Parametric Functions and Motion

Two functions, one moving point

A parametric function feeds one input $t$ into two outputs at once: $x=f(t)$ and $y=g(t)$. The point $(f(t),g(t))$ traces a curve as $t$ runs through its interval — but unlike the graph of $y=h(x)$, a parametric curve carries extra information: where the point is at each instant, which direction it travels, and how the curve can cross itself or fail the vertical line test.

Three questions define the topic:

• Position: plug in a value of $t$ to get one point.
• Direction of motion: track how $x(t)$ and $y(t)$ increase or decrease as $t$ increases — right/left comes from $x(t)$, up/down from $y(t)$.
• Eliminating the parameter: solve one equation for $t$, substitute into the other. The Cartesian equation describes the road; the parametrization describes the trip — including any restriction on $x$ or $y$ inherited from the range of $f$ and $g$.

Standard example: $x=3\cos t$, $y=2\sin t$ gives $\left(\frac{x}{3}\right)^2+\left(\frac{y}{2}\right)^2=1$, an ellipse traced counterclockwise starting from $(3,0)$.