Curves in Motion

Calculus 2 · Axiom Academy

Let time do the drawing. When a point's x and y are each their own function of t, the path it leaves behind is a parametric curve. A curve is the trail a moving point leaves You already graph y as a function of x. But how do you describe a thrown ball, a planet's orbit, or a looping racetrack — paths that double back, where one x can have two y's? You hand the wheel to a third variable: time. Give x and y each their own rule in terms of t, and as t ticks forward the point sketches the whole curve for you. Watch a single point ride the rules x(t) = 15t and y(t) = 20t − 5t², the same pair that governs a tossed ball. As the clock t climbs from 0 to 4 seconds, the point moves and the curve is simply the trail it leaves behind. The curve was never drawn all at once — it is the record of where the point has been. Scrub time, and read off the point Drag the time slider yourself. At every instant t, the rules hand you one horizontal value x(t) and one vertical value y(t) — and the point lands exactly there on the curve. The dashed lines show how each coordinate is read straight off the axes. One input t in, one point (x, y) out — that is exactly what a parametrization is. Two simple motions, secretly combined

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