1. The formula for inverse variation is y = k/x^n, where k is a non-zero constant and n is greater than 0. 2. For inverse variation, when one variable increases the other variable decreases, and vice versa.

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