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- 1. Warm-Up • Read page 71.
- 2. Chapter 2: Variations Sections 2.1 and 2.2 Direct and Inverse Variation
- 3. Essential Question: • What are the differences between direct and inverse variation?
- 4. Direct Variation
- 5. Direct Variation y = kx where k is a nonzero constant n and n is a positive number
- 6. Direct Variation y = kx where k is a nonzero constant n and n is a positive number We say this “y is directly proportional to x”
- 7. Direct Variation y = kx where k is a nonzero constant n and n is a positive number We say this “y is directly proportional to x” When one variable increases then the other variable increases
- 8. Direct Variation y = kx where k is a nonzero constant n and n is a positive number We say this “y is directly proportional to x” When one variable increases then the other variable increases also the opposite - one decreases the other decreases
- 9. Examples: 1. The cost of gas for a car varies directly as the amount of gas purchased. C = cost Equation: A = amount k depends on the economy
- 10. Examples: 1. The cost of gas for a car varies directly as the amount of gas purchased. C = cost Equation: A = amount k depends on the economy C = kA
- 11. Examples: 1. The cost of gas for a car varies directly as the amount of gas purchased. C = cost Equation: A = amount k depends on the economy C = kA 2. The volume of a sphere varies directly as the cube of its radius.
- 12. Examples: 1. The cost of gas for a car varies directly as the amount of gas purchased. C = cost Equation: A = amount k depends on the economy C = kA 2. The volume of a sphere varies directly as the cube of its radius. Equation:
- 13. Examples: 1. The cost of gas for a car varies directly as the amount of gas purchased. C = cost Equation: A = amount k depends on the economy C = kA 2. The volume of a sphere varies directly as the cube of its radius. Equation: V = kr3
- 14. Inverse Variation
- 15. Inverse Variation k y = n where k ≠ 0 and n > 0. x € €
- 16. Inverse Variation k y = n where k ≠ 0 and n > 0. x € € “y is inversely proportional to x” We say
- 17. Inverse Variation k y = n where k ≠ 0 and n > 0. x € € “y is inversely proportional to x” We say When one variable increases then the other variable decreases or vice versa
- 18. Examples
- 19. Examples 3. m varies inversely with n2
- 20. Examples 3. m varies inversely with n2 k m= 2 n €
- 21. Examples 3. m varies inversely with n2 k m= 2 n 4. The weight W of a body varies inversely with the square of its distance d € the center of the earth. from
- 22. Examples 3. m varies inversely with n2 k m= 2 n 4. The weight W of a body varies inversely with the square of its distance d € the center of the earth. from k W = 2 d €
- 23. Four Steps to Predict the Values of Variation Functions:
- 24. Four Steps to Predict the Values of Variation Functions: 1. Write an equation that describes the variation
- 25. Four Steps to Predict the Values of Variation Functions: 1. Write an equation that describes the variation 2. Find the constant of variation (k)
- 26. Four Steps to Predict the Values of Variation Functions: 1. Write an equation that describes the variation 2. Find the constant of variation (k) 3. Rewrite the variation function using k.
- 27. Four Steps to Predict the Values of Variation Functions: 1. Write an equation that describes the variation 2. Find the constant of variation (k) 3. Rewrite the variation function using k. 4. Evaluate the function for the desired value of the independent variable.
- 28. Examples:
- 29. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3.
- 30. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn
- 31. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12)
- 32. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) k=4
- 33. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) k=4
- 34. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4
- 35. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4
- 36. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6.
- 37. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k (1.) y = 3 x €
- 38. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k (1.) y = 3 (2.) 5 = 3 x 2 € €
- 39. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k (1.) y = 3 (2.) 5 = 3 x 2 k 5= € 8 € €
- 40. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k (1.) y = 3 (2.) 5 = 3 x 2 k 5= € 8 € k = 40 €
- 41. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k 40 (1.) y = 3 (2.) 5 = 3 (3.) y = 3 x 2 6 k 5= € 8 € € k = 40 €
- 42. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k 40 40 (1.) y = 3 (2.) 5 = 3 (3.) y = 3 (4.) y = x 2 6 216 k 5= € 8 € € € k = 40 €
- 43. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k 40 40 (1.) y = 3 (2.) 5 = 3 (3.) y = 3 (4.) y = x 2 6 216 k 5 5= y= € 8 27 € € € k = 40 € €
- 44. Examples: 5. m is directly proportional to n. If m = 48 when n = 12, ﬁnd m when n = 3. (1.) m = kn (2.) 48 = k(12) (3.) m = 4(3) (4.) m = 12 k=4 6. y varies inversely as the cube of x. If y = 5 when x = 2, ﬁnd y when x = 6. k k 40 40 (1.) y = 3 (2.) 5 = 3 (3.) y = 3 (4.) y = x 2 6 216 k 5 5= y= € 8 27 € € € k = 40 € €
- 45. Last ONE!
- 46. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9.
- 47. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx €
- 48. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 € €
- 49. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 63 = 9k € € €
- 50. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 63 = 9k k=7 € € €
- 51. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 (3.) y = 7(9) 2 63 = 9k k=7 € € € €
- 52. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 (3.) y = 7(9) (4.) y = 7(81) 2 63 = 9k k=7 € € € € €
- 53. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 (3.) y = 7(9) (4.) y = 7(81) 2 63 = 9k y = 567 k=7 € € € € €
- 54. Last ONE! 7. y varies directly as the square of x. If y = 63 and x = 3, ﬁnd y when x = 9. 2 (1.) y = kx (2.) 63 = k32 (3.) y = 7(9) (4.) y = 7(81) 2 63 = 9k y = 567 k=7 € € € € €
- 55. Summarizer:
- 56. Summarizer: 1. What is the formula for inverse variation?
- 57. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x € €
- 58. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes _________? € €
- 59. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € €
- 60. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation?
- 61. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation? y = kx n €
- 62. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation? y = kx n 4 2 4. For V = πr : 3 € €
- 63. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation? y = kx n 4 2 4. For V = πr : 3 € a. What is the constant of variation? € What is the independent variable? b. c. What is the dependent variable?
- 64. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation? y = kx n 4 2 4. For V = πr : 3 € 4 a. What is the constant of variation? π 3 € What is the independent variable? b. c. What is the dependent variable? €
- 65. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation? y = kx n 4 2 4. For V = πr : 3 € 4 a. What is the constant of variation? π 3 € What is the independent variable? b. r c. What is the dependent variable? €
- 66. Summarizer: 1. What is the formula for inverse variation? k y = n where k ≠ 0 and n > 0. x 2. For inverse, when one variable goes down the other variable goes up _________? € € 3. What is the formula for direct variation? y = kx n 4 2 4. For V = πr : 3 € 4 a. What is the constant of variation? π 3 € What is the independent variable? b. r c. What is the dependent variable? € V
- 67. Homework: 2.1 A Worksheet and 2.2 A Worksheet

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