L6.7 - Distances and Parabolas
Learning Intentions and Success Criteria
- Comprehend (in spoken and written language) that a parabola is the set of points equidistant from a given focus and directrix.
- Know that a parabola is the set of points equidistant from a given point and line.
7.1: Notice and Wonder: Distances
What do you notice? What do you wonder?
7.2: Into Focus
Open BOOK - Geo.6.7 Distances and Parabolas. Scroll down to 7.2: Into Focus to use the applet to complete the questions below. The applet shows a parabola. In the applet, move point F (the focus) and the line (the directrix) and observe how the shape of the parabola changes. 1. What happens as the focus and directrix move farther apart?
2. 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?
3. 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?
4. Move the directrix to lie on the x-axis and move the focus to be on the point (2, 2). Plot a point P, with coordinates (6, 5). It should lie on the parabola. a. What is the distance between point P and the directrix?
b. What does this tell you about the distance between P and F?
7.3: On Point
1. The point (11, 5) looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning.
2. The point (14, 10) looks like it might be on the parabola. Determine if it really is on the parabola. Explain or show your reasoning.
3. In general, how can you determine if a particular point (x, y) is on the parabola?
Learning Intentions and Success Criteria
- Comprehend (in spoken and written language) that a parabola is the set of points equidistant from a given focus and directrix.
- Know that a parabola is the set of points equidistant from a given point and line.
Cool-Down: A Point and a Line
The image shows a point and a line. Suppose we create a parabola using the point as the focus and the line as the directrix. Decide whether each point on the list is on this parabola. Explain your reasoning. 1. (-1, 5)
2. (3, 3)
3. (5, 5)
