10. Parametric Equations

Given the parametric equations $x=t^2$ and $y=t+1$, eliminate the parameter to find a Cartesian equation of the curve.

Eliminate the parameter to rewrite the following as a rectangular equation.

$x\left(t\right)=2t-1$

$y\left(t\right)=t^5-2$

First eliminate the parameter, then graph the plane curve of the parametric equations.

$x\left(t\right)=2+\cos t$, $y\left(t\right)=-1+\sin t$; $0\le t\le2\pi$

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 = 2 sin t, y = 2 cos t; 0 ≤ t < 2π

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 = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

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 = 5 sec t, y = 3 tan t

In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.

A hyperbola: x = h + a sec t, y = k + b tan t