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.

$x=\sqrt{t}+4,y=\sqrt{t}-4t\ge 0$

Short answer

The answer is$y=x-8.$

Step-by-step solution

Step. 1. Given information.

$x=\sqrt{t}+4,y=\sqrt{t}-4$is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

t	x	y	(x, y)
$0$	$4$	$-4$	$\left(4,-4\right)$
$1$	$5$	$-3$	$\left(5,-3\right)$
$2$	$\sqrt{2}+4$	$\sqrt{2}-4$	$\left(\sqrt{2}+4,\sqrt{2}-4\right)$
$3$	$\sqrt{3}+4$	$\sqrt{3}-4$	$\left(\sqrt{3}+4,\sqrt{3}-4\right)$

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

$$\begin{gathered} x=\sqrt{t}+4 \\ x-4=\sqrt{t} \\ {(x-4)}^2=tt={(x-4)}^2 \end{gathered}$$

Put the value of t-

$$\begin{gathered} y=\sqrt{t}-4 \\ y=\sqrt{{(x-4)}^2}-4 \\ y=x-4-4 \\ y=x-8 \end{gathered}$$

Thus, the equation is$y=x-8$.