Question 567835
You must be given at least 3 points, call them (x1,y1),(x2,y2),(x3,y3).  Then we know the equation has the form f(x)=ax^2+bx+c.  We get a system of 3 equations in 3 unknowns.  Solving for a,b,c in,
{{{f(x1)=a(x1)^2+b(x1)+c=y1}}}
{{{f(x2)=a(x2)^2+b(x2)+c=y2}}}
{{{f(x3)=a(x3)^2+b(x3)+c=y3}}}
gives you the equation for f(x).

I'll give an example,
If we know that three points are (-1,-2),(1,4),(2,13).  Then we set up three equations about a,b,c that we know:
{{{f(-1)=a(-1)^2+b(-1)+c=-2}}}
{{{f(1)=a(1)^2+b(1)+c=4}}}
{{{f(2)=a(2)^2+b(2)+c=13}}}
So we have,
{{{a-b+c=-2}}}
{{{a+b+c=4}}}
{{{4a+2b+c=13}}}
We need to solve for a,b,c.  So, we subtract eqn1 from eqn2 to get,
{{{a-b+c=-2}}}
{{{2b=6}}}
{{{4a+2b+c=13}}}
Divide eqn2 by 2 to get b=3.  Then, we have
{{{a-3+c=-2}}}
{{{4a+2(3)+c=13}}}
So,
{{{a+c=1}}}
{{{4a+c=7}}}
Subtract the first eqn from the second to get,
{{{a+c=1}}}
{{{3a=6}}}
Divide the second eqn by 3 to get a=2.  Then we have
{{{2+c=1}}}
So,
{{{c=-1}}}
Therefore we find that
{{{f(x)=2x^2+3x-1}}}