Question

Identify the vertex, focus, and directrix of the parabola with the equation

Answer 1

I know first step is complete the square so it makes the equation vertex form but idk how to complete the square

Answer 2

\[x^2+kx+(\frac{k}{2})^2=(x+\frac{k}{2})^2\]

Answer 3

\[x^2+6x+(?)^2=(x+?)^2\]
what goes in place of the question marks

Answer 4

6?

Answer 5

k is 6
so
k/2 is ?

Answer 6

6/2

Answer 7

3

Answer 8

yes!

Answer 9

\[x^2+6x+(3)^2=(x+3)^2\]

Answer 10

So back to your equation above

Answer 11

\[x^2-6x+(3)^2-(3)^2-8y+49=0\]
we can rewrite the first three terms on the left side
so our equation becomes
\[(x-3)^2-3^2-8y+49=0\]

Answer 12

you can simplify some of what you have on the left hand side

Answer 13

-3^2+49?

Answer 14

yes you can simplify that part

Answer 15

40

Answer 16

cool so you have this so far:
\[(x-3)^2-8y+40=0\]

Answer 17

yes

Answer 18

any thoughts where to go now

Answer 19

isolate y?

Answer 20

Well there are two forms some prefer you put it in:
This form:
\[(x-h)^2=4p(y-k)\]
or this form:
\[y=a(x-h)^2+k\]

Answer 21

so it seems like you are wanting to put in that second form

Answer 22

yeah

Answer 23

ok so yeah
do you want to see if you can solve for y yourself
or do you need some hints

Answer 24

is that similar?

Answer 25

yep a=1/4c

Answer 26

just learned something new

Answer 27

well y=1/4c(x-h)^2+k
would be the same as
y=a(x-h)^2+k
if a=1/4c right?

Answer 28

yeah

Answer 29

:)

Answer 30

beautiful!
\[y=\frac{1}{8} \cdot (x-3)^2+5 \\ \text{ compare \it \to } \\ y=\frac{1}{4 c}(x-h)^2+k \\ 4c=8 \text{ and } (h,k)=(3,5)\]
can you solve for c

Answer 31

would c be 2

Answer 32

yep
the vertex is (h,k)=(3,5)
the focus is (h,k+c)=(3,5+2)

Answer 33

wow I understand it a lot better

Answer 34

cool stuff

Answer 35

if it was a sideways parabola
you would have to do focus as (h+c,k)

Answer 36

how would I know if its vertical or horizontal?

Answer 37

y=1/(4c)(x-h)^2+k is vertical parabola
x=1/(4c)(y-k)^2+h is horizontal parabola

Answer 38

i see

Answer 39

thank you! @freckles

Answer 40

np