2. Lesson 2
Find the vertex, focus, and
directrix, and draw a graph of a
parabola, given its equation.
3. As you may or may not know, a parabola is the locus of points in a plane
equidistant from a fixed line and a fixed point on the plane. We know this
fixed line to be the directrix and the fixed point to be the focus.
To see an animated picture of the above description, you need to have
Geometer's SketchPad for either Macintosh or PC loaded on your
computer. If you have GSP, click here. To download the script of this
picture so you can create it yourself, click here.
Let's now take a look at a parabola that has all of the elements that we
will be looking for:
the vertex
the focus
the directrix
4. From the above picture, I have labeled three items that we need to pay close
attention to. The highest point of the parabola is the vertex (and the
maximum). The plus sign that is directly under the vertex is the focus. The
green line that is above the parabola (and directly above the vertex) is the
directrix. You may be able to see, by eyeballing, that the distance from the
focus to the vertex is the same distance as the vertex to the directrix.
5. Example
Let's take a look at the equation (y + 3)2
= 12 (x - 1).
We can easily identify that the parabola is opening left or right. Since the
coefficient in front of the x term is positive, we can say that the parabola will
open to the right. The focus will be to the right of the vertex, and the directrix
will be a vertical line that is the same distance to the left of the vertex that
the focus is to the right.
The vertex is (1, -3), the axis of symmetry (now horizontal) is y = -3, and we
don't recognize "max's and min's" for parabolas that open left or right.
The term in front of the x term is a 12. This is what our 4p term is equal to.
So 4p = 12, making p = 3. So we now need to move the focus 3 units right
from the the origin. This means that the coordinate for the focus is (4, -3),
and the directrix will be a vertical line going through the point (-2, -3).
This problem is illustrated in the picture on the next page.
6. Our green line represents the directrix and the plus sign represents the
aforementioned focus.
7. Our green line represents the directrix and the plus sign represents the
aforementioned focus.
