Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 3.5 Rational Functions and Their Graphs.

Slides:

Advertisements

Similar presentations

Warm Up Find the zeros of the following function F(x) = x2 -1

Advertisements

Rational Expressions GRAPHING.

Rational function A function  of the form where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

Warm-Up: January 12, 2012  Find all zeros of. Homework Questions?

3.6: Rational Functions and Their Graphs

Rational Functions Sec. 2.7a. Definition: Rational Functions Let f and g be polynomial functions with g (x ) = 0. Then the function given by is a rational.

Rational Functions and Their Graphs. Example Find the Domain of this Function. Solution: The domain of this function is the set of all real numbers not.

4.4 Rational Functions Objectives:

Section 7.2.  A rational function, f is a quotient of polynomials. That is, where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

2.7 – Graphs of Rational Functions. By then end of today you will learn about……. Rational Functions Transformations of the Reciprocal function Limits.

3 Polynomial and Rational Functions © 2008 Pearson Addison-Wesley. All rights reserved Sections 3.5–3.6.

Rational Functions. 5 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros.

2.7 Rational Functions By: Meteor, Al Caul, O.C., and The Pizz.

The exponential function f with base a is defined by f(x) = ax

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

2.6 & 2.7 Rational Functions and Their Graphs 2.6 & 2.7 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph.

5.1 Polynomial Functions Degree of a Polynomial: Largest Power of X that appears. The zero polynomial function f(x) = 0 is not assigned a degree.

9.3 Rational Functions and Their Graphs Rational Function – A function that is written as, where P(x) and Q(x) are polynomial functions. The domain of.

Rational Functions and Their Graphs

Chapter 3 – Polynomial and Rational Functions Rational Functions.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1.

Rational Functions - Rational functions are quotients of polynomial functions: where P(x) and Q(x) are polynomial functions and Q(x)  0. -The domain of.

Section 2.6 Rational Functions Part 1

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec

Graphing Rational Functions. 2 xf(x)f(x) xf(x)f(x) As x → 0 –, f(x) → -∞.

Copyright © 2011 Pearson Education, Inc. Slide More on Rational Functions and Graphs Asymptotes for Rational Functions Let define polynomials.

Copyright © 2011 Pearson Education, Inc. Rational Functions and Inequalities Section 3.6 Polynomial and Rational Functions.

Section 5.2 Properties of Rational Functions

ACTIVITY 35: Rational Functions (Section 4.5, pp )

Class Work Find the real zeros by factoring. P(x) = x4 – 2x3 – 8x + 16

Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Chapter 7 Polynomial and Rational Functions with Applications Section 7.2.

Rational Functions and Their Graphs

Graphing Rational Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 xf(x)f(x)

Lesson 3.5 – Finding the domain of a Rational Function To find the domain set the denominator to zero and solve for x. The domain will be all real number.

Rational Functions and Their Graphs. Example Find the Domain of this Function. Solution: The domain of this function is the set of all real numbers not.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

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

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Rational Functions and Their Graphs.

Section 4.5 Rational Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Alg 2 Warm Up – Wed (5/15)-Thurs (5/16) 1.List the possible roots. Then find all the zeros of the polynomial function. f(x) = x 4 – 2x 2 – 16x -15 Answers:

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

Rational Functions and Their Graphs Objectives Find the domain of rational functions. Find horizontal and vertical asymptotes of graphs of rational functions.

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Rational Functions and Their Graphs.

Rational Functions Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as where p(x) and q(x)

Section 2.6 Rational Functions and their Graphs. Definition A rational function is in the form where P(x) and Q(x) are polynomials and Q(x) is not equal.

Essential Question: How do you find intercepts, vertical asymptotes, horizontal asymptotes and holes? Students will write a summary describing the different.

Date: 1.2 Functions And Their Properties A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain.

2.6. A rational function is of the form f(x) = where N(x) and D(x) are polynomials and D(x) is NOT the zero polynomial. The domain of the rational function.

Chapter 2 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Rational Functions and Their Graphs.

Rational Functions. 6 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros 6)Slant Asymptotes.

Copyright © 2007 Pearson Education, Inc. Slide 4-1.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph Sketching: Asymptotes and Rational Functions OBJECTIVES  Find limits.

Chapter 2 – Polynomial and Rational Functions 2.6/7 – Graphs of Rational Functions and Asymptotes.

Rational Functions Algebra III, Sec. 2.6 Objective Determine the domains of rational functions, Find asymptotes of rational functions, and Sketch the graphs.

College Algebra Chapter 3 Polynomial and Rational Functions Section 3.5 Rational Functions.

 Find the horizontal and vertical asymptotes of the following rational functions 1. (2x) / (3x 2 +1) 2. (2x 2 ) / (x 2 – 1) Note: Vertical asymptotes-

Math Rational Functions Basic Graph Characteri stics Transforma tions Asymptotes vs Holes Solving Equations.

4.5 Rational Functions  For a rational function, find the domain and graph the function, identifying all of the asymptotes.

Rational Functions…… and their Graphs

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

College Algebra Chapter 3 Polynomial and Rational Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Rational Functions and Their Graphs

Rational functions are quotients of polynomial functions.

Rational Expressions and Functions

3.3: Rational Functions and Their Graphs

3.3: Rational Functions and Their Graphs

Rational Functions A rational function f(x) is a function that can be written as where p(x) and q(x) are polynomial functions and q(x) 0 . A rational.

Presentation transcript:

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Rational Functions and Their Graphs

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Rational Functions Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as where p and q are polynomial functions and The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Example: Finding the Domain of a Rational Function Find the domain of the rational function:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Finding the Domain of a Rational Function Find the domain of the rational function:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Finding the Domain of a Rational Function Find the domain of the rational function:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Arrow Notation We use arrow notation to describe the behavior of some functions.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Definition of a Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Definition of a Vertical Asymptote (continued) The line x = a is a vertical asymptote of the graph of a function f if f(x) increases or decreases without bound as x approaches a.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Locating Vertical Asymptotes If is a rational function in which p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Finding the Vertical Asymptotes of a Rational Function Find the vertical asymptotes, if any, of the graph of the rational function:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Definition of a Horizontal Asymptote

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Locating Horizontal Asymptotes

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Finding the Horizontal Asymptote of a Rational Function Find the horizontal asymptote, if any, of the graph of the rational function:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Basic Reciprocal Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Using Transformations to Graph a Rational Function Use the graph of to graph

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Strategy for Graphing a Rational Function 1. Determine whether the graph of f has symmetry. y-axis symmetry origin symmetry 2. Find the y-intercept (if there is one) by evaluating f(0). 3.Find the x-intercepts (if there are any) by solving the equation p(x) = 0. 4.Find any vertical asymptote(s) by solving the equation q(x) = 0.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Strategy for Graphing a Rational Function (continued) 5. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6.Plot at least one point between and beyond each x-intercept and vertical asymptote. 7.Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Graphing a Rational Function Graph:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. In general, if p and q have no common factors, and the degree of p is one greater than the degree of q, find the slant asymptotes by dividing q(x) into p(x).

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Finding the Slant Asymptotes of a Rational Function Find the slant asymptote of