Rational Functions

1. Rational Functions Macon State College Gaston Brouwer, Ph.D. June 2010

2. Georgia Performance Standards Mathematics 4 MM4A1. Students will explore rational functions. • Investigate and explain characteristics of rational • functions, including domain, range, zeros, points of • discontinuity, intervals of increase and decrease, rates • of change, local and absolute extrema, symmetry, • asymptotes, and end behavior. b. Find inverses of rational functions, discussing domain and range, symmetry, and function composition. c. Solve rational equations and inequalities analytically, graphically, and by using the appropriate technology.

3. Rational Functions • Basics • What is a Rational Function? • Domain • Horizontal & Vertical Asymptotes • Zeros • Graphing a Rational Function • Solving Rational Equations • Inverses • Range • Solving Rational Inequalities

4. Basics Multiplying fractions: Adding fractions: Simplifying fractions:

5. Simplifying fractions with variables

6. Rational Functions Definition A rational function can be written in the form Where and are both polynomial functions and

7. Examples Rational function Rational function Not a rational function

8. Domain of a Rational Function The domain of a rational function is given by: Examples Domain: Domain:

9. End Behavior Let be a rational function. The line is a horizontal asymptote (HA) if:

10. How to find a horizontal asymptote (1) 1. Divide and by the highest power of that shows up in . Call the resulting functions and . 2. HA:

11. HA Examples HA:

12. HA Examples (Continued) HA: No Horizontal Asymptote

13. How to find a horizontal asymptote (2) • If degree( ) < degree( ), the HA is given by 2. If degree( ) = degree( ), the HA is given by 3. If degree( ) > degree( ), there is no HA.

14. HA Examples Degree( ) = degree( )=2, so: HA:

15. HA Examples (Continued) Degree( ) > degree( ), so there is no HA.

16. General end behavior Let be a rational function and let Then the end behavior of is the same as the end behavior of:

17. End behavior example Consider the function Degree( ) > degree( ), so there is no HA. Its end behavior is the same as

18. Vertical Asymptotes Let be a rational function. The line is a vertical asymptote (VA) if:

19. How to find vertical asymptotes 1. Reduce the function to lowest terms. 2. The vertical asymptote(s) is (are): where is (are) the solution(s) to

20. VA Example Solve: VA: (Note that is not a vertical asymptote!)

21. How to find zeros of a rational function 1. Reduce the function to lowest terms. 2. The zeros of the rational function are the solutions to

22. Example Find the zeros of 1. Reduce the function to lowest terms. 2. Set the numerator equal to zero and solve

23. Graphing a Rational Function Graph: 1. Reduce to lowest terms: 2. Find y-intercepts (set x=0): 3. Find zeros/x-intercepts (solve f(x)=0): 4. Find the horizontal asymptote: 5. Find the vertical asymptote(s):

24. Graphing a Rational Function (Cont’d) 6. Create a table for Not in the domain! (open circle)

25. Graphing a Rational Function (Cont’d)

26. Solving a Rational Equation Solve Multiply both sides by On the TI83/84 calculator:

27. Solving a Rational Equation Solve Multiply both sides by No solution

28. Inverses Find the inverse of 1. Write the function in the form y=… 2. Interchange x and y 3. Solve for y 3. Write in the form

29. Range of a Rational Function 1. Read from graph, or 2. Use the fact that:

30. Range of a Rational Function Find the range of Previously we found that Domain of : Range of :

31. Solving a Rational Inequality Solve Write the equation in the form: On a number line, mark all the points where with a “0” and all the points where with a “?”. Then determine the sign of on each interval by using test points. On the TI83/84:

32. Questions?