Quadratic Relations and Conic Sections #2

Question

The equation of a parabola is  . Identify the vertex, focus, and directrix of the parabola.

Answer 1

Use the form 4p(y)= a(x-h)^2 +k where (h,k) are the coordinates of the vertex and 4p is the focal distance

So for 12y= (x-1)^2 -48. 12= 4p p=3 which is the distance from the focus to the vertex

Dividing both sides of the equation by 12 gives us

Y= 1/12 (x-1)^2 - (48/12)

y = 1/12(x-1)^2 -4 so the vertex is at (1,-4) The focus would be 3 units from the vertex at (1,-1)

I added 3 to the y value of the vertex

The directrix will be the horizontal line 3 units below the vertex which would be y = -7