Chapter 9.4, Problem 14E

(a)

To determine

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

Explanation of Solution

Given information: Parametric equations $x=t;y=\frac{1}{2}t$

Calculation:

Parametric equations are given as

  $x=t;y=\frac{1}{2}t$

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

$t$	$x=t$	$y=\frac{1}{2}t$
-3	-3	$\frac{-3}{2}$
-2	-2	-1
-1	-1	$-\frac{1}{2}$
0	0	0
1	1	$\frac{1}{2}$
2	2	$1$
3	3	$\frac{3}{2}$

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 14E

The rectangular equation is $x-2y=0$

Explanation of Solution

Given information: Parametric equations $x=t;y=\frac{1}{2}t$

Calculation:

Parametric equations are given as

  $x=t;y=\frac{1}{2}t$

From equation $x=t$

  $t=x$ -------(1)

From equation $y=\frac{1}{2}t$

  $t=2y$ ------(2)

Equating eq (1) and (2),

  $\begin{array}{l}x=2y\\ x-2y=0\end{array}$

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

Conclusion:

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