Answer
$y=x^2-3$
Work Step by Step
We have to determine $a,b,c$ so that the graph of $y=ax^2+bx+c$ passes through the points $(-1,-2),(2,1),(-2,1)$.
We get the system:
$\begin{cases}
a(-1)^2+b(-1)+c=-2\\
a(2)^2+b(2)+c=1\\
a(-2)^2+b(-2)+c=1
\end{cases}$
$\begin{cases}
a-b+c=-2\\
4a+2b+c=1\\
4a-2b+c=1
\end{cases}$
We will use the addition method. Multiply Equation 1 by -1 and add it to Equation 2 to eliminate $c$. Also multiply Equation 2 by -1 and add it to Equation 3 to eliminate $c$:
$\begin{cases}
4a+2b+c-a+b-c=1-(-2)\\
4a-2b+c-4a-2b-c=1-1
\end{cases}$
$\begin{cases}
3a+3b=3\\
-4b=0
\end{cases}$
$\begin{cases}
a+b=1\\
-4b=0
\end{cases}$
$b=0$
$a+0=1$
$a=1$
$a-b+c=-2$
$1-0+c=-2$
$c=-3$
The system's solution is:
$(1,0,-3)$
The function is:
$y=x^2-3$
