Separate a path from the way it is traveled

Try

A point follows x(t)=2t−1 and y(t)=t² for −1≤t≤2. Calculate the paired coordinates at t=−1, 0, 1 and 2. They are (−3,1), (−1,0), (1,1) and (3,4). The same parameter value must feed both functions; pairing coordinates from different times creates points that may not belong to the trajectory.

Predict

Predict whether eliminating t gives every point on the resulting parabola. Solving t=(x+1)/2 gives y=(x+1)²/4, but the parameter interval also implies −3≤x≤3. The unrestricted parabola includes points this motion never reaches.

Experiment

Sketch x=3cos t, y=2sin t for 0≤t≤2π, using quarter-turn samples.

t	0	π/2	π	3π/2	2π
x	3	0	−3	0	3
y	0	2	0	−2	0

Draw arrows between successive points. The ellipse is traced counterclockwise, beginning and ending at (3,0). Replace t with −t to reverse direction. Replace it with 2t to complete two turns over the same parameter interval. The geometric ellipse remains the same.

Understand

A parametrization specifies a point (x(t),y(t)) for every allowed parameter t. The parameter may represent time, but it can also be an abstract indexing quantity. Its domain and units belong to the definition. A Cartesian equation for the point set generally does not retain traversal order, repeated visits or timing information.

For a segment from A to B, use A+t(B−A) with 0≤t≤1. At t=0 the point is A, and at t=1 it is B. Intermediate t forms a weighted average. Allowing all real t gives the entire line; restricting to t≥0 gives a ray unless A=B. A zero difference vector produces one constant point, not a line with a direction.

Elimination can also merge branches. For x=t²,y=t with t≥0, the relation is x=y² with y≥0. Dropping that sign condition adds the lower half of the parabola. Squaring coordinate equations can similarly lose sign information, so check the reached point set after eliminating.

Equal parameter steps need not produce equal traveled distances. For x=t,y=t², motion becomes steeper as t grows. Derivative-based velocity and speed require the later calculus of parametric curves; a picture of the locus alone cannot determine them. This lesson establishes representation and traversal, not full motion analysis.

Apply

In practice, state coordinate formulas and the parameter interval together. Verify endpoints and at least one interior point. In a same-path comparison, explain whether the points, direction and number of traversals agree separately.

Continue

Conics provide important curve families that admit both implicit equations and parametrizations. The final lesson introduces their geometric meaning and the restrictions needed when completing squares or squaring distance equations.