# 1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses the following rules to calculate parking fees: A flat rate of \$5. for an amount of time up to and including the first hour A flat rate of \$1.5 for an amount of time over 1 h and up to and including h A flat rate of \$13 plus \$3 per hour for each hour after h? How can ou describe the function for parking fees in terms of the number of hours parked? EXAMPLE 1 Representing the problem using a graphical model Use a graphical model to represent the function for parking fees. Solution Time (h) Parking Fee (\$) Create a table of values. 1. Piecewise Functions NEL

2 1. Plot the points in the table of values. Use a solid dot to include a value in an interval. Use an open dot to eclude a value from an interval. Cost (\$) Time (h) 5 The domain of this piecewise function is \$. The function is linear over the domain, but it is discontinuous at 5, 1, and. There is a solid dot at (, ) and an open dot at (, 5) because the parking fee at h is \$.. There is a closed dot at (1, 5) and an open dot at (1, 1.5) because the parking fee at 1 h is \$5.. There is a closed dot at (, 1.5) and an open dot at (, 13) because the parking fee at h is \$1.5. The last part of the graph continues in a straight line since the rate of change is constant after h. piecewise function a function defined b using two or more rules on two or more intervals; as a result, the graph is made up of two or more pieces of similar or different functions Each part of a piecewise function can be described using a specific equation for the interval of the domain. EXAMPLE Representing the problem using an algebraic model Use an algebraic model to represent the function for parking fees. Solution if 5 if, # 1 if 1, # if. Write the relation for each rule. NEL Chapter 1 7

3 f () 5 μ, if 5 5, if, # 1 1.5, if 1, # , if. Combine the relations into a piecewise function. The domain of the function is \$. The function is discontinuous at 5, 1, and because there is a break in the function at each of these points. Reflecting A. How do ou sketch the graph of a piecewise function? B. How do ou create the algebraic representation of a piecewise function? C. How do ou determine from a graph or from the algebraic representation of a piecewise function if there are an discontinuities? APPLY the Math EXAMPLE 3 Representing a piecewise function using a graph Graph the following piecewise function.,if, f () 5 e 1 3, if \$ Solution Create a table of values. f () 5 f () f() f() From the equations given, the graph consists of part of a parabola that opens up and a line that rises from left to right. Both tables include 5 since this is where the description of the function changes. 1. Piecewise Functions NEL

4 1. 1 = f() Plot the points, and draw the graph. A solid dot is placed at (, 7) since 5 is included with f() An open dot is placed at (, ) since 5 is ecluded from f() 5. f () is discontinuous at 5. EXAMPLE = f () Representing a piecewise function using an algebraic model Determine the algebraic representation of the following piecewise function. Solution 1 f () 5,if #, if. The graph is made up of two pieces. One piece is part of the reciprocal function defined b 5 1 when #. The other piece is a horizontal line defined b 5 when.. The solid dot indicates that point Q, 1 R belongs with the reciprocal function. NEL Chapter 1 9

5 EXAMPLE 5 Reasoning about the continuit of a piecewise function Is this function continuous at the points where it is pieced together? Eplain. 1 1, if # g() 5 1 1, if,, 3, if \$ 3 Solution The function is continuous at the points where it is pieced together if the functions being joined have the same -values at these points. Calculate the values of the function at 5 using the relevant equations: () Calculate the values of the function at 5 3 using the relevant equations: (3) The function is discontinuous, since there is a break in the graph at 5 3. The graph is made up of three pieces. One piece is part of an increasing line defined b when #. The second piece is an increasing line defined b when,, 3. The third piece is part of a parabola that opens down, defined b 5 when \$ 3. The two -values are the same, so the two linear pieces join each other at 5. The two -values are different, so the second linear piece does not join with the parabola at 5 3. Tech Support For help using a graphing calculator to graph a piecewise function, see Technical Appendi, T-1. Verif b graphing Piecewise Functions NEL

6 1. In Summar Ke Ideas Some functions are represented b two or more pieces. These functions are called piecewise functions. Each piece of a piecewise function is defined for a specific interval in the domain of the function. Need to Know To graph a piecewise function, graph each piece of the function over the given interval. A piecewise function can be either continuous or not. If all the pieces of the function join together at the endpoints of the given intervals, then the function is continuous. Otherwise, it is discontinuous at these values of the domain. CHECK Your Understanding 1. Graph each piecewise function., if, 1 1,if #1 a) f () 5 e d) f () 5 e 3,if \$ 1 1, if.1, if, f () 5 e e) 1, if \$ 1, if # c) f () 5 e f) f () 5,if, 1,if.,if \$ 1. State whether each function in question 1 is continuous or not. If not, state where it is discontinuous. 3. Write the algebraic representation of each piecewise function, using function notation. a) = f () f () 5 e!,if,,if \$ = f (). State the domain of each piecewise function in question 3, and comment on the continuit of the function. NEL Chapter 1 51

7 PRACTISING 5. Graph the following piecewise functions. Determine whether each K function is continuous or not, and state the domain and range of the function., if,1 a) f () 5 e c) 3, if \$1 f () 5 e 1 1, if, 1 1, if \$ 1, if,1 f () 5 e,if # d) f () 5 1, if 1 # # 3,if. 5, if. 3. Graham s long-distance telephone plan includes the first 5 min per A month in the \$15. monthl charge. For each minute after 5 min, Graham is charged \$.. Write a function that describes Graham s total long-distance charge in terms of the number of long distance minutes he uses in a month. 7. Man income ta sstems are calculated using a tiered method. Under a certain ta law, the first \$1 of earnings are subject to a 35% ta; earnings greater than \$1 and up to \$5 are subject to a 5% ta. An earnings greater than \$5 are taed at 55%. Write a piecewise function that models this situation.. Find the value of k that makes the following function continuous. T Graph the function. f () 5 e k,if,1 1, if \$1 9. The fish population, in thousands, in a lake at an time,, in ears is modelled b the following function:,if # # f () 5 e 1, if. This function describes a sudden change in the population at time 5, due to a chemical spill. a) Sketch the graph of the piecewise function. Describe the continuit of the function. c) How man fish were killed b the chemical spill? d) At what time did the population recover to the level it was before the chemical spill? e) Describe other events relating to fish populations in a lake that might result in piecewise functions Piecewise Functions NEL

8 1. Create a flow chart that describes how to plot a piecewise function with two pieces. In our flow chart, include how to determine where the function is continuous An absolute value function can be written as a piecewise function that C involves two linear functions. Write the function f () as a piecewise function, and graph our piecewise function to check it. 1. The demand for a new CD is described b 1 D(p) 5 p,if, p # 15, if p. 15 where D is the demand for the CD at price p, in dollars. Determine where the demand function is discontinuous and continuous. Etending 13. Consider a function, f (), that takes an element of its domain and rounds it down to the nearest 1. Thus, f (15.) 5 1, while f (1.7) 5 and f (3) 5 3. Draw the graph, and write the piecewise function. You ma limit the domain to P[, 5). Wh do ou think graphs like this one are often referred to as step functions? 1. Eplain wh there is no value of k that will make the following function continuous. 5, if,1 f () 5 1 k, if 1 # # 3,if The greatest integer function is a step function that is written as f () 5 3, where f () is the greatest integer less than or equal to. In other words, the greatest integer function rounds an number down to the nearest integer. For eample, the greatest integer less than or equal to the number [5.3] is 5, while the greatest integer less than or equal to the number 35.3 is. Sketch the graph of f () a) Create our own piecewise function using three different transformed parent functions. Graph the function ou created in part a). c) Is the function ou created continuous or not? Eplain. d) If the function ou created is not continuous, change the interval or adjust the transformations used as required to change it to a continuous function. NEL Chapter 1 53

9 11. No; several students could have the same grade point average. 1. a) f 1 () 5 1 ( ) 3 h 1 () 5 c) g 1 () 5 " 3 d) m 1 () a) 5 ( 3) 1 1 c) 5 Å 1 d) (., 3.55), (.,.), (3.55,.), (3., 3.) e) \$ 3 because a negative square root is undefined. f) g() 5 5, but g 1 (5) 5 or ; the inverse is not a function if this is the domain of g. 1. For 5" 1, D 5 5PR \$ and R 5 5PR #. For 5, D 5 5PR and R 5 5PR \$. The student would be correct if the domain of 5 is restricted to D 5 5PR #. 15. Yes; the inverse of 5 " 1 is 5 so long as the domain of this second function is restricted to D 5 5PR \$. 1. John is correct. Algebraic: ; ; ( ) 5 3 ; 5 " 3 ( ). Numeric: Let Graphical: " 3 (1) 5 " ; 5 " 3 ( ) 5 " 3 (1 ) The graphs are reflections over the line f () 5 k works for all kpr. 5 k Switch variables and solve for : 5 k 5 k So the function is its own inverse. 1. If a horizontal line hits the function in two locations, that means there are two points with equal -values and different -values. When the function is reflected over the line 5 to find the inverse relation, those two points become points with equal -values and different -values, thus violating the definition of a function. Lesson 1., pp a) c) d) e) f). a) Discontinuous at 5 1 Discontinuous at 5 c) Discontinuous at 5 d) Continuous e) Discontinuous at 5 f) Discontinuous at 5 1 and 5 3. a) 3 1 f() 5 e, if # 1 1 1, if. 1, if, 1 f () 5 e!, if \$ 1. a) D 5 5PR; the function is discontinuous at 5 1. D 5 5PR; the function is continuous. 5. a) The function is discontinuous at 51. D 5 5PR R 5 5, 3 The function is continuous. D 5 5PR R 5 5f ()PR f () \$ 1 Answers NEL

10 c) Answers ma var. For eample: Plot the function for the left interval. Plot the function for the right interval. 1. Answers ma var. For eample: 1 3, if,1 a) f () 5 1 1, if 1 # #! 1 1, if. 5 1 The function is continuous. D 5 5PR R 5 5f ()PR f () \$ 1 d) The function is continuous. D 5 5PR R 5 5 f ()PR 1 # f () # 5 15, if # # 5. f () 5 e 15 1., if \$ 5 7. f () 5.35, if # # 1.5 1, if 1, # 5.55, if Determine if the plots for the left and right intervals meet at the -value that serves as the common end point for the intervals; if so, the function is continuous at this point. Determine continuit for the two intervals using standard methods. 1 3, if \$3 f () e 3, if,3 1. discontinuous at p 5 and p 5 15; continuous at, p, 15 and p c) The function is not continuous. The last two pieces do not have the same value for , if,1 d) f () 5 1 1, if 1 # # 1! 1 1, if. 1 Lesson 1.7, pp a) 5(, ), (, 5), (1, 5), (, 1) 5(, ), (, 3), (1, 1), (, ) c) 5(, ), (, 3), (1, 1), (, ) d) 5(, ), (, ), (1, ), (, ). a) 1. k 5 9. a) The function is discontinuous at 5. c) 3 fish d) 1 5 ; 5 5; 5 1 e) Answers ma var. For eample, three possible events are environmental changes, introduction of a new predator, and increased fishing. 13., if #, 1 1, if 1 #, f () 5 f, if #, 3 3, if 3 #,, if #, It is often referred to as a step function because the graph looks like steps. 1. To make the first two pieces continuous, 5(1) 51 1 k, so k 5. But if k 5, the graph is discontinuous at a) 1 Answers NEL Answers 19

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### Clovis Community College Core Competencies Assessment 2014 2015 Area II: Mathematics Algebra

Core Assessment 2014 2015 Area II: Mathematics Algebra Class: Math 110 College Algebra Faculty: Erin Akhtar (Learning Outcomes Being Measured) 1. Students will construct and analyze graphs and/or data

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

### YOU CAN COUNT ON NUMBER LINES

Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and

### Algebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range:

Name: Date: Block: Functions: Review What is a.? Relation: Function: Domain: Range: Draw a graph of a : a) relation that is a function b) relation that is NOT a function Function Notation f(x): Names the

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering

Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

The Quadratic Formula Target Audience: Introduction to Algebra Class at Bel Air High School, 90 minutes Lesson Topic: The Quadratic Formula NCTM Standards: (Algebra Standard for Grades 9-12) Understand

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 22, 2013 9:15 a.m. SAMPLE RESPONSE SET

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesday, January 22, 2013 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 31.......................

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### UNIT PLAN: EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT PLAN: EXPONENTIAL AND LOGARITHMIC FUNCTIONS Summary: This unit plan covers the basics of exponential and logarithmic functions in about 6 days of class. It is intended for an Algebra II class. The

Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

### x 2 if 2 x < 0 4 x if 2 x 6

Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) =

### Algebraic Concepts Algebraic Concepts Writing

Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 9-12 GRADE 9-12 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### MATH 110 College Algebra Online Families of Functions Transformations

MATH 110 College Algebra Online Families of Functions Transformations Functions are important in mathematics. Being able to tell what family a function comes from, its domain and range and finding a function

Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

### SECTION P.5 Factoring Polynomials

BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

### SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

### Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

### Algebra 2: Q1 & Q2 Review

Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### Chapter 13 Introduction to Linear Regression and Correlation Analysis

Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

### Chapter 8. Lines and Planes. By the end of this chapter, you will

Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes

### PoW-TER Problem Packet A Phone-y Deal? (Author: Peggy McCloskey)

PoW-TER Problem Packet A Phone-y Deal? (Author: Peggy McCloskey) 1. The Problem: A Phone-y Deal? [Problem #3280] With cell phones being so common these days, the phone companies are all competing to earn

### ( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:

Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (

### MATH 185 CHAPTER 2 REVIEW

NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..

### Administrative - Master Syllabus COVER SHEET

Administrative - Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for

### Indiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.

STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE

LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE MAT 119 STATISTICS AND ELEMENTARY ALGEBRA 5 Lecture Hours, 2 Lab Hours, 3 Credits Pre-

### STUDENT TEXT AND HOMEWORK HELPER

UNIT 4 EXPONENTIAL FUNCTIONS AND EQUATIONS STUDENT TEXT AND HOMEWORK HELPER Randall I. Charles Allan E. Bellman Basia Hall William G. Handlin, Sr. Dan Kenned Stuart J. Murph Grant Wiggins Boston, Massachusetts

### Mohawk Valley Community College MVCC MA115 Mr. Bauer

Mohawk Valley Community College MVCC MA115 Course description: This is a dual credit course. Successful completion of the course will give students 1 VVS Credit and 3 MVCC Credit. College credits do have

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### March 2013 Mathcrnatics MATH 92 College Algebra Kerin Keys. Dcnnis. David Yec' Lscture: 5 we ekly (87.5 total)

City College of San Irrancisco Course Outline of Itecord I. GENERAI- DESCRIPI'ION A. Approval Date B. Departrnent C. Course Number D. Course Title E. Course Outline Preparer(s) March 2013 Mathcrnatics

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### Prerequisites: TSI Math Complete and high school Algebra II and geometry or MATH 0303.

Course Syllabus Math 1314 College Algebra Revision Date: 8-21-15 Catalog Description: In-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems

### THIS CHAPTER INTRODUCES the Cartesian coordinate

87533_01_ch1_p001-066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms

### Online Algebra 2 Syllabus Fairfax County Public Schools-Online Campus

Part 1: Course Information Course Description Algebra 2 provides a thorough treatment of algebraic concepts through the study of functions, polynomials, rational expressions, complex numbers, exponential

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### 8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz

8-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.

### Roots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)

Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### EL CAMINO COLLEGE COURSE OUTLINE OF RECORD. Grading Method: Letter Credit/No Credit Both No Grade

EL CAMINO COLLEGE COURSE OUTLINE OF RECORD I. COURSE DESCRIPTION Course Title and Number: Mathematics 43 Descriptive Title: Extended Elementary Algebra, Part II Discipline: Mathematics Division: Mathematical

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### PHILOSOPHY OF THE MATHEMATICS DEPARTMENT

PHILOSOPHY OF THE MATHEMATICS DEPARTMENT The Lemont High School Mathematics Department believes that students should develop the following characteristics: Understanding of concepts and procedures Building

### Prerequisite: MATH 0302, or meet TSI standard for MATH 0305; or equivalent.

18966.201610 COLLIN COLLEGE COURSE SYLLABUS Course Number: MATH 0305.XS1 Course Title: Beginning Algebra Course Description: With an emphasis on developing critical thinking skills, a study of algebraic

### Warm-Up Oct. 22. Daily Agenda:

Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

### FSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers

FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

### COLLEGE ALGEBRA LEARNING COMMUNITY

COLLEGE ALGEBRA LEARNING COMMUNITY Tulsa Community College, West Campus Presenter Lori Mayberry, B.S., M.S. Associate Professor of Mathematics and Physics lmayberr@tulsacc.edu NACEP National Conference

### Straightening Data in a Scatterplot Selecting a Good Re-Expression Model

Straightening Data in a Scatterplot Selecting a Good Re-Expression What Is All This Stuff? Here s what is included: Page 3: Graphs of the three main patterns of data points that the student is likely to

### The majority of college students hold credit cards. According to the Nellie May

CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials

### MATH 110: College Algebra

MATH 110: College Algebra Introduction Required Materials Course Components Final Exam Grading Academic Policies Study Suggestions Course Outline and Checklist Introduction Welcome to Math 110. This course

### Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph

The Effect of Taxes on Equilibrium Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph Solution to part a: Set the

### 4/27/2010 ALGEBRA 1 CHAPTER 8 FACTORING. PAR Activities Cara Boening

4/27/2010 ALGEBRA 1 CHAPTER 8 FACTORING PAR Activities Cara Boening Table of Contents Preparation Strategies... 3 Scavenger Hunt... 4 Graphic Organizer Chapter 8.... 6 Word Inventory... 8 TEASE... 12 Anticipation

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### Algebra II New Summit School High School Diploma Program

Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,

### MTH124: Honors Algebra I

MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

### Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)

Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland) POLYNOMIALS Aim To demonstrate how the TI-8 can be used

### Mathematics INDIVIDUAL PROGRAM INFORMATION 2014 2015. 866.Macomb1 (866.622.6621) www.macomb.edu

Mathematics INDIVIDUAL PROGRAM INFORMATION 2014 2015 866.Macomb1 (866.622.6621) www.macomb.edu Mathematics PROGRAM OPTIONS CREDENTIAL TITLE CREDIT HOURS REQUIRED NOTES Associate of Arts Mathematics 62

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### Math 1314 College Algebra Course Syllabus

Math 1314 College Algebra Course Syllabus Instructor: Mahmoud Basharat; E-mail: Please use email within Eagle-Online Alternating email: basharatah@hotmail.com or mahmoud.basharat@hccs.edu. (Please use

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Questions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number?

Content Skills Essential August/September Number Theory Identify factors List multiples of whole numbers Classify prime and composite numbers Analyze the rules of divisibility What is meant by a number

### Find the Relationship: An Exercise in Graphing Analysis

Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.

### High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.

Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations

### The Big Picture. Correlation. Scatter Plots. Data

The Big Picture Correlation Bret Hanlon and Bret Larget Department of Statistics Universit of Wisconsin Madison December 6, We have just completed a length series of lectures on ANOVA where we considered

### High School Functions Interpreting Functions Understand the concept of a function and use function notation.

Performance Assessment Task Printing Tickets Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures using algebra.

### Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions

### Interpretation of Test Scores for the ACCUPLACER Tests

Interpretation of Test Scores for the ACCUPLACER Tests ACCUPLACER is a trademark owned by the College Entrance Examination Board. Visit The College Board on the Web at: www.collegeboard.com/accuplacer

### Learning Objectives 9.2. Media Run Times 9.3

Unit 9 Table of Contents Unit 9: Factoring Video Overview Learning Objectives 9.2 Media Run Times 9.3 Instructor Notes 9.4 The Mathematics of Factoring Polynomials Teaching Tips: Conceptual Challenges

### -- Martensdale-St. Marys Community School Math Curriculum

-- Martensdale-St. Marys Community School Standard 1: Students can understand and apply a variety of math concepts. Benchmark; The student will: A. Understand and apply number properties and operations.

### Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

### El Paso Community College Syllabus Instructor s Course Requirements Summer 2015

Syllabus, Part I Math 1324, Revised Summer 2015 El Paso Community College Syllabus Instructor s Course Requirements Summer 2015 I. Course Number and Instructor Information Mathematics 1324-31403, From

### Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties.

Polynomial functions mc-ty-polynomial-2009-1 Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties. In order to master the techniques

### Requisite Approval must be attached

Requisite Approval must be attached CITRUS COMMUNITY COLLEGE DISTRICT DEPARTMENT Mathematics COURSE NUMBER MATH 148 TITLE Intermediate Algebra I THIS COURSE IS CLASSIFIED AS: DEGREE APPLICABLE UNIT VALUE

### 2-2 Linear Relations and Functions. So the function is linear. State whether each function is a linear function. Write yes or no. Explain.

1. 2. 3. 4. State whether each function is a linear function. Write yes or no. Explain. The function written as. is linear as it can be + b. cannot be written in the form f (x) = mx So the function is

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

### 1 The Concept of a Mapping

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing

### Transition To College Mathematics

Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### Entry Level College Mathematics: Algebra or Modeling

Entry Level College Mathematics: Algebra or Modeling Dan Kalman Dan Kalman is Associate Professor in Mathematics and Statistics at American University. His interests include matrix theory, curriculum development,

### PRE-CALCULUS with TRIGONOMETRY MTH 166 Online

PRE-CALCULUS with TRIGONOMETRY MTH 166 Online INSTRUCTOR INFORMATION Name: Dr. Pablo Chalmeta Phone: 540-674-3600, ext. 4266 (or 4115) Email: pchalmeta@nr.edu Office: Godbey Hall, Room 48 (or Mall 115A)

### Students will model a real-world situation by using information and mathematical data on companies to build a stock portfolio.

Title: Exploring the Stock Market Link to Outcomes: Problem Solving Communication Reasoning Connections Functions Statistics Technology Cooperation Interdiscipline Students will model a real-world situation