In this task, students look at the relationship between the positioning of the focus and directrix and the size and shape of the resulting parabola. This visual understanding of parabolas will be helpful when students write an equation for a parabola, in an upcoming activity.

As students consider the positioning of the vertex, encourage them to use precise vocabulary. Monitor for responses such as, “The vertex is halfway between the focus and the directrix.” Point out that if they mean the vertex is an equal distance from the focus and directrix, this is true for all points on the parabola. Ask these students how the vertex differs from other points on the parabola. As students carefully describe the relationship between points on the parabola, the focus, and the directrix, they are increasing their precision of language (MP6).

In the digital version of the activity, students use an applet to change the parameters of a parabolic equation. The applet allows students to see how changing the focus, directrix, and vertex of a parabola affects the shape of its graph. The digital version may be helpful for connecting, in a dynamic way, the parts of an equation to the graph of a parabola.

The goal of this discussion is to deepen students’ understanding of the role distances play in the definition of a parabola. Direct students’ attention to the reference created using Collect and Display. Ask students to share their descriptions of the relationship between the parabola, focus, and directrix. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Display this image. Ask students to identify the location of the focus (point ) and the directrix (the horizontal line ).

Here are some questions for discussion:

• “The vertex is a special point on a parabola. How can we use distances to describe how it’s unique?” (The vertex appears to be the point that is closest to both the focus and the directrix. We measure distance from a point to a line by drawing a segment from the point perpendicular to that line. So, the vertex is the midpoint of the segment drawn from the focus perpendicular to the directrix.)
• “What is the distance from point to the directrix? What does that tell us about the distance from to ?” ( is 6.5 units from the directrix. That means must also be 6.5 units from .)
• “How could we verify that really is 6.5 units from ?” (Draw in a right triangle in which segment is the hypotenuse. Find the lengths of the legs by subtracting coordinates. Substitute into the Pythagorean Theorem and check that the result is 6.5.)

Students use distance calculations to test whether points are on a parabola. In an upcoming activity, students will generalize this process when they write an equation for a parabola given a focus and directrix.

Monitor for students who draw right triangles on the image to help calculate distances.

If students are stuck, suggest they refer to the definition of a parabola on their reference chart. Ask them what must be true about the points in order for them to be on the parabola.

Invite a previously selected students to share their picture of a right triangle. Ask them to describe how they found the lengths of the triangle’s legs.

Then invite students to share their strategies for testing a general point . If possible, ask some students who have used words and others who have used equations or expressions to share, and draw connections between these two strategies. For example, as students discuss “the distance between the point and the directrix,” ask students how they know that the distance between the point and the directrix is . (Since the directrix is the -axis, the distance is the height of the point, or the -coordinate value).

Students may be unsure how to find the distance between point and the directrix. Remind them that the distance from a point to a line is the length of the segment between the point and the line and perpendicular to the line.