1. Short Run Behavior of Rational Functions Lesson 9.5

2. Zeros of Rational Functions We know that So we look for the zeros of P(x), the numerator Consider  What are the roots of the numerator?  Graph the function to double check

3. Zeros of Rational Functions Note the zeros of the function when graphed r(x) = 0 when x = ± 3

4. Vertical Asymptotes A vertical asymptote happens when the function R(x) is not defined  This happens when the denominator is zero Thus we look for the roots of the denominator Where does this happen for r(x)?

5. Vertical Asymptotes Finding the roots of the denominator View the graph to verify

6. Summary The zeros of r(x) are where the numerator has zeros The vertical asymptotes of r(x) are where the denominator has zeros

7. Drawing the Graph of a Rational Function Check the long run behavior  Based on leading terms  Asymptotic to 0, to a/b, or to y=(a/b)x Determine zeros of the numerator  These will be the zeros of the function Determine the zeros of the denominator  This gives the vertical asymptotes Consider

8. Given the Graph, Find the Function Consider the graph given with tic marks = 1 What are the zeros of the function? What vertical asymptotes exist? What horizontal asymptotes exist? Now … what is the rational function?

9. Look for the Hole What happens when both the numerator and denominator are 0 at the same place? Consider We end up with which is indeterminate  Thus the function has a point for which it is not defined … a “hole”

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

11. Assignment Lesson 9.5 Page 420 Exercises 1 – 41 EOO