Question

More College Algebra! yay...
I'm still having trouble with finding the equation of a parabola when the vertex is not at the origin.
Example problem below

Answer 1

consider <y> = a <x>^2

Answer 2

<x>,<y> represent the 'modified origin'

Answer 3

but hurry up with the example :)

Answer 4

Question: Find an equation of the parabola with vertex at (-2, 3) and focus at (0, 3).

Solution Given: The vertex (-2, 3) and focus (0, 3) both lie on the horizontal like y = 3 (the axis of symmetry). The distance \(\it a\) from the vertex to the focus is a = 2. Also, because the focus lies to the right of the vertex, the parabola opens to the right. Consequently, the form of the equation is \((y-k)^2 = 4a(x-h)\)

Answer 5

Sorry, was having some internet issues :)

Answer 6

How did they get that a = 2?

Answer 7

do you know how to move a point?

Answer 8

I don't think so

Answer 9

add something to it ...

Answer 10

Do I just need to make the -2 in the vertex 0 to find \(\it a\)?

Answer 11

we are given 2 points

vertex at (-2, 3) and focus at (0, 3)

spose we move them so the focus is at the origin (0,0) ... we would translate all points as (x+0,y-3)

vertex at (-2+0, 3-3) and focus at (0+0, 3-3)
vertex at (-2,0) and focus at (0,0)

how far is the vertex -2 from the focus 0?

Answer 12

or zeroing out the vertex is fine too ... the distance would remain the same

Answer 13

2, so however far away the x-coordinate of the vertex is from the x-coordinate of the focus would be a?

Answer 14

depends on direction, but yes

Answer 15

Okay, I think I can do this now :)

Answer 16

if we zero out one point, the other will tell us how far they are from one another

Answer 17

v=(2,5) f=(2,-2)

whats the distance and direction between them?

Answer 18

zero one of them out (move it to the origin)

v=(2, 5) f=(2,-2)
-2-5 -2 -5
-----------------
v=(0,0) f=(0,-7)

-7 tells us the focus is under the vertex, and is 7 units away

Answer 19

ooohhh, that seems really simple, I was looking at the formulas for the equations and going "what am I supposed to use???"

Answer 20

Random problem from the book:
v (4, -2) f (6, -2)

v = (4, -2) f = (6, -2)
-4, +2 -4, +2
---------------------------
v = (0, 0) f = (2, 0)

So the focus is 2 units from the vertex? And 2 would be a?

Answer 21

very good, and the parabola opens up ... the focus is 2 above the vertex

Answer 22

Yay! :)

Answer 23

err, 2 to the right ... opens to the right

Answer 24

haha that's what I was about to say :P

Answer 25

Thank you :3

Answer 26

:) the voices in my head like to tell me when im wrong

Answer 27

goodluck

Answer 28

Well they must not speak very often since you're so smart :)