Study Guide
Learning Objectives
- Verify (x(t), y(t)) satisfies an implicit equation on its domain (4.7.A.1).
- Parametrize y = f(x) and invertible inverses (4.7.A.2).
- Parametrize conics by solving for a variable or using standard trig forms (4.7.B.1-B.3).
Course scope only (Unit 4 additional topic; not on the AP Exam).
Suggested Practice Sequence
1. Start with implicit equation F(x, y) = 0. 2. Propose x(t), y(t) and substitute to verify. 3. For y = f(x), try (t, f(t)). 4. For ellipse/hyperbola, use standard trig parametrizations when given. 5. Graph both forms with technology and compare (2.A).
Classroom / Check-In Moves
- "Substitute first."
- "Which variable becomes t?"
- "Domain where sec/tan defined?"
Misconception Watch-List
- Parametrization that fails substitution for some t.
- Swapping sec/tan patterns for hyperbola openings.
- Using full [0, 2pi] without noting branch coverage.
- Confusing parametrization with implicit equation itself.
Vocabulary to Own
- parametrization; verify by substitution
- replace x with t; replace y with t
- ellipse cos/sin; hyperbola sec/tan
- domain of t; equivalent form
Materials Map
- Practice: verify, build, conic parametrizations
- Word: paths described implicitly then parametrically
- Mixed / multi-*: combine 4.5-4.6 with 4.7
- Review & practice test: parametrization fluency
- Interactive: substitution and form checks