Find turning points

[Lars Kollmeier]

Dear all,

for my current work, I need to display turning points of my quadratic function. My dependent var Y depends on X, X² , G and the interaction term X*G
Y= a₁X² + a₂X + a₃G + a₄G*X

For this function, I need to calculate the turning point (Vertex). I am aware of the formula where the turning point for y = ax² + bx + c is at x = -b/2a. However, I still don't know how/whether to include the coefficient for the interaction term and how to deal with two explanatory variables, nor do I know how to smartly calculate the turning points in Stata. Maybe the "margins" command can be helpful?
I would appreciate any help, in explaining the rationales and also hints for my state code.

I did not provide any details about my dataset because I think it should not be relevant. If I am mistaken here, please let me know. I use Stat 16.
Many thanks in advance
Lars

[Nick Cox]

What is G? If G is a binary variable, say with values {0, 1}, then that defines two points, if I understand correctly; otherwise I think the turning point is a ridge within the space of those variables. Otherwise put, you are there in 3-dimensional space.

[Lars Kollmeier]

Thank you for the quick answer. G is numeric with a range from 0 to 1.
The three-dimensional space seems inappropriate than. My thoughts are treating G as a simple control variable. Still, I am unsure about how to deal with the interaction term.

[Maarten Buis]

A binary variable is a numeric variable, but not all numeric variables are binary. So you have still not answered Nick's question. Does G take only the values 0 or 1 (and maybe missing) or can it also take values like 0.1, 0.2, 0.3333, etc.?

[Maarten Buis]

Here is an example with a binary G:

sysuse nlsw88, clear
reg wage c.tenure##c.tenure i.south i.south#c.tenure
margins , at(south=(0/1)) ///
   expression(-(_b[tenure]+_b[1.south#c.tenure]*1.south)/(2*_b[c.tenure#c.tenure]))

[Lars Kollmeier]

Thank you for the response Maarten, I am not too familiar with the terms, excuse me. Your latter suggestion was right, it can take values like 0.1; 0.5 etc.
Could you indicate an example with a non-binary G please?