Lesson 4 Directing Our Focus Develop Understanding
Use a diagram to find the distance between the point and the line or the two points given. Distance between a point and a line is defined as the length of the perpendicular segment from the line to the point.
Develop a geometric definition for a familiar shape.
Circles can be made from all the points that are the same distance away from the center. What shape is made from all the points that are the same distance away from a point and line?
Open Up the Math: Launch, Explore, Discuss
On a board in your classroom, your teacher has set up a point and a line like this:
We’re going to call the line a directrix and the point a focus. They’ve been labeled on the drawing. Similar to the circles task, the class is going to construct a geometric figure using the focus (point
Cut a piece of string longer than
Mark the midpoint of the string with a marker or pen. If your teacher instructs you to, tie a slip knot on one end of the string before finding the midpoint.
Position the string on the board so that the midpoint is equidistant from the focus (point
As your classmates post their strings, what geometric figure do you predict will be made by the midpoints (the collection of all points like (
Consider the following construction with focus point
Where is the vertex of the figure located? How do you know?
Where is the line of symmetry located? How do you know?
You have just constructed a parabola based upon the definition: A parabola is the set of all points
How would the parabola change if the focus were moved up, away from the directrix?
How would the parabola change if the focus were to be moved down, toward the directrix?
How would the parabola change if the focus were to be moved down, below the directrix?
Ready for More?
In this unit, we have looked at the set of all points in a plane that are equidistant from a given point (a circle), and the set of all points in a plane that are equidistant from a point and a line (a parabola). Now your challenge is to find the figure that is formed by the set of all points equidistant from two points. Try using some of the same ideas that you used for parabolas and circles.
Definition of a parabola:
Relationships between features of a parabola:
- parabola: conic definition, geometric definition
- Bold terms are new in this lesson.
In this lesson, we learned the geometric definition of a parabola. Much like circles, parabolas are a geometric shape that can be constructed from a definition and as a set of points generated from an equation. In the same way that the defining features of a circle are the center and radius, the defining features of a parabola are the focus and directrix.
Graph each set of functions on the same coordinate axes. Describe in what way the graphs are the same and in what way they are different.
Rewrite the equation