I am trying to make a curve that can go straight or change direction completely when equations are change. In the attached file there is sketch with two curves made of two arcs each and a spline whos control handles are made collinear with construction lines that would be tangent with those arcs for a given y distance of the end point from the start point.

So if you change that y dimension, I want the spline to match the curve that would be made with two arcs using that start and end point. The only thing that stands in my way is I have no idea what tangent weighting actually means or how to calculate them?

If anyone has any insight or can think of another way to do it which doesn't include just making different components for each case of curve direction, I'd greatly appreciate it.

(using solidworks 2020)

Sounds like you want curvature continuity. Why not apply that instead of wrestling with tangent weight?