Parametric equations

Definition

Parametric equations define a set of related quantities as explicit functions of an independent parameter, often denoted as $t$. These equations are commonly used to describe curves and motion in the plane.

5 Must Know Facts For Your Next Test

1. Parametric equations for a curve generally consist of two functions: $x(t)$ and $y(t)$.
2. The parameter $t$ often represents time, but can represent any other variable.
3. You can convert parametric equations into a single Cartesian equation by eliminating the parameter $t$.
4. The derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are used to find the slope of the tangent line to the curve at any point.
5. The arc length of a curve defined by parametric equations can be found using $$\int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$.

Review Questions

• What are the parametric equations for a circle of radius 3 centered at the origin?
• How do you eliminate the parameter from the parametric equations $x = t^2 + 1$ and $y = 2t + 3$?
• How do you find the slope of a curve at a specific point given its parametric equations?

Related terms

Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and a distance from a reference point.

Arc Length: The distance measured along the path of a curve.

Tangent Line: A straight line that touches a curve at one point without crossing it, representing the instantaneous direction of change.