Step 1
When x=0, y=-2
[tex]\text{The general equation of a quadratic function is y=ax}^2+bx+c[/tex]
Substitute for x=0,y=-2
[tex]\begin{gathered} -2=a(0)^2+b(0)+c \\ -2=c \\ c=-2 \end{gathered}[/tex]
Step 2
When x=-1,y=-8
[tex]\begin{gathered} -8=a(-1)^2+b(-1)+c \\ -8=a-b+c \\ -8=a-b-2 \\ a-b=-8+2 \\ a-b=-6--(1) \end{gathered}[/tex]
When x=3,y=-8
[tex]\begin{gathered} -8=a(3)^2+b(3)+c \\ -8=9a+3b-2 \\ 9a+3b=-8+2 \\ 9a+3b=-6---(2) \end{gathered}[/tex]
Step 3
From 1;
[tex]\begin{gathered} a=-6+b \\ \text{Substituting for a in equation gives; 9(-6+b)+3b=-6} \\ -54+9b+3b=-6 \\ 12b=54-6 \\ 12b=48 \\ \frac{12b}{12}=\frac{48}{12} \\ b=4 \end{gathered}[/tex]
From equation 1;
[tex]\begin{gathered} a-b=-6 \\ a-4=-6 \\ a=-6+4 \\ a=-2 \end{gathered}[/tex]
Therefore, the equation will be;
[tex]y=-2x^2+4x-2[/tex]
