chapter 2. a polynomial function has the form where are real numbers and n is a nonnegative...

21

2-1 QUADRATIC FUNCTION Chapter 2

Upload: cameron-richard

Post on 03-Jan-2016

222 views

Category:

Documents

1 download

Report

Download

Tags:

Embed Size (px):

TRANSCRIPT

Page 1: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

2-1 QUADRATIC FUNCTIONChapter 2

Page 2: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

WHAT IS A POLYNOMIAL FUNCTION?

A polynomial function has the form

where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one or more monomials with real coefficients and nonnegative integer exponents. The degree of the polynomial function is the highest value for n where an is not equal to 0. Polynomial functions of only one term are called monomials or power functions.

Page 3: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

CLASSIFICATION OF POLYNOMIALS

Polynomials are classify based on the leading exponent and that leading exponent is what we called degree.

has degree 0 and is called constant function.

has degree 1 and is called linear function

has degree 2 and is called quadratic function

Page 4: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

QUADRATIC FUNCTION

quadratic function is a function of the form

where ,a, b and c and are real numbers and not equal to zero. The graph of the quadratic function is called a parabola. It is a "U" shaped curve that may open up or down depending on the sign of coefficient a .

Page 5: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

EXAMPLES OF QUADRATIC FUNCTIONS Lets just graph the examples :

Page 6: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

PROPERTIES OF THE QUADRATIC FUNCTION

Page 7: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

THE STANDARD FORM OF A QUADRATIC FUNCTION

Any quadratic function can be written in the standard form

where h and k and are given in terms of coefficients a, b and c .

The graph of f is a parabola whose axis is the vertical line x=h and whose vertex is the point(h ,k ). When a>o, the parabola opens upward, and when a<0, the parabola opens downward.

Page 8: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

IDENTIFYING THE VERTEX OF A QUADRATIC FUNCTION

Write the quadratic function given by in standard form and find the vertex of the graph.

Page 9: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

IDENTIFYING THE VERTEX OF A QUADRATIC FUNCTION

Write the quadratic function given by in standard form and find the vertex of the graph.

Page 10: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

STUDENT GUIDED PRACTICE

Do problems 23-25 from book page 96

Page 11: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

IDENTIFYING THE X-INTERCEPTS OF THE QUADRATIC FUNCTION

The intercepts of the graph of a quadratic function given by

are the real solutions, if they exist, of the quadratic equation

Page 12: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

EXAMPLE OF FINDING THE X-INTERCEPTS Find the x intercepts for the graph of

each function given below a) b) )

Page 13: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

STUDENT GUIDED PRACTICE

Do problems 49 and 51 from book page 97

Page 14: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

MAXIMUM AND MINIMUM

What is the minimum value? When the parabola opens upward , the

y-value of the vertex is the minimum value.

What is the maximum value? When the parabola opens downward

the y-value of the vertex is the maximum value.

Page 15: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

Page 16: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

FINDING THE MINIMUM AND MAXIMUM Find the minimum or maximum value of

f(x) = –3x2 + 2x – 4.

Solution:

Step 1 Determine whether the function has minimum or maximum value.

Because a is negative, the graph opens downward and has a maximum value.

Step 2 Find the x-value of the vertex.

Page 17: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

CONTINUE

Step 3 Then find the y-value of the vertex,

The maximum value is -11/3

Page 18: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

STUDENT GUIDED PRACTICE

Find the minimum or maximum value of f(x) = 6x2 + 5x – 4.

Page 19: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

STUDENT GUIDED PRACTICE

Find the maxima or minima from the following function

Page 20: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

HOMEWORK

Do problems 24-27 and 33 and 36 from page 96

Page 21: Chapter 2. A polynomial function has the form where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one

CLOSURE

Today we learned about quadratic equations

Next time we are going to continue with 2.3

Integer and Polynomial Factoring Ideas

math.hawaii.eduyuen/Putnam-2016-questions.pdfFind the smallest positive integer j such that for every polynomial p(x) with integer coefficients and for every integer k, the integer

Nonnegative approximations of nonnegative tensors

On Polynomial Kernels for Integer Linear Programs ... · ity of integer linear programs (ILPs), and for ﬁnding good solutions for covering and packing ILPs. Our main results are

Lec4p1, ORF523 - Princeton Universityaaa/Public/Teaching/ORF523/S17/ORF523... · 2017. 2. 18. · •The set of nonnegative polynomials in variables and of degree (A polynomial )

Integer Wavelet-Based Image Interpolation in Lifting ... · Image Compression, Image Interpolation, Wavelet Transform, Integer Wavelet Transform, Polynomial Curve Fitting 1. Introduction

Chapter 4dog/Math1300/Chapter 4/1300_Ch4.doc · Web viewIf the polynomial can not be rewritten as factors with integer coefficients, then write the original polynomial as your answer

Page 1 Undecibability Hilbert's 10th Problem: Give an ...karlan/AoK/F11_16.pdf · Hilbert's 10th Problem: Give an algorithm that given a polynomial decides if the polynomial has integer

Integer Programming Techniques for the Nurse Rostering Problem · In this work we present a monolithic compact Integer Programming for-mulation, i.e. a formulation with a polynomial

scholar.princeton.edu · Abstract The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial

Perturbation of Bound States of the Radial Equation by ... · Repulsive Singular Potentials-First-Order Asymptotics 1%‘. ... INTRODUCTION ... nonnegative integer or half-integer

Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions Definition of a polynomial function Let n be a nonnegative integer so n={0,1,2,3…}

3.1 - 1 Polynomial Function A polynomial function of degree n, where n is a nonnegative integer, is a function defined by an expression of the form where

A Polynomial-Time Approximation Algorithm for the ...vigoda/Permanent.pdf · A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix with Nonnegative Entries MARK

Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:

Rational Functions With Nonnegative Integer …people.brandeis.edu/~gessel/homepage/slides/nonneg.pdfRational Functions With Nonnegative Integer Coeﬃcients Ira Gessel Brandeis University

1. 2 Polynomial Function A polynomial function is a function of the form where n is a nonnegative integer and each a i (i = 0,1,…, n) is a real number

Polynomial Versions of Integer Partitions and Their Zerosrboyer/talks/talk_rutgers_2012.pdf · Polynomial Versions of Integer Partitions and Their Zeros Robert Boyer Rutgers University

HW: Pg. 175 #7-16. Polynomial Functions- ◦ Let n be a nonnegative integer and let a 0, a 1, a 2,…, a n-1, a n be real numbers with a n ≠0. The functions

Integer and Polynomial Arithmetic First Steps towards ...hansc/vena/webalt.pdf · Integer and Polynomial Arithmetic First Steps towards Abstract Algebra Arjeh M. Cohen Hans Cuypers

A Strongly Polynomial Algorithm for Bimodular Integer ... · A Strongly Polynomial Algorithm for Bimodular Integer Programming Rico Zenklusen ETH Zurich joint work with Stephan Artmann

MATHEMATICS · positive integer, Binomial series, approximation, ... sum, difference product, quotient of two functions, constant function, linear function, polynomial

Nonnegative Matrix Factorization - Complexity, Algorithms ... · Nonnegative Matrix Factorization, Neural Computation 2012. Householder XIX Nonnegative Matrix Factorization: Complexity,

Project PEN The IMO Compendium Group...Project PEN The IMO Compendium Group 18 Given a positive integer a, how many nonnegative integer solutions does the equation £ x a ⁄ = h x

Parameterised Integer Programming, Integer Cones, and ... · At last, we propose other integer programming formulations for the cuttingstock problem, which appearto have polynomial

Start Up Day 14 WRITE A POLYNOMIAL FUNCTION OF MINIMUM DEGREE WITH INTEGER COEFFICIENTS GIVEN THE FOLLOWING ZEROS:

Toolkit of Functions · A function P is called a polynomial if 12 ( ) ... 1 2 1 0 nn P x a x a x a x a x a n n Where n is a nonnegative integer and the numbers a 0, a 1, a 2, …

Integer programming techniques for Polynomial Optimization

Integer Programming in Strongly Polynomial Oracle Time

Integer programming (part 1)ryantibs/convexopt-F16/lectures/integer1.pdf · Integer programming is NP-hard. There are no known polynomial-time algorithms for solving integer programs

COMPUTING A NONNEGATIVE MATRIX SANJEEV ARORA AND · COMPUTING A NONNEGATIVE MATRIX SANJEEV ARORA nonnegative matrix nonnegative matrices of size, denotes the Frobenius norm; we refer

Linear recurrences with polynomial coeﬃcients and ...eschost/publications/pollard.pdf · Linear recurrences with polynomial coeﬃcients and application to integer factorization

Wide partitions, Latin tableaux, and Rota’s basis conjecturetheory.stanford.edu/~jvondrak/data/partition.pdfA nonnegative integer square matrix whose rows and columns all sum tonmay

Anewexceptionalpolynomialfortheinteger ...archive.numdam.org/article/JTNB_2003__15_3_847_0.pdf · 847 A new exceptional polynomial for the integer transfinite diameter of [0,1] par

Section 4.1 Polynomial Functions. A polynomial function is a function of the form a n, a n-1,…, a 1, a 0 are real numbers n is a nonnegative integer D: