Introduction to Rational Equations

1. Introduction to Rational Equations

2. 2 Types of Functions • Continuous • Discontinuous

3. Continuous

4. Continuous • Keeps going • No breaks in graph • Smooth

5. Discontinuous

6. Discontinuous • Stops • Graph has breaks or holes

7. Examples

8. Continuous Graphs → Polynomials • Discontinuous Graphs → Rational Equations

9. Examples

10. Your Turn: Be Prepared to Share!!! • Complete problems 1 – 6 on the Introduction to Rational Equations handout • Remember, you need to: • Classify the graph as either continuous or discontinuous • Classify the graph as either a polynomial or a rational equation • Justify your reasoning!!!

11. Sharing Activity • I will gently throw the ball to a student. • That student answers the first question. • Then the student will gently throw the ball to another student. • That student answers the next question. • Repeat until we’ve answered all the questions. Say the student’s name before you throw him/her the ball!

12. Polynomial • Polygamy • Polytheism • Polydactyl • Polyglot • Monomial • Binomial • Trinomial

13. Polynomials, cont. • A polynomial is an algebraic expression that can be written in the form anxn + an-1xn-1 + … + a2x2 + a1x1 + a0 • An equation or an expression with a single variable raised to (usually many) powers • All exponents are whole numbers • an ≠ 0 (Leading Coefficient ≠ 0)

14. Polynomial Examples • Generally a long list of variables • f(x) = x4 – 4x3 + 2x2 – 3x + 11 • f(x) = x11 + 7x5 – 4x3 + x – 12 • But we can also have a short list of variables • f(x) = x5 + x • f(x) = x2 – 1 • Or even no variables at all! • f(x) = 10 • f(x) = ½

15. Rational Equation

16. Rational Equations, cont. • Rational equations are fractions in which both the numerator and the denominator are polynomials • We don’t need variables in the numerator, but we must have them in the denominator!!!

17. Rational Examples

18. Polynomials vs. Rational Equations 7. f(x) = x8 – 7x2 + 4 8. f(x) = 11 9. 10. 11.

19. Your Turn: Be Prepared to Share!!! • Complete problems 12 – 17 on the Introduction to Rational Equations handout. • Remember, you need to: • Classify the equation as either a polynomial or a rational equation. • Justify your reasoning

20. Compare – Contrast – Summarize Graphic Organizer Continuity → Continuous or Discontinuous

21. How Alike?

22. How Different?

23. How Different?

24. How Different?

25. Summarize:

26. Discontinuous Graphs Discontinuities Rational Graphs

27. *Discontinuities • Discontinuity – a point or a line where the graph of an equation has a hole, a jump, a break, or a gap • Affect the shape, domain and range of an equation

28. Discontinuities, cont. • Three major types of discontinuities: • Vertical Asymptotes • Horizontal Asymptotes • Holes Asymptotes Point (Removable) Discontinuity

29. Lines that the graph approaches but (almost) never crosses Represented by a dashed line Not part of the equation We don’t draw them if they happen on either the x-axis or the y-axis Type of Discontinuities – Asymptotes

30. Vertical Asymptotes (1st Column) • Occur when the numerator is a non-zero # and the denominator equals zero • Can never be crossed • Always in the form x = • Abbreviated VA

31. Vertical Asymptotes, cont. Hand Drawn Calculator Drawn The calculator doesn’t draw the asymptotes!!!!

32. Experiment • Graph in your graphing calculator

33. Calculators and Vertical Asymptotes

34. Occur when the degree of the denominator is ≥ the degree of the numerator Ex. Can be crossed when |x| is very small Describes the end behavior of a rational equation Always in the form y = Abbreviated HA Horizontal Asymptotes (2nd Column)

35. Horizontal Asymptotes, cont. Hand Drawn Calculator Drawn The calculator doesn’t draw the asymptotes!!!!

36. Gaps in the graph at a single point Occurs when Always in the form x = Represented by an open circle (or hole) in the graph Point (Removable) Discontinuities – Holes (3rd Column)

37. Holes, cont. Hand Drawn

38. Graphing calculators have difficulty showing removable discontinuities ****Check the table for errors! Graphing Calculators and Holes

39. Example #1 • x-int = • y-int = • VA: • HA: • Holes:

40. Example #2 • x-int = • y-int = • VA: • HA: • Holes:

41. Your Turn: • Complete problems 1 – 6 on the Identifying Features of Rational Equations Practice handout. • Don’t answer the domain and range questions!

42. 1. 2.3. 4.

43. 5. 6.

44. Discontinuities and Domain and Range • Discontinuities affect the domain and range of a rational equation • Vertical Asymptotes → Domain • Horizontal Asymptotes →Range • Holes → Domain and Range

45. Domain: Range: Example 1:

46. Domain: Range: Example 2:

47. Your Turn: • Answer the domain and range questions for problems 1 – 6 on the Identifying Features of Rational Equations Practice handout.

48. 1. 2.3. 4.5. 6.

49. Homework • Complete problems 1 – 6 on the Identifying the Features of Rational Equations Homework handout.

50. Exit Ticket • Identify the following features of the graph on the right: • x-int. = • y-int. = • VA: • HA: • Holes: • Domain: • Range: