MISSION · Algebra
Parametric Equations
Letting a third variable drive both coordinates
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The idea in one sentence
Instead of tying \(y\) directly to \(x\), give both of them their own formula in terms of a third variable — and suddenly you can describe paths no function could.
\[x = f(t), \qquad y = g(t)\]Why ordinary functions aren’t enough
\(y = f(x)\) demands one \(y\) for each \(x\). That rules out circles, spirals, loops, and anything that doubles back — which is most of the interesting curves, and every path taken by a moving object.
Parametrically, a circle is effortless:
\[x = \cos t, \qquad y = \sin t\]As \(t\) runs from 0 to \(2\pi\), the point walks once around the unit circle. No branches, no \(\pm\), no awkwardness at the top and bottom where the slope goes vertical.
The parameter usually means something
Most often \(t\) is time, and that reading turns the equations into a description of motion. A projectile launched at speed \(v\) and angle \(\theta\):
\[x = v\cos(\theta)\,t, \qquad y = v\sin(\theta)\,t - 4.9t^2\]Horizontal motion is steady; vertical motion is pulled down by gravity. Between them they trace the arc.
And now the equations say more than the curve does. They say where the object is at each instant, and how fast. Two projectiles can follow the identical arc at different speeds: same picture, different parametrisation. The path is what you see; the parametrisation is the schedule for walking it — including which direction you go.
Eliminating the parameter
To recover an ordinary equation, solve one for \(t\) and substitute into the other.
\[x = t + 1, \quad y = t^2 \quad \Rightarrow \quad t = x - 1 \quad \Rightarrow \quad y = (x-1)^2\]For the circle, use an identity instead of solving: square both and add, and \(\cos^2 t + \sin^2 t = 1\) gives \(x^2 + y^2 = 1\) immediately.
Be careful about what gets lost. If \(t\) was restricted — say \(0 \le t \le 2\) — the parametric version draws only part of the curve, while the eliminated equation draws all of it. The restriction has to be carried across as a restriction on \(x\).
What it buys you
- Curves that fail the vertical line test: circles, ellipses, figure-eights, spirals.
- Motion problems, where time is the natural variable.
- Computer graphics and animation, where everything drawn along a path is parametrised.
- Three dimensions, where \(x\), \(y\), and \(z\) all take their own formula in \(t\) and no “\(y\) in terms of \(x\)” exists at all.
Common mix-ups
- Forgetting the direction of travel. Reversing \(t\) traces the same curve backwards, and that’s a different answer to a motion question.
- Dropping the range of \(t\) when eliminating, which draws too much curve.
- Assuming equal \(t\) steps mean equal distances. Speed varies unless the parametrisation is built to keep it constant.
- Treating \(t\) as an axis. It isn’t plotted; it’s the driver.
Check yourself
What curve is \(x = 2\cos t\), \(y = 3\sin t\)? Eliminate \(t\) to confirm.
Divide and square: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = \cos^2 t + \sin^2 t = 1\). That’s an ellipse with semiaxes 2 and 3 — 4 units wide and 6 units tall.
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