Essay

Geometric Properties of a Parabolic Reflector Path

You are a junior drafting technician at an automotive design firm. Your team is designing a custom parabolic headlight reflector whose cross-sectional profile is represented on a coordinate grid by the equation x=2y2x = 2y^2. Before importing the coordinate path into the CNC milling software, your supervising engineer asks you to write a brief technical memo.

In your memo, recall and state the fundamental geometric properties of this horizontal parabola directly from its algebraic form. Specifically, your memo must:

  1. Identify the direction in which the parabola opens and explain the rule based on the coefficient.
  2. State the equation of the axis of symmetry and the formula used to determine it.
  3. State the coordinates of the vertex and explain how they relate to the intercepts on the coordinate axes.
  4. List at least two distinct pairs of symmetric points on the graph (four points total, excluding the vertex) that can be used to accurately sketch the curve, showing the calculation for each pair.

Write a clear, structured response that addresses all four requirements.

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Updated 2026-06-05

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