Chapter 9.4, Problem 15E

(a)

To determine

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

Explanation of Solution

Given information: Parametric equations $x=3t-3;y=2t+1$

Calculation:

Parametric equations are given as

  $x=3t-3;y=2t+1$

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

$t$	$x=3t-3$	$y=2t+1$
-3	-12	-5
-2	-9	-3
-1	-6	-1
0	-3	1
1	0	3
2	3	5
3	6	7

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 15E

The rectangular equation is $2x-3y+9=0$

Explanation of Solution

Given information: Parametric equations $x=3t-3;y=2t+1$

Calculation:

Parametric equations are given as

  $x=3t-3;y=2t+1$

From equation $x=t$

  $\begin{array}{l}x=3t-3\\ 3t=x+3\\ t=\frac{x+3}{3}----(1)\end{array}$

From equation $y=2t+1$

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

Equating eq (1) and (2),

  $\begin{array}{l}\frac{x+3}{3}=\frac{y-1}{2}\\ 2\left(x+3\right)=3\left(y-1\right)\\ 2x+6=3y-3\\ 2x-3y+6+3=0\\ 2x-3y+9=0\end{array}$

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

Conclusion:

Hence, rectangular equation is $2x-3y+9=0$