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 is called an *inverse variation* (or *indirect variation*). That is, *as x increases, y decreases.* And *as y increases, x decreases.* You may see the equation *xy* = *k* representing an inverse variation, but this is simply a rearrangement of .

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 : Use the first set of information and substitute 4 for *y* and 9 for *x*, then find *k.*

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.*