Parametric Form of a Parabola

Parameter, defined. If the coordinates of a moving point on a curve are expressed as functions of a single third variable, that variable is called the parameter for the curve. Parametric descriptions are often algebraically simpler to work with than the direct $(x,y)$ equation, especially for tangent-line problems.

Parametric form of $y^2=4ax$. Consider the pair of expressions

where $t$ ranges over all real numbers. Substituting these into the left- and right-hand sides of the parabola's equation:

So the point $(at^2, 2at)$ satisfies $y^2=4ax$ for every real value of $t$ — meaning every such point genuinely lies on the parabola. Conversely, every point on the parabola can be written this way for some value of $t$.

We denote this point $P(t) \equiv (at^2, 2at)$, calling $t$ the parameter of the point. This single-variable description is what makes the tangent-line and locus derivations in sections 7.1.11–7.1.13 so much cleaner than working with $x,y$ directly: instead of two coupled variables satisfying one equation, we have one free variable $t$ ranging over all reals. …