Types of VariationDirect Variation: y varies directly as x. As x increases, y also increases. As x decreases, y also decreases.

Equation for Direct Variation: y = kx

Inverse Variation: y varies inversely as x. As x increases, y decreases. As x decreases, y increases. The product of x and y is always constant.

Equation for Inverse Variation: xy = k

Joint Variation: z varies jointly as x and y. As x and y increases, z also increases. As x and y decrease, z also decreases.

Equation for Joint Variation: y = kxz

k is the constant of variation

The graph of Direct Variation is a straight line that passes through the origin whose slope, k,

is the constant of variation.

Let’s look at the graph of y = 2x

The graph of y = 2x is a straight line passing through the origin with a slope of 2.

k is the constant of variation

The graph of Inverse Variation is a rectangular hyperbola such that the product

of x and y is k, the constant of variation.

Let’s look at the graph of xy = 2

The graph of xy = 2 is a rectangular hyperbola whose product of x and y is always 2.

Direct Variation Example # 1

If three men earn $180 in one day, how much will 15 men earn at the same rate of pay?

First write an equation 180 = k(3)

Find the constant of variation 180/3 = kk = 60

Use the constant of variationalong with the given information

y = 60(15)y = 900

Therefore, 15 men will earn $900 in one day at the same rate of pay

That was easy

Direct Variation Example # b

If a boat travels 132 miles in 11 hours, how far an it travel in 38.5 hours traveling at the same rate of speed?

First write an equation 132 = k(11)

Find the constant of variation 132/11 = kk = 12

Use the constant of variationalong with the given information

y = 12(38.5)y = 462

Therefore, the boat can travel 462 miles traveling at the same rate of speed.

That was easy

Inverse Variation Example # a

The speed of a gear varies inversely as the number of teeth. If a gear which has 36 teeth makes 30 revolutions per minute, how many revolutions per minute will a gear which has 24 teeth make?First write an equation (30)(36) = k

Find the constant of variation k = 1080

Use the constant of variationalong with the given information

y(24) = 1080y = 1080/24y = 45

Therefore, a gear with 24 teeth will make 45 revolutions per minute.

That was easy

Inverse Variation Example # 2

If 8 snow plows can plow the airport in 6 hours, how many hours will it take 3 snow plows to plow the same airport?

First write an equation (6)(8) = k

Find the constant of variation k = 48

Use the constant of variationalong with the given information

y(3) = 48y = 48/3y = 16

Therefore, it will take 3 snow plows 16 hours to plow the airport.

That was easy

Joint Variation Example # a

Suppose y varies jointly as x and z. Find y when x = 9 and z = 2, if y = 20 when z = 3 and x = 5.

Method 1

1

2 2 2

1

2

1, , 35

, 9 2,

20

y

y

z

xy

x

z

11 2221 an dy yzx zkxk

52 )3(0 )(k2015

k43

k

2 (943

2)( )y

2 24y

Method 2

1

1

1

yx z

k2

2

2

yx z

k

2

1

1 2

2

1

y yzx x z

22( )( ) ( )( )

02935

y

215 360y

2 24y

Asi De Facil

Joint Variation Example # 2

Suppose y varies jointly as x and z. Find y when x = 9 and z = -3, if y = -50 when z = 5 and x = -10.

Method 1

11

222 2

1 5, ,

, ,

5 10

9 3

0y

y y

zx

x z

11 2221 an dy yzx zkxk

( )( )50 510k

0505

k

1k

2 9( )1 )3(y

2 27y

Method 2

1

1

1

yx z

k2

2

2

yx z

k

2

1

1 2

2

1

y yzx x z

2

( )( )5

1 ( )5 )3(00

9y

250 3501y

2 27y

Asi De Facil

Combined Variation # 1

When one quantity varies directly and inversely as two or more quantities.Suppose f varies directly as g and f varies inversely as h. Find g when f = 18 and h = -3, if g = 24 when h = 2 and f = 6.

11

1

fkgh

and

22

2

fkgh

1

1

1kf hg

and

2

2

2kf hg

2

1 1 22

1

ffgh h

g

2

( )( ) ( )(24

)826 1 3g

212 1296g

2 108g

Direct Variation

y xk

Inverse Variationxy k

yxk

f varies directly as g so g goes in the numerator.

f varies inversely as h so h goes in the denominator.

That was easy

Combined Variation # b

When one quantity varies directly and inversely as two or more quantities.Suppose a varies directly as b and a varies inversely as c. Find b when a = 7 and c = -8, if b = 15 when c = 2 and a = 4.

11

1

akbc

and

22

2

akbc

1

1

1kacb

and

2

2

2ka cb

2

1 1 22

1

a abc c

b

2

( )( ) ( )( )2 715

4 8b

28 840b

2 105b

Direct Variation

y xk

Inverse Variationxy k

yxk

a varies directly as b so b goes in the numerator.

a varies inversely as c so c goes in the denominator.

That was easy

This variation stuff is pretty

easy.I feel like jumping for joy!

Oh my! I think I’ll just push the easy button. Tha

t was

easy

Variation Word ProblemsThe volume of a gas varies v varies inversely

as the pressure p and directly as the temperature t.

a) Write an equation to represent the volume of a gas in terms of pressure and temperature.

b) Is your equation a direct, joint, inverse, or combined variation?

c) A certain gas has a volume of 8 liters, a temperature of 275 Kelvin, and a pressure of 1.25 atmospheres. If the gas is compressed to a volume of 6 liters and is heated to 300 Kelvin, what will the pressure be?