2.6 Rational Functions

1. 2.6Rational Functions JMerrill,2010

2. Domain Find the domain of Think: what numbers can I put in for x???? Denominator can’t equal 0 (it is undefined there)

3. You Do: Domain Find the domain of Denominator can’t equal 0

4. You Do: Domain Find the domain of Denominator can’t equal 0

5. Vertical Asymptotes At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below. none

6. Vertical Asymptotes The figure below shows the graph of The equation of the vertical asymptote is

7. Vertical Asymptotes Definition: The line x = a is a vertical asymptote of the graph of f(x) if or as x a from either the left or the right. Look at the table of values for

8. x x f(x) f(x) -3 -1 1 -1 -2.5 -1.5 -2 2 -2.1 -1.9 -10 10 -1.99 -2.01 100 -100 -2.001 -1.999 -1000 1000 Vertical Asymptotes As x approaches____ from the _______, f(x) approaches _______. As x approaches____ from the _______, f(x) approaches _______. -2 -2 right left Therefore, by definition, there is a vertical asymptote at

9. x x f(x) f(x) -2 -4 1 -1.333 -2.5 2.2222 -3.5 -2.545 -2.9 11.837 -3.1 -12.16 -2.99 119.84 -3.01 -120.2 -2.999 1199.8 -3.001 -1200 Vertical Asymptotes Describe what is happening to x and determine if a vertical asymptote exists, given the following information: Therefore, a vertical asymptote occurs at x = -3. As x approaches____ from the _______, f(x) approaches _______. As x approaches____ from the _______, f(x) approaches _______. -3 -3 left right

10. Vertical Asymptotes • Set denominator = 0; solve for x • Substitute x-values into numerator. The values for which the numerator ≠ 0 are the vertical asymptotes

11. Example • What is the domain? • x ≠ 2 so • What is the vertical asymptote? • x = 2 (Set denominator = 0, plug back into numerator, if it ≠ 0, then it’s a vertical asymptote)

12. You Do • Domain: x2 + x – 2 = 0 • (x + 2)(x - 1) = 0, so x ≠ -2, 1 • Vertical Asymptote: x2 + x – 2 = 0 • (x + 2)(x - 1) = 0 • Neither makes the numerator = 0, so • x = -2, x = 1

13. The graph of a rational function NEVER crosses a vertical asymptote. Why? • Look at the last example: Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0!

14. Horizontal Asymptotes Definition:The line y = b is a horizontal asymptote if as or Look at the table of values for

15. x x f(x) f(x) 1 -1 1 .3333 10 -10 -0.125 .08333 -100 100 -0.0102 .0098 -1000 1000 -0.001 .0009 Horizontal Asymptotes 0 0 y→_____ as x→________ y→____ as x→________ Therefore, by definition, there is a horizontal asymptote at y = 0.

16. Examples Horizontal Asymptote at y = 0 Horizontal Asymptote at y = 0 What relationships exists between the numerator and the denominator in each of these problems? The degree of the denominator is larger than the degree of the numerator.

17. Examples Horizontal Asymptote at Horizontal Asymptote at y = 2 What relationships exists between the numerator and the denominator in each of these problems? The degree of the numerator is the same as the degree or the denominator.

18. Examples No Horizontal Asymptote No Horizontal Asymptote What relationships exists between the numerator and the denominator in each of these problems? The degree of the numerator is larger than the degree of the denominator.

19. Asymptotes: Summary 1. The graph of f has vertical asymptotes at the _________ of the denominator.  2. The graph of f has at most one horizontal asymptote, as follows:  a)If n < d, then the ____________ is a horizontal asymptote. b)If n = d, then the line ____________ is a horizontal asymptote (leading coef. over leading coef.) c)If n > d, then the graph of f has ______ horizontal asymptote. zeros line y = 0 no

20. Asymptotes • Some things to note: • Horizontal asymptotes describe the behavior at the ends of a function. They do not tell us anything about the function’s behavior for small values of x. Therefore, if a graph has a horizontal asymptote, it may cross the horizontal asymptote many times between its ends, but the graph must level off at one or both ends. • The graph of a rational function may or may not cross a horizontal asymptote. • The graph of a rational function NEVER crosses a vertical asymptote. Why?

21. You Do Find all vertical and horizontal asymptotes of the following function Vertical Asymptote: x = -1 Horizontal Asymptote: y = 2

22. You Do - Again Find all vertical and horizontal asymptotes of the following function Vertical Asymptote: none Horizontal Asymptote: y = 0

23. The Last You-Do (for now :) Find all asymptotes of the following function Vertical Asymptote: x = 2 Horizontal Asymptote: none

24. Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes. The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both. When doing long division, we do not care about the remainder.

25. Example Find all asymptotes. Vertical Horizontal Slant none x = 1 y = x

26. Example • Find all asymptotes: Vertical asymptote at x = 1 y = x + 1 n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote.

27. Graphing Rational Functions • To graph a rational function, you find all asymptotes • You must show your work • You must identify the domain and range • You must identify the x- and/or y-intercepts • You may have to “blow up” part of the graph (Zoom:Box) to actually see how the graph fits next to the asymptote.