1. Types of Variation
Direct 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

2. 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.

3. 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.

4. 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 = k
k = 60
Use the constant of variation
along 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

5. 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 = k
k = 12
Use the constant of variation
along 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

6. 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 variation
along with the given information
y(24) = 1080
y = 1080/24
y = 45
Therefore, a gear with 24 teeth will make 45 revolutions per minute.
That was easy

7. 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 variation
along with the given information
y(3) = 48
y = 48/3
y = 16
Therefore, it will take 3 snow plows 16 hours to plow the airport.
That was easy

8. 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
Method 2
Asi De Facil

9. 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
Method 2
Asi De Facil

10. Combined Variation # 1 That was easy
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.
f varies directly as g so g goes in the numerator.
Direct Variation
f varies inversely as h so h goes in the denominator.
Inverse Variation
and
and
That was easy

11. Combined Variation # b That was easy
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.
a varies directly as b so b goes in the numerator.
Direct Variation
a varies inversely as c so c goes in the denominator.
Inverse Variation
and
and
That was easy

12. This variation stuff is pretty easy.
I feel like jumping for joy!
Oh my! I think I’ll just push the easy button.
That was easy

13. Variation Word Problems
The volume of a gas varies v varies inversely as the pressure p and directly as the temperature t.
Write an equation to represent the volume of a gas in terms of pressure and temperature.
Is your equation a direct, joint, inverse, or combined variation?
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?