Chapter 10.7, Problem 9AYU

To determine

The graph of the curve of a parametric equation $x=t+2,y=\sqrt{t};t\ge 0$, show its orientation. Find the rectangular equation of curve.

Answer to Problem 9AYU

Solution:

The graph of the parametric equations $x=t+2,y=\sqrt{t};t\ge 0$ is

The rectangular equation of the parametric equation $x=t+2,y=\sqrt{t};t\ge 0$ is $x-y^2=2$.

Explanation of Solution

Given information:

The parametric equation $x=t+2,y=\sqrt{t};t\ge 0$

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 $t\ge 0$

$\begin{array}{cccc}t& x=t+2& y=\sqrt{t}& \left(x,y\right)\\ 0& 2& 0& \left(2,0\right)\\ 1& 3& 1& \left(3,1\right)\\ 2& 4& \sqrt{2}& \left(4,\sqrt{2}\right)\\ 3& 5& \sqrt{3}& \left(5,\sqrt{3}\right)\\ 4& 6& \sqrt{4}& \left(6,2\right)\end{array}$

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 $\left(2,0\right)$ to $\left(6,2\right)$

Use points in the table to sketch the curve.

The graph of the parametric equation $x=t+2,y=\sqrt{t};t\ge 0$ is

Let $x=t+2$ ——- (1)

$y=\sqrt{t}$ ——– (2)

Solve equation (1) for the variable $t$ ,

$\Rightarrow x-2=t$

Substitute this result for $t$ into equation (2),

$\Rightarrow x-2=y^2$

$\Rightarrow x-y^2=2$

The rectangular equation of the parametric equation $x=t+2,y=\sqrt{t};t\ge 0$ is $x-y^2=2$.