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Question 1103734: I am trying to help a student, but all of the online resources have x and y on opposite sides of the equation. Why is that? How do we change it?
Here is what he has been asked to solve: Identify the vertex, axis of symmetry, focus and directrix of the given parabola.
x = 1/3(y+2)^2+3
Found 2 solutions by richwmiller, greenestamps:
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! The equation for a horizontal (sideways) parabola has the form x = a(y-k)² + h
Where the vertex is (h, k).
The axis of symmetry is y = k.
If a<0, the parabola opens to the left.
If a>0, the parabola opens to the right.
The focus is (h+1/(4a), k).
The directrix is (h-1/(4a), k).
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
If a parabola is in the "usual" orientation -- opening up or down, with a vertical axis of symmetry, then the equation can be written in the form
A parabola of this type opens up if p is positive or down if p is negative.
If a parabola is "lying on its side" -- opening right or left, with a horizontal axis of symmetry, then the equation can be written in the form
A parabola of this type opens to the right if p is positive or tothe left if p is negative.
I think that is the difference you are talking about ("...x and y on opposite sides").
The equation of your parabola is  , so the parabola is lying on its side, with a horizontal axis of symmetry.
From the standard form, we see that the vertex (h,k) is (3,-2). And (1/4p) is equal to 1/3, so 4p=3, so p = 4/3.
The parameter p in either of these forms for the equation of a parabola is the distance from the vertex to the focus and from the vertex to the directrix.
With p being positive for this parabola, the graph opens to the right; so the focus is 4/3 units to the right of the vertex, and the directrix is 4/3 units to the left of the vertex.
So now we have everything we need to answer the questions:
vertex: (3,-2)
axis of symmetry: y = -2 [the horizontal line passing through the vertex]
focus: (13/3,-2) [4/3 units to the right of the vertex]
directrix: x = 5/3 [4/3 units to the left of the vertex]
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