Question

Find the vertex focus and directrix of the parabola y^2+3y=-x-2

Answer 1

I would put the equation into general form:
x= -y^2 -3y - 2
do you know how to complete the square, so that you can change this into vertex form?

Answer 2

No i am not sure this is where i am stuck

Answer 3

see
http://www.khanacademy.org/math/trigonometry/conics_precalc/parabolas_precalc/v/parabola-focus-and-directrix-1
for info on this.

Answer 4

can you show me what to do next? i watched the video and still confused

Answer 5

well the equation is a parabola in the form

\[(y-k)^2 = 4a(x-h) \]

where ((h, k) is the vertex and a is the focal length

so complete the square in y by adding a value to both sides of the equation

it basically looks like

Answer 6

this video is shorter and closer to your problem

http://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/ex3-completing-the-square

Answer 7

though in your problem, we switch the x and y 's
we want to change
x= -y^2 -3y - 2
to the form
x = a(y-h)^2 + k
(h,k) will be the vertex

Answer 8

-y(y+3)-2=x

Answer 9

?

Answer 10

y^2 + 3y = -x - 2
complete the square. Take coefficient of y term, divide by 2: 3/2
3/2 will go inside the square and you need to add (3/2)^2 to the right side:
(y + 3/2)^2 = -x - 2 + (3/2)^2
= -x - 2 + 9/4
= -x - 8/4 + 9/4
= -x + 1/4
(y + 3/2)^2 - 1/4 = -x
multiply both sides by -1
-(y + 3/2)^2 + 1/4 = x
x = -(y + 3/2)^2 + 1/4
compare it to
x = a(y - k)^2 + h

a = -1; h = 1/4; k = -3/2

Vertex at (1/4, -3/2)

If p is the distance from the vertex to the focus then, p = |1/4a| = 1/4

Since a is negative, this is a parabola that opens in the negative x direction.
The focus will be to the left of the vertex. Subtract 1/4 from the x value of the vertex.
Focus will be at (1/4 - 1/4, -3/2) = (0, -3/2)

The directrix will be at distance p to the right of the vertex.
(remember the parabola lways opens away from the directrix. It will never intersect with the directrix).
Add 1/4 to the x value of the vertex: 1/4 + 1/4 = 1/2
The equation of the directrix is x = 1/2 (a vertical line at x = 1/2)

Answers:
Vertex at (1/4, -3/2)
Focus at (0, -3/2)
Equation of directrix: x = 1/2

Answer 11

so would this parabola open to the left?

Answer 12

yes.