IM Geo.6.7 Lesson: Distances and Parabolas
What do you notice? What do you wonder?
What happens as the focus and directrix move farther apart?
Try to make the parabola open downward (that is, to look like a hill instead of a valley). What needs to be true for this to happen?
The vertex of the parabola is the lowest point on the curve if it opens upward, or the highest point if it opens downward. Where is the vertex located in relationship to the focus and the directrix?
Move the directrix to lie on the -axis and move the focus to be on the point . Plot a point , with coordinates . It should lie on the parabola. What is the distance between point and the directrix?
What does this tell you about the distance between and ?
The image shows a parabola with focus and directrix (the -axis).
The point looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning.
The point looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning.
In general, how can you determine if a particular point is on the parabola?
The image shows a parabola with directrix y=0 and focus at F=(2,5). Imagine you moved the focus from F to F'=(2,2). Sketch the new parabola.
How does decreasing the distance between the focus and the directrix change the shape of the parabola?
Suppose the focus were at , on the directrix. What would happen?