Question

what is the equation of a parabola with focus at F (2,-2) and directrix at y=4

Answer 1

@jim_thompson5910 Please help!

Answer 2

@dan815 help please!

Answer 3

I need to find the equation

Answer 4

The y coordinate of the focus is y = -2

how far is y = -2 from y = 4 ?

Answer 5

6?

Answer 6

correct

Answer 7

y = 1/12 x^2 -1/3x-2/3?

Answer 8

cut this in half to get 6/2 = 3

the distance from the focus to the vertex is 3 units, so p = 3 (p is the focal distance)

Answer 9

what is the midpoint of (2,-2) and (2,4) ?

Answer 10

4?

Answer 11

plot the two points (2,-2) and (2,4)

which point is directly in the middle of the two of them?

Answer 12

2,1?

Answer 13

that's the vertex

(h,k) = (2,1)

Answer 14

we will plug p = 3, h = 2, k = 1 into the "vertical parabola" formula shown here

http://www.mathwords.com/f/focus_parabola.htm

Answer 15

y = -1/12 x^2 -1/3x-2/3?

Answer 16

4p(y-k) = (x-h)^2

4*3(y-1) = (x-2)^2

12(y-1) = (x-2)^2

y-1 = (1/12)*(x-2)^2

y = (1/12)*(x-2)^2+1

Answer 17

one final adjustment

the parabola opens downward, so the leading coefficient 1/12 needs to be negative

so the actual equation is y = (-1/12)*(x-2)^2+1

Answer 18

hopefully you can see how I isolated y

Answer 19

thanks so it is y = -1/12 x^2 +1/3x+2/3?

Answer 20

yes it is once you expand out y = (-1/12)*(x-2)^2+1

Answer 21

thanks can you help with a few more?

Answer 22

I'll do 2 more

Answer 23

a parabola has a focus at F(-2,4)and a directrix at y=0. what is the location of the vertex?

Answer 24

what formula should be used to derive the equation of a parabola given its focus and directrix

Answer 25

what is the general equation for a parabola with its vertex at the origin?

Answer 26

we will use 4p(y-k) = (x-h)^2 again

p = focal distance
(h,k) = vertex

Answer 27

to find the focal distance, you find the distance from directrix to focus, then you cut that distance in half

Answer 28

so what is the focal distance?

Answer 29

-2?

Answer 30

y = 0 is the directrix

y = 4 is the y coordinate of the focus

what is the change in y?

Answer 31

2?

Answer 32

(-1,4)

Answer 33

y = 0 to y = 4 is a change of 4

Answer 34

4-0 = 4

Answer 35

cut this distance in half

4/2 = 2

so p = 2