A security camera does not only care that a person walked along a hallway. It cares about the timestamp. A game engine does not only store a curve; it asks “where is the character on this frame?” That shared idea is a parameter — often called t for time.
Two stories, one clock
A parametric description of a plane path looks like:
x = x(t) y = y(t)
Both coordinates are driven by the same clock. At each t you get one point (x(t), y(t)). As t runs, the point draws a path.
Why not only y = f(x)?
A graph y = f(x) can only show one height per x. It cannot describe:
- A vertical segment (x fixed, y changing)
- A loop that crosses itself
- Going back along the same road later
Parametric form can. It separates shape of the journey from schedule of the journey.
A simple straight-line example
Suppose x = 2t and y = t. At t = 0 you are at the origin. At t = 3 you are at (6, 3). Eliminating t gives y = x/2 — the path is a line — but the parametric form also tells you how fast you move along that line.
Circles and wheels
A ferris-wheel seat is a classic parametric motion:
x = R cos t y = R sin t
As t increases, the seat goes around a circle of radius R. Ordinary y = f(x) cannot represent the full loop as a single function of x.
Common trap: Eliminating t shows the path but throws away timing. Two runners can share the same track equation and still finish at different times if their t-schedules differ.
Practice in the lab
AP → Parametric Equations: robots, tosses, and circular motion with a time parameter.
Open Hyper-Scale LabRelated guides
- Polar coordinates
- Rates and limits — how fast something changes
- Trigonometry basics