Parabola with given points

Question

Let if there is a parabola passing through some points eg $(0,1)$ , $(-1,3)$ , $(3,3)$ & $(2,1)$ Then if have we to find vertex and directrix . As there are two parrallel chords then the abcissa of vertex will be $1$ as it lie on line $x=1$ because the mid pojnts of chord lie on the axis . But how can we find other information

Answer 1

Hint:

from the given points it seems that the axis of the parabola is the straight line $x=1$, parallel to the $y$ axis. If so, than use the coordinates of three points in the generic equation $y=ax^2+bx+c$ and find a system of three linear equations in the three unknowns $a,b,c$ solve , and you have the equation of the parabola. Verify that the other, not used, point stay on this parabola, an if so, you can use the equation to find all the geometric informations about it.

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Substituting $(0,1) $ in $y=ax^2+bx+c$ we have : $1=c$

Substituting $(-1,3) $ in $y=ax^2+bx+1$ we have :$3=a-b+1 \rightarrow a=2+b$

Substituting $(2,1) $ in $y=(2+b)x^2+bx+1$ we have : $b=-\frac{4}{3}$

so the equation is: $$ y=\frac{2}{3}x^2-\frac{4}{3}x+1 $$