1. January 23, 2012 At the end of today, you will be able to understand the asymptotes and domain of rational functions. Warm-up: Solve without notes or calculator! 1.Name 2 angles for x where 2. Sketch the graph of sin x and csc x. HW 2.6: Pg. 194 #29- 39 odd, #55, 57

2. Lesson 2.6 Rational Functions and their Asymptotes How the Domain and Vertical Asymptote relate… Find the domain of a rational function. The domain is all real numbers, except x = 0. Graph: Vertical asymptote is x = 0 What is the vertical asymptote? The line that the graph approaches, but never touches.

3. Example 1: Find the domain and vertical asymptote for Domain: x 2 – 1 ≠ 0 x 2 ≠ 1 Domain is all real numbers, except x = 1, x = - 1 Vertical Asymptotes (x = ?): Vert Asymp: x = 1 and x = - 1

4. Finding the Horizontal Asymptotes of a Rational Function For N(x) = numerator n = degree of numerator D(x) = denominator m = degree of denominator Rules for Horizontal Asymptotes (y = ?) n horizontal asymp: y = 0 (x-axis) n = m ------------ > horiz asymp: n > m ----------- > No horizontal asymp The leading coefficient of the numerator over the denominator.

5. Example 2: Find the horizontal asymptotes for: Look at the degree: 2 = 2 Since n = m ------------ > horiz asymp: So, write the leading coefficients as a fraction…

6. Sketch Vertical asymptotes: x = 1, x = -1 Horizontal asymptotes: y = 2 X-intercept: (let y =0) x= 0 Y-intercept: (let x = 0) y = 0 Sketch everything and finish it by graphing it in your calculator.

7. Let’s put everything together… Example 3: Find the domain of the function and identify any horizontal and vertical asymptotes. Domain: All real #’s except x = 3 Vertical Asymp: x = 3 Horizontal Asymp: y = 1 *Factoring will help find the domain.

8. When there isn’t a Horizontal Asymptote, there may be a Slanted Asymptote! Example 3: Domain: All real #s, except x = -1 Vertical Asymp: x = -1 n > m, so no Horizontal Asymptote So, use long division to find the slant asymptote! x x 2 + x − − -2x + 0 -2 -2x + 1 1 Remainder So FINALLY! The slant asymptote is: y = x – 2 + −

9. Find the slanted asymptote

10. HW 2.6: Pg. 194 #29-39 odd, #55, 57 Find the domain Find all intercepts: x-int --> when y = 0, y-int --> when x = 0 Find the vertical and horizontal/slanted asymp Sketch graph from calculator