Chapter 9.4, Problem 9E

a.

To determine

To create a table of $x-$ and $y-$ values using the $t=0,1,2,3,4$ for the given parametric equations.

Answer to Problem 9E

  $\begin{array}{cccccc}t& 0& 1& 2& 3& 4\\ x& 0& 1& \sqrt{2}& \sqrt{3}& 2\\ y& 2& 1& 0& -1& -2\end{array}$

Explanation of Solution

Given: The parametric equations $x=\sqrt{t}$ and $y=2-t$ .

Calculation:

When $t=0$ , $x=\sqrt{0}=0,y=2-0=2$ .

When $t=1$ , $x=\sqrt{1}=1,y=2-1=1$ .

When $t=2$ , $x=\sqrt{2},y=2-2=0$ .

When $t=3$ , $x=\sqrt{3},y=2-3=-1$ .

When $t=4$ , $x=\sqrt{4}=2,y=2-4=-2$ .

So, the table becomes as shown below.

  $\begin{array}{cccccc}t& 0& 1& 2& 3& 4\\ x& 0& 1& \sqrt{2}& \sqrt{3}& 2\\ y& 2& 1& 0& -1& -2\end{array}$

b.

To determine

To plot the points $\left(x,y\right)$ generated in part (a) and graph the parametric equation. Also, describe the orientation of the curve.

Explanation of Solution

Given: The parametric equations $x=\sqrt{t}$ and $y=2-t$ .

Plot:

From part (a), the table is given below.

  $\begin{array}{cccccc}t& 0& 1& 2& 3& 4\\ x& 0& 1& \sqrt{2}& \sqrt{3}& 2\\ y& 2& 1& 0& -1& -2\end{array}$

The points become $\left(0,2\right),\left(1,1\right),\left(\sqrt{2},0\right),\left(\sqrt{3},-1\right),\left(2,-2\right)$ .

Now, plot these points on the coordinate axes as shown below.

  

Now, join the points to graph the parametric equations.

  

The curve is moving towards negative infinity as the value of x increases.

c.

To determine

To graph the parametric equations by using graphing utility.

Explanation of Solution

Given: The parametric equations $x=\sqrt{t}$ and $y=2-t$ .

Graph:

The graph of the parametric equations using graphing utility can be given as shown below.

  

The curve is moving towards negative infinity as the value of x increases.

d.

To determine

To find the rectangular equation by eliminating the parameter $t$ . Then, sketch its graph and describe how the graph differs from the graphs in part (b) and (c).

Answer to Problem 9E

  $y=2-x^2,x\ge 0$

Explanation of Solution

Given: The parametric equations $x=\sqrt{t}$ and $y=2-t$ .

Calculation:

  $x=\sqrt{t}\Rightarrow t=x^2$ .

Then, $y=2-t\Rightarrow y=2-x^2$ .

So, the rectangular equation for the given parametric equations is $y=2-x^2,x\ge 0$ .

The graph of this equation can be given as shown below.

  

The curve is moving towards negative infinity as the value of x increases.

The graph is same as obtained in part (b) and (c).