you think this happens? Repeat with Y ( x 3) ( x ), Y ( x 5) ( x 8), etc. 3 3 Repeat with Y ( x 5) ( x 8), Y ( x ) ( x 4) (use viewing window [-5,5] by 4 [-00,00] for this one), Y ( x ) ( x 3) Be sure students understand WHY the graph turns back at zeros with even multiplicities (why does a parabola have a turning point?). Then put these function rules in the Y= menu on your calculator and ask students to predict the zeros and whether the graphs cross or touch at these zeros. Y ( x ) ( x 3) ; 4 3 Y x x ; ( ) ( ) Y ( x 3)( x ) Then you might put the graphs of these on the viewscreen one at a time and let students name the factors and a possible function rule. Y ( x 4)( x ), [-0,0] for this one); and Suggested HW: MLP HW 4. Y x x. ( ) ( ) 3 Y x x (use viewing window [-5,5] by ( ) ( ) 4. Graphing Polynomial Functions Sketch by hand a graph of a polynomial function You might begin class by asking students to do # in small groups or all together as a class. Then, putting together the class discussion from yesterday, take your time as you work through or 3 examples in detail. The sign chart will be new for some students. Remind students that you will ask them to graph a polynomial by hand (with no graphing calculator) on a quiz or exam. Suggest that they work through the HW problems on MyLabsPlus BY HAND. Many of the MLP problems in this section are multiple choice.

4 Algebra for Calculus Fall 0 The Intermediate Value Theorem is optional. Suggested HW: MLP HW Polynomial Division; Remainder and Factor Theorems Perform long division with polynomials and determine whether one polynomial is a factor of another Use synthetic division to divide a polynomial by x c. Understand and use the Remainder and Factor Theorems Work through Example using long division, relating the process to long division for numbers. Be sure to set up the checks by multiplication: DIVIDEND = DIVISOR x QUOTIENT + REMAINDER, i.e. x 3 x 5x 6 ( x )( x x 6) 0 (I wouldn t take the time to multiply this out, though); eventually use P( x) ( x c) Q( x) R( x). You might want to foreshadow the Remainder Theorem here by asking them the value of P( ). Synthetic division is straightforward and easy for students, though they sometimes forget to put in a 0 as a placeholder for missing terms. Example 6 from start to finish is a good way to end the lesson. Suggested HW: MLP HW Theorems about Zeros of Polynomial Functions Given a polynomial function with integer coefficients, use the Rational Zero Test to name the possible rational zeros. Understand WHY the Rational Zero Test works (relate to factors of a quadratic). Given a polynomial with integer coefficients, find all the zeros. Write a polynomial function, given its zeros In this section, we use the skills learned in the previous section and pull everything together. Though the text begins with a statement of the Fundamental Theorem of Algebra, I d save that for last kind of a culmination of our work.

5 Algebra for Calculus Fall 0 Intuitively develop the Rational Zeros Theorem. You might begin be recalling the process of factoring a quadratic such as f ( x) x x 6. How do you find the factors? In the factored form ( x a)( x b ), what does the product ab =? Similarly for other easy quadratics such as Then you might ask them f ( x) x x 5, f x x x ( ) 7 0 In the polynomial there would be? 4 3 f ( x) x 5x 7x 3x 0, how many (linear) factors do you think Then I d write ( x?)( x??)( x???)( x????). What does the product of all the?????????? have to be? So what are the possible rational zeros? Similar questioning, then, with a polynomial with a leading coefficient not equal to, such as: 4 3 f ( x) 3x 4x x 6x which would factor into the form: ( ax?)( bx??)( cx???)( dx????) How many linear factors (at most) would there be? What does the product?????????? have to be? What are possible values for a, b, c, d? How does this affect the zeros? How do you find a zero once you know a factor? Then state the Rational Zeros Theorem. Pull everything together then and summarize the steps for finding zeros of a polynomial:. Use the Rational Zeros Theorem to list possible rational zeros. Use your graphing calculator to help narrow down the choices 3. Use synthetic division to find the zeros and list factored formula Work through examples 5 and 6 with students. I insist that they show me the newly factored form every time they find a zero. Descartes Rule of Signs is optional. Suggested HW: MLP HW 4.4

6 Algebra for Calculus Fall Rational Functions Graph by hand a rational function, identifying all asymptotes You might begin by asking students to make a table of values for the simple rational function y, asking a student to put the table on the board and sketch the function. Your table might include x these input values: y x x y x x and their negatives Discuss the shape of the graph, its domain, and why it looks like it does. Note the asymptotes and relate to the asymptotes of the exponential functions we studied in MATH You might then ask students to make a table of values (perhaps on the calculator this time) and hand graph y, noting in particular the end behavior and the behavior close to x. x Be sure to suggest appropriate input values. Start: t = 3; = -0.. Then Start: t =.5; = -0.0 Extend the idea to more complicated rational functions, emphasizing domains and asymptotes. Students will intuit the location of vertical asymptotes with proper questioning. Horizontal asymptotes are a little more subtle; the approach in example 4 on p foreshadows the limit concept. You also might want to point out that we can graph the function g in example 6 on page 308 by writing it like this: gx ( ), then using a vertical translation of the graph of y x x Summarize and be sure to do a complete example from start to finish. Example 8 is good. Example 9 time permitting. You might note that a graph can cross a horizontal asymptote, but not a vertical one. Oblique asymptotes are optional. Suggested HW: MLP HW

7 Algebra for Calculus Fall Rational Functions, cont d (Day ) You ll probably have many questions on the homework. You might use any left over time and by allowing students to work on some of the even exercises, or correct the ones on the homework they couldn t get. Introduce oblique asymptotes if time. This topic provides is a good review of long division. 4.6 Polynomial and Rational Inequalities ( days) Solve polynomial and rational inequalities After reviewing yesterday s homework, you might discuss #84 on page 358. Then as a motivation for section 4.6, problems #77 8 on pp are good. (You needn t solve these quite yet, just set up one or two). Calc teachers assume that students know how to use sign charts, so teach carefully! Please refer to the MATH 0035 e-book, available at for ideas for teaching this section. Note especially the prep assignments at the beginning of the section both the skill prep and concept prep. Feel free to print and use with your students. Be sure that students can solve these both graphically and using sign charts. Reading the graph is essential, but not sufficient. You might do polynomial inequalities on day and rational inequalities on day, though our MATH 0035 students have seen quadratic inequalities. Another option is to begin with polynomials and finish with rational inequalities in one day, then use the second day as a work session. Conceptually the problems are the same, though of course the rational inequalities involve more algebraic manipulation. Suggested HW: Day : MLP HW 4.6A (polynomial inequalities 5 questions) Day : MLP HW 4.6B Exam Ch 3 and MLP has a set of review problems and a practice exam. Please note that I did NOT include sections 4.5 and 4.6 on the exam. I ll quiz on these sections separately in class. All in-class exams are paper and pencil exams and a sample is available for you. Please see Bev for access.

### Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions

### Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring.

GSE Advanced Algebra Polynomial Functions Polynomial Functions Zeros of Polynomial Function Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. In the video,

### 3.3 Real Zeros of Polynomial Functions

71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2

### Polynomial and Synthetic Division

Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1

### Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition

Math 115 Syllabus (Spring 2017 Edition) By: Elementary Courses Committee Textbook: Intermediate Algebra by Aufmann & Lockwood, 9th Edition Students have the options of either purchasing the loose-leaf

### . As x gets really large, the last terms drops off and f(x) ½x

Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be

Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application

### Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers

### Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

### Course Outline and Objectives. MA 1453 Precalculus with Graphing Calculators

Effective Fall 2011 Course Outline and Objectives MA 1453 Precalculus with Graphing Calculators TEXT: Precalculus, 5 th Edition, by Faires and DeFranza ISBN 978-0-8400-6862-0 NOTE: A graphing calculator

### Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational

### A Add, subtract, multiply, and simplify polynomials and rational expressions.

ED 337 Paul Garrett Selected Response Assessment 10 th Grade Mathematics Examination Algebra II Clear Purpose: The purpose of this selected response is to ensure an understanding of expressions, manipulation

### Polynomial Functions and Models

1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

### Intermediate Algebra

Intermediate Algebra The purpose of this course is to strengthen students foundational conceptual and procedural skills in order to prepare students for college-and-career readiness. The course textbook,

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard

### Looking Ahead to Chapter 10

Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.

### Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

### Math 70 and 95 Master Syllabus Beginning and Intermediate Algebra, 5th ed., Miller, O Neill, Hyde with ALEKS.

Math 70 and 95 Master Syllabus 2017-2018 Course Coordinator: Tammy Nezol (tnezol@uoregon.edu) Textbook and ALEKS: Beginning and Intermediate Algebra, 5th ed., Miller, O Neill, Hyde with ALEKS. We have

### Algebra I Assessment. Eligible Texas Essential Knowledge and Skills

Algebra I Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra I Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.

### Algebra II Assessment. Eligible Texas Essential Knowledge and Skills

Algebra II Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra II Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.

# %

### Algebra 1. Correlated to the Texas Essential Knowledge and Skills. TEKS Units Lessons

Algebra 1 Correlated to the Texas Essential Knowledge and Skills TEKS Units Lessons A1.1 Mathematical Process Standards The student uses mathematical processes to acquire and demonstrate mathematical understanding.

### College Algebra Notes

Metropolitan Community College Contents Introduction 2 Unit 1 3 Rational Expressions........................................... 3 Quadratic Equations........................................... 9 Polynomial,

### ( 3) ( ) ( ) ( ) ( ) ( )

81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root

### Math 5a Reading Assignments for Sections

Math 5a Reading Assignments for Sections 4.1 4.5 Due Dates for Reading Assignments Note: There will be a very short online reading quiz (WebWork) on each reading assignment due one hour before class on

### MATH 1130 Exam 1 Review Sheet

MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This

### Algebra III. Mathematics Curriculum Framework. Revised 2004

Algebra III Mathematics Curriculum Framework Revised 2004 Title: Algebra III (Fourth-year Course) Course/Unit Credit: 1 Course Number: Teacher Licensure: Secondary Mathematics Pre-requisite: Algebra II

### Lesson 11: The Special Role of Zero in Factoring

Lesson 11: The Special Role of Zero in Factoring Student Outcomes Students find solutions to polynomial equations where the polynomial expression is not factored into linear factors. Students construct

### To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

### Precalculus 1, 161. Spring 2018 CRN Section 009. Time: S, 12:30 p.m. - 3:35 p.m. Room BR-11

Precalculus 1, 161 Spring 2018 CRN 11996 Section 009 Time: S, 12:30 p.m. - 3:35 p.m. Room BR-11 SYLLABUS Catalog description Functions and relations and their graphs, transformations and symmetries; composition

### 3.4 The Fundamental Theorem of Algebra

333371_0304.qxp 12/27/06 1:28 PM Page 291 3.4 The Fundamental Theorem of Algebra Section 3.4 The Fundamental Theorem of Algebra 291 The Fundamental Theorem of Algebra You know that an nth-degree polynomial

### FLORIDA STANDARDS TO BOOK CORRELATION

FLORIDA STANDARDS TO BOOK CORRELATION Florida Standards (MAFS.912) Conceptual Category: Number and Quantity Domain: The Real Number System After a standard is introduced, it is revisited many times in

### Course Name: MAT 135 Spring 2017 Master Course Code: N/A. ALEKS Course: Intermediate Algebra Instructor: Master Templates

Course Name: MAT 135 Spring 2017 Master Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 01/15/2017 End: 05/31/2017 Course Content: 279 Topics (207

### Midterm 3 Review. Terms. Formulas and Rules to Use. Math 1010, Fall 2011 Instructor: Marina Gresham. Odd Root ( n x where n is odd) Exponent

Math 1010, Fall 2011 Instructor: Marina Gresham Terms Midterm 3 Review Exponent Polynomial - Monomial - Binomial - Trinomial - Standard Form - Degree - Leading Coefficient - Constant Term Difference of

### ALGEBRA 2 CURRICULUM Course 17006

ALGEBRA 2 CURRICULUM Course 17006 The main portion of this course broadens the topics that were first seen in Algebra I. The students will continue to study probability and statistics along with a variety

### Lesson 19 Factoring Polynomials

Fast Five Lesson 19 Factoring Polynomials Factor the number 38,754 (NO CALCULATOR) Divide 72,765 by 38 (NO CALCULATOR) Math 2 Honors - Santowski How would you know if 145 was a factor of 14,436,705? What

### INSTRUCTIONS USEFUL FORMULAS

MATH 1100 College Algebra Spring 18 Exam 1 February 15, 2018 Name Student ID Instructor Class time INSTRUCTIONS 1. Do not open until you are told to do so. 2. Do not ask questions during the exam. 3. CAREFULLY

### Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

and PreAP Scope and Sequence with NMSI s Laying the Foundation lessons and Quality Modules Unit 1: Functions, Equations, Absolute Value and Inequalities 15 days of instruction plus assessment time- 3.5

### Exploring and Generalizing Transformations of Functions

Exploring and Generalizing Transformations of Functions In Algebra 1 and Algebra 2, you have studied transformations of functions. Today, you will revisit and generalize that knowledge. Goals: The goals

### SYLLABUS FOR [FALL/SPRING] SEMESTER, 201x

SYLLABUS FOR [FALL/SPRING] SEMESTER, 201x Course Title: College Algebra and Trigonometry Instructor: [Instructor Name] Credit Hours: 5 Office: [Office Location] Course Number: MATH 1340-00x Hours: [Office

### Algebra Summer Review Packet

Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills

### Algebra 2 Secondary Mathematics Instructional Guide

Algebra 2 Secondary Mathematics Instructional Guide 2009-2010 ALGEBRA 2AB (Grade 9, 10 or 11) Prerequisite: Algebra 1AB or Geometry AB 310303 Algebra 2A 310304 Algebra 2B COURSE DESCRIPTION Los Angeles

### This document has been made possible by funding from the GE Foundation Developing Futures grant, in partnership with Atlanta Public Schools.

Atlanta Public Schools Teacher s Curriculum Supplement Web-Edition Common Core Georgia Performance Standards Mathematics III Unit 2: Polynomial Functions of Higher Degree This document has been made possible

### Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to:

What students need to know for AQR Students expecting to take Advanced Qualitative Reasoning should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every

### Math 0031, Final Exam Study Guide December 7, 2015

Math 0031, Final Exam Study Guide December 7, 2015 Chapter 1. Equations of a line: (a) Standard Form: A y + B x = C. (b) Point-slope Form: y y 0 = m (x x 0 ), where m is the slope and (x 0, y 0 ) is a

### Algebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2

Algebra 2 (2006) Correlation of the ALEKS Course Algebra 2 to the California Content Standards for Algebra 2 Algebra II - This discipline complements and expands the mathematical content and concepts of

### Function Junction: Homework Examples from ACE

Function Junction: Homework Examples from ACE Investigation 1: The Families of Functions, ACE #5, #10 Investigation 2: Arithmetic and Geometric Sequences, ACE #4, #17 Investigation 3: Transforming Graphs,

### MATHia Unit MATHia Workspace Overview TEKS

1 Function Overview Searching for Patterns Exploring and Analyzing Patterns Comparing Familiar Function Representations Students watch a video about a well-known mathematician creating an expression for

### MCPS Algebra II Pacing Guide

Units to be covered 1 st Semester: Units to be covered 2 nd Semester: Vocabulary Semester Class: 10 days Year Long: 20 days OR Unit: Functions & Graphs Include content with individual units SOL AII.6 Recognize

### COURSE OUTLINE CHAFFEY COLLEGE

COURSE OUTLINE CHAFFEY COLLEGE Discipline: Mathematics 1. COURSE IDENTIFICATION: MATH 425 2. COURSE TITLE: Intermediate Algebra 3. UNITS: 4 Lecture Hours: Normal: 72 Range: 64-76 4. GRADING: Letter Grade

### This format is the result of tinkering with a mixed lecture format for 3 terms. As such, it is still a work in progress and we will discuss

Version 1, August 2016 1 This format is the result of tinkering with a mixed lecture format for 3 terms. As such, it is still a work in progress and we will discuss adaptations both to the general format

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Modeling & Problem Solving Common Core Standard: A-REI.4a: Solve quadratic equations in one

### Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,

### SUBJECT: ADDITIONAL MATHEMATICS CURRICULUM OUTLINE LEVEL: 3 TOPIC OBJECTIVES ASSIGNMENTS / ASSESSMENT WEB-BASED RESOURCES. Online worksheet.

TERM 1 Simultaneous Online worksheet. Week 1 Equations in two Solve two simultaneous equations where unknowns at least one is a linear equation, by http://www.tutorvista.com/mat substitution. Understand

### Agile Mind Algebra I Scope and Sequence, Texas Essential Knowledge and Skills for Mathematics

In the three years prior to Algebra I, students have already begun their study of algebraic concepts. They have investigated variables and expressions, solved equations, constructed and analyzed tables,

### SCOPE & SEQUENCE. Algebra I

Year at a Glance August September October November December January February March April May 1. Functions, 4.Systems Expressions, 2. Polynomials and 6. Exponential STAAR SLA 3. Linear Functions of Break

### n The coefficients a i are real numbers, n is a whole number. The domain of any polynomial is R.

Section 4.1: Quadratic Functions Definition: A polynomial function has the form P ( x ) = a x n+ a x n 1+... + a x 2+ a x + a (page 326) n n 1 2 1 0 The coefficients a i are real numbers, n is a whole

### Semester Review Packet

MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

### PUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION ALGEBRA 2 - HONORS. Date Approved: August 25, 2014

PUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION ALGEBRA 2 - HONORS Length of Course: Elective/Required: Schools: Term Required High School Eligibility: Grades 9-12 Credit Value:

### STUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition

STUDY GUIDE Math 0 To the students: To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition When you study Algebra, the material is presented to you in a logical sequence.

### Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

### Instructional Materials for the WCSD Math Common Finals

201-2014 Algebra 2 Semester 1 Instructional Materials for the WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Math Common Final blueprint for

### Solving Equations Quick Reference

Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

### Unit 4: Polynomial and Rational Functions

50 Unit 4: Polynomial and Rational Functions Polynomial Functions A polynomial function y px ( ) is a function of the form p( x) ax + a x + a x +... + ax + ax+ a n n 1 n n n 1 n 1 0 where an, an 1,...,

### Section 0.2 & 0.3 Worksheet. Types of Functions

MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Calculus I - Homework Chapter 2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the graph is the graph of a function. 1) 1)

### Lesson 9 Exploring Graphs of Quadratic Functions

Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point

### Algebra 1.5. Content Skills Learning Targets Assessment Resources & Technology CEQ:

St. Michael Albertville High School Teacher: Mindi Sechser Algebra 1.5 September 2015 Content Skills Learning Targets Assessment Resources & Technology CEQ: Chapters 1 3 Review Variables, Function Patterns,

### Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

### Integer Division. Student Probe

Student Probe What is 24 3? Answer: 8 Integer Division Lesson Description This lesson is intended to help students develop an understanding of division of integers. The lesson focuses on using the array

### HW# Date In Class Homework. Sections 11.1 and 11.2: Simplifying Rational Expressions, Multiplying and Dividing Rational Expressions

Algebra 1 Chapter 11 Notes Spragg Spring 010 Name: Algebra Homework: Chapter 11 (Homework is listed by date assigned; homework is due the following class period) HW# Date In Class Homework 33 Th 6/ Sections

### MATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.

MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)

### Algebra 1. Mathematics Course Syllabus

Mathematics Algebra 1 2017 2018 Course Syllabus Prerequisites: Successful completion of Math 8 or Foundations for Algebra Credits: 1.0 Math, Merit The fundamental purpose of this course is to formalize

### SUPPORTING INFORMATION ALGEBRA II. Texas Education Agency

SUPPORTING INFORMATION ALGEBRA II Texas Education Agency The materials are copyrighted (c) and trademarked (tm) as the property of the Texas Education Agency (TEA) and may not be reproduced without the

### Mission 1 Simplify and Multiply Rational Expressions

Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following

### COMMON CORE STATE STANDARDS TO BOOK CORRELATION

COMMON CORE STATE STANDARDS TO BOOK CORRELATION Conceptual Category: Number and Quantity Domain: The Real Number System After a standard is introduced, it is revisited many times in subsequent activities,

### _Algebra 2 Marking Period 1

_Algebra 2 Marking Period 1 Topic Chapters Number of Blocks Dates Equations and Inequalities 1 8 9/9-9/27 PRE-TEST 1 9/27-10/2 Linear Relations and Functions 2 10 12/3-10/25 System of Equations and Inequalities

### Algebra 2 A/B Curriculum Pacing Guide

Algebra A/B is a yearlong course that incorporates a math lab elective. During the lab students use Imagine Math as a resource. Anderson School District Five Page 1 017-018 Unit 1 Functions A.AAPR.1* A.ACE.*

### Pre Calculus 1. Guiding question: How are quadratic functions related to polynomial functions?

Pre Calculus 1 Polynomial and Rational Functions Day 1: Quadratic Functions Guiding question: How are quadratic functions related to polynomial functions? Students will begin by writing everything they

### SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION

2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides

### 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

### 171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### California Common Core State Standards for Mathematics Standards Map Algebra I

A Correlation of Pearson CME Project Algebra 1 Common Core 2013 to the California Common Core State s for Mathematics s Map Algebra I California Common Core State s for Mathematics s Map Algebra I Indicates

### How can you solve a multistep. How can you solve an absolute value equation? How can you solve and absolute value. inequality?

WDHS Curriculum Map Course: Algebra 1 June 2015 Time Interval/ Content Standards/ Strands Essential Questions Skills Assessment Unit 1 Transfer Goal: Recognize and solve practical or theoretical problems

### Modeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2

Name: Period: Pre-Cal AB: Unit 1: Polynomials Monday Tuesday Block Friday 11/1 1 Unit 1 TEST Function Operations and Finding Inverses 16 17 18/19 0 NO SCHOOL Polynomial Division Roots, Factors, Zeros and

### Mathematics Student Workbook. Lesson 1: Polynomial Functions Approximate Completion Time: 3 Days

Mathematics 30- Student Workbook Unit 2 3 Lesson : Polynomial Functions Approximate Completion Time: 3 Days - 3-4 -5 2 3-7 2 3-7 2 0 Lesson 2: Polynomial Division Approximate Completion Time: 3 Days x

### Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations

Class: Date: Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations 1 Express the following polynomial function in factored form: P( x) = 10x 3 + x 2 52x + 20 2 SE: Express the following

### Polynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions

CCGPS Advanced Algebra Polynomial Functions Polynomial Functions Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the

### Important Math 125 Definitions/Formulas/Properties

Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient

### MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline

MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 1-75 odd) A. Simplify fractions. B. Change mixed numbers

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

### COWLEY COLLEGE & Area Vocational Technical School

COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR Student Level: This course is open to students on the college level in their freshman or sophomore year. Catalog Description: INTERMEDIATE

Summer 2015 Units: 10 high school credits UC Requirement Category: c General Description: ALGEBRA II Grades 9-12 Algebra II is a course which further develops the concepts learned in Algebra I. It will

### Mathematical Focus 1 Complex numbers adhere to certain arithmetic properties for which they and their complex conjugates are defined.

Situation: Complex Roots in Conjugate Pairs Prepared at University of Georgia Center for Proficiency in Teaching Mathematics June 30, 2013 Sarah Major Prompt: A teacher in a high school Algebra class has

### HMH Fuse Algebra correlated to the. Texas Essential Knowledge and Skills for Mathematics High School Algebra 2

HMH Fuse Algebra 2 2012 correlated to the Texas Essential Knowledge and Skills for Mathematics High School Algebra 2 111.33. Algebra II (b) Knowledge and skills. (1) Foundations for functions. The student

### Algebra , Martin-Gay

A Correlation of Algebra 1 2016, to the Common Core State Standards for Mathematics - Algebra I Introduction This document demonstrates how Pearson s High School Series by Elayn, 2016, meets the standards

### ALGEBRA 2/MATH 3 COURSE 1

ALGEBRA 2/MATH 3 COURSE 1 TABLE OF CONTENTS NUMBER AND QUANTITY 6 THE REAL NUMBER SYSTEM (N.RN) 6 EXTEND THE PROPERTIES OF EXPONENTS TO RATIONAL EXPONENTS. (N.RN.1-2) 6 Expectations for Learning 6 Content