2. Rational Functions and Their Graphs

3. Why Should You Learn This? Rational functions are used to model and solve many problems in the business world. Some examples of real-world scenarios are:  Average speed over a distance (traffic engineers)  Concentration of a mixture (chemist)  Average sales over time (sales manager)  Average costs over time (CFO’s)

4. Introduction to Rational Functions What is a rational number? So just for grins, what is an irrational number? A rational function has the form A number that can be expressed as a fraction: A number that cannot be expressed as a fraction:

5. What is Asymptote? A line that a curve approaches as it heads towards infinity:

6. Types of Asymptote: Horizontal Asymptotes as x goes to infinity (or to -infinity) then the curve approaches some fixed constant value "b"

7. Types of Asymptote: Vertical Asymptotes as x approaches some constant value "c" (from the left or right) then the curve goes towards infinity (or -infinity)

8. Types of Asymptote: Oblique Asymptotes as x goes to infinity (or to -infinity) then the curve goes towards a line defined by y=mx+b (note: m is not zero as that would be horizontal).

9. The important point is that: The distance between the curve and the asymptote tends to zero as they head to infinity Got it!!!!

10. Parent Function The parent function is The graph of the parent rational function looks like……………………. The graph is not continuous and has asymptotes

11. Transformations The parent function How does this move?

12. The parent function How does this move?

13. The parent function And what about this?

14. The parent function How does this move?

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

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

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

19. 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

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

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

22. 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)

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

24. 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!

25. Points of Discontinuity (Holes) Set denominator = 0. Solve for x Substitute x-values into numerator. You want to keep the x-values that make the numerator = 0 (a zero is a hole) To find the y-coordinate that goes with that x: factor numerator and denominator, cancel like factors, substitute x-value in.

26. Example Function: Solve denom. Factor and cancel Plug in -2: Hole is

27. Asymptotes Some things to note: H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. TThe graph of a rational function may or may not cross a horizontal asymptote. TThe graph of a rational function NEVER crosses a vertical asymptote. Why?

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

29. Horizontal Asymptotes xf(x) 1.3333 10.08333 100.0098 1000.0009 y → _____ as x → ________ 0 xf(x) 1 -10-0.125 -100-0.0102 -1000-0.001 y → ____ as x → ________ 0

30. Examples What similarities do you see between problems? The degree of the denominator is larger than the degree of the numerator. Horizontal Asymptote at y = 0

31. Examples What similarities do you see between problems? The degree of the numerator is the same as the degree or the denominator. Horizontal Asymptote at y = 2 Horizontal Asymptote at

32. Examples What similarities do you see between problems? The degree of the numerator is larger than the degree of the denominator. No Horizontal Asymptote

33. Slant asymptotes Parabolic asymptotes Slant or Oblique asymptotes only occur when the numerator of f(x) has a degree that is one higher than the degree of the denominator. Parabolic asymptotes only occur when the numerator of f(x) is more than one higher than the degree of the denominator. When you have this situation, simply divide the numerator by the denominator, using polynomial long division or synthetic division. The quotient (set equal to y) will be the oblique asymptote. Note that the remainder is ignored.

34. Asymptotes: Summary 1. The graph of f has vertical asymptotes at the _________ of q(x). 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

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

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

37. Oblique/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.

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

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