Question

Anyone know an easy way to do this?
If a parabola has its directrix at y = 14 and its focal point at (-4, 4), then what are the coordinates of the parabola's vertex?

Answer 1

yes

Answer 2

How!?!?

Answer 3

plug and chug into the general equation

Answer 4

details ...please >.< general equation?

Answer 5

(x - h)^2 = 4p(y-k) is the general equation , where p is the distance from vertex to directrix

Answer 6

just sketch and you'll see

Answer 7

but how do you plug in p if you don't know distance from the vertex to directrix?

Answer 8

can you draw a quick sketch?

Answer 9

sure :o

Answer 10

I'll post my solution after you two are done as an alternative method.

Answer 11

So whats the first step ? sorry i learn new things everyday. I'm barely learning this today so bare with me..

Answer 12

draw the porabola for me

Answer 13

draw it ._. how!?

Answer 14

click on "Draw" button, then click "copy previous drawing" , then draw

Answer 15

i tried but it's blank?

Answer 16

?!?!

Answer 17

Okay, I'm just going to post my solution now. Give me a minute.

Answer 18

@sourwing :o

Answer 19

Okay, so basically, you are given two points \((-4, 4)\) and \((x, 14)\). Plug them in to this formula:

\((x - x_1)^2 + (y - y_1)^2 = ((x - x_2)^2 + (y - y_2)^2\)

\((x - (-4))^2 + (y - 4)^2 = (x - x)^2 + (y - 14)^2\)

\((x + 4)^2 + (y - 4)^2 = (y - 14)^2\)

Expand \((y - 4)^2\) and \((y - 14)^2\)

\((x + 4)^2 + y^2 - 8y + 16 = y^2 - 28y + 196\)

\(y^2\) cancels on both sides leaving just:

\((x + 4)^2 - 8y + 16 = y^2 - 28y + 196\)
Add 8y to both sides; Subtract 196 from both sides:

\((x + 4)^2 + 16 - 196 = -28y + 8y\)
\((x + 4)^2 - 180 = -20y\)

Divide both sides by -20
\(-\dfrac{(x + 4)^2}{20} + 9 = y\)

or

y = \(-\dfrac{1}{20}(x + 4)^2 + 9\)

Notice that the parabola is of the form \(y = a(x - h)^2 + k\) where \((h, k)\) is the vertex.
Thus, the coordinates of the vertex of the parabola is \((-4, 9)\).

Answer 20

Okay this looks a little easier for me.. I barely learned how to graph parabolas today so let me try...

Answer 21

That's because it's a completely algebraic method. No graphing necessary.

Answer 22

thanks for your time^.^ im gonna try it.. so starting off, is it (x-(-4)^2 + (y-4)^2 = ....

Answer 23

Yes, (x - (-4))^2 + (y - 4)^2 = ...
keep going

Answer 24

yay ok.....

Answer 25

ok but see.. this is where i get stuck.. what is (x-x2)^2 supposed to be?

Answer 26

Notice that the second point which represents the directix is \((x_2, y_2) = (x , 14)\). For \((x - x_2)^2\) You replace \(x_2\) with \(x\) to get \((x - x)^2\)

Answer 27

okay got it. so , so far so good right ? just keep going.. (x-x2)^2 then becomes (x-x)^2 ?

Answer 28

You're probably wondering why we're writing the directrix as a point instead of a line. It's because the point \((x , 14)\) is the same as the line y = 14 because no matter what the value of x, y will always be 14.

Answer 29

nope i got it.... kinda weird for a second but then i figured it was the same as y=14 lol

Answer 30

Which makes it very convenient to use the distance formula for this.

Answer 31

yes!! im glad you gave me this formula

Answer 32

You can use it no matter what the direction of the parabola is.

Answer 33

I made a slight mistake in my steps above. I said \(y^2\) cancels on both sides at a certain point, which it does, but I cancelled on one side and forgot to cancel on the other side. Please ignore that typo. The rest of it is corect as written.

Answer 34

ok! ... so i got (x+4)^2+(y-4)^2=(x-x)^2+(y-14)^2 ... correct so far?

Answer 35

Yes correct. Notice that \((x - x)^2 = (0)^2 = 0\) since \(x - x = 0\)

Answer 36

oh thanks....i was just gonna say how do you solve that...

Answer 37

I got (2x+16)+(2y+16)=(2y+196) ?

Answer 38

Omg you wrote the same thing ._. i didn't know you were actually writing the same problem

Answer 39

I don't know how you got that. There's no way to get that from your previous step.

Answer 40

I'm confusing myself

Answer 41

I hate math i give up :(

Answer 42

\((y - 4)^2 = (y - 4)(y - 4) \)

\(\\= y(y - 4) - 4(y - 4)
\\= y^2 - 4y - 4y + 16
\\= y^2 - 8y + 16\)

Answer 43

Never give up

Answer 44

Now what >.<

Answer 45

Just review the steps I posted (minus the typo)

Answer 46

I'm going to study your steps allllll night

Answer 47

thanks so much for thoroughly explaining. i appreciate it soooooo much!

Answer 48

You're welcome