Question

Find the vertex, focus, and directrix of the parabola
y^2+6y+4x+1=0

Answer 1

so far I have y^2 + 6y = -4x-1 = 4(-x+1)

Answer 2

but cannot factor y^2+6y.... SO I might need to go by a diff approach.

Answer 3

okay complete the square for the left side.
by adding and subtracting 9 on left.

Answer 4

Ha funny because I just entered that into wolfram to see if it would factor :P.

(y^2 + 6y +9) -9 = ....
Y+3)^2 - 9 = ....

Answer 5

sorry openstudy just crashed me :(.

Answer 6

okay now you have
(y+3)^2=-4x+8
(+3)^2=-4(x-2)
compare with standard form
(y-b)^2=4c(x-a)
where (a,b) is vertex and c is the distance between focus and vertex.
hope you can do this now.

Answer 7

Thanks, I can finish this now :D

Answer 8

you 're welcome :)

Answer 9

Very useful that formula... Can you tell me the ones for ellipse, and hyperbola too please?

Answer 10

Oh and how do we find x for the focus? c?

Answer 11

in this case x = 1, but in another case c = 1/4 and the x was 0.

Answer 12

yeah sure.
for ellipse
\[\Large \frac{(x-a)^2}{h^2}+\frac{(y-b)^2}{k^2}=1\]
it is centered at (a,b) and h,k are length of major and minor axis.
for hyperbola
\[\Large \frac{(x-a)^2}{h^2}-\frac{(y-b)^2}{k^2}=1\]
(a,b ) is center and h,k are major and minor axis respectively.

Answer 13

x for the focus is found by
first find distance c
-4c=4
c=-1 |c|=1
since vertex is (2,-3) subtract c from the x coordinate
(1,-3) is focus.

Answer 14

hmmm.. ok gotta read over this thanks :).

Answer 15

oic that makes sense thanks :D.

Answer 16

and how do we find the directrix?

Answer 17

I got it as x = 3, it seems like it's just the value of y.

Answer 18

so 2--1 = 3? :)

Answer 19

directrix is just at the same distance from vertex as focus is but is at the right side of vertex so since C=1 so

x=1 is directrix

Answer 20

http://www.wolframalpha.com/input/?i=y%5E2%2B6y%2B4x%2B1%3D0 ??

Answer 21

from the example and what you said it seems like if your'e doing -c for the focus and c for directrix... In the example it's -1/4 so if you do +1/4 you get 0 which is correct for the focus... But if you do -1/4 - 1/4 you get -2/4 which is -1/2 directrix which is what the correct answer is SO Idk..

Answer 22

sorry the directx is x=3
because 1 should be added to x coordinate .
x=3

Answer 23

so it's x + c?

Answer 24

yes but remember c is distance so take it as positive..

Answer 25

hmm?

Answer 26

So how does it work in the other case?

Answer 27

x+c is equation of directx . c is positive.

Answer 28

y2 − x − y = 0

Answer 29

Vertex is (-1/4,1/2) Directix is x = -1/2

Answer 30

c = 1/4 if it's -1/4 + 1/4 it should be 0... which is the focus x.

Answer 31

Maybe I'm just confused haha.

Answer 32

you should be able to do this now !!!
you should look at the graph .if it opens leftward then add c to x coordinate.
if opens rightward subtract c from x coordinate.
just look at the graph dirextx should always be behind the vertex at the distance c . by looking to graph make appropriate decision.

Answer 33

oic :).

Answer 34

One more thing how do we find the vertices of a parabola?