Given data:
The given table.
The expression for the quadratic equation is,
[tex]y=a(x-b)^2+c[/tex]Substitute 0 for x and 3 for y in the above expression.
[tex]\begin{gathered} 3=a(0-b)^2+c \\ 3=ab^2+c \\ c=3-ab^2 \end{gathered}[/tex]Substitute 1 for x and 1.75 for y in the above expression.
[tex]\begin{gathered} 1.75=a(1-b)^2+c \\ 1.75=a(1+b^2-2b)+c \end{gathered}[/tex]Substitute (3-ab^2) for c in the above expression.
[tex]\begin{gathered} 1.75=a(1+b^2-2b)+3-ab^2 \\ 1.75=a-2ab+3 \\ -1.25=a-2ab \\ 1.25=a(2b-1) \\ a=\frac{1.25}{2b-1} \end{gathered}[/tex]Substitute -1 for x and 3.75 for y in the given quadratic equation.
[tex]\begin{gathered} 3.75=a(-1-b)^2+c \\ 3.75=a(1+b^2+2b)+c \end{gathered}[/tex]Substitute (3-ab^2) for c in the above expression.
[tex]\begin{gathered} 3.75=a+ab^2+2ab+3-ab^2 \\ 3.75=a+2ab+3 \\ 0.75=a(1+2b) \end{gathered}[/tex]Substitute 1.25/(2b-1) for a in the above expression.
[tex]\begin{gathered} 0.75=\frac{1.25}{2b-1}(1+2b) \\ 1.5b-0.75=1.25+2.5b \\ -b=2 \\ b=-2 \end{gathered}[/tex]The value of a is.
[tex]\begin{gathered} a=\frac{1.25}{2(-2)-1} \\ =-0.25 \end{gathered}[/tex]The value of c is,
[tex]\begin{gathered} c=3-(-0.25)(-2)^2 \\ =3+1 \\ =4 \end{gathered}[/tex]Thus, the given quadratic equationn is -0.25(x-(-2))^2 +4