How to assess the effect of individual difference measure on quadratic fit of a within-subject factor? In my task, there are 8 levels of a within-subjects/repeated measure factor. The overall relationship that this factor has with  the DV is in the shape of an inverted-U, such that their DV scores at levels 1 and 8 are generally lowest, and those at 4 and 5 are highest. 

What I want to test is whether my individual difference measure interacts with the within-subjects factor to make the inverted U (quadratic) shape fit better or worse. My hypothesis is that those high on my indiv diff measure will have more of a straight line than the curvey one in the overall sample. And I'd like to keep the indiv diff measure continuous.

Input and insights on good statistical approaches to this question are greatly appreciated!

Emmett
By default your 8 levels are probably coded linearly: e.g., 

1, 2, 3, 4, 5, 6, 7, 8


Or

-3.5,-2.5, ..., 2.5,  3.5


To examine quadratic effects you could simply square these values. Interpretation of linear and quadratic parameters will be clearer if you first center the linear variable before squaring.

So you could enter (-3.5)^2 = 12.25; (-2.5)^2 = 6.25, etc.

Equally, some data analysis software will allow you to specify polynomial contrasts (i.e., linear, quadratic, cubic). And in such software you can specify interactions.

In R, you would specify that the 8-level variable is an ordered factor.

In SPSS, the GLM - Repeated measures dialog allows you specify interactions and polynomial contrasts.