Concept preview

What you will be able to do

1. 01Describe a curve given parametrically by $x=x(t)$ and $y=y(t)$, and explain why such a curve need not be the graph of a single function $y=f(x)$.
2. 02Compute the slope of a parametric curve with the chain-rule formula $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ — dividing the vertical rate by the horizontal rate, not multiplying them.
3. 03Locate horizontal tangents where $\dfrac{dy}{dt}=0$ (and $\dfrac{dx}{dt}\neq 0$) and vertical tangents where $\dfrac{dx}{dt}=0$ (and $\dfrac{dy}{dt}\neq 0$), and evaluate $\dfrac{dy}{dx}$ at a chosen value of $t$.

A parametric curve is a pair of equations $x=x(t)$ and $y=y(t)$ sharing one parameter $t$. Think of $t$ as time and the pair as the recipe for a point that moves through the plane. For example, $x=3\cos t$ and $y=2\sin t$ with $t$ running over $[0,\,2\pi]$ traces an ellipse: at $t=0$ the point sits at $(3,\,0)$, at $t=\tfrac{\pi}{2}$ it is at $(0,\,2)$, at $t=\pi$ it is at $(-3,\,0)$, and it returns to the start at $t=2\pi$. No single $y=f(x)$ describes this whole loop — for most $x$ there are two $y$ values — which is exactly why we need the parameter.