1. Variation
Variation describes the relationship between two or more variables.
Two variables, x and y, can vary directly or inversely.
Three or more variables can vary jointly.
Combined variation involves a combination of direct, inverse and/or joint variation.

2. Direct Variation A direct variation is a relationship where…
as x increases, y increases
or as x decreases, y decreases
Variation occurs at a CONSTANT RATE (k). x y 2 4 8 16 x y 20 10 5
2.5

3. Direct Variation
Each equation below is an example of direct variation where k is the “constant of proportionality” or the “constant of variation”.
y varies directly as x.
y varies directly as the square of x.
y varies directly as the square root of x.
y varies directly as the cube of x.

4. Inverse Variation An inverse variation is a relationship where…
as x increases, y decreases
or as x decreases, y increases
Variation occurs at a CONSTANT RATE (k). x y 4 8 2 16 1 x y 30 2 20 3 10 6

5. Inverse Variation
Each equation below is an example of inverse variation where k is the “constant of proportionality” or the “constant of variation”.
y varies inversely as x.
y varies inversely as the square of x.
y varies inversely as the square root of x.
y varies inversely as the cube of x.

6. MODELS FOR DIRECT AND INVERSE VARIATION
DIRECT VARIATION k x = y
y = kx
k > 0
k > 0
Direct variation always goes through the origin. Inverse variation does not go through the origin.

7. Joint Variation
Joint variation involves the product of two or more variables.
y varies jointly as x and z
w varies jointly as x , y and z
z varies jointly as the square of x and the cube of y

8. Combined Variation Two or more types of variation in the same problem.
y varies directly as x and inversely as z
m varies directly as the square of t and inversely as the cube of n
p varies jointly as m and n and inversely as h