Lesson 8

Students find distances from points on a parabola to the parabola’s focus using the fact that this distance is equal to the distance between the point and the parabola’s directrix. This will be helpful in an upcoming activity in which students write an equation for a parabola.

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.

The goal is for students to summarize their observations about the general point \((x,y)\). Display this image for all to see.

Ask students what we know about these two distances. (The distances are equal. Each distance is \(y\) units.)

Students use the concepts they’ve explored in recent activities to write an equation for a parabola and rewrite it in vertex form. 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 \((3,2)\) and \((x,y)\), 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 \((y-2)^2\). Then rearrange the terms so \(y\) 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.

\((x-3)^2 + (y-2)^2 = y^2\)

\(y=\frac 14 (x-3)^2+1\)

Here are some questions for discussion:

• “There is an \((x-3)^2\) in each equation. What does this mean in each one?” (In the first equation, the 3 came from the \(x\)-coordinate of the focus. In the second equation, it’s the \(x\)-coordinate of the vertex.)
• “Was it just coincidence that the \(x\)-coordinate of the focus and vertex were the same? That is, if we tried another example, could we find one in which the \(x\)-coordinate is different?” (If the directrix is a horizontal line, then the vertex and the focus will always be on the same vertical of line—the axis of symmetry.)

Tell students that it’s possible to create parabolas using directrices that are vertical or even slanted. A vertical directrix is relatively easy to work with, but for slanted directrices, we need to use more complicated trigonometry than what we’ve yet learned. Explain that because of these complications, we won’t create a general form for the equation of a parabola like we did for a circle—we will just create equations for specific parabolas, as we did in this activity.

A sorting activity gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

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

Arrange students in groups of 2. Distribute one set of cards to each group. Give students time to work with their partner, followed by a whole-class discussion.

Select groups to share how they sorted their cards. 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.