Defining Parametric Equations

Trigonometry · Axiom Academy

A powerful way to describe curves using a parameter that traces the path point by point 1. What Are Parametric Equations? Unlike traditional equations where we solve for y in terms of x , parametric equations express both coordinates as functions of a third variable. The parameter t often represents time, but it can represent any continuous quantity. As t increases, we can think of a point moving along the curve. Each value of t gives us one point ( x , y ) on the curve. Let's see this with a simple example: The same curve can be described in different ways. Let's compare the two approaches: Sometimes we want to convert parametric equations back to Cartesian form by eliminating the parameter . The strategy is to solve for t in one equation and substitute into the other. 5. Why Use Parametric Equations? Parametric equations offer several advantages over traditional Cartesian equations: Motion and Time: The parameter t can represent time, making it perfect for describing motion and trajectories. Multiple y-values: Parametric equations can describe curves that fail the vertical line test (like circles). Direction: Parametric equations naturally indicate the direction of motion along the curve. Complex Curves: Some curves (like spirals, loops, and cycloids) are extremely difficult or impossible to express as y = f(x) .

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