Answer
$y=2x^2+x-5$
Work Step by Step
We have to determine $a,b,c$ so that the graph of the function $y=ax^2+bx+c$ passes through the points $(-1,-4),(1,-2),(2,5)$.
Use the fact that each of the given points $(x,y)$ satisfies the equation $y=ax^2+bx+x$.
We find the system:
$\begin{cases}
a(-1)^2+b(-1)+c=-4\\
a(1)^2+b(1)+c=-2\\
a(2)^2+b(2)+c=5
\end{cases}$
$\begin{cases}
a-b+c=-4\\
a+b+c=-2\\
4a+2b+c=5
\end{cases}$
We will use the addition method. Multiply Equation 1 by -1 and add it to Equation 2 and Equation 3 to eliminate $c$:
$\begin{cases}
-a+b-c=4\\
a+b+c=-2\\
4a+2b+c=5
\end{cases}$
$\begin{cases}
a+b+c-a+b-c=-2+4\\
4a+2b+c-a+b-c=5+4
\end{cases}$
$\begin{cases}
2b=2\\
3a+3b=9
\end{cases}$
$\begin{cases}
b=1\\
a+b=3
\end{cases}$
$a+b=3$
$a+1=3$
$a=2$
Determine $c$ by substituting $a,b$ in Equation 1:
$a-b+c=-4$
$2-1+c=-4$
$1+c=-4$
$c=-5$
The system's solution is:
$(2,1,-5)$
The function is fully determined:
$y=2x^2+x-5$
