From single variable function to parametric equation

In single-variable calculus, we usually describe a curve by

y = f(x).

For example, the exponential function is easy to represent in this form:

A standard single-variable graph: y = e^x

This model is powerful, but it has a limitation: a single $x$ can only produce one $y$ .

That means many geometric trajectories cannot be represented by one equation $y=f(x)$ , especially curves that loop back or self-intersect.

For example, this figure-eight style trajectory is easy to write parametrically but cannot be written as one single-valued function $y=f(x)$ :

Self-intersection example (parametric curve)

Parametric equations and parametric curves

Precise definition

Instead of defining $y$ directly from $x$ , we introduce a parameter $t$ and define both coordinates as functions of $t$ :

x = f(t), \quad y = g(t), \quad t \in I.

The parametric curve is the set (trajectory) of points

\big(x(t), y(t)\big)

as $t$ varies in the interval $I$ .

How the graph is generated from points

For the curve

x(t)=\sin t, \quad y(t)=\sin(2t), \quad t\in[-\pi,\pi],

we evaluate at sample parameter values from $-\pi$ to $\pi$ with step size $\pi/4$ :

$t$	$x(t)=\sin t$	$y(t)=\sin(2t)$	point
$-\pi$	$0$	$0$	$(0,0)$
$-\frac{3\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$1$	$\left(-\frac{\sqrt{2}}{2},1\right)$
$-\frac{\pi}{2}$	$-1$	$0$	$(-1,0)$
$-\frac{\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$-1$	$\left(-\frac{\sqrt{2}}{2},-1\right)$
$0$	$0$	$0$	$(0,0)$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$1$	$\left(\frac{\sqrt{2}}{2},1\right)$
$\frac{\pi}{2}$	$1$	$0$	$(1,0)$
$\frac{3\pi}{4}$	$\frac{\sqrt{2}}{2}$	$-1$	$\left(\frac{\sqrt{2}}{2},-1\right)$
$\pi$	$0$	$0$	$(0,0)$

In the visualization below, the sample points from the table are marked in blue and each one is labeled with its $(x,y)$ coordinate. The graph is also zoomed in so these sampled points are easier to read.

Point sampling creates the parametric trajectory

Then we connect these sampled points in increasing order of $t$ . As sampling gets denser, the plotted polyline approaches the smooth parametric curve.

Elimination of parameters

Sometimes we want an equation that only involves $x$ and $y$ . The process is called eliminating the parameter.

For the curve

x = \cos t, \quad y = \sin t,

we use the identity

\cos^2 t + \sin^2 t = 1.

Substituting $x$ and $y$ gives

x^2 + y^2 = 1.

So after eliminating $t$ , the $xy$ -relation is the unit circle.

Important note: this algebraic relation gives the geometric set of points, but the parameter interval controls which part of the set is traced and how it is traced.

Example: $x=\cos t,\ y=\sin t$ with different parameter ranges

Ponder this question: what changes when $t$ varies over different intervals?

$t \in [0,2\pi]$
$t \in [2\pi,4\pi]$
$t \in [0,\pi]$

Case 1: $t$ from $0$ to $2\pi$

x = cos t, y = sin t, t in [0, 2pi]

This traces the full circle exactly once.

Case 2: $t$ from $2\pi$ to $4\pi$

x = cos t, y = sin t, t in [2pi, 4pi]

This gives the same geometric circle, again traced once.

Case 3: $t$ from $0$ to $\pi$

x = cos t, y = sin t, t in [0, pi]

This traces only the upper semicircle.

Summary

$y=f(x)$ describes many curves, but not all trajectories.
Parametric equations $x=f(t), y=g(t)$ remove the single-valued restriction.
A parametric curve is the trajectory traced by $(x(t),y(t))$ as $t$ changes.
Computationally, plotting is: sample $t$ values -> compute points -> connect points.
Eliminating parameters can produce an $xy$ equation, but parameter ranges still determine how much of the curve is traced.

Parametric Curves