  In this paper, we introduce a generalized quadratic functional equation $f(rx
+ sy) = rf(x) + sf(y) - rsf(x - y)$ where $r, s$ are nonzero real numbers with
$r + s = 1.$ We show that this functional equation is quadratic if $r, s$ are
rational numbers. We also investigate its stability problem on restricted
domains. These results are applied to study of an asymptotic behavior of these
generalized quadratic mappings.
