Finding a general equation for a quadratic curve passing through three points.

Question

I have three points (250, 0), (500,500) and (750, 0). To find a curve passing through these points all I have to do is plug-in these values into the general quadratic equation:

f(x) = ax^2 + bx + c

getting:

f(x) = -x^2/125 + 8x -1500

But, this is specific to this curve.

How would I generalize this to a curve that passes through (j, k), (p,q) and (u,v) such that the equation

f(x) = ax^2 + bx + c

has a, b and c expressed in terms of j,k,p,q,u and v; thus allowing me to plug-in arbitrary point values into this new general equation to get a specific equation for a curve?

I tried solving and eliminating variables, but all I end up with is an unsimplifiable mess that becomes too large to handle and I can't get very far.

Answer 1

If you plug j,k,p,q,u and v into the quadratic equation, you can get a system of 3 linear equations with 3 variables a,b,c and 6 parameters.

$$\begin{cases} k=j^2a+jb+c\\ q=p^2a+pb+c\\ v=u^2a+ub+c \end{cases}$$

It can be solved by various methods. The most suitable in this case is Cramer's rule. Look for the section called Explicit formulas for small systems.

Answer 2

Essentially, you have 3 linear equations in 3 variables $a$, $b$ and $c$ as follows:

$aj^2 + bj + c = k$

$ap^2 + bp + c = q$

$au^2 + bu + c = v$

This is a system of linear equations and you can solve them using any of the methods given.