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Understanding Rational Expressions: Asymptotes and Holes

Learn how to find vertical asymptotes, holes, and horizontal asymptotes in rational expressions with real-world applications. Explore examples and discover slant asymptotes in this comprehensive guide.

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Understanding Rational Expressions: Asymptotes and Holes

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  1. Warm Up

  2. Rational Expressions, Vertical Asymptotes, and Holes

  3. Objective Find Asymptotes and Holes

  4. Relevance Learn how to evaluate data from real world applications that fit into a quadratic model.

  5. Rational Expression • It is the quotient of two polynomials. • A rational function is a function defined by a rational expression. Examples: Not Rational:

  6. Find the domain: Graph it:

  7. Find the domain: Graph it:

  8. Find the domain: Graph it:

  9. Find the domain: Graph it:

  10. Vertical Asymptote • If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.

  11. Find the domain: Graph it using the graphing calculator. What do you see?

  12. Find the domain: Graph it using the graphing calculator. What do you see?

  13. Hole (in the graph) • If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote. • The exact point of the hole can be found by plugging b into the function after it has been simplified.

  14. Find the domain and identify vertical asymptotes & holes.

  15. Find the domain and identify vertical asymptotes & holes.

  16. Find the domain and identify vertical asymptotes & holes.

  17. Find the domain and identify vertical asymptotes & holes.

  18. Find the domain and identify vertical asymptotes & holes.

  19. Find the domain and identify vertical asymptotes & holes.

  20. Find the domain and identify vertical asymptotes & holes.

  21. Find the domain and identify vertical asymptotes & holes.

  22. Horizontal Asymptotes & Graphing

  23. Horizontal Asymptotes • Degree of numerator = Degree of denominator • Degree of numerator < Degree of denominator • Degree of numerator > Degree of denominator Horizontal Asymptote: Horizontal Asymptote: Horizontal Asymptote:

  24. Find all asymptotes & holes & then graph:

  25. Find all asymptotes & holes & then graph:

  26. Find all asymptotes & holes & then graph:

  27. Find all asymptotes & holes & then graph:

  28. Find all asymptotes & holes & then graph:

  29. Find all asymptotes & holes & then graph:

  30. Find all asymptotes & holes & then graph:

  31. Find all asymptotes & holes & then graph:

  32. Find all asymptotes & holes & then graph:

  33. Find all asymptotes & holes & then graph:

  34. Find all asymptotes & holes & then graph:

  35. Find all asymptotes & holes & then graph:

  36. Find all asymptotes & holes & then graph:

  37. Find all asymptotes & holes & then graph:

  38. Slant Asymptotes (Oblique) • Find a slant asymptote when the H.A. DNE. • Occurs when the degree of the top is exactly 1 more than the degree of the bottom. • To find the slant asymptote, we divide them and the answer is the asymptote.

  39. Slant Asymptote If the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the function has a slant (or oblique) asymptote.

  40. Find the oblique (slant) asymptote: The line x + 1 is an oblique asymptote

  41. Find the slant asymptote: The line x - 1 is an oblique asymptote

  42. Find the slant asymptote: The line x +3 is an oblique asymptote

  43. Slant Asymptote Sketch the graph:

  44. Slant Asymptote Sketch the graph:

  45. Assignments CW: Polynomial Review HW: Textbook p.148 (17 – 23) Odd and (37 – 41) Odd Textbook p. 157 (18, 22, 24, 32, 52, 54)

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