PARABOLAS

Lesson 2

Find the vertex, focus, and directrix, and draw a graph of a parabola, given its equation.

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

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.

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.

Our green line represents the directrix and the plus sign represents the aforementioned focus.