Chapter 2.6 Rational Functions and Asymptotes

1. Chapter 2.6 Rational Functions and Asymptotes Feb. 2011

2. Section Objectives:Students will know how to determine the domains and findthe asymptotes of rational functions.

3. Concept Textbook pp.142-143 A rational function is a function of the form f(x) = N(x)/D(x), where N and D are both polynomials. The domain of f is all x such that D(x) ≠ 0. Example1 Find the domain of The domain is

4. Concept Textbook pp.143 II. Horizontal and Vertical Asymptotes Let’s sketch the graph of   Now, let’s just plug in some values of x and see what we get.

5. Concept Textbook pp.143 II. Horizontal and Vertical Asymptotes Let’s sketch the graph of   Now, let’s just plug in some values of x and see what we get.

6. Concept Textbook pp.143 II. Horizontal and Vertical Asymptotes Let’s sketch the graph of   Now, let’s just plug in some values of x and see what we get. As x approaches zero from the left, y decreases without bound. + As x approaches zero from the right, y increases without bound. − The line x = 0 is a vertical asymptote of .

7. Concept Textbook pp.143-146 II. Horizontal and Vertical Asymptotes Let’s sketch the graph of   Now, let’s just plug in some values of x and see what we get. You can see that the graph of f also has a horizontal asymptote, the line y = 0 . This means that the values of f (x) approach zero as x increases or decreases without bound.

8. Concept Textbook pp.143 II. Horizontal and Vertical Asymptotes

9. Concept Textbook pp.144 II. Horizontal and Vertical Asymptotes Determining asymptotes is actually a fairly simple process.  First, let’s start with the rational function where n is the largest exponent in the numerator and m is the largest exponent in the denominator.   The graph of has vertical asymptotes at the zeros of 2. The graph of has at most one horizontal asymptote determined by comparing the degrees of . a. If the graph of has the line (the -axis) as a horizontal asymptote. b. If , the graph of has the line as a horizontal asymptote where is the leading coefficient of the numerator and is the leading coefficient of the denominator. c. If , the graph of has no horizontal asymptote.

10. Example1 Textbook pp.144

11. Example2 Textbook pp.144 The horizontal asymptote: y= 1/2, the vertical asymptote: x = 3/2. No horizontal asymptote the verticalasymptote: x = -1

12. Example3 Textbook pp.144 Find all horizontal and vertical asymptotes of the graph of each rational function. a) b) c) Horizontal Asymptotes Horizontal Asymptotes Horizontal Asymptotes Vertical Asymptotes Vertical Asymptotes Vertical Asymptotes

13. Practice HW: Page 148-151; #s 1-12; 15, 19, 23, 27-45, odd.