Constraining parameters for a quadratic equation

[kungfuscious]

Hi there! I'm working on analyzing some data from an experiment, and you can see a graph of some of the results in the attached .jpg image. I've done both a linear and a quadratic fit on the data points, weighted to error bars (the red lines are the fits). I use percents in the graph, but I'd rather use more general values between 0 and 1.

I'm trying to write out constraints on the parameters of a quadratic fit in more general terms.

Constraints:
1) I know that the quadratic fit needs to pass through the point x=1, y=0.

2) For the domain, 0$\leq$x$\leq$1

3) The range is similar: 0$\leq$y$\leq$1

4) The quadratic must fit below the line going from the top left to the bottom right. (the line y=-x+1)

Different quadratics could be drawn in here, and I'm trying to generalize. y=ax$^{2}$+bx+c

What I've figured out so far is that for 1), 0=a+b+c

Then, for 4), I can say that 0$\leq$y$\leq$1 , and
y$\leq$-x+1

I guess that also means I can say that ax$^{2}$+bx+c$\leq$-x+1

I can plug in for a or b or c in that equation (using 0=a+b+c), but I just end up running around in a circle.

I get the feeling there's a lot more I can do with this to write out how a and b and c might be related or constrained, but I'm a bit stuck. Can anyone help?

Cheers,

Kungfuscious

 

[gsal]

If you want the quadratic curve to fit under the -x+1 line even right at the x=1,y=0 point, then you could add an additional constraint where the tangent of the quadratic curve has a slope of -1 or less at that x=1,y=0 point...otherwise, the curve could excursion beyond your -x+1 line

so dy/dx = 2ax + b and so 2ax + b < -1 and at x=1, 2a+b < -1

no?

 

[kungfuscious]

Thanks a lot! I had figured out another few things, but I think your suggestion will work. Thank you so much!

-Kungfuscious