Warm Up .

Slides:

Advertisements

Similar presentations

9.3 Rational Functions and Their Graphs

Advertisements

Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Objectives Identify and evaluate rational functions. Graph a rational function, find its.

We will find limits algebraically

Rational Expressions, Vertical Asymptotes, and Holes.

Rational Expressions GRAPHING.

2.6 Rational Functions.

Warm Up - Factor the following completely : 1. 3x 2 -8x x x x 3 +2x 2 -4x x 2 -x x (3x-2)(x-2) 11(x+3)(x-3)

3.4 Rational Functions and Their Graphs

Section 5.2 – Properties of Rational Functions

4.4 Rational Functions Objectives:

Objectives: Find the domain of a Rational Function Determine the Vertical Asymptotes of a Rational Function Determine the Horizontal or Oblique Asymptotes.

Section4.2 Rational Functions and Their Graphs. Rational Functions.

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

RATIONAL FUNCTIONS A rational function is a function of the form: where p and q are polynomials.

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.

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.

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

Introducing Oblique Asymptotes Horizontal Asymptote Rules: – If numerator and denominator have equal highest power, simplified fraction is the H.A. – If.

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

Section 9.2/9.3 Rational Functions, Asymptotes, Holes.

2.6 Rational Functions and Asymptotes 2.7 Graphs of Rational Functions Rational function – a fraction where the numerator and denominator are polynomials.

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

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.

Section 2.7. Graphs of Rational Functions Slant/Oblique Asymptote: in order for a function to have a slant asymptote the degree of the numerator must.

Algebra 2 Ch.9 Notes Page 67 P Rational Functions and Their Graphs.

Rational Functions Intro - Chapter 4.4.  Let x = ___ to find y – intercepts A rational function is the _______ of two polynomials RATIO Graphs of Rational.

Complete the table of values for the function: 1 x / f(x) x21.51½ f(x)

I can graph a rational function.

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

MAT 150 – Class #16 Topics: Graphing Rational Functions Asymptotes Vertical Slanted Horizontals Holes.

Ch : Graphs of Rational Functions. Identifying Asymptotes Vertical Asymptotes –Set denominator equal to zero and solve: x = value Horizontal Asymptotes.

Graphing Rational Expressions. Find the domain: Graph it:

APC Unit 3 CH-4.5 Real Zeros, Long And synthetic division Remainder theorem, Rational Zero Test.

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

Graphing Rational Functions Day 3. Graph with 2 Vertical Asymptotes Step 1Factor:

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

Lesson 21 Finding holes and asymptotes Lesson 21 February 21, 2013.

Graphs of Rational Functions Section 2.7. Objectives Analyze and sketch graphs of rational functions. Sketch graphs of rational functions that have slant.

Rational Functions A rational function has the form

Unit 3 – Rational Functions

Section 2.6 Rational Functions Part 2

Rational Functions.

Section 2.7B Slant Asymptotes

Horizontal Asymptotes

4.4 Rational Functions A Rational Function is a function whose rule is the quotient of two polynomials. i.e. f(x) = 1

Graphing Rational Functions

8.2 Rational Functions and Their Graphs

28 – The Slant Asymptote No Calculator

Rational functions are quotients of polynomial functions.

Lesson 2.7 Graphs of Rational Functions

Section 3.5 Rational Functions and Their Graphs

26 – Limits and Continuity II – Day 2 No Calculator

Which is not an asymptote of the function

“This is the most magnificent discarded living room set I've ever seen

Warm-Up: FACTOR x2 – 36 5x x + 7 x2 – x – 2 x2 – 5x – 14

RATIONAL FUNCTIONS A rational function is a function of the form:

Graphing Rational Functions

Section 5.2 – Properties of Rational Functions

Factor completely and simplify. State the domain.

Holes & Slant Asymptotes

Simplifying rational expressions

5-Minute Check Lesson 3-7.

2.6 Rational Functions and Their Graphs

Graphing Rational Expressions

Which is not an asymptote of the function

4.4 Rational Functions Rational functions are the quotient of two polynomials. Analyzing rational functions with many properties. Find Domain Find vertical.

Find the zeros of each function.

“This is the most magnificent discarded living room set I've ever seen

Presentation transcript:

Warm Up

Rational Expressions, Vertical Asymptotes, and Holes

Objective Find Asymptotes and Holes

Relevance Learn how to evaluate data from real world applications that fit into a quadratic model.

Rational Expression It is the quotient of two polynomials. A rational function is a function defined by a rational expression. Examples: Not Rational:

Find the domain: Graph it:

Find the domain: Graph it:

Find the domain: Graph it:

Find the domain: Graph it:

Vertical Asymptote If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.

Find the domain: Graph it using the graphing calculator. What do you see?

Find the domain: Graph it using the graphing calculator. What do you see?

Hole (in the graph) If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote. The exact point of the hole can be found by plugging b into the function after it has been simplified.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Find the domain and identify vertical asymptotes & holes.

Horizontal Asymptotes & Graphing

Horizontal Asymptotes Degree of numerator = Degree of denominator Degree of numerator < Degree of denominator Degree of numerator > Degree of denominator Horizontal Asymptote: Horizontal Asymptote: Horizontal Asymptote:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Find all asymptotes & holes & then graph:

Slant Asymptotes (Oblique) Find a slant asymptote when the H.A. DNE. Occurs when the degree of the top is exactly 1 more than the degree of the bottom. To find the slant asymptote, we divide them and the answer is the asymptote.

Slant Asymptote If the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the function has a slant (or oblique) asymptote.

Find the oblique (slant) asymptote: The line x + 1 is an oblique asymptote

Find the slant asymptote: The line x - 1 is an oblique asymptote

Find the slant asymptote: The line x +3 is an oblique asymptote

Slant Asymptote Sketch the graph:

Slant Asymptote Sketch the graph:

Assignments CW: Polynomial Review HW: Textbook p.148 (17 – 23) Odd and (37 – 41) Odd Textbook p. 157 (18, 22, 24, 32, 52, 54)