Answer
$y=x^2-9x+22$
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 $(2,8),(3,4),(4,2)$.
Use the fact that each of the given points $(x,y)$ satisfies the equation $y=ax^2+bx+x$.
We find the system:
$\begin{cases}
a(2)^2+b(2)+c=8\\
a(3)^2+b(3)+c=4\\
a(4)^2+b(4)+c=2
\end{cases}$
$\begin{cases}
4a+2b+c=8\\
9a+3b+c=4\\
16a+4b+c=2
\end{cases}$
We will use the addition method. Multiply Equation 1 by -1 and add it to Equation 2 and Equation 3 to eliminate $c$:
$\begin{cases}
9a+3b+c-4a-2b-c=4-8\\
16a+4b+c-4a-2b-c=2-8
\end{cases}$
$\begin{cases}
5a+b=-4\\
12a+2b=-6
\end{cases}$
$\begin{cases}
5a+b=-4\\
6a+b=-3
\end{cases}$
Multiply Equation 1 by -1 and add it to Equation 2 to eliminate $b$ and determine $a$:
$\begin{cases}
-5a-b=4\\
6a+b=-3
\end{cases}$
$-5a-b+6a+b=4-3$
$a=1$
Determine $b$ using the equation $5a+b=-4$:
$5(1)+b=-4$
$5+b=-4$
$b=-9$
Determine $c$ by substituting $a,b$ in Equation 1:
$4a+2b+c=8$
$4(1)+2(-9)+c=8$
$-14+c=8$
$c=22$
The system's solution is:
$(1,-9,22)$
The function is fully determined:
$y=x^2-9x+22$
