Which of the following parametric equations represents a circle of radius $3$ centered at the origin, traced counterclockwise as $t$ increases from $0$ to $2\pi$?

Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.

x = 1 + cos t , y = sin t ― 1

Which of the following best describes the graph of the parametric equations $x=t^2+2t$ and $y=-t$ as $t$ varies over all real numbers?

Which of the following best describes the graph of the parametric equations $x=1-t^2$ and $y=2t$?

Given the parametric equations $x=t\cos \left(t\right)$, $y=t$, $z=t\sin \left(t\right)$ for $t\ge 0$, which of the following best describes the graph of these equations?

Given the parametric equations $x=cos\left(8t\right)$, $y=sin\left(8t\right)$, $z=e^{0.8t}$, $t\ge 0$, which of the following best describes the graph of these equations?

Which of the following best describes the shape traced by the curve with parametric equations $x=\sin \left(t\right)$, $y=3\sin \left(2t\right)$, $z=\sin \left(3t\right)$ as $t$ varies from $0$ to $2\pi$?

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range. x = t² + t + 1, y = 2t