CPhill

The focus of a parabola is (0,−2) . The directrix of the parabola is the line y=−3

The y coordinate of the focus is > the y value of the directrix....thus this opens upward

Vertex    ( 0, (-3 +   -2) /2 )   =   ( 0,  -5/2)   =  (h, k)

"p"  =   l  -5/2 - - 2 l  =  l  - 5/2 + 2 l  =   l -5/2 + 4/2 l  =  l -1/2  l = 1/2

The form is

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

4(1/2) ( y -  - 5/2) = (x - 0)^2

2(y + 5/2) = x^2        multiply both sides by  1/2

y + 5/2  =  (1/2)x^2

y = (1/2)x^2  - 5/2

24 +  7  =   31

The directrix of a parabola is the line y=7 . The focus of the parabola is (2,3)

Remember jenny....if the y coordinate of the focus is less than the directrix....the parabola turns downward

Vertex     ( 2,  (3+7)/2)  = (2, 10/2) = ( 2, 5)  = (h, k)

"p"  =  l  5 - 3 l = l 2 l = 2

We have the form

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

-4(2) ( y - 5) = ( x - 2)^2

-8 ( y - 5) = ( x - 2)^2        divide both sides by -1/8

y - 5 =  (-1/8)( x - 2)^2

y = -(1/8) ( x - 2)^2 + 5

Second one.....the dirctrix is below the focus....this turns upward

Vertex    ( -2 ,  (-6 - 8)/2 )  = (-6, -14/2)  = (-2, -7)    = (h, k)

"p" =  l -7 - - 6 l =   l -1 l   =   1

We have the form

4p(y - h ) =  (x - k)^2

4(1) ( y - - 7) =  ( x - - 2)^2

4 (y + 7) = ( x + 2)^2            multiply both sides by 1/4

y + 7 =  (1/4) ( x + 2)^2       subtract 7 from both sides

y =  (1/4)(x + 2)^2 - 7

Looks good  !!!

OK....

The y coordinate of the focus is below the directrix.....thus....this turns downward

Find the vertex =  ( -3 , (-5 + 2)/2 ) =  (-3, -3/2)

Find p  ..... l -3/2 -  - 5 l  = l  5 - 3/2 l  = [ 10/2 - 3/2 l  =  7/2

So.....we have the form

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

-4(7/2)(y - -3/2) = (x - - 3)^2      simplify

-14 ( y + 3/2) = (x + 3)^2             mutiply both sides by -1/14

y + 3/2 = -(1/14) ( x + 3)^2        subtract 3/2 from both sides

y =  -(1/14)(x + 3)^2 - 3/2

Looks like you are correct, jenny  !!!

Oops....sorry Jenny...fixed it...

Her's a checlklst to help you with these  [ this just applies to parabolas opening upward/downward]

1. Look at the y coordinate of the focus.....fi its numerical value is greater than that of the directrix, the parabola opens upward......if its value is less....the parabola opens downward.....this latter situation also implies that we will have a negative sign involved in our equation

Since 1 < 5....this opens downward

2. Find the vertex.....for parabolas opening upward/downward.....the x coordinate of the focus = the x coordinate of the  vertex......so.....we only need to determine the y coordinate of the vertex....to do this.....add the diectrix value and the y coordinate of the focus  and divide this sum by 2......so we have  [ 5 + 1]/ 2 = 6/2 = 3

So....the vertex =  ( 2, 3)  = (h, k)

3.  Find "p".....this is the distance between the focus and vertex.....again.....we are just concerned with the y coordinates of the vertex and focus.....to find p, take the absolute value of their difference.....so we have

l 3 - 1 l =  l 2 l   =  2

So......we have the form

4p (y - k) = -(x - h)^2              note the "- "   ....now....  fill in what we know

4 (2)   (y - 3) = -(x - 2)^2

8 ( y - 3) = - (x  -2)^2          multiply both sides by  1/8

(y - 3) = -(1/8) ( x - 2)^2      add 3 to both sides

y = -(1/8)(x - 2)^2 + 3

Unit rate = 25 calories per hour

Since the y coordinate of the focus is -4  and the directrix is y = -6  .....the focus lies above the directrix .....so....this parabola opens upward

The vertex lies 1/2 of the way between the focus and directrix......to find this    we have ( -2,  [ -4 +  - 6] /2 ) =

(-2, -10/2 )   =   (-2, -5)  = (h, k)

"p' is the distance between the vertex and focus  =  l -5 - (-4) l = l -5 + 4 l =  l -1 l  = 1

So....we have the form

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

4(1) [ y - - 5 ] =  (x - - 2)^2

4 ( y + 5)  =  (  x + 2)^2           multiply both sides by 1/4

y + 5 = (1/4)(x + 2)^2           subtract 5 from both sides

y =  (1/4)(x + 2)^2  - 5