Question

How do you determine the constant of variation for the direct variation given by (1,8) (2,4) (4,2) (8,1)?

Answer 1

The set given does not represent a direct variation;
it is an inverse variation with a constant of `8`

Explanation:

If `(x,y) in` direct variation set
`rarr y=c*x` for some constant `c` and all pairs `(x,y)`
If we consider only the first 2 pairs of the defining set
we have:
`color(white)("XXX")8=c*1`
and
`color(white)("XXX")4=c*2`
Obviously both can not be true for any single value of `c`.

However, if `y` varies inversely with `x` then
`rarr x*y = c` for some constant `c` and all pairs `(x,y)`
and looking at all given `(x,y)` pairs:
`(1,8)rarr 1xx8 = 8`
`(2,4)rarr 2xx4 = 8`
`(4,2)rarr 4xx2 = 8`
`(8,1)rarr 8xx1 = 8`
We appear to have an inverse variation with a constant of `8`

For a direct variation if you multiply the value of one of your variables by a number, it must result in the value of the other variable being multiplied by that same number.

For an inverse variation if you multiply the value of one of your variables by a number, it must result in the value of the other variable being divided by that number.