Has to be free. Has to combine all the stupidly separated modes the Windows 8 calculator has. Has to have better history. Can have a learning curve, but should still feel like an elementary tool. Should be lightweight.
Bonus 1: be able to resolve formulas into graphs
Bonus 2: be able to sketch polygons on a grid, be able to view lengths, angles and name vertices; being able to color stuff by shape would be nice
Bonus 3: based on bonuses 1 and 2, an offline save/load function would be nice
It's 2013 - I don't think it's that much to ask. Apparently it is from Google, anyway.