1. Sketching graph of a rational funtion Rational Functions Domain, Horizontal Assymptote, and Vertical Assymptote

2. Rational Functions? 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. Examples of Rational Functions

3. Find the Domain of: x 2 − 4 = 0 ( x + 2 )( x − 2 ) = 0 x + 2 = 0 ⇒ x = −2 x − 2 = 0 ⇒ x = 2 The domain is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞).

4. Rules of Horizontal Asymptotes of Rational Functions a. If n < m, then y = 0 b. If n = m, then y = a n /b m c. If n > m, then there is no HA m

5. Rules of VERTICAL Asymptotes of Rational Functions The graph of f has a vertical asymptote at x = b if D ( b ) = 0 and N ( a ) ≠ 0.

6. Graph of a Rational Funtion HA: y=0 VA: x=-1, x=1 Note! VA: x=-1 will have a hole since -1 will make the numerator equals zero therefore we ’ re not using it

7. Your turn! HA: y=½ VA: x=3/2

8. 2 Vertical Asymptotes HA: y=0 D(x) x 2 − 2x − 8 = 0 (x+2)(x-4) therefore VA: x=-2, x=4 X-int: x=2 Note! X-int of x=2 is found by equating the NUMERATOR to zero to solve for x ●

9. HA: y=0 X-int: x=0 ● Your turn! D(x) x 2 + 3x − 4 = 0 (x+4)(x-1) therefore VA: x=-4, x=1

10. Rules of Horizontal Asymptotes of Rational Functions a. If n < m, then y = 0 b. If n = m, then y = a n /b m c. If n > m, then there is no HA m

11. Rules of VERTICAL Asymptotes of Rational Functions The graph of f has a vertical asymptote at x = b if D ( b ) = 0 and N ( a ) ≠ 0.

12. Graph of a Rational Funtion HA: y=0 VA: x=-1, x=1 Note! VA: x=-1 will have a hole since -1 will make the numerator equals zero therefore we ’ re not using it

13. Your turn! HA: y=½ VA: x=3/2

14. 2 Vertical Asymptotes HA: y=0 D(x) x 2 − 2x − 8 = 0 (x+2)(x-4) therefore VA: x=-2, x=4 X-int: x=2 Note! X-int of x=2 is found by equating the NUMERATOR to zero to solve for x ●

15. HA: y=0 X-int: x=0 ● Your turn! D(x) x 2 + 3x − 4 = 0 (x+4)(x-1) therefore VA: x=-4, x=1