The equation for this parabola satisfies
[tex]\begin{cases}a-b+c=9\\9a+3b+c=-19\\16a-4b+c=-12\end{cases}[/tex]
(in other words, plug in each given point's coordinates [tex](x,y)[/tex] into the equation [tex]y=ax^2+bx+c[/tex])
Now,
[tex](a-b+c)-(9a+3b+c)=9-(-19)\implies-8a-4b=28\implies2a+b=-7[/tex]
[tex](a-b+c)-(16a-4b+c)=9-(-12)\implies-15a+3b=21\implies5a-b=-7[/tex]
and
[tex](2a+b)+(5a-b)=(-7)+(-7)\implies7a=-14\imples a=-2[/tex]
[tex]2(-2)+b=-7\implies b=-3[/tex]
[tex](-2)-(-3)+c=9\implies c=8[/tex]
So the equation of the parabola is
[tex]y=-2x^2-3x+8[/tex]
