PART A
The equation of the parabola in vertex form is given by the formula,
[tex]y - k = a {(x - h)}^{2} [/tex]
where
[tex](h,k)=(1,-2)[/tex]
is the vertex of the parabola.
We substitute these values to obtain,
[tex]y + 2 = a {(x - 1)}^{2} [/tex]
The point, (3,6) lies on the parabola.
It must therefore satisfy its equation.
[tex]6 + 2 = a {(3 - 1)}^{2} [/tex]
[tex]8= a {(2)}^{2} [/tex]
[tex]8=4a[/tex]
[tex]a = 2[/tex]
Hence the equation of the parabola in vertex form is
[tex]y + 2 = 2 {(x - 1)}^{2} [/tex]
PART B
To obtain the equation of the parabola in standard form, we expand the vertex form of the equation.
[tex]y + 2 = 2{(x - 1)}^{2} [/tex]
This implies that
[tex]y + 2 = 2(x - 1)(x - 1)[/tex]
We expand to obtain,
[tex]y + 2 = 2( {x}^{2} - 2x + 1)[/tex]
This will give us,
[tex]y + 2 = 2 {x}^{2} - 4x + 2[/tex]
[tex]y = {x}^{2} - 4x[/tex]
This equation is now in the form,
[tex]y = a {x}^{2} + bx + c[/tex]
where
[tex]a=1,b=-4,c=0[/tex]
This is the standard form
