Parametric Equations and Motion

Objective: Explain eliminating a parameter, apply the rule, and justify relevant restrictions.

Parametric equations x=f(t), y=g(t) describe coordinates using a common parameter. Eliminating t may produce a Cartesian relation, but the permitted t interval must be translated into coordinate restrictions. For x=at+b with a≠0, substitute t=(x−b)/a into y. Squaring during elimination can conceal a restriction or introduce parts of a curve not traced by the original rule.

Objective: Explain direction and rate along a curve, apply the rule, and justify relevant restrictions.

The parameter order specifies how a curve is traversed, which a Cartesian equation alone does not record. For x=at+b and y=ct, with a≠0, elimination gives a line of slope c/a; increasing t moves right if a>0 and left if a<0. A geometric slope compares coordinate changes, while speed compares distance with elapsed time. These are distinct quantities.