9-1/2-3 NotesDirect and Inverse Variation

Direct VariationY varies directly to x, when x and y are

related by the equation: y=kx. Here k is the constant of variation.

In other words…generally as x increases y will also increase and vice versa. Also, x and y are directly proportional so:

1 2

1 2

y y

x x

x y5 115 325 5

D P100 642,640 ?

Ex1: Determine from the table if y varies directly to x. If so, find the constant of variation.

Ex2: D varies directly with P. Find the missing value.

X and y are directly related because if you plug into the equation y=kx, you will get the same k value each time…k=1/5.

To solve for the missing value, set up a proportion…solve by cross multiplying.100 64

2640100 168960

1689.6

xx

x

x y5 115 325 5

D P100 642,640 ?

Ex1: Determine from the table if y varies directly to x. If so, find the constant of variation.

Ex2: D varies directly with P. Find the missing value.

X and y are directly related because if you plug into the equation y=kx, you will get the same k value each time…k=1/5.

To solve for the missing value, set up a proportion…solve by cross multiplying.100 64

2640100 168960

1689.6

xx

x

Do numbers 1 to 4…Compare answers as a class.

Key #1-41. Yes, x and y vary directly, because

2.

3. y = -64. y = 56/3

15 30 15 30 55

3 6 3 6 1

1

1

5y

kx

1

1

5.4 27

3.8 19

yk

x

Inverse VariationY varies inversely to x, when x and y are

related by the equation In other words…generally as x increases y

will decrease and vise versa. Also x and y are inversely proportional so: (basically the x1 and y1 are diagonally across from each other, as are the x2 and y2)

This also means that

ky

x

1 2

2 1

x x

y y

1 1 2 2 ...x y x y k

x y-1 -6-2 -31 62 3

S T150 5? 4

Ex3: Determine from the table if y is inversely proportional to x. If so, find the constant of variation.

Ex4: S varies inversely with T. Find the missing value.

X and y are inversely related because when plugged into the equation y=k/x, you get the same thing every time for k. k=6.

To solve for x, set up an inverse proportion and cross multiply.

150 5

4150 20

7.5

xx

x

x y-1 -6-2 -31 62 3

S T150 5? 4

Ex3: Determine from the table if y is inversely proportional to x. If so, find the constant of variation.

Ex4: S varies inversely with T. Find the missing value.

X and y are inversely related because when plugged into the equation y=k/x, you get the same thing every time for k. k=6.

To solve for x, set up an inverse proportion and cross multiply.

150 5

4150 20

7.5

xx

x

Do numbers 5 to 8…Compare answers as a class.

KEY #5-85. Yes, x and y are inversely proportional

because

6. y = 400

7. k = .6 (.4) = .24

8. x = 8

( 4)( 6) ( 3)( 8) 3(8) 4(6)

Joint variationZ varies jointly with x and y, when x, y , and z

are related by the equation z=kxy.‘Varies’ tells you where to put the equal sign.

k always comes after the equal sign.

Example 5Z varies jointly as x and y, if z=56 when x=7

and y=10, find the constant of variation.To solve use the joint variation equation z=kxy

and solve for k.56 7 10

560.8

70

k

k

To solve you need to make an equation that relates x, y, and z. Remember the varies tells you where to put the equal sign, k always comes after the equal, directly means multiply, and inversely means divide. This means your equation should be:

Plug in the values they give for x, y, and z to solve for k.

Use this k value to solve for z when x=6 and y=4.

3

kxz

y

3

83

28

38

3

k

k

k

3

3 6

418

649

32

z

z

z

Example 6 Z varies directly with x and inversely with the cube of y. When x=8 and y=2, z=3. Find z when x=6 and y=4.

Example 7 Describe the variation that is modeled by each formula. Remember the

equal sign is represented by varies when you are describing a variation!

If you are describing a variable that is multiplied you will say directly.

If you are describing a variable the is divided you will say inversely.

If you are describing two variables that are both multiplied say jointly.

If there is a number then it is the constant of variation!

0.5A bh3

BhV A varies jointly with b

and h, when 0.5 is the constant of variation.

V varies jointly with B and h, when 1/3 is the constant of variation.

Example 8 z varies jointly with x and y and inversely with w. When x = 5, y = 6, and w = 2, z = 45. Write a function that models this relationship, then find z when x = 4, y = 8, and w = 16.

To solve you need to make an equation that relates x, y, and z. Remember the varies tells you where to put the equal sign, k always comes after the equal, directly means multiply, inversely means divide, jointly means multiply by both variables. This means your equation should be:

Then use the first set of values to solve for the k value:

Then plug in the second set of values with k to solvefor z:

kxyz

w

5 645

230

452

45 15

3

k

k

k

k

3 4 8

166

z

z

Do numbers 9 to 20…Compare as a class.

Answers.1. yes; k=52. k=27/193. y=-64. y=56/35. yes; k=246. y=4007. k=6/258. x=89. k=-110. z=(0.5y)/x11. directly; k=512. b

13. y=-2214. x=100/715. k=316. 5 hours17. 14 days18. l varies directly

with V and inversely with the product of w and h.

19. z=4/320. k=4186