## Section 1.1 | An Introduction to Parametric Equations

**DEFINITION | PARAMETRIC EQUATION**

Let \(f\) and \(g\) be functions defined on an interval \(I\). The equations \[x = f(t), \quad y = g(t),\]where \(t \in I\), are called

**parametric equations**with

**parameter**\(t\).

**DEFINITION | PARAMETRIC CURVE**

If \(f\) and \(g\) are functions defined on an interval \(I\) such that \(x = f(t)\) and \(y = g(t)\), then the set of points \((f(t),g(t))\) is called a

**parametric curve**(also known as a

**plane curve**).

This last definition is just a fancy way of saying that the parametric curve is the path carved out by parametric equations. This path indicates

*when*a point was in a specific location and the*direction*it was heading.**DEFINITION | INITIAL AND TERMINAL POINTS OF A PARAMETRIC CURVE**

Suppose we are given the parametric equations \(x = f(t)\) and \(y = g(t)\). If the parameter \(t\) has been restricted to, say, \(a \leq t \leq b\), then the

**initial point**is \((f(a),g(a))\) and the

**terminal point**is \((f(b),g(b))\).

**DEFINITION | ELIMINATING THE PARAMETER**

If it is possible, the process of developing a relationship between \(x = f(t)\) and \(y = g(t)\) to resolve to a single equation involving only

(x\) and \(y\) is called

**eliminating the parameter**.

**EXAMPLE 1 [online #1] | Sketching a parametric curve**

Sketch the curve defined by the parametric equations \[x(t) = e^{-t} + t, \quad y(t) = e^t - t, \quad -2 \leq t \leq 2 .\]Indicate with an arrow the direction in which the curve is traced as \(t\) increases.

Parametric Curves in 2D from the Wolfram Demonstrations Project by Abby Brown

**EXAMPLE 2 [online #2] | Sketching a parametric curve**

Sketch the parametric curve defined by \[x(t) =\sqrt{t}, \quad y(t) = 8 - t\]without eliminating the parameter. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. Then eliminate the parameter to find the Cartesian equation.

**IMPORTANT NOTE:**

Unfortunately, there is no general rule for choosing what values of \(t\) to use when tracing out a parametric curve. It is usually a good idea to try both negative and positive values, but you may also consider large and small values of \(t\) as well.

**EXAMPLE 3 [online #3] | Sketching a parametric curve**

Consider the parametric equations \[x(\theta) = \cos{\left(\frac{\theta}{2}\right)}, \quad y(\theta) = \sin{\left(\frac{\theta}{2}\right)}, \quad -\pi \leq \theta \leq \pi .\]Eliminate the parameter to find a Cartesian equation of the curve, and sketch the curve. Be sure to note the direction of travel along the curve.

Parametric Trace from the Wolfram Demonstrations Project by Richard Mercer

**EXAMPLE 4 [online #4] | It's a sinh**

Consider the parametric equations \[x = \sinh{(t}), \quad y = \cosh{(t)} .\]Eliminate the parameter to find a Cartesian equation of the curve, and sketch the curve. Be sure to note the direction of travel along the curve.

**EXAMPLE 5 | Parametric fun with technology**

Use WinPlot to sketch \[x(t) = \cos{(et)}, \quad y(t) = \sin{(\sqrt{3} t)}, \quad 0 \leq t \leq 200\]by setting the equations in WinPlot to \(\cos{(eAt)}\), \(\sin{(\sqrt{3}At)}\), \(0 \leq A \leq 1\), "slides" = 200, and "window" \([-1,1]\times[-1,1]\).

**EXAMPLE 6 | Inverses in parametrics**

Suppose \(f(x) = x^3 + x + 2\).

- How could we write this equation using parametric equations?
- How could we write the inverse of this using parametric functions?

A Parametric Plot of a Spiral on a Paraboloid from the Wolfram Demonstrations Project by Michael Chang

**EXAMPLE 7* | The cycloid**

Set the ratio in the Wolfram Demonstration below to 1 and derive the formulas for the parametric equations of the path that is being carved out.

Cycloid Curves from the Wolfram Demonstrations Project by Sean Madsen