Chapter 9.4, Problem 19E

(a)

To determine

To calculate: To sketch the curve represented by the parametric equations

Explanation of Solution

Given information: Parametric equations $x=t+2;y=t^2$

Calculation:

Parametric equations are given as

  $x=t+2;y=t^2$

Calculating the values of $x$ and $y$ at various values of $t$

$t$	$x$	$y$
-3	-1	9
-2	0	4
-1	1	1
0	2	0
1	3	1
2	4	4
3	5	9

Graph of the function is as follows:

The arrows on the curve indicates its orientation as $t$ increases from $-3$ to $3$

  

Conclusion:

Hence, graph is sketched

(b)

To determine

To calculate: To write the rectangular equations for the parametric equations

Answer to Problem 19E

The rectangular equation is $y-x^2+4x-4=0$

Explanation of Solution

Given information: Parametric equations $x=t+2;y=t^2$

Calculation:

Parametric equations are given as

  $x=t+2;y=t^2$

From equation $x=t+2$

  $\begin{array}{l}x=t+2\\ t=x-2--(1)\end{array}$

Substitute $t=x-2$ in $y=t^2$ ,

  $\begin{array}{l}y={\left(x-2\right)}^2\\ y=x^2-4x+4\\ y-x^2+4x-4=0\end{array}$

Here, domain is $-\infty <t<\infty$

Conclusion:

Hence, rectangular equation is $y-x^2+4x-4=0$