2. MAT 150 Module 8 – Rational Functions Lesson 1 – Rational Functions and their Graphs https://sites.google.com/site/conicsectionshyp erbolas/_/rsrc/1433254688151/home/real- life-example- 2/Hyperbola%20Smokestack.png?height=188& width=303

3. Rational Functions A Rational Function R(x) is the quotient of two polynomials. The equation of the function is where P(x) and Q(x) are polynomials.

4. Rational Functions The graph of a rational function looks strange because it usually has asymptotes. Asymptotes are boundaries that the graph does not cross. Think of them as invisible fences that the graph can get close to but not cross. HorizontalVerticalLinear

5. Example 1 Graph the functions using your graphing calculator. What types of asymptotes does the graphs appear to have?

6. Calculator Note When graphing or evaluating rational functions on a scientific calculator, always put parentheses around the entire numerator and the entire denominator. Y = (x+5)/(x^2 + 2x + 1) Y = x+5/x^2 + 2x + 1 CORRECT INCORRECT

7. Example 1 – Solution – Part a X = -2 Vertical Y = 0 Horizontal

8. Example 1 – Solution – Part b X = -1 Vertical Y = 1 Horizontal

9. Vertical Asymptotes Domain Exclude x-values that make the denominator = 0 Vertical Asymptotes occur at these x- values – usually!

10. Vertical Asymptotes X-value makes denominator = 0 Also makes numerator= 0 Produces a “hole in the graph” instead of a vertical asymptote Does not make numerator = 0 There is a vertical asymptote at this x value

11. “Hole in the Graph” Hole Vertical Asymptote

12. Example 2 Find the vertical asymptotes or holes in the graph for the following rational functions:

13. Factor: Example 2 - Solution Find the vertical asymptotes or holes in the graph for the following rational functions: Denominator = 0 when x = -8 Nothing cancels Vertical Asymptote at x = -8

14. Example 2 - Solution

15. Factor: Example 2 - Solution Find the vertical asymptotes or holes in the graph for the following rational functions: Denominator = 0 when x = 6 or x = 2 Nothing cancels Vertical Asymptotes at x = 6 and x = 2

16. Example 2 - Solution

17. Factor: Example 2 - Solution Find the vertical asymptotes or holes in the graph for the following rational functions: Denominator = 0 when x = -4 or x = -1 The factor x + 4 cancels Hole in the graph at x = -4 Vertical asymptote at x= -1

18. Example 2 - Solution

19. Horizontal Asymptotes A horizontal asymptote is a horizontal line that also forms an “invisible fence” on the graph of a rational function. A horizontal asymptote represents the limit of the function for very large x values. When a function has reached a limit, this means the value of f(x) effectively stops changing even as x gets bigger. While the graph will never cross a vertical asymptote, the graph may cross a horizontal asymptote near the origin.

20. Horizontal Asymptotes Horizontal asymptotes are determined by the ratio of the degree of the numerator to the degree of the denominator. There are three cases: Degree of Numerator LESS THAN Degree of Denominator Degree of Numerator EQUAL TO Degree of Denominator Degree of Numerator GREATER THAN Degree of Denominator Horizontal asymptote at y = 0 No Horizontal asymptote Horizontal asymptote at

21. Example 3 Find the horizontal asymptotes, if any, of the graph:

22. Example 3 - solution Find the horizontal asymptotes, if any, of the graph: Degree 1 Degree 2 Degree of Numerator LESS THAN Degree of Denominator Horizontal asymptote at y = 0

23. Example 3 - solution

24. Find the horizontal asymptotes, if any, of the graph: Degree 2 Degree 1 Degree of Numerator GREATER THAN Degree of Denominator NO Horizontal asymptote

25. Example 3 - solution

26. Find the horizontal asymptotes, if any, of the graph: Degree 3 Degree of Numerator EQUAL TO Degree of Denominator Horizontal asymptote at = 2

27. Example 3 - solution

28. Slant Asymptotes If the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. In this case we need to look for a slant asymptote, which is an asymptote in the form of a line, y = mx + b. Remember, there can only be a slant asymptote when there is no horizontal asymptote – a function cannot have both a horizontal and a slant asymptote at the same time.

29. Slant Asymptotes To find the equation of the oblique asymptote, use synthetic division to divide the numerator of the function by the denominator. Ignore the remainder, if there is one. The quotient will be a linear equation in the form y = mx + b. This is the equation of the slant asymptote.

30. Example 4 Examples: Find the equation of the slant asymptotes:

31. Example 4 – Solution – Part a We want to use synthetic division to divide x 2 + 4x + 4 by x - 3. x 2 x C 31 4 4 3 21 1 7 25 The quotient is x+7 and the remainder is 25. We ignore the remainder and the equation of the slant asymptote is y = x+7.

32. Example 4 – Solution – Part a

33. Example 4 – Solution – Part b We want to use synthetic division to divide 4x 2 -16 by x +6. x 2 x C -6 4 0 -16 -24 144 4 -24 128 The quotient is 4x-24 and the remainder is 128. We ignore the remainder and the equation of the slant asymptote is y = 4x-24.

34. Example 4 – Solution – Part b