Rational Functions and Their Graphs Section 3.5

1. Rational Functions and Their GraphsSection 3.5 JMerrill,2005 Revised 08

2. 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)

3. 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:

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

5. Transformations • The parent function • How does this move?

6. Transformations • The parent function • How does this move?

7. Transformations • The parent function • And what about this?

8. Transformations • The parent function • How does this move?

9. Transformations

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

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

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

13. 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

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

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

16. 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

17. 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 - 4 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

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

19. 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)

20. 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

21. 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!

22. 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.

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

24. 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?

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

26. 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.

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

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

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

30. 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

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

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

33. 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.

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

35. 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.

36. Solving and Interpreting a Given Scenario

37. The Average Cost of Producing a Wheelchair • A company that manufactures wheelchairs has costs given by the function C(x) = 400x + 500,000, where the x is the number of wheelchairs produced per month and C(x) is measured in dollars. The average cost per wheelchair for the company is given by …

38. Original: C(x) = 400x + 500,000 • C(x) = 400x + 500,000 x • Find the interpret C(1000), C(10,000), C(100,000). • C(1000) = 900; the average cost of producing 1000 wheelchairs per month is $900.

39. C(x) = 400x + 500,000 x • Find the interpret C(10,000) C(10,000) = 450; the average cost of producing 10,000 wheelchairs per month is $450. • Find the interpret C(100,000) C(100,000) = 405; the average cost of producing 100,000 wheelchairs per month is $405.

40. C(x) = 400x + 500,000 x • What is the horizontal asymptote for the average cost function? • Since n = d (in degree) then y = 400 • Describe what this represents for the company.

41. C(x) = 400x + 500,000 x The horizontal asymptote means that the more wheelchairs produced per month, the closer the average cost comes to $400. Lower prices take place with higher production levels, posing potential problems for small businesses.