Examine the graph at the
right:
The linear equation graph at the
right shows that as the x value increases, so does the
y value increase for the coordinates that lie on this line
.
For instance, if x = 2,
y = 4.
If x = 6 (multiplied
by 3),
then y = 12 (also multiplied by 3). 

This is a graph of
direct variation.
If the value of x is increased, then y increases as well.
Both variables change in the same manner.
If x decreases, so does the value of y. We say that
y varies directly as the value of x.
A direct
variation between 2 variables, y and x, is a
relationship that is expressed as:
where the variable k is called the
constant of
proportionality.
In most problems, the k value
needs to be found using the first set of data given.
Examples:
Typical Direct Variation Problem:

In a factory, the profit,
P, varies directly with
the inventory, I.
If P = 100 when I = 20, find P when
I = 50. 
It will be
necessary to use the "first" set of data to find
the value for the constant, k.
As in most
variation problems, it is necessary to do this problem
in three steps:
1. Set up the
formula.


2. Find the
missing constant, k,
using the first set of data given.


3. Using the
formula and constant, k,
find the missing value in the problem. 


Joint Variation Problem:
Sometimes more than one
variable is involved in a direct variation problem. In
these cases, the problem is referred to as a
joint
variation. The formula remains the same, with
the additional variables included in the product.

For example: If P
varies jointly as the values of R and S,
then the formula will be:

Joint Variation Example: 
Variable M
varies jointly as the values of
p and q.
If
M = 88 when
p = 4 and q = .4,
find M when p = 8 and q
= 1.2. 
Use the same three process steps:
1.
Set up the formula.


2.
Find the missing
constant
of proportionality, k.


3.
Using the formula and
constant, k, find the new
value in the problem.



