1. ## Number of parabollas

How many parabollas can pass through 3 points $\displaystyle A(x_0,y_0),B(x_1,y_1),C(x_2,y_2)$

only one

why?

4. ## Re: Number of parabollas

Originally Posted by TheodorMunteanu
How many parabollas can pass through 3 points $\displaystyle A(x_0,y_0),B(x_1,y_1),C(x_2,y_2)$
$\displaystyle y=ax^2+bx+c$

Solve:

$\displaystyle \left\{\begin{matrix}y_0=ax_0^2+bx_0+c\\ y_1=ax_1^2+bx_1+c\\ y_2=ax_2^2+bx_2+c\end{matrix}\right$

And find $\displaystyle a,b,c$.

5. ## Re: Number of parabollas

only one parabola if the axis of symmetry is parallel to the y-axis. (not really sure about this, yet)

if the axis of symmetry can be any line in the xy plane ... how many ???

6. ## Re: Number of parabollas

Originally Posted by TheodorMunteanu
How many parabollas can pass through 3 points $\displaystyle A(x_0,y_0),B(x_1,y_1),C(x_2,y_2)$
What if $\displaystyle A,~B,~\&,~C$ are collinear?

7. ## Re: Number of parabollas

Originally Posted by Plato
What if $\displaystyle A,~B,~\&,~C$ are collinear?
A line is a degenerate form of a parabola, so it's still only one parabola.