How is it possible to obtain a unique quadratic equation given one ordered pair (x,y)?
I am working on a mathematic study for my work and need your advice on quadratic equation. I will use a modified and simplified example to explain the problem I am facing.
To put it simply, I now have say 5 quadratic equations (in the form of y=ax^{2}+bx+c) which give five similar but not identical parabolas:
- y_{1}=-0.07x_{1}^{2}+27.85x_{1}+11600
- y_{2}=-0.05x_{2}^{2}+58.39x_{2}+14000
- y_{3}=-0.03x_{3}^{2}+50.53x_{3}+32400
- y_{4}=-0.02x_{4}^{2}+51.83x_{4}+43850
- y_{5}=-0.01x_{5}^{2}+47.54x_{5}+74780
All equations have negative "a" values, and I only need right half part of each of the parabola i.e. x > -b/(2a). Plotting the five equations, the plots will appear like the parabolas are moving toward 2 o'clock direction and also open wider slowly.
What I need to create is a "universal" quadratic equation so that if I am given a set of (x, y) values I can get a new quadratic equation which meets the trend of the other parabolas. In other words, I need to be able to obtain the coefficients a, b and c by only one set of (x, y) values.
This would have been much easier if these were linear equations. However in my case these are quadratic equations with all three coefficients i.e. a, b and c vary from equation to equation and not simply shifting horizontally or vertically.
I probably will need to do regression analysis but at the moment I am not sure how to apply the analysis on a number of quadratic equations so as to predict a new similar equation on which a new set of (x, y) values would pass the plot.
At a glance this should be doable but I haven't figured out how to do this mathematically. Any advice would be greatly appreciated.
Regards,
Sam