How many parabolas fit through any given three points?

Question

What happens when the x-values are/are not distinct? What about when the y-values are/are not distinct? Can this be shown using matrices?

Can this be proved generally? (i.e. where the x and y values aren't specific numbers)

Answer 1

EXACTLY One parabola fits through three points.
If the x-values are not distinct, then we do not have three points.. only two.. or if the y-val are dift for the same x, then it is NOT a function (every input must have one output)
Okay for the y-values not to be extinct.. that is still three points
YES: show with matrices using the coefficient matrix... three equations and three unknowns: y = a x^2 + b x + c
Plug in each point and get three equations.. three unknowns a,b,c

Answer 2

If the three points are not collinear then an infinite number of parabolas can be fitted thru them. If the x and y values are distinct there will be one equation of the form

y = ax² + bx + c

and one equation of the form

x = a'y² + b'y + c'

In addition an infinite number of parabolas that are on an angle and not aligned with either axis can be fitted to three non-collinear points.

Answer 3

Given (x1,y1), (x2,y2), (x3,y3) three distinct points can we fit a parabola though it? In general the answer contrary to what has been said to you is no. But you can do this with a matrix:

we know all parabolas look like: y = ax^2 +bx +c
So we wish to determine a,b,c for a fixed set of three points.

Plug in points we get three equations and three unknowns (recall we know (x1,y1), (x2,y2), (x3,y3)

y1 = a(x1)^2 +b(x1) + c
y2 = a(x2)^2 +b(x2) + c
y3 = a(x3)^2 +b(x3) + c

this gives us an augmented matrix:
(x1)^2 x1 1 y1
(x2)^2 x2 1 y2
(x3)^2 x3 1 y3

which then we can row reduce and maybe find a solution.

So do three points always determine a parabola? no!
(1,0) (2,0) (3,0), these are three distinct points but there is no way to find a parabola that will go through them. You are welcome to try but i claim it is impossible
a line goes thought all three and that's easy to see but suppose that you wanted to describe a polynomial that went though the points then the polynomial would have 3 zeros one at 1, another at 2 and the last one at 3. Where as a parabola can have at most two