The Hyperbola

I've looked at the wikipedia entry for hyperbola but I can't think how to rotate one or shift its origin.

If I have three coordinates, how do I express the hyperbola that passes through all of them?

Let's say

x,y=-1,0
x,y=0,1
x,y=infinity,10

What form of notation should I be looking for?

Well yes, Denis. What I doubtless had drummed into me as a teenager is how to rotate by 45 degrees and shift an axis one bit to the right, but it' evaporated. It's why I came here for a clue. What would you suggest?

It's a practical question, I'm writing a simulation page to show, among other things, the effects of alternative progressive tax systems and an assortment of poverty traps. I'm stuck without this hyperbolic approach to a maximum figure, and the variable initial gradient at which the tax is applied. I do wish I had a comprehensive text book to crib from but it's frustrating to know there are people who could get me back on track but they choose not to for reasons of principle.

So you want a hyperbola that passes through the points (1, 0) and (0, 1) and such that y approaches 10 as x goes to infinity. Any hyperbola with vertical and horizontal asymptotes is of the form y= B/(x- a)+ C The fact that "y approaches 10 as x goes to infinity means that C= 10. So we have y= B/x- a)+ 10. y= 0 when x= 1 means B/(1- a)+ 10= 0 and y= 1 when x= 0 means B/(0- a)+ 10= 1. That last equation gives -B/a= -9 or B= 9a. Putting that into the first equation, 9a/(1- a)+ 10= 0. 9a/(1- a)= -10. 9a= 10(1- a)= 10- 10a. Adding 10a to both sides, 19a= 10 so a= 10/19. And since B= 9a, B= 90/19. The equation is y= (90/19)/(x- 10/19)+ 10.

Thank you, that seems to cover exactly what I need - I'll slip those bits into javascript and see what plots I can generate. I think you have a typo at "through the points (1, 0) and (0, 1)" which ought to read "through the points (-1, 0) and (0, 1)". I'll be a few hours working on my code. I'm very grateful.