A curve with a clock attached.

An ordinary equation tells you where a curve is. A parametric one also tells you when each point is reached, which is exactly what you want for anything that moves.

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A ball is kicked from the ground. After $t$ seconds it is

metres along and up. Each value of the parameter $t$ gives one point of the path.

The dots mark every half second. Their shadows on the ground are evenly spaced: $x$ grows by $10$ m each time. Their shadows on the pole bunch up near the top: $y$ slows, stops and falls back. Each coordinate has its own story in $t$.

It lands when $y = 0$: $t(15 - 4.9t) = 0$, so $t = 3.061\ldots$ and $x = 61.2$ m. The height is greatest halfway between the roots, at $t = 1.530\ldots$, where $y = 11.5$ m.

Getting rid of $t$ gives the Cartesian equation of the path, a parabola, for $0 \le x \le 61.2$. It says where the ball goes, but no longer when: the clock is gone.

Two equations in t. One path.

Eliminating the parameter with trig identities, circles, restricted domains, where curves meet lines, and twelve graded exam questions with full worked solutions are waiting.

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