Given the three end points (3,8), (5,20/3) and (6,5) which are written in the form of (x1,y1), (x2,y2) and (x3,y3).
Write out the formula to obtain the equation of the parabola
[tex]y=ax^2+bx+c[/tex]
Write out the given coordinates
[tex]\begin{gathered} x_1=3,y_1=\text{ 8} \\ x_2=5,y_2=\text{ }\frac{20}{3} \\ x_3=6,y_3=\text{ 5} \end{gathered}[/tex]
Substitute the coordinates into the given equation above to obtain the three equations
[tex]\begin{gathered} y_1=ax^2_1+bx_1+c \\ 8=a(3)^2+\text{ b(3)+ c} \\ 8=\text{ 9a +3b+c }\ldots\ldots\ldots\ldots equation\text{ 1} \end{gathered}[/tex][tex]\begin{gathered} y_2=ax^2_{^{}2}+bx_2+c \\ \frac{20}{3}=a(5)^2_{}+\text{ b(5)+ c} \\ \frac{20}{3}=\text{ 25a+5b+c} \\ \text{Multiply all through by 3, we have} \\ 20=\text{ 3}\times25a+3\times5b+3\times c \\ 20=\text{ 75a+15b+3c }\ldots\ldots\ldots\ldots\ldots equation\text{ 2} \end{gathered}[/tex][tex]\begin{gathered} y_3=ax^2_3+bx_3+\text{ c} \\ 5_{}=a(6)^2+\text{ b(6)+c} \\ 5=\text{ 36a+ 6b+c }\ldots\ldots\ldots\ldots\ldots\ldots\ldots equation\text{ 3} \end{gathered}[/tex]
Combining the three equations together
[tex]\begin{gathered} 9a+3b+c=8\ldots\ldots\ldots\text{.equation 1} \\ 75a\text{ +15b+3c=20}\ldots\ldots\ldots equation\text{ 2} \\ 36a+6b+c=5\ldots\ldots\ldots\text{.equation 3} \end{gathered}[/tex]
Hence, the three equations above are the equations of the parabola with three unknowns a,b and c.
