# The Method of Least Squares

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The Method of Least Squares Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract The Method of Least Squares is a procedure to determine the best fit line to data; the proof uses simple calculus and linear algebra. The basic problem is to find the best fit straight line y = ax + b given that, for n {,..., }, the pairs (x n, y n ) are observed. The method easily generalizes to finding the best fit of the form y = a f (x) + + c K f K (x); (0.) it is not necessary for the functions f k to be linearly in x all that is needed is that y is to be a linear combination of these functions. Contents Description of the Problem 2 Probability and Statistics Review 2 3 The Method of Least Squares 4 Description of the Problem Often in the real world one expects to find linear relationships between variables. For example, the force of a spring linearly depends on the displacement of the spring: y = kx (here y is the force, x is the displacement of the spring from rest, and k is the spring constant). To test the proposed relationship, researchers go to the lab and measure what the force is for various displacements. Thus they assemble data of the form (x n, y n ) for n {,..., }; here y n is the observed force in ewtons when the spring is displaced x n meters.

2 Figure : 00 simulated observations of displacement and force (k = 5). Unfortunately, it is extremely unlikely that we will observe a perfect linear relationship. There are two reasons for this. The first is experimental error; the second is that the underlying relationship may not be exactly linear, but rather only approximately linear. See Figure for a simulated data set of displacements and forces for a spring with spring constant equal to 5. The Method of Least Squares is a procedure, requiring just some calculus and linear algebra, to determine what the best fit line is to the data. Of course, we need to quantify what we mean by best fit, which will require a brief review of some probability and statistics. A careful analysis of the proof will show that the method is capable of great generalizations. Instead of finding the best fit line, we could find the best fit given by any finite linear combinations of specified functions. Thus the general problem is given functions f,..., f K, find values of coefficients a,..., a K such that the linear combination is the best approximation to the data. y = a f (x) + + a K f K (x) (.) 2 Probability and Statistics Review We give a quick introduction to the basic elements of probability and statistics which we need for the Method of Least Squares; for more details see [BD, CaBe, Du, Fe, Kel, LF, MoMc]. Given a sequence of data x,..., x, we define the mean (or the expected value) to be 2

3 (x + + x )/. We denote this by writing a line above x: thus x = x n. (2.2) The mean is the average value of the data. Consider the following two sequences of data: {0, 20, 30, 40, 50} and {30, 30, 30, 30, 30}. Both sets have the same mean; however, the first data set has greater variation about the mean. This leads to the concept of variance, which is a useful tool to quantify how much a set of data fluctuates about its mean. The variance of {x,..., x }, denoted by σ 2 x, is σ 2 x = (x i x) 2 ; (2.3) the standard deviation σ x is the square root of the variance: σ x = (x i x) 2. (2.4) ote that if the x s have units of meters then the variance σ 2 x has units of meters 2, and the standard deviation σ x and the mean x have units of meters. Thus it is the standard deviation that gives a good measure of the deviations of the x s around their mean. There are, of course, alternate measures one can use. For example, one could consider (x n x). (2.5) Unfortunately this is a signed quantity, and large positive deviations can cancel with large negatives. In fact, the definition of the mean immediately implies the above is zero! This, then, would be a terrible measure of the variability in data, as it is zero regardless of what the values of the data are. We can rectify this problem by using absolute values. This leads us to consider x n x. (2.6) While this has the advantage of avoiding cancellation of errors (as well as having the same units as the x s), the absolute value function is not a good function analytically. It is not differentiable. This is primarily why we consider the standard deviation (the square root of the variance) this will allow us to use the tools from calculus. 3

4 We can now quantify what we mean by best fit. If we believe y = ax+b, then y (ax+b) should be zero. Thus given observations we look at {(x, y ),..., (x, y )}, (2.7) {y (ax + b),..., y (ax + b)}. (2.8) The mean should be small (if it is a good fit), and the variance will measure how good of a fit we have. ote that the variance for this data set is σy (ax+b) 2 = (y n (ax n + b)) 2. (2.9) Large errors are given a higher weight than smaller errors (due to the squaring). Thus our procedure favors many medium sized errors over a few large errors. If we used absolute values to measure the error (see equation (2.6)), then all errors are weighted equally; however, the absolute value function is not differentiable, and thus the tools of calculus become inaccessible. 3 The Method of Least Squares Given data {(x, y ),..., (x, y )}, we may define the error associated to saying y = ax + b by E(a, b) = (y n (ax n + b)) 2. (3.0) This is just times the variance of the data set {y (ax +b),..., y n (ax +b)}. It makes no difference whether or not we study the variance or times the variance as our error, and note that the error is a function of two variables. The goal is to find values of a and b that minimize the error. In multivariable calculus we learn that this requires us to find the values of (a, b) such that E a = 0, E b = 0. (3.) ote we do not have to worry about boundary points: as a and b become large, the fit will clearly get worse and worse. Thus we do not need to check on the boundary. Differentiating E(a, b) yields E a E b = = 2 (y n (ax n + b)) ( x n ) 2 (y n (ax n + b)). (3.2) 4

5 Setting E/ a = E/ b = 0 (and dividing by 2) yields (y n (ax n + b)) x n = 0 (y n (ax n + b)) = 0. (3.3) We may rewrite these equations as ( ) ( ) a + x n b = x 2 n ( ) ( ) x n a + b = x n y n y n. (3.4) We have obtained that the values of a and b which minimize the error (defined in (3.0)) satisfy the following matrix equation: x2 n x n a x ny n =. (3.5) x n b y n We will show the matrix is invertible, which implies a x2 n x n = b x n As Denote the matrix by M. The determinant of M is det M = x 2 n x n we find that x = det M = x ny n y n. (3.6) x n. (3.7) x n, (3.8) x 2 n (x) 2 ( = 2 = 2 5 ) x 2 n x 2 (x n x) 2, (3.9)

6 where the last equality follows from simple algebra. Thus, as long as all the x n are not equal, det M will be non-zero and M will be invertible. Thus we find that, so long as the x s are not all equal, the best fit values of a and b are obtained by solving a linear system of equations; the solution is given in (3.6). Remark 3.. The data plotted in Figure was obtained by letting x n = 5 +.2n and then letting y n = 5x n plus an error randomly drawn from a normal distribution with mean zero and standard deviation 4 (n {,..., 00}). Using these values, we find a best fit line of y = 4.99x +.48; (3.20) thus a = 4.99 and b =.48. As the expected relation is y = 5x, we expected a best fit value of a of 5 and b of 0. While our value for a is very close to the true value, our value of b is significantly off. We deliberately chose data of this nature to indicate the dangers in using the Method of Least Squares. Just because we know 4.99 is the best value for the slope and.48 is the best value for the y-intercept does not mean that these are good estimates of the true values. The theory needs to be supplemented with techniques which provide error estimates. Thus we want to know something like, given this data, there is a 99% chance that the true value of a is in (4.96, 5.02) and the true value of b is in (.22,.8); this is far more useful than just knowing the best fit values. If instead we used E abs (a, b) = y n (ax n + b), (3.2) then numerical techniques yield that the best fit value of a is 5.03 and the best fit value of b is less than 0 0 in absolute value. The difference between these values and those from the Method of Least Squares is in the best fit value of b (the least important of the two parameters), and is due to the different ways of weighting the errors. Exercise 3.2. Generalize the method of least squares to find the best fit quadratic to y = ax 2 + bx+c (or more generally the best fit degree m polynomial to y = a m x m +a m x m + +a 0 ). While for any real world problem, direct computation determines whether or not the resulting matrix is invertible, it is nice to be able to prove the determinant is always non-zero for the best fit line (if all the x s are not equal). Exercise 3.3. If the x s are not all equal, must the determinant be non-zero for the best fit quadratic or the best fit cubic? Looking at our proof of the Method of Least Squares, we note that it was not essential that we have y = ax + b; we could have had y = af(x) + bg(x), and the arguments would have 6

7 proceeded similarly. The difference would be that we would now obtain f(x n) 2 f(x n)g(x n ) a f(x n)y n = f(x n)g(x n ) g(x n) 2 b g(x n)y n. (3.22) Exercise 3.4. Consider the generalization of the Method of Least Squares given in (3.22). Under what conditions is the matrix invertible? Exercise 3.5. The method of proof generalizes further to the case when one expects y is a linear combination of K fixed functions. The functions need not be linear; all that is required is that we have a linear combination, say a f (x) + + a K f K (x). One then determines the a,..., a K that minimize the variance (the sum of squares of the errors) by calculus and linear algebra. Find the matrix equation that the best fit coefficients (a,..., a K ) must satisfy. Exercise 3.6. Consider the best fit line from the Method of Least Squares, so the best fit values are given by (3.6). Is the point (x, y), where x = n x n and y = y n, on the best fit line? In other words, does the best fit line go through the average point? References [BD] [CaBe] P. Bickel and K. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Holden-Day, San Francisco, 977. G. Casella and R. Berger, Statistical Inference, 2nd edition, Duxbury Advanced Series, Pacific Grove, CA, [Du] R. Durrett, Probability: Theory and Examples, 2nd edition, Duxbury Press, 996. [Fe] [Kel] [LF] [MoMc] W. Feller, An Introduction to Probability Theory and Its Applications, 2nd edition, Vol. II, John Wiley & Sons, ew York, 97. D. Kelley, Introduction to Probability, Macmillan Publishing Company, London, 994. R. Larson and B. Farber, Elementary Statistics: Picturing the World, Prentice-Hall, Englewood Cliffs, J, D. Moore and G. McCabe, Introduction to the Practice of Statistics, W. H. Freeman and Co., London,

### The Method of Least Squares

The Method of Least Squares Steven J. Miller Department of Mathematics and Statistics Williams College Williamstown, MA 01267 Abstract The Method of Least Squares is a procedure to determine the best fit

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### DRAFT. Algebra 1 EOC Item Specifications

DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

### Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

### Bracken County Schools Curriculum Guide Math

Unit 1: Expressions and Equations (Ch. 1-3) Suggested Length: Semester Course: 4 weeks Year Course: 8 weeks Program of Studies Core Content 1. How do you use basic skills and operands to create and solve

### Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### 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

### PROBABILITY AND STATISTICS. Ma 527. 1. To teach a knowledge of combinatorial reasoning.

PROBABILITY AND STATISTICS Ma 527 Course Description Prefaced by a study of the foundations of probability and statistics, this course is an extension of the elements of probability and statistics introduced

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

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

### 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

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### CURVE FITTING LEAST SQUARES APPROXIMATION

CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

### Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS

Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS The following is a list of terms and properties which are necessary for success in Math Concepts and College Prep math. You will

### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

### Roots of Polynomials

Roots of Polynomials (Com S 477/577 Notes) Yan-Bin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x

### COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Algebra II Semester Exam Review Sheet

Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph

### 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

### Algebra. Indiana Standards 1 ST 6 WEEKS

Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

### Applications to Data Smoothing and Image Processing I

Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

### SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

### MATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS

* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

### Advanced Algebra 2. I. Equations and Inequalities

Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Math 2331 Linear Algebra

2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

### REVIEW EXERCISES DAVID J LOWRY

REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and

### Polynomials Classwork

Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### ALGEBRA I (Created 2014) Amherst County Public Schools

ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies

### North Carolina Math 1

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.

### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### 5-6 The Remainder and Factor Theorems

Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 Divide the function by x 4. The remainder is 58. Therefore, f (4) = 58. Divide the function by x + 2.

### Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}

Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following

### Multiplicity. Chapter 6

Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

### A Systematic Approach to Factoring

A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool

### 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,

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### 3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Expressions and Equations Understand the connections between proportional relationships, lines, and linear equations.

Performance Assessment Task Squares and Circles Grade 8 The task challenges a student to demonstrate understanding of the concepts of linear equations. A student must understand relations and functions,

### Chapter 14: Analyzing Relationships Between Variables

Chapter Outlines for: Frey, L., Botan, C., & Kreps, G. (1999). Investigating communication: An introduction to research methods. (2nd ed.) Boston: Allyn & Bacon. Chapter 14: Analyzing Relationships Between

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### 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.

### Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

### Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then

CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,

### 5. Factoring by the QF method

5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

### Unit 2 Quadratic Equations and Polynomial Functions Algebra 2

Number of Days: 29 10/10/16 11/18/16 Unit Goals Stage 1 Unit Description: Students will build on their prior knowledge of solving quadratic equations. In Unit 2, solutions are no longer limited to real

### AND RATIONAL FUNCTIONS. 2 Quadratic. Polynomials

UNIT 5 POLYNOMIAL In prior units of Core-Plus Mathematics, you developed understanding and skill in the use of linear, quadratic, and inverse variation functions. Those functions are members of two larger

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### SCHOOL DISTRICT OF THE CHATHAMS CURRICULUM PROFILE

CONTENT AREA(S): Mathematics COURSE/GRADE LEVEL(S): Honors Algebra 2 (10/11) I. Course Overview In Honors Algebra 2, the concept of mathematical function is developed and refined through the study of real

### Multiplying Polynomials 5

Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive

### Section 3.1 Quadratic Functions and Models

Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

### CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of