The Main Idea 

Traditional functions [latex]y = f(x)[/latex] work great for many curves, but what about loops, vertical lines, or paths that double back on themselves? Parametric equations solve this by letting both [latex]x[/latex] and [latex]y[/latex] depend on a third variable called a parameter.

In parametric equations, we write [latex]x = x(t)[/latex] and [latex]y = y(t)[/latex], where [latex]t[/latex] is the parameter. Think of [latex]t[/latex] as time on a stopwatch—as [latex]t[/latex] changes, both coordinates change simultaneously, tracing out a path through the plane.

The parameter [latex]t[/latex] gives curves an orientation—a direction of travel. As [latex]t[/latex] increases, you can follow the path from start to finish with arrows showing which way you’re moving.

The Process: Make a table with [latex]t[/latex] values, calculate corresponding [latex]x[/latex] and [latex]y[/latex] coordinates, then plot and connect the points. The arrows show the direction as [latex]t[/latex] increases.