Parabolas

In this section we shall study the graph of the equation

y = ax² + bx + c,

which is a U-shaped curve called a vertical parabola. We begin with the general definition of a parabola in the plane.

Recall that the distance between a point P and a line L is the length of the perpendicular line from P to L, as in Figure 5.4.1. If we are given a line L and a point F not on L, the set of all points equidistant from L and F will form a U-shaped curve that passes midway between L and F. This curve is a parabola, shown in Figure 5.4.2.

Figure 5.4.1

Figure 5.4.2: Parabola = set of points equidistant from L and F.

The line through the focus perpendicular to the directrix is called the axis of the parabola. The point where the parabola crosses the axis is called the vertex. These are illustrated in Figure 5.4.3.

Figure 5.4.3

As we can see from the figure, the parabola is symmetric about its axis. That is, if we fold the page along the axis, the parabola will fold upon itself. The vertex is just the point halfway between the focus and directrix. It is the point on the parabola which is closest to the directrix and focus.