PreCalculus
Activity:
Exploring the Parabola
Posted on May 19, 2008
Topic: Conics
This activity explores the key features of the parabola, both geometrically and algebraically. A variety of interactive representations support student learning as they build their understanding of this important curve and its real world applications.
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Activity Key Steps:
In this activity, students take a sheet of paper and mark a point somewhere near the center of the page. They produce a series of folds from different points along one of the longest edges that pass through the point.
When students trace along these straight-line folds they produce an envelope similar to the one shown. They see the connection between the crease in the paper and the perpendicular bisector.
Students use their handheld’s capabilities to manipulate different parts of the construction. They can move the focus and directrix to observe their roles in the locus construction of a parabola. Students may observe that the perpendicular bisector becomes tangent to the parabola at point P.
Students are introduced to the algebraic notation for the quadratic function and engage in manual regression. They translate the fundamental curve y = x² to match the envelope.
Students will learn that this process acts to model the curve. They explore the relationship between various components of the algebraic form and the physical features of the curve such as that modeled in the original parabola locus by the students.
To build an algebraic understanding, students can use algebra to determine an ‘equation’ for the curve, based on these properties. This amounts to using the formula for the distance between two points.
At the end of this activity, students will be able to write an equation of a parabola with vertex at (h, k) and axis of symmetry x = h or y = k and graph it. Students will also be able to derive the formula for any conic.