Given the table of points that lie on a parabola.
[tex]\begin{center}
\begin{tabular}
{|c||c|c|c|c|}
x&-3&-2&-1&0\\
y&-1&-4&-1&8
\end{tabular}
\end{center}[/tex]
The equation of a parabola is given by [tex]y=ax^2+bx+c[/tex], where a, b and c are constants. Since the equation of a data point has three constants, three data points/equations will be needed to obtain the value of the constants and hence the required equation.
when x = 0, y = 8 and we have:
[tex]8=a(0)^2+b(0)+c \\ \\ \Rightarrow c=8[/tex]
When x = -1, y = -1 and we have:
[tex]-1=a(-1)^2+b(-1)+c \\ \\ \Rightarrow-1=a-b+8 \\ \\ \Rightarrow a-b=-1-8=-9[/tex]
When x = -2, y = -4 and we have:
[tex]-4=a(-2)^2+b(-2)+c \\ \\ \Rightarrow-4=4a-2b+8 \\ \\ \Rightarrow4a-2b=-4-8=-12[/tex]
Solving the two equations simultaneously, we have:
[tex]a-b=-9\Rightarrow a=b-9 \\ \\ 4a-2b=-12 \\ \Rightarrow4(b-9)-2b=-12 \\ \Rightarrow4b-36-2b=-12 \\ \Rightarrow2b-36=-12 \\ \Rightarrow2b=-12+36=24 \\ \Rightarrow b=24/2=12. \\ \\ a=b-9=12-9=3.[/tex]
Thus, a = 3, b = 12 and c = 8
Therefore, the required equation of the parabola is given by:
[tex]y=3x^2+12x+8[/tex]
