Question

(x-1)^2=12(y-1)
Find vertex, focus, and directrix.

Answer 1

can you sketch it?

Answer 2

Sure

Answer 3

ok, where is the vertex?

Answer 4

I believe it is at (1,1)

Answer 5

good. how far away is vertex from the focus?

Answer 6

Sadly, I don't know where the focus is.

Answer 7

what is the standard equation for this parabola?

Answer 8

\[(x-1)^2=12(y-1)\]

Answer 9

very good.
So you have the case where
(x - h)^2 = 4p(y-k), and we already found that (h,k) = (1,1) yes?
Can you tell me what p stands for?

Answer 10

focus?

Answer 11

no, it's not. But we'll use it to find the focus. Remember focus is a point where as p is just a number

p stands for the *distance* between the vertex and the focus

Answer 12

any ideas how to find p?

Answer 13

Divide 4p by 12? which = 1/3

Answer 14

no, but you're on the right track. Look at the formula and the given equation.
(x - k)^2 = 4p (y-k)
(x - 1)^2 = 12 (y-1)

what have to be the same?

Answer 15

k has to be the same

Answer 16

yeah but already found (h,k) to be (1,1). What else have to be the same?

Answer 17

4p and 12? or x and y?

Answer 18

x and y are just variables. They change. So what does that tell you?

Answer 19

that 4p and 12 have to be the same.

Answer 20

which means p must be?

Answer 21

3

Answer 22

how did you know?

Answer 23

12 divided by 4

Answer 24

4p and 12 must be the same, so 4p = 12
divide by 4, gives p = 3. Very good

so what does 3 stand for again?

Answer 25

the distance between the vertex and focus

Answer 26

very good. But how do you know if it's a vertical distance or horizontal distance? or maybe a slant distance?

Answer 27

I'm sorry, I do not know.

Answer 28

Look at the graph. Where do you think the focus is at?

Answer 29

(0,1)?

Answer 30

Nope.

Answer 31

what do you think that dotted line is?

Answer 32

axis of symmetry?

Answer 33

very good. The focus will always lies on the line of symmetry. But where though?

Answer 34

but first, where do you think that line is at?