Common Mistakes
Frequent Errors
1. Accepting (x(t), y(t)) without substitution check (4.7.A.1). 2. Using (f(t), t) when equation was solved for y incorrectly (4.7.B.1). 3. Swapping (t, f(t)) and (f(t), t) for inverse cases (4.7.A.2). 4. Ellipse form applied to hyperbola (4.7.B.2 vs 4.7.B.3). 5. Horizontal hyperbola written with tan/sec on x incorrectly (4.7.B.3). 6. Domain includes t where sec t or tan t is undefined. 7. Assuming one parametrization covers all branches of implicit curve. 8. Skipping technology comparison of forms (2.A).
Why These Happen
Students memorize trig templates without verification or domain checks.
How to Fix Them
- Always substitute x(t), y(t) into F(x, y) = 0.
- Match hyperbola opening to sec/tan assignment.
- Restrict domain to valid t values.
Avoidance Scripts
- "Substitute before you trust."
- "Solve for one variable; the other becomes t."
- "Ellipse cos/sin; hyperbola sec/tan by opening."
Quick Self-Check
- Did substitution simplify to true on the domain?
- Is the t-domain valid for all trig functions used?
- Does the parametric graph match the implicit curve intended?