+1993 The directrix of a parabola is the line y=5 . The focus of the parabola is (2,1)
.
What is the equation of the parabola?
y=1/8(x−2)^2 +3
y=1/8(x−2)^2 −3
y=−1/8(x−2)^2 +3
y=−1/8(x−2)^2 −3
Question 2
The focus of a parabola is (0,−2) . The directrix of the parabola is the line y=−1
.
What is the equation of the parabola?
y=−1/2x^2−3/2
y=1/4x^2−2
y=1/2x^2−3/2
y=−1/4x^2+2
+36916 These are VERY similar to the question cphill answered a few minutes ago ....try his methodology on these before you ask for answers....I think you should be able to do that....give it a try to see what you can find ! 
+36916 I get the THIRD option.....see if you can figure out where you steered off-course.....
+1993 Oh wait no , it would be the third .I thought A was -1/8 , it would be the third option since it is -1;8
+36916 I came up with A...... look again for where you might have gotten off-course
+1993 I dont see how it wouldve been negatitive. I am sorry I guess I am not quite getting it. because -1/2 + 1= 1/2
+36916 To plaigerize cphill's answer from earlier:
Since the y coordinate of the focus is -2 and the directrix is y = -1 .....the focus lies below the directrix ..
...so....this parabola opens downward
The vertex lies 1/2 of the way between the focus and directrix......to find this we have (0, [ -2 + - 1] /2 ) =
(0, -3/2 ) = (h, k)
"p' is the distance between the vertex and focus = l -3/2 - (-2) l = 1/2
So....we have the form
4(-p)[ y - k ] = (x - h)^2
the sign of 'p' tells us which way the parabola faces......we determined that this is a DOWNWARD facing parabola so - p goes in the equation
4(-1/2) [ y - - 3/2)] = (x - 0)^2
-2 (y+3/2)= x^2
y+3/2 = -1/2 x^2
y= -1/2 x^2 - 3/2
Maybe it would be clearer at THIS step:
"p' is the distance between the vertex and focus = l -3/2 - (-2) l = 1/2 " to say p = -1/2 becuase it is a downward opening parabola
+128474 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
