1. Chapter 12 Rational Expressions

2. Section 12-1: Inverse Variation Algebra I June 26, 2016

3. Sect. 12-1 Warm-up Sunday, June 26, 2016 Write the equation of the direct variation that includes the given point. Identify the constant of variation. (y = kx) 1.(–4, 8)2. (1, 3)3. (12, –16) 4.The maximum weight you can lift with a lever varies directly with the amount of force you apply. Suppose you can lift a 50-lb weight by applying 20 lb of force to a certain lever. How much force will you need to lift 130 lb.? y = -2x y = 3x y = -4/3x 52

4. June 26, 2016Sect. 12-1 Objectives Solve inverse variation problems Compare direct and inverse variation

5. June 26, 2016Sect. 12-1 RECALL: Direct Variation : an equation in the form of y = kx Constant of Variation : the “k” value – the slope of the equation Given a point, write the equation of direct variation: (-3,4) (3,6) (-2,-1) (8,-4) Graph – a line

6. June 26, 2016Sect. 12-1 Inverse Variation : an equation in the form of y = Constant of Variation : the “k” value Given a point, write the equation of inverse variation: (-3,4) (3,6) (-2,-1) (8,-4)

7. June 26, 2016Sect. 12-1 Inverse Variation an equation of the form xy = k or

8. June 26, 2016Sect. 12-1 Writing an equation Suppose y varies inversely with x. Write an equation for the inverse variation. 1.y = 3 & x = 6 2. y = 10 & x = 5 3. y = 9 & x = 2

9. June 26, 2016Sect. 12-1 To find the missing coordinate if the graph is a direct variation we use the ratios ex: (x, 2) and (-3, 5) Cross multiply and solve 5x = -6 x = -6/5 RECALL: Finding a missing coordinate

10. June 26, 2016Sect. 12-1 2 ways to find a missing coordinate with inverse variation 1. Use the complete coordinate given to find k. Then use the other coordinate to find the missing value. 2. use the coordinates to fill in the ratio for inverse variation and solve for the missing value Finding a missing coordinate

11. June 26, 2016Sect. 12-1 Finding a missing coordinate Each pair of points is on the graph of an inverse variation. Find the missing value. 1. ( x, 11 ) and ( 1, 66 ) 2. ( 6, 12 ) and ( 9, y ) 3. ( 5, 6 ) and ( 3, y ) 6 8 10

12. June 26, 2016Sect. 12-1 Application… The weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. A weight of 120 lb is 6 ft from the fulcrum. How far away from the fulcrum must a 150 lb weight be placed to balance the lever? 6 ft x fulcrum 120 lb 150 lb x = 4.8

13. June 26, 2016Sect. 12-1 Comparing direct and inverse variation Is the data direct or inverse variation? xy 25 410 25 xy 520 10 254 xy 312 520 832 Try each table using direct or inverse ratios. Cross multiply. Both sides of the equation must be equal.

14. June 26, 2016Section 12-3 Identifying the asymptotes Asymptotes: an imaginary line that the graph of the function gets closer to as x or y gets larger in absolute value Horizontal Asymptote Vertical Asymptote Horizontal asymptote Vertical Asymptote

15. June 26, 2016Section 12-3 Graph each equation and determine the vertical and horizontal asymptotes

16. June 26, 2016Section 12-3 Families of Functions Linear FunctionAbsolute Value Function Quadratic FunctionExponential Function Radical FunctionRational Function

17. Section 12-3: Simplifying Rational Expressions Algebra I June 26, 2016

18. Warm Up Sunday, June 26, 2016  Check your homework answers 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. xy = 18 y = 18/x

19. June 26, 2016Section 12-3 Objectives  Simplify rational expressions

20. June 26, 2016Section 12-3 Simplify the rational expression Simplify by factoring the GCF

21. June 26, 2016Section 12-3 Simplify the rational expression 1.Factor the numerator and denominator completely 2.Cancel any common factors 1) 2) 3) Your Turn 2

22. June 26, 2016Section 12-3 Simplify By Factoring a Polynomial Your Turn

23. June 26, 2016Section 12-3 Try these:

24. June 26, 2016Section 12-3 Recognizing opposite factors Your Turn Rewrite the denominator so that the x 2 term is first, then factor out a negative 1

25. June 26, 2016Section 12-3 Warm-up Sunday, June 26, 2016  Simplify each expression: 1. 2. 3.  Factor each polynomial: 4. y 2 + 8y 5. 4t 3 – 10t 2 6. x 2 + 7x – 18 y ( y + 8) 2t 2 ( 2t – 5) (x + 9)( x – 2)

26. June 26, 2016Section 12-3 Find the Probability of hitting the shaded area 3x 6x Area of the shaded Area of the whole 2y 3y 8 8

27. Section 12-4: Multiplying/Dividing Rational Expressions Algebra I June 26, 2016

28. Objectives  Multiply and divide rational expressions When multiplying rational expressions – factor – reduce – simplify When dividing rational expressions – change to multiplication and flip the second term – factor – reduce – simplify

29. Multiplying radical expressions  Your turn Multiply top with top and bottom with bottom

30. A little more work…  Your turn Factor as much as you can – reduce - simplify

31. Multiplying by a polynomial  Your turn Second polynomial can be written as a fraction by placing a one under it – factor – reduce - simplify

32. Dividing rational expressions  Your turn When dividing rational expressions – change to multiplication – flip second term – factor – reduce - simplify

33. Last time…

34. Review for tomorrow Suppose y varies inversely with x. Write an equation for the inverse variation. 1.y = 2 & x = –7 2.y = –3 & x = –6 Each pair of points is on the graph of an inverse variation. Find the missing value. 1. ( x, 10 ) and ( 22, 5 ) 2. ( 7, 12 ) and ( 4, y )

35. Comparing direct and inverse variation  Is the data direct or inverse variation? xy 34 1824 3648 xy 215 65 301

36. June 26, 2016Section 12-3 Identifying Asymptotes

37. Simplifying rational expressions

38. Section 12-5: Dividing Simple Polynomials Algebra I June 26, 2016

39. Simplify Warm-up Sunday, June 26, 2016

40. Warm Up Workbook p. 164 #21-26 1. 2. 3. 4. 5. 6.

41. Workbook p. 166 1 st column

42. June 26, 2016Sect. 12-1 Objectives Divide polynomials by monomials Divide polynomials by binomials

43. June 26, 2016Sect. 12-1 RECALL: Dividing by a monomial

44. June 26, 2016Sect. 12-1 Dividing by a Monomial Each term in the numerator must be divided by the term in the denominator Another way to write it is: Reduce each fraction

45. TRY: When you have a remainder it is placed in the solution as a fraction

46. TRY: When you have a remainder it is placed in the solution as a fraction

47. June 26, 2016Sect. 12-1 Dividing by a Binomial You should set this type of problem up by using the long division format Base what you multiply by, using the first term in the dividend ?What should I multiply x by to get 2x 2 ?

48. June 26, 2016Sect. 12-1 TRY

49. June 26, 2016Sect. 12-1 Dividing by Polynomials with a Zero Coefficient * Notice in the dividend there is no x 2 term. * You must take up the space with a 0x 2

50. June 26, 2016Sect. 12-1 TRY Do these

51. WARM UP Simplify: Solve:

52. June 26, 2016Sect. 12-1 12.6A Add and Subtract Simple Rational Expressions with Like Denominators If fractions have common denominators – add or subtract the numerators TRY:

53. Once you have subtracted the numerators – factor – see if you can reduce the fractional solution - simplify

54. June 26, 2016Sect. 12-1 Let’s try some harder problems Once you have subtracted the numerators – factor – see if you can reduce the fractional solution - simplify

55. TRY:

56. 12-7A Solving Rational Equations Check for extraneous solutions