8−4
Rational Expressions and Equations: Direct, Joint, and Inverse Variation
Main Idea:
Recognize and solve direct and joint variation problems.
Recognize and solve inverse variation problems.
Notes:
1. Direct variation: y varies directly as x if there is some nonzero constant k such
that y = kx.
a. The constant k is called the constant of variation.
b. The graph of a direct variation will be a straight line passing through the
origin; if there is a nonzero additive constant term, as in y = kx + b, the
graph will not pass through the origin and the equation is not a direct
variation.
c. Direct variation problems can be solved using the proportion
y1 y 2

.
x1 x 2
2. Joint variation: y varies jointly as x and z if there is some nonzero constant k
such that y = kxz.
a. Joint variation occurs when one quantity varies directly with the product of
two or more other quantities.
b. Joint variation problems can be solved using the proportion
y1
y
 2 .
x1z1 x 2 z2
3. Inverse variation: y varies inversely as x if there is some nonzero constant k
such that xy = k or y = k/x.
a. Inverse variation equations will have graphs similar to rational equations.
b. Inverse variation problems can be solved using the proportion
x1 x 2

.
y 2 y1