Tables and Graphs of Parametric Curves

Think of t as a stopwatch. At every tick, x(t) tells you how far left or right the dot sits, and y(t) tells you how far up or down it sits. A table just records these snapshots: pick several t values, plug each into x(t) and y(t) separately, and write down the resulting (x, y) pair. Because t is doing the same job in both equations at once, the two coordinates are synced to the same clock even though they came from different formulas.

To turn the table into a graph, plot each (x, y) pair as a point, then connect them in the order t increases. That order matters: parametric curves have a direction of travel built in, something a plain y = f(x) graph never shows you. Skipping ahead or connecting points out of t-order gives a shape that looks wrong or crosses itself in the wrong place.

This table-and-plot method is the foundation for everything else in parametric work: finding where a curve self-intersects, matching a parametric curve to its underlying conic, or later describing motion with vectors. Changing the domain of t does not change the formulas for x(t) and y(t), so it does not change the shape or size of the full curve. It only changes how much of that curve gets drawn, since the path stops wherever the t-interval ends.