This document discusses different types of variation, including direct variation, inverse variation, and joint variation. It provides examples of solving variation problems by writing the general relationship as an equation using the constant of variation k, substituting given values to find k, and then substituting values into the original equation to solve for the unknown. Types of problems include direct variation where y varies directly with x, inverse variation where y varies inversely with x, and joint variation where y varies jointly as functions of x and z.

1. 3.6 Variation
Unit 3 Polynomial and Rational Functions

2. Concepts and Objectives
 Variation
 Solve for the constant of variation
 Solve direct and inverse variation problems
 Solve joint variation problems

3. Direct Variation
 y varies directly as x, or y is directly proportional to x,
if there exists a nonzero real number k, called the
constant of variation, such that
y = kx.
 Let n be a positive real number. Then y varies directly
as the nth power of x, or y is directly proportional to
the nth power of x, if there exists a nonzero real
number k such that n
y kx

4. Inverse Variation
 y varies inversely as x, or y is inversely proportional
to x, if there exists a nonzero real number k, such that
 Let n be a positive real number. Then y varies inversely
as the nth power of x, or y is inversely proportional to
the nth power of x, if there exists a nonzero real
number k such that
k
y
x

n
k
y
x


5. Solving Variation Problems
1. Write the general relationship among the variables as
an equation. Use the constant k.
2. Substitute given values of the variables and find the
value of k.
3. Substitute this value and the remaining values into the
original equation and solve for the unknown.

6. Examples
1. If y varies directly as x, and y = 24 when x = 8, find y
when x = 12.
2. If y varies inversely as x, and y = 7 when x = 4, find y
when x = 14.

7. Examples
1. If y varies directly as x, and y = 24 when x = 8, find y
when x = 12.
2. If y varies inversely as x, and y = 7 when x = 4, find y
when x = 14.
 24 8k
3k 
  3 12y 
36y 
7
4
k

28k 
28
14
y 
2y 

8. Joint Variation
 Let m and n be real numbers. Then y varies jointly as
the nth power of x and the mth power of z if there exists
a nonzero real number k such that
 Note: If n or m is negative, then the variable is said to
vary inversely.
n m
y kx z

9. Example
3. If u varies jointly as v and w, and u = 48 when v = 12 and
w = 8, find u when v = 10 and w = 6.
4. If z varies directly as x and inversely as y2, and z = 8
when x = 6 and y = 3, find z when x = 10 and y = 4.

10. Example
3. If u varies jointly as v and w, and u = 48 when v = 12 and
w = 8, find u when v = 10 and w = 6.
4. If z varies directly as x and inversely as y2, and z = 8
when x = 6 and y = 3, find z when x = 10 and y = 4.
  48 12 8k
0.5k 
   0.5 10 6u
30u
 
2
6
8
3
k

12k 
  
2
12 10
4
z 
7.5z 

11. Classwork
 College Algebra
 Page 384: 12-20 (even), page 352: 48-54 (even), page
338: 56-72 (4)