If students are stuck, suggest that they refer to the definition of a parabola on their reference chart. Ask if there are any distances they can easily find without needing the Pythagorean Theorem.

In this activity, students create an equation for parabolas with a directrix of . As students build from work they have done in earlier activities, they are using repeated reasoning to generalize about parabolas (MP8).  

If students ask, share that while it is possible to create parabolas with a directrix that is vertical or slanted, this is beyond the scope of student work at this time. Students will create equations for specific parabolas in this and later activities, but they will not create a general equation for a parabola similar to the general equation of a circle. 

If many students struggle with the algebraic manipulation, consider working through some of the steps as a class.

If students struggle to write the initial equation, suggest they draw in a right triangle whose hypotenuse is the distance between and , and write expressions for the lengths of the triangle’s legs. Then, suggest they look back to the Warm-up for a way to label the hypotenuse.

If students don’t know how to start rewriting the equation into vertex form, suggest they begin by squaring the expression . Then rearrange the terms so is alone on one side of the equation.

Display the two forms of the equation. If students ask about the name of the original form, tell them that there is no particular name for that form. It is simply a starting point.

Here are some questions for discussion:

• “There is an in each equation. What does this mean in each one?” (In the first equation, the 3 came from the -coordinate of the focus. In the second equation, it’s the -coordinate of the vertex.)
• “Was it just coincidence that the -coordinate of the focus and vertex were the same? That is, if we tried another example, could we find one in which the -coordinate is different?” (If the directrix is a horizontal line, then the vertex and the focus will always be on the same vertical line—the axis of symmetry.)

In this activity, students match representations of parabolas, including equations and graphs. They compare the features of the graph such as the focus and directrix with the structure of the equation. 

A sorting activity gives students opportunities to analyze representations and structures closely and make connections (MP7).

As students work, encourage them to refine their descriptions of the parabolas using more precise language and mathematical terms (MP6).

The purpose of this discussion is for students to notice connections between the features of the graph and the structure of the equation. 

Once all groups have completed the Card Sort, discuss the following: 

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Attend to the language that students use to describe the graphs and equations, giving them opportunities to describe the parabolas more precisely. Highlight the use of terms like focus, directrix, and distance.