Chapter 9.4, Problem 18E

(a)

To determine

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

Explanation of Solution

Given information: Parametric equations $x=\frac{1}{4}t;y=t^2$

Calculation:

Parametric equations are given as

  $x=\frac{1}{4}t;y=t^2$

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

$t$	$x$	$y$
-3	-0.75	9
-2	-0.5	4
-1	-0.25	1
0	0	0
1	0.25	1
2	0.5	4
3	0.75	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 18E

The rectangular equation is $y-16x^2=0$

Explanation of Solution

Given information: Parametric equations $x=\frac{1}{4}t;y=t^2$

Calculation:

Parametric equations are given as

  $x=\frac{1}{4}t;y=t^2$

From equation $x=\frac{1}{4}t$

  $\begin{array}{l}x=\frac{1}{4}t\\ t=4x---(1)\end{array}$

Substitute $t=4x$ in $y=t^2$ ,

  $\begin{array}{l}y={\left(4x\right)}^2\\ y=16x^2\\ y-16x^2=0\end{array}$

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

Conclusion:

Hence, rectangular equation is $y-16x^2=0$