# Parabolas Lesson 5 Episode 6 (Teachers)

### Exploring

Sasha and Keoni use the Pythagorean theorem and the definition of a parabola to derive the equation for a parabola with a vertex at the origin and a distance of p between the focus and vertex.

### Episode Supports Focus Questions

For use in a classroom, pause the video and ask these questions:

1. [Pause video at 2:25]. What are the lengths of the vertical lines that Sasha and Keoni just drew?

2. [Pause video at 6:09]. Finish writing the equation and then solve for y. [Then start the video again and stop at 7:58]. How did your solution method compare with Sasha and Keoni’s?

Supporting Dialogue

Provide opportunities for all your students to express their ideas verbally, by asking them to talk with a partner.

1. [Pause the video at 3:58]. Talk with your neighbor. Where does the term y – p come from and what does it mean?

2. [Pause the video at 7:58]. Talk with your neighbor. Where does the equation y = x2/(4p)  come from? Where does the 4p come from?

Math Extensions

1. Examine the parabola with a vertex at the origin and a focus at (0, -2). A general point on the parabola is labeled (x, y). A right triangle was formed so that the hypotenuse connects the (x, y) and the focus. The lengths of the three sides of the right triangle are x,
-y + 2, and -y – 2. Explain why:

• the distance from (x, y) to the x-axis is -y.
• the length of the vertical side of the right triangle is -y – 2.
• the length of the hypotenuse of the right triangle is -y + 2.
• the length of the horizontal side of the right triangle is x.

2a. Using the Pythagorean Theorem and the definition of a parabola, derive the equation of the parabola with a vertex at the origin and a focus at (0,-2).

2b. Compare your equation with the equation that Keoni and Sasha derived for a parabola with a vertex at the origin and a focus at (0,2). What do you notice? 