Question 1103734
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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<br>
{{{y = (1/4p)(x-h)^2+k}}}<br>
A parabola of this type opens up if p is positive or down if p is negative.<br>
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<br>
{{{x = (1/4p)(y-k)^2+h}}}<br>
A parabola of this type opens to the right if p is positive or tothe left if p is negative.<br>
I think that is the difference you are talking about ("...x and y on opposite sides").<br>
The equation of your parabola is {{{x = (1/3)(y+2)^2+3}}}, so the parabola is lying on its side, with a horizontal axis of symmetry.<br>
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.<br>
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.<br>
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.<br>
So now we have everything we need to answer the questions:<br>
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]<br>
{{{graph(400,200,-2,10,-10,5,sqrt(3(x-3))-2,-sqrt(3(x-3))-2)}}}