A variation is a relation between a set of values of one variable and a set of values of other variables.
Direct variation
In the equation y = mx + b, if m is a nonzero constant and b = 0, then you have the function y = mx (often written y = kx), which is called a direct variation. That is, you can say that y varies directly as x or y is directly proportional to x. In this function, m (or k) is called the constant of proportionality or the constant of variation. The graph of every direct variation passes through the origin.
Example 1
Graph y = 2 x.
x
y
| 0 | 0
| 1 | 2
| 2 | 4
Example 2
If y varies directly as x, find the constant of variation when y is 2 and x is 4.
Because this is a direct variation,
y = kx (or y = mx)
Now, replacing y with 2 and x with 4,
The constant of variation is
Example 3
If y varies directly as x and the constant of variation is 2, find y when x is 6.
Since this is a direct variation, simply replace k with 2 and x with 6 in the following equation.
A direct variation can also be written as a proportion.
This proportion is read, “ y 1 is to x 1 as y 2 is to x 2.” x 1 and y 2 are called the means, and y 1 and x 2 are called the extremes. The product of the means is always equal to the product of the extremes. You can solve a proportion by simply multiplying the means and extremes and then solving as usual.
Example 4
r varies directly as p. If r is 3 when p is 7, find p when r is 9.
Method 1. Using proportions: Set up the direct variation proportion
Now, substitute in the values.
Multiply the means and extremes (cross multiplying) give
Method 2. Using y = kx:
Replace the y with p and the x with r.
p = kr
Use the first set of information and substitute 3 for r and 7 for p, then find k.
Rewrite the direct variation equation as
Now use the second set of information that says r is 9, substitute this into the preceding equation, and solve for p.
Inverse variation (indirect variation)
A variation where
This function is also referred to as an inverse or indirect proportion. Again, m (or k) is called the constant of variation.
Example 5
If y varies indirectly as x, find the constant of variation when y is 2 and x is 4.
Since this is an indirect or inverse variation,
Now, replacing y with 2 and x with 4,
The constant of variation is 8.
Example 6
If y varies indirectly as x and the constant of variation is 2, find y when x is 6.
Since this is an indirect variation, simply replace k with 2 and x with 6 in the following equation.
As in direct variation, inverse variation also can be written as a proportion.
Notice that in the inverse proportion, the x 1 and the x 2 switched their positions from the direct variation proportion.
Example 7
If y varies indirectly as x and y = 4 when x = 9, find x when y = 3.
Method 1. Using proportions: Set up the indirect variation proportion.
Now, substitute in the values.
Multiply the means and extremes (cross‐multiplying) gives
Method 2. Using
Rewrite the direct variation equation as
Now use the second set of information that says y is 3, substitute this into the preceding equation and solve for x.