M 1310 3.3 Variation 1
Direct Variation: If a situation is described by an equation in the form y = kx where k is a nonzero constant, we say that y varies directly as x or y is directly proportional to x . The number k is called the constant of variation or the constant of proportionality.
Example: Sales Tax varies directly as the sale price of an item.
The larger the sale, the more tax to be paid.
The smaller the sale, the less tax to be paid.
Other ways to say direct variation:
y varies directly as x.
y varies with x.
y is directly proportional to x.
y is proportional to x.
Inverse Variation: If a situation is described by an equation in the form y = k/x where k is a nonzero constant, we say that y varies inversely as x or y is inversely proportional to x . The number k is called the constant of variation or the constant of proportionality.
Example: In the formula, T = D/R, time varies inversely as the rate, given that D is a constant.
The faster you go, less time it takes to get there.
The slower you go, the more time it takes to get there.
Joint Variation: If a situation is described by an equation in the form y = kxy where k is a nonzero constant, we say that y varies jointly as x and y . The number k is called the constant of variation or the constant of proportionality.
Example:
In the formula, I = PRT (Simple Interest Formula), the simple interest, I, varies jointly as the simple interest rate, R, and time,
T, given that the principal, P, is a constant.

M 1310 3.3 Variation 2
Example 1:
P varies directly as t. If t = 6, then P = 120. Find the constant of proportionality and write the formula to express this statement.
Example 2:
y varies inversely as the square of x. If x = 5, then y = –3.
Find the constant of proportionality and write the formula to express this statement.
Example 3:
C varies directly with x and inversely with the square root of w. If x = 4 and w = 36, then C = 10. Find the constant of proportionality and write the formula to express this statement.

M 1310 3.3 Variation 3
Example 4:
T varies jointly with x and then inversely with the cube of z.
If x = –2 and z = 3, then T = 6. Find the constant of proportionality and write the formula to express this
statement.
Example 5:
The cost of printing a magazine is jointly proportional to the number of pages in the magazine and the number of magazines printed. Find the expression to express this statement.
Find the constant of proportionality if the printing cost is
$60,000 for the 4,000 copies of a 120 page magazine.
How much would the printing cost be for 5,000 copies of a
92 page magazine?

M 1310 3.3 Variation 4
Example 6:
The distance that an object falls varies directly as the square of the time it has been falling. An object falls 16 feet in 1 second.
How far will it fall in 3 seconds? How long will it take an object to fall 64 feet?
Example 7:
The time it takes you to get to campus varies inversely as your driving rate. Averaging 20 miles per hour in terrible traffic, it takes 1.5 hours to get to campus. How long would the trip take averaging 60 miles per hour?

M 1310 3.3 Variation 4
Example 6:
The distance that an object falls varies directly as the square of the time it has been falling. An object falls 16 feet in 1 second.
How far will it fall in 3 seconds? How long will it take an object to fall 64 feet?
Example 7:
The time it takes you to get to campus varies inversely as your driving rate. Averaging 20 miles per hour in terrible traffic, it takes 1.5 hours to get to campus. How long would the trip take averaging 60 miles per hour?