Determining the Orientation of the Parabola
Any time we are given only one or more of the following:
- Axis of Symmetry
and are asked to find one or more of the others, we must first determine the orientation of the parabola, if we are not already told what it is (as we usually are not).
In the cases when we are given either the axis of symmetry or directrix, we will be able to tell right away the parabola orientation because vertically oriented parabolas have vertical axes of symmetry and horizontal directrices. Therefore we will be given equations for a vertical line (“x=[a constant]”) in the case of the axis of symmetry, and an equation of a horizontal line (“y=[a constant]”) in the case of the directrix. When the parabola is horizontally oriented, we will have been given horizontal axes of symmetry (“y=[a constant]”) and/or vertical directrices (“x=[a constant]”).
In the case that we are given either only the focus or only the vertex of a parabola, we will not be able to determine the parabola’s orientation — we need both, or at least some other piece of information in addition only one of these. When we are given both, we can quickly tell the orientation of the parabola by the repeated value in either the x-coordinates of the focus and vertex or the y-coordinates of the focus and vertex.
If the repeated value occurs in the x-coordinates of the focus and vertex, this means the axis of symmetry is vertical, and the parabola is vertically oriented. (Try to visualize this. See Figure A, Example 1.)
If the repeated value occurs in the y-coordinates of the focus and vertex, the axis of symmetry is horizontal, and the parabola is horizontally oriented. (Try to visualize this. See Figure A, Example 2.)
So for example, if we are given that a parabola’s focus is F(3,-2) and vertex is V(-3,-2), then the repeated value occurs in the y-coordinates; the axis of symmetry is therefore horizontal, and the parabola is likewise oriented horizontally.
Deriving the Directrix Equation from the Vertex and Focus Coordinates
Once we have found the orientation of the parabola, we can find the directrix in a couple of ways. Possibly the most straight forward way is to use the midpoint formula, given that the vertex is midpoint between the focus and a collinear point on the directrix. See Figure B. So for instance, if we have V(3,1) and F(3,4), we find d in the point D(3,d):
(d+4)/2 = 1; d+4=2; and d=-2.
Thus, D(3,-2) is a point lying on the directrix. Because we determined the parabola was vertically oriented, the directrix is the horizontal line passing through D(3,-2), which is y=-2. Here is another example:
Example: If the vertex of a parabola is V(3,1) and its focus is F(1,1), find the directrix of the parabola.
Solution: We are asked to find the directrix given the vertex and focus of the parabola, so first we determine the orientation of the parabola — which we find through identifying the repeated component value present in the vertex and focus; this being y=1, the y-coordinate, we know the axis of symmetry is horizontal, the parabola is horizontally oriented, and the directrix is vertical.
Next, we find the directrix by using the fact that the vertex V(3,1) is the midpoint between the focus F(1,1) and a collinear point on the directrix, D(d,1) Thus:
3=(1+d)/2; 6=1+d; and d=5.
Because we determined the directrix is vertical, the point D(5,1) on the directrix gives us the equation of the directrix x=5.
Have something to add to this walkthrough? Share it in the comments.
(1) Explain how we know that F(-1,2) and V(-1,4) are from a vertically oriented parabola.
(2) How do we know that the parabola with axis of symmetry y=-3/4 is horizontally oriented?
(3) Why do we know that the directrix x=9 is from a horizontally oriented parabola?
(4) Show that if the focus of a parabola is at F(2,3) and its directrix is at y=-3, the vertex is at V(2,0).
(5) Show that if F(2,1/2) and V(3,1/2) the directrix is given by x=4.
(6) Show that if F(-5,1) and V(-5,10), the directrix is given by y=19.