The general equation of a parabola, is:
[tex]y=ax^2+bx+c[/tex]
Then, to find the parabola that passes through the given points using system of equations, replace the value of the coordinates of the points to get a 3x3 system whose unkowns are a, b and c.
(3,-29)
[tex]\begin{gathered} -29=a(3)^2+b(3)+c \\ \Rightarrow-29=9a+3b+c \end{gathered}[/tex]
(2,-12)
[tex]\begin{gathered} -12=a(2)^2+b(2)+c \\ \Rightarrow-12=4a+2b+c \end{gathered}[/tex]
(-1,-9)
[tex]\begin{gathered} -9=a(-1)^2+b(-1)+c \\ \Rightarrow-9=a-b+c \end{gathered}[/tex]
Then, the system of equations that we get, is:
[tex]\begin{gathered} 9a+3b+c=-29 \\ 4a+2b+c=-12 \\ a-b+c=-9 \end{gathered}[/tex]
Solving this system yields:
[tex]\begin{gathered} a=-4 \\ b=3 \\ c=-2 \end{gathered}[/tex]
Therefore, the equation of the parabola that passes through the given points, is:
[tex]y=-4x^2+3x-2[/tex]
