Lesson 4 Directing Our Focus Develop Understanding

Jump Start

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.

a.

Point $(4, 5)$ Line: $y = 1$

b.

Point $(x, y)$ Line: $y = 1$

c.

Point $(4, 5)$ Point $(0, 3)$

d.

Point $(x, y)$ Point $(0, 3)$

Learning Focus

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 $A$ ) and directrix (line $ℓ$ ).

1.

Cut a piece of string longer than $12$ inches.

2.

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.

3.

Position the string on the board so that the midpoint is equidistant from the focus (point $A$ ) and the directrix (line $ℓ$ ), which means that it must be perpendicular to the directrix. You may choose to put your string on either the right or left side of the focus. While holding the string in this position, put a pin through the midpoint and at each endpoint. If you are using tape, tape down each of the endpoints and the midpoint as shown. Depending on the size of your string, it will look something like this:

4.

As your classmates post their strings, what geometric figure do you predict will be made by the midpoints (the collection of all points like ( $x, y$ ) shown in the figure above)? Why?

5.

Consider the following construction with focus point $A$ and the $x$ -axis as the directrix. Use a ruler to complete the construction of the parabola in the same way that the class constructed the parabola with string.

6.

Where is the vertex of the figure located? How do you know?

7.

Where is the line of symmetry located? How do you know?

8.

You have just constructed a parabola based upon the definition: A parabola is the set of all points $(x, y)$ equidistant from a line $ℓ$ (the directrix) and a point not on the line (the focus). Use this definition to write the equation of the parabola sketched in problem 5, using the point $(x, y)$ to represent any point on the parabola.

9.

How would the parabola change if the focus were moved up, away from the directrix?

10.

How would the parabola change if the focus were to be moved down, toward the directrix?

11.

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.

Takeaways

Definition of a parabola:

Relationships between features of a parabola:

Vocabulary

directrix
focus
parabola: conic definition, geometric definition
Bold terms are new in this lesson.

Lesson Summary

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.

Retrieval

1.

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.

$y = x^{2}$
$y = \frac{1}{3} x^{2}$
$y = - \frac{1}{3} x^{2}$
$y = {(x - 3)}^{2}$

2.

Rewrite the equation $x^{2} + y^{2} - 2 x + 10 y + 21 = 0$ so that it shows the center $(h, k)$ and radius $r$ of the circle. This is called the standard form of a circle. ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$