Question

Find the vertex, focus, directrix, and axis of the parabola: x^2 -4x+3y-5=0
I understand how to find everything the question is asking, but not when the parabola is in that form. How do I get that particular parabola to be in standard form?

Answer 1

complete the square (hope i don't steer you wrong this time)
btw i saw my mistake on the last one, it was \(20+40=60\) i believe
do you know how to complete the square?

Answer 2

\[x^2 -4x+3y-5=0 \] complete the square on the \(x\) term to get
\[(x-2)^2+3y-5-4=0\] or
\[(x-2)^2+3y-9=0\]

Answer 3

so how do we complete the square?

Answer 4

you have
\[x^2-4x\] lets ignore everything else

Answer 5

okay so far I'm following

Answer 6

half of \(-4\) is \(-2\) so you write
\[x^2-4x=(x-2)^2-4\]

Answer 7

the reason being if you square \(x-2\) you get \[x^2-4x+4\]so you have added 4, now you need to subtract it off to make it equal to what you started with

Answer 8

that is how we ended up with
\[(x-2)^2+3y-9=0\]

Answer 9

now make it
\[-3y+9=(x-2)^2\]

Answer 10

factor out the \(-3\) (carefully) and get
\[-3(y-3)=(x-2)^2\] and you can read off the vertex from that one

Answer 11

you good from there? it should be in the form you want it now

Answer 12

how did you go from summing a geometric series to this? quite random

Answer 13

Yes I'm just writing everything out. Hahaha, I doing a practice test for my test tomorrow so it covers a couple sections

Answer 14

I'm*

Answer 15

i see
i hope you don't get a bouncing ball problem
if you do, remember to take the distance that it falls outside the series
also remember that
\[a+ar+ar^2+ar^3+...=\frac{a}{1-r}\]

Answer 16

in any case you should be done here \[-3(y-3)=(x-2)^2\]gives you the vertex of \((2,3)\)

Answer 17

yes I got that :)

Answer 18

then since \(4p=3\) you get \(p=\frac{3}{4}\) and the directix is \(\frac{3}{4}\) units above the vertex, focus is \(\frac{3}{4}\) units below

Answer 19

let me make sure that is right so i don't steer you wrong again

Answer 20

no, it is right
directrix \(y=3+\frac{3}{4}=\frac{15}{4}\)

Answer 21

and since \(3-\frac{3}{4}=\frac{9}{4}\) the focus is \((2,\frac{9}{4})\)

Answer 22

Hahaha I was just about to check that with you, thank you. I got them right

Answer 23

great!