How to read a negative voltage

After plugging that formula into Excel, this is what I get

V Ref Ra Vsig Rb Vin 5 10000 0 10000 -5 5 10000 2.5 10000 0 2.5 10000 -1.25 10000 -5 2.5 10000 1.25 10000 0 10 10000 2.5 10000 -5 10 10000 5 10000 0

So the 5v and the 10v ref voltage give me the same range. I played with the resitor values a bit and it looks like about the best I can get is a 2.5 volt range.

Hmmm, what about maybe doing something like this to get a full range, tell me if this concept is ok

What about measuring the positive voltage, and whenever the voltage hits the 0 which means it will be going to the negative, you use transistor to switch + with -, and then programaticly you just substract the 1023 value (because the reading will be positive again), and then you have the negative reading?