Chapter 9, Problem 59CR

a.

To determine

To graph the curve represented by the parametric equation and indicate its orientation.

Explanation of Solution

Given information:

The given parametric equation is

  $\begin{array}{l}x=t^3\\ y=\frac{1}{2}{t}^2\end{array}$

Graph:

Make the table of values

t	0	1	$-1$	2
x	0	1	$-1$	8
y	0	$\frac{1}{2}$	$\frac{1}{2}$	$2$

The graph of the parametric equation.

  

Interpretation: the orientation of the graph is $y=\frac{1}{2}{x}^{\frac{2}{3}}$

b.

To determine

To eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.

Answer to Problem 59CR

The required rectangular equation is $y=\frac{1}{2}{x}^{\frac{2}{3}}$

Explanation of Solution

Given information:

The given parametric equation is

  $\begin{array}{l}x=t^3\\ y=\frac{1}{2}{t}^2\end{array}$

Calculation:

Evaluate t in terms of x

  $\begin{array}{l}x=t^3\\ t=\sqrt[3]{x}\\ \text{substitute }t\text{ in second equation}\\ y=\frac{1}{2}{\left(\sqrt[3]{x}\right)}^2\\ y=\frac{1}{2}{x}^{\frac{2}{3}}\end{array}$

The required rectangular equation is $y=\frac{1}{2}{x}^{\frac{2}{3}}$