Find the Vertex Focus Directrix of the parabola y12x2x2 15 P

Find the Vertex, Focus, Directrix of the parabola y+12x-2x² =15

Please shop all steps. Thank you.

We have given parabola y+12x-2x²=15

Writing the equation of the parabola in vertex form

y=2x²-12x+15

y-15=2x²-12x

y-15=2(x²-6x)

y-15+18=2(x²-6x)+18 since adding 18 both side of the equation

y+3=2(x²-6x+9)

y+3=2(x-3)²

(1/2)*(y+3)=(x-3)²----(1)

We know the equation of a vertical parabola with vertex (h, k) is:
4p(y - k) = (x - h)² ---(2)

By comparing both the equations 1 and 2 we get

4p=1/2 implies p=1/8

k=-3, h=3

This particular parabola is vertical and opens upwards.
The vertex (h, k) = (3, -3)

Since the parabola is vertical the line of symmetry is also vertical and passes thru the vertex. Its equation is

x=h

x=3

The focus is also on the line of symmetry at a directed distance of p from the vertex. Its coordinates are

(h, k + p) =(3,-3+1/8) =(3,-23/8)

The directrix is a horizonal line at a directed distance of -p from the vertex. Its equation is

y = k - p =-3-1/8=-25/8