Is the relationship between the variables in the table a direct variation, an inverse variation, or neither? If it is a direct or inverse variation, write a function to model it.

x - 6,-4,-3,1

y - 72,-47,-36,12

Answer: neither direct variation nor inverse variation.

Explanation:

1) The relation between two variables, y and x, is a direct variation if and only if the quotient between them, y/x, is constant. This is: y / x = k or, equivalently, y = kx.

Note that if y / x is constant, x / y is also constant.

In a direct variation, when x increases, y increases, and when x decreases, y decreases.

2) The relation between two variables, y and x, is an inverse variation if and only if their product, y*x is constant. This is: y * x = k or, equivalently y = k / x or x = k / y.

In an inverse variation when one of the variables increases the other decreases.

3) Writhe the given table and study whether the conditions for direct or inverse variation are met:

x y y/x y*x

-6 - 72 - 72 / (-6) = 12 (-6) (-72) = 432

-4 - 47 - 47 / (-4) = 11.75 (-4) (-47) = 188

-3 - 36 - 36 / (-3) = 12 (-3) (-36) = 108

1 12 12 / 1 = 12 (1) (12) = 12

Conclusions;

a) Since neither y / x nor y*x have the same result for every pair, the relation is neither direct nor inverse.

b) Note that if the third pair were (-4, - 48) instead of (-4, - 47), y / x would be 12, which make a direct variation. In this case the function that modeled it would be:

y = 12x.