Question

In problem graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=2e^t,y=1+e^t;t\ge 0.$

Short answer

The answer is$2y=2+x.$

Step-by-step solution

Step 1. Given information.

$x=2e^t,y=1+e^t$ is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$t$	$x$	$y$	$(x,y)$
$0$	$2$	$2$	$(2,2)$
$1$	$5.436$	$3.718$	$(5.436,3.718)$
$2$	$14.77$	$8.38$	localid="1648387282180" $(14.77,8.38)$

We find the rectangular equation by eliminating the parameter t from the parametric equations.

localid="1648387616129" $$\begin{gathered} x=2e^t \\ {e}^t=\frac{x}{2} \end{gathered}$$

Put the value of t-

$$\begin{gathered} y=1+e^2 \\ y=1+\frac{x}{2} \\ 2y=2+x \end{gathered}$$

Thus, the equation is$2y=2+x.$