Finding an equation of a curve from y-icpt, max point.
I want to find an equation for a curve with certain properties (I realise there will be serveral million/infinite)
The curve is essentially a cubic, negative gradient, (from top left to bottom right, at any rate) with a y-intercept at (0,1) and a single positive root.
There is a minimum point at (-x,+y) and a maximum at (2,3).
I want you to either tell me an equation of a curve which passes through those points or, even better explain to me how to do it.
I came up with a straight line that goes through those points. My best idea is to find a dy/dx that gives x=2 as a turning point, and that gives y=3 when x=2 but its a bit trial-and-error (mainly error) and very unscientific, there must be an actual way of doing it. Or a computer program that does it.
The full story (if desired) - In an A level maths paper, (Edexcel C3 Jan 2010 Q6) there is a diagram of the graph as described above and you have to perform various transformations on it. That's easy but as a side point (not required for the question) I was wondering if there is a way to work out a rough equation of the curve so that I could put it on my graphical calculator and perform the transformations on it. I was half thinking that I could check my answers this way to the actual question and I was half just interested really.