Parametrization of Implicitly Defined Functions
Unit 4 is not assessed on the AP exam; schools include it based on local requirements, and everything here still builds directly toward calculus. Parametrizing an implicit curve means building a journey that travels it: cos/sin templates cover circles and ellipses, x = t covers any explicit graph, and interpolation covers segments. Substitution into the implicit equation is the correctness test.
The mistakes split cleanly in two. Coverage errors: substitution succeeding is read as the whole curve being traced, when the interval or the formulas may cover only a piece. Uniqueness errors: assuming a curve owns one parametrization, when reversing, rescaling, and re-anchoring all produce equally valid covers. The lesson drills verify-then-audit as a two-step habit.
The work
3 ways in · any order
Lesson
Parametrization of Implicitly Defined Functions
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Substitute to verify, audit to trust. The lesson builds cos/sin covers for circles and ellipses, trivial covers for function graphs, and interpolated segments, then stress-tests coverage and direction. Ten scenarios close it out, half of them about journeys that stay on the curve without covering it.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 4.7 misconceptions: substitution success mistaken for full coverage, and the one-true-parametrization assumption. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.