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AP Calculus Question of the Month: October

by Lin McMullin
Educational Consultant and Writer
Niantic, Connecticut

What's Normal for a Quadratic?
A normal to a curve is the line perpendicular to the line tangent to the curve at the point of tangency. Consider the parabola given by with . Let  be the family of nonvertical lines normal to this parabola.
  1. Show that the vertical distance between the y-intercept of the normal line and the point where the normal intersects the parabola is constant, and find, in terms of p, that distance.

  2. Discuss how to use the distance found in (A) to draw the normal at any point.

  3. What is the relationship between this vertical distance and the distance between the vertex and the focus of the parabola?

  4. What is the range of the y-intercepts of the nonvertical normal lines?

Complete the question before viewing the answers and explanation!
Lin McMullin, an educational consultant and writer with extensive experience teaching AP Calculus, lives in Niantic, Connecticut. As a College Board consultant he has presented AP Calculus institutes and workshops in the United States and Europe, and is a Table Leader at the Reading. His work as a writer includes co-authorship of the popular D&S Marketing Systems review books for the AP Calculus Exam and Teaching AP® Calculus, a book especially for AP Calculus teachers.

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