Family Overview

In Plain Language

Topic 4.7 parametrizes curves given implicitly or as conics.

• Verification: (x(t), y(t)) must satisfy the implicit equation for every t in the domain (4.7.A.1).
• Explicit function: y = f(x) gives (t, f(t)); invertible inverse gives (f(t), t) (4.7.A.2).
• General conics: solve for x or y and replace the other variable with t (4.7.B.1).
• Ellipse: x(t) = h + a cos t, y(t) = k + b sin t on [0, 2pi] (4.7.B.2).
• Hyperbola: sec/tan or tan/sec forms by opening direction (4.7.B.3).

Course scope only (Unit 4 additional topic; not on the AP Exam).

Why It Matters

Parametrization links implicit graphs (4.5), conics (4.6), and motion models (4.1-4.4).

How Families Can Help

• Substitute x(t) and y(t) into the equation to verify.
• Ask: "Can we solve for y and let x = t?"
• Compare technology graphs of implicit vs parametric forms.

What Not to Do

• Do not assume a parametrization works without substitution check.
• Do not use ellipse trig form for a hyperbola without the correct sec/tan pattern.
• Do not ignore t-domain restrictions (especially sec/tan).