1.  FOR A RATIONAL FUNCTION, FIND THE DOMAIN AND GRAPH THE FUNCTION, IDENTIFYING ALL OF THE ASYMPTOTES.  SOLVE APPLIED PROBLEMS INVOLVING RATIONAL FUNCTIONS. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 4.5 Rational Functions

2. Rational Function Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley A rational function is a function f that is a quotient of two polynomials, that is, where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x)  0.

3. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Consider. Find the domain and graph f. The graph of the function is the graph of y = 1/x translated to the left 4 units.

4. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine the domain of each of the functions shown below. a.b.

5. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine the domain of each of the functions shown below. c.d.

6. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine the domain of each of the functions shown below. e.f.

7. Vertical Asymptotes Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

8. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine the vertical asymptotes of the function.

9. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Determine the vertical asymptotes of the function. Hole located at (0, 4)

10. Horizontal Asymptotes

11. Determining a Horizontal Asymptote Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

12. Notice how the horizontal asymptote intersects the graph. Examples of Graphs with Horizontal Asymptotes and without

13. Horizontal Asymptote Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example: Find the horizontal asymptote:

14. Graphing Rational Functions Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 1. Determine the domain of the function. 2. Find the real zeros of the denominator and sketch any vertical asymptotes. 3. Find the horizontal or oblique asymptote, if there is one, and sketch it. 4. Use the graph on a calculator to sketch in the function.

15. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graph

16. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graph

17. More Examples Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graph the following functions. a) b) c)

18. More Practice Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley