1. Short Run Behavior of Rational Functions Lesson 9.5

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

3. 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. 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. 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. 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. 7 Example

8. 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. 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. 10 zFor some reason this is making a shadow when I copy…but I think you get the point. zAs

11. 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. 12

13. 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. 14 Assignment zGraph by hand. zNO CALCULATOR!!!!!!!!!!!!!!!!!!!