Parametric Equations Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

Eliminate the parameter from

x = t^2 - 1

and

y = t^3 - t

and describe the curve.

Solution

1
Note $y = t^3 - t = t(t^2 - 1) = t \cdot x$ , so $t = \frac{y}{x}$ when $x \neq 0$ . Also $x = t^2 - 1$ , so $x + 1 = t^2 = \frac{y^2}{x^2}$ .
2
Therefore $y^2 = x^2(x + 1)$ , or equivalently $y^2 = x^3 + x^2$ . This is a cubic curve with a node at the origin (the curve crosses itself when $t = \pm 1$ both give $x = 0, y = 0$ ).

Answer

y^2 = x^3 + x^2

This is a nodal cubic curve — it has a self-intersection (node) at the origin where

t = 1

and

t = -1

both map to

(0, 0)

. Parametric representations can describe curves that fail the vertical line test, showing one advantage of parametric form over rectangular equations.

About Parametric Equations

A way of defining a curve by expressing both $x$ and $y$ as separate functions of a third variable (parameter), typically $t$ : $x = f(t)$ , $y = g(t)$ .

Learn more about Parametric Equations →

More Parametric Equations Examples

Example 1 easy

Eliminate the parameter from [formula] and [formula] to find the rectangular equation.

Example 2 medium

Eliminate the parameter from [formula] and [formula].

Example 3 medium

Find parametric equations for the line through [formula] and [formula].

← All Parametric Equations Examples Parametric Equations Concept

Related Concepts

Function Trigonometric Functions Parametric Graphs Polar Coordinates