Hi all thank you for your attention

I have used logistic regression to test whether a predictor and it's squared form would predict a binary outcome. Now that both variables are significant and the model seems to be fitting well, I would also like to know whether the predictor would change its direction in predicting the outcome once it has reached the peak (since the range of X in the current data might not fully cover the quadratic curve, it is possible that it would just become insignificant after the peak, but not necessarily change its direction). One example is that a nutrients is beneficial to health in a linear fashion until a certain amount, but we would like to know if it is detrimental to health after this amount or that it is neither beneficial nor detrimental after this amount.

Give that vertex is x=−β1/(2β2) in a quadratic equation, I calculated the peak of the quadratic curve, which is supposedly where the predicted probability starts to change. Now, is it okay to separate the data into two groups based on this vertex, and run a moderator analysis and see if this grouping would moderate the association between the predictor variable and the outcome?

Since in moderation analysis the calculation is y = B0 + B1x + B2m(moderator) + B3x*m, but in quadratic regression it is y = B0 + B1x + B2x_squared, (logisitc regression uses different equations, but you get my idea), they don't seem entirely identical. I wonder if I will make the correct conclusion if I use this approach?

Thank you.