Parametrizing an Implicit Curve

A curve like y = x² - 4 is a set of points, drawn all at once. Give it a parameter and it becomes a trip: at each time t, x(t) and y(t) tell you exactly where you are. Nothing about the shape changes, but now you can ask which point comes first, how fast you move, and which direction you travel. That extra structure is the whole point of parametrizing.

Checking a parametrization is mechanical: take x(t) and y(t), plug them into the original equation in place of x and y, and see if the equation becomes a true statement for every t. If it simplifies to something always true, like 0=0, the parametrization is valid. For x(t) = t and y(t) = t² - 4, substitute into y = x² - 4: the right side becomes t² - 4, and that matches y(t) exactly, so the check passes for every value of t.

This same idea connects directly to inverses. If x(t) and y(t) trace out y = f(x), then swapping the two formulas, letting x(t) be the old y(t) and y(t) be the old x(t), traces out the inverse relation x = f(y). You are not solving for a new formula; you are just relabeling which expression drives the horizontal position and which drives the vertical one. That swap is the same reflection over the line y = x that defines inverses in the first place, now seen through the motion of a point.