Short Run Behavior of Rational Functions Lesson 9.5.

Slides:



Advertisements
Similar presentations
9.3 Rational Functions and Their Graphs
Advertisements

Rational Expressions, Vertical Asymptotes, and Holes.
Short Run Behavior of Rational Functions Lesson 9.5.
Discussion X-intercepts.
Homework Check – have homework ready! Learning Goals: Find the Domain of a Rational Function Find the equation of the Vertical and Horizontal Asymptotes.
Polynomial and Rational Functions
Section4.2 Rational Functions and Their Graphs. Rational Functions.
ACT Class Openers:
Rational Functions. 5 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros.
WARM UP: Factor each completely
AP CALCULUS 1003 Limits pt.3 Limits at Infinity and End Behavior.
2.6 Rational Functions & Their Graphs
1 § 1-4 Limits and Continuity The student will learn about: limits, infinite limits, and continuity. limits, finding limits, one-sided limits,
Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:
Infinite Limits Lesson 1.5.
Rational Functions and Models Lesson 4.6. Definition Consider a function which is the quotient of two polynomials Example: Both polynomials.
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.
Graphing Rational Functions. 2 xf(x)f(x) xf(x)f(x) As x → 0 –, f(x) → -∞.
Section 5.2 Properties of Rational Functions
Asymptotes Objective: -Be able to find vertical and horizontal asymptotes.
Class Work Find the real zeros by factoring. P(x) = x4 – 2x3 – 8x + 16
Graphing Rational Functions Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 xf(x)f(x)
Rational Functions Intro - Chapter 4.4.  Let x = ___ to find y – intercepts A rational function is the _______ of two polynomials RATIO Graphs of Rational.
Do Now: Explain what an asymptote is in your own words.
As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or.
Rational Functions Standard 4a Find Zeros, Vertical Asymptotes, and Points of Exclusion of a Rational Function and distinguish them from one another.
Graphing Rational Functions Objective: To graph rational functions without a calculator.
Polynomial inequalities Objective –To Solve polynomial inequalities.
 Review:  Graph: #3 on Graphing Calc to see how it looks. › HA, VA, Zeros, Y-int.
1 Limits at Infinity Section Horizontal Asymptotes The line y = L is a horizontal asymptote of the graph of f if.
Lesson 8-3: Graphing Rational Functions
Limits at Infinity Lesson 4.5. What Happens? We wish to investigate what happens when functions go … To infinity and beyond …
Calculus Section 2.5 Find infinite limits of functions Given the function f(x) = Find =  Note: The line x = 0 is a vertical asymptote.
Limits at Infinity: End behavior of a Function
Unit 7 –Rational Functions Graphing Rational Functions.
Section 5.3 – Limits Involving Infinity. X X Which of the following is true about I. f is continuous at x = 1 II. The graph of f has a vertical asymptote.
CHAPTER 9 SECTION 3 RATIONAL FUNCTIONS AND GRAPHS Algebra 2 Notes May 21, 2009.
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:
Calculus Section 2.5 Find infinite limits of functions Given the function f(x) = Find =  Note: The line x = 0 is a vertical asymptote.
Rational Functions and Models
Graph Sketching: Asymptotes and Rational Functions
Unit 4: Graphing Rational Equations
Rational functions are quotients of polynomial functions.
Limits and Continuity The student will learn about: limits,
Polynomial and Rational Functions
Section 3.5 Rational Functions and Their Graphs
Graphing Polynomial Functions
3.5: ASYMPTOTES.
Objective: Section 3-7 Graphs of Rational Functions
Copyright © Cengage Learning. All rights reserved.
Sec. 2.2: Limits Involving Infinity
Graphing Rational Functions
2.2 Limits Involving Infinity
Short Run Behavior of Rational Functions
Essential Questions Solving Rational Equations and Inequalities
Limit as x-Approaches +/- Infinity
Limits at Infinity 3.5 On the agenda:
Graphing Rational Functions
2.2 Finding Limits Numerically
Introduction to Rational Equations
Rational Functions Lesson 9.4.
5-Minute Check Lesson 3-7.
Graphing Rational Expressions
26 – Limits and Continuity II – Day 1 No Calculator
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.
Graphing Rational Functions
EQ: What other functions can be made from
2.5 Limits Involving Infinity
Limits Involving Infinity
Presentation transcript:

Short Run Behavior of Rational Functions Lesson 9.5

2 Zeros of Rational Functions zWe know that zSo we look for the zeros of P(x), the numerator

3 Vertical Asymptotes zA vertical asymptote happens when the function R(x) is not defined yThis happens when the denominator is zero zThus we look for the roots of the denominator zWhere does this happen for r(x)?

4 Summary zThe zeros of r(x) are where the numerator has zeros zThe vertical asymptotes of r(x) are where the denominator has zeros

5 Drawing the Graph of a Rational Function zCheck the long run behavior yBased on leading terms yAsymptotic to 0, to a/b, or to y=(a/b)x zDetermine zeros of the numerator yThese will be the zeros of the function zDetermine the zeros of the denominator yThis gives the vertical asymptotes zConsider the behavior near the asymptote.

6 Near an asymptote... zAs x approaches a vertical asymptote y will approach either positive or negative infinity. zYou will need to plug in a point very near the asymptote ON EITHER SIDE to determine the SIGN of the output. zDo you need to know the actual value? NO

7 Example

8 zVertical Asymptote at x=-5. zChose a point to the right. Say…x=-4.9 zOnly Decide the sign because we know it will go to infinity!! zSo y will approach neg. infinity.

9 Check other side... zChose a point to the right. Say…x=-5.1 zOnly Decide the sign because we know it will go to infinity!! zSo y will approach pos. infinity.

10 zFor some reason this is making a shadow when I copy…but I think you get the point. zAs

11 Look for the Hole zWhat happens when both the numerator and denominator are 0 at the same place? zConsider zWe end up with which is indeterminate yThus the function has a point for which it is not defined … a “hole”

12

13 Look for the Hole zNote that when graphed and traced at x = -2, the calculator shows no value zNote also, that it does not display a gap in the line

14 Assignment zGraph by hand. zNO CALCULATOR!!!!!!!!!!!!!!!!!!!