Parabola is the locus of equi-distant point to a given point and a line. Now, what would happen if the line is changed to a segment of line?
The a and b are the parameters to change the size and position the line segment.
The c is the parameter to change the position of the given point C.
The curve consists of three parts.
On the left (blue), try to move the point H to the left and right. See what is going on.
On the middle (red), try to move the point D to the left and right. See what is going on.
On the right (green), try to move the point F to the left and right. See what is going on.

Could you describe the properties of the curve traced by the points?