Parametric Functions

Ordinary functions y = f(x) tie each x to exactly one y, so their graphs can never fold back on themselves — that's the vertical line test. Parametric equations break out of that cage by adding a third variable, t, often read as time. Instead of y depending directly on x, both x and y depend separately on t: x = x(t), y = y(t). As t sweeps through its interval, the point (x(t), y(t)) traces a path. Because x and y are each free to rise, fall, or repeat values independently as t moves, the resulting curve can cross itself, loop, or retrace a section — something impossible for y = f(x).

To graph a parametric curve, build a table: pick values of t, compute x(t) and y(t) for each, then plot the points (x, y) in order. For x(t) = t² - 1 and y(t) = t with t from -2 to 2, t = -2 gives (3, -2), t = -1 gives (0, -1), t = 0 gives (-1, 0), t = 1 gives (0, 1), and t = 2 gives (3, 2). Plotting these in order traces a sideways parabola opening rightward — and since x = y² - 1, it really is that curve, just traced with a time stamp on every point.

This sets up everything else in the unit: vectors describe motion along such paths using separate horizontal and vertical components, and matrices later transform these same (x, y) points all at once. Parametric thinking — tracking x and y as twin outputs of one input — is the thread connecting all three.