10. Parametric Equations

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

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 $2\pi$?

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=2t-1$; $y\left(t\right)=2\sqrt{t}$

$t\ge 0$

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

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

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

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 = 4 sin t , y = 3 cos t

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 3 − 5t, y = 4 + 2t; t = 1

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2

In Exercises 57–58, the parametric equations of four plane curves are given. Graph each plane curve and determine how they differ from each other. x = t and y = t² − 4

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 7 − 4t, y = 5 + 6t; t = 1

In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = t² + 1, y = 5 − t³; t = 2