Question

What are the vertex, focus and directrix of `y=-15+12x-2x^2`?

Answer 1

`(3,3),(3,23/8),y=25/8`

Explanation:

`"since the equation has an "x^2" term, this is a"`
`"vertically opening parabola"`

`"the equation of a vertically opening parabola is"`

`•color(white)(x)(x-h)^2=4a(y-k)`

`"where "(h,k)" are the coordinates of the vertex and a"`
`"is the distance from the vertex to the focus and directrix"`

`"if "a>0" then opens upwards"`

`"if "a< 0" then opens downwards"`

`"to obtain this form "color(blue)"complete the square"`

`y=-2(x^2-6x+15/2)`

`color(white)(y)=-2(x^2+2(-3)x+9-9+15/2)`

`color(white)(y)=-2(x-3)^2+3`

`(x-3)^2=-1/2(y-3)`

`4a=-1/2rArra=-1/8" parabola opens down"`

`"vertex "=(3,3)`

`"focus "=(h,a+k)=(3,23/8)`

`"directrix is "y=-a+k=1/8+3=25/8`
graph{(y+2x^2-12x+15)(y-0.001x-25/8)=0 [-10, 10, -5, 5]}