Answer
See below
Work Step by Step
The standard form of the equation is: $y=ax^2+bx+c$
For $f(1)=0 \rightarrow 0=a(1)^2+b(1)+c$
For $f(2)=-3 \rightarrow -3=a(2)^2+b(2)+c$
For $f(3)=-8 \rightarrow -8=a(3)^2+b(3)+c$
We have the system: $a+b+c=0\\4a+2b+c=-3\\9a+3b+c=-8$
Multiply the first equation by $-1$ and add it to the second equation
$3a+b=-3$ (1)
Multiply the first equation by $-1$ and add it to the third equation
$8a+2b=-8\\
\rightarrow 4a+b=-4$
We have the new system:
$a+b+c=0\\3a+b=-3\\4a+b=-4$
Multiply the second new equation by $-1$ and add it to the third equation:
$a=-1$
Find $b$:
$3(-1)+b=-3\\
\rightarrow b=0$
Find $c$:
$-1+0+c=0\\
\rightarrow c=1$
Hence, $a=-1\\b=0\\c=1$
Substitute back to the initial equation: $y=-x^2+1$
