Parametric Equations
Define parametric equations x = f(t) and y = g(t). Graph parametric curves by plotting points, eliminate the parameter to find a Cartesian equation, and understand parametrics as a way to represent motion with explicit time dependence.
A lesson of Pre-Calculus, in Applications of Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to graph curves defined by parametric equations, eliminate the parameter to find the rectangular equation, and use parametric equations for projectile motion.
The Big Idea
Parametric equations describe a curve by expressing $x$ and $y$ separately as functions of a third variable $t$ (the parameter, often time). This allows curves that can't be described as $y = f(x)$ (like circles, spirals, and paths where the same $x$ has multiple $y$-values). Physics uses parametric equations constantly — position, velocity, and trajectory are all parametric.
Definition
A parametric curve is given by:
$$x = f(t), \quad y = g(t), \quad t \in I$$
The parameter $t$ traces out the curve as it varies over interval $I$.
Orientation: the direction traced as $t$ increases (shown by arrows on the graph).
Graphing Parametric Curves
- Create a table of $(t, x, y)$ values
- Plot the $(x, y)$ points
- Draw the curve with arrows showing direction of increasing $t$
Example 1 — Linear Parametric Curve
$x = 2t + 1$, $y = t - 3$, $t \in [-2, 4]$
| $t$ | $x$ | $y$ |
|---|---|---|
| $-2$ | $-3$ | $-5$ |
| $0$ | $1$ | $-3$ |
| $2$ | $5$ | $-1$ |
| $4$ | $9$ | $1$ |
From $t = y + 3$: $x = 2(y+3)+1 = 2y+7 \implies y = (x-7)/2$ — a line.
Eliminating the Parameter
Method 1 (Algebraic): Solve one equation for $t$, substitute into the other.
Method 2 (Trig): Use $\sin^2 t + \cos^2 t = 1$ when equations involve $\sin t$ and $\cos t$.
Example 2 — Circle
$x = 3\cos t$, $y = 3\sin t$, $t \in [0, 2\pi]$
$x^2 + y^2 = 9\cos^2 t + 9\sin^2 t = 9$ → circle $x^2 + y^2 = 9$.
The curve traces the circle counterclockwise starting at $(3, 0)$ when $t = 0$.