Chapter 10.7, Problem 12AYU

To determine

The graph of the curve of a parametric equation $x\left(t\right)=\sqrt{t}+4,$ $y\left(t\right)=\sqrt{t}-4;t\ge 0$, show its orientation. Find the rectangular equation of curve.

Answer to Problem 12AYU

Solution:

The graph of the parametric equations $x\left(t\right)=\sqrt{t}+4,$ $y\left(t\right)=\sqrt{t}-4;t\ge 0$ is

The rectangular equation of the parametric equation $x\left(t\right)=\sqrt{t}+4,$ $y\left(t\right)=\sqrt{t}-4;t\ge 0$ is $x-y=8$.

Explanation of Solution

Given information:

The parametric equation $x\left(t\right)=\sqrt{t}+4,y\left(t\right)=\sqrt{t}-4;t\ge 0$

Explanation:

To draw the given parametric equation, plug some values of $t$ in the given equation and find few points on the curve.

The interval for $t$ is $t\ge 0$

$\begin{array}{cccc}t& x\left(t\right)=\sqrt{t}+4& y\left(t\right)=\sqrt{t}-4& \left(x,y\right)\\ 0& 4& -4& \left(4,-4\right)\\ 1& 5& -3& \left(5,-3\right)\\ 9& 7& -1& \left(7,-1\right)\\ 16& 8& 0& \left(8,0\right)\\ 25& 9& 1& \left(9,1\right)\end{array}$

The orientation is curve traced out in a certain direction by the corresponding succession of points $\left(x,y\right)$

The arrow shows the orientation along the curve as $t$ varies from $t\ge 0$.

As $t$ takes values from $t\ge 0$, the corresponding orientation is from $\left(4,-4\right)$ to $\left(9,1\right)$.

Use points in the table to sketch the curve.

The graph of the parametric equation $x\left(t\right)=\sqrt{t}+4,y\left(t\right)=\sqrt{t}-4;t\ge 0$ is

Let $x=\sqrt{t}+4$ and $y=\sqrt{t}-4$

$\Rightarrow x-4=\sqrt{t}$

$\Rightarrow y=x-4-4$

$\Rightarrow y=x-8$

$\Rightarrow x-y=8$

The rectangular equation of the parametric equation $x\left(t\right)=\sqrt{t}+4,y\left(t\right)=\sqrt{t}-4;t\ge 0$ is $x-y=8$.