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Common Mistakes

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Frequent Errors

1. Using t in [0, pi] for a full CCW unit circle revolution (4.4.A.1). 2. Swapping cos and sin without adjusting direction/domain (4.4.A.1). 3. Forgetting to add center (h, k) in circle parametrizations (4.4.A.2). 4. Applying radius to only one component incorrectly (4.4.A.2). 5. Line rates that do not reach (x2, y2) (4.4.A.3). 6. Checking only one endpoint of a segment (4.4.A.3). 7. Assuming a unique parametrization for a line segment (4.4.A.3). 8. Not stating domain of t for the intended trace (4.4.A.1).

Why These Happen

Students memorize formulas without endpoint/domain checks.

How to Fix Them

  • Always evaluate at t = 0 and t = domain end.
  • For circles, write center and radius before shifting/scaling.
  • For lines, solve for rates/T so both endpoints match.

Avoidance Scripts

  • "Domain first, endpoints second."
  • "Standard CCW unit circle starts at (1, 0)."
  • "Many parametrizations can describe the same segment."

Quick Self-Check

  • Did I verify both endpoints for a line?
  • Did I use [0, 2pi] for one full CCW unit circle trip?
  • Did I include center shift for a non-origin circle?

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