Chapter 1.7, Problem 25E

To determine

To find: The sketch for the parametric curve,

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 1

Calculation:

From the graph the value of $x$ and $y$ for the different values of $t$ is,

For $t=0$

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

For $t=\frac{1}{2}$

  $\begin{array}{l}x=0\\ y=1\end{array}$

For $t=1$

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

The range of the curve are,

  $\begin{array}{l}-1\le x\le 1\\ 0\le y\le 1\end{array}$

This shows that the curve lies between the $x=-1$ and $x=1$ . Where the origin of the curve is $\left(0,0\right)$ . As the value of $t$ increases the value of $x$ oscillates between $\left(-1,1\right)$ .

For the range $0<t<\frac{1}{2}$ . The value of $x$ increases up to $1$ and then for the range $\frac{1}{2}<t<1$ the value of $y$ reduces to $0$ .

The sketch for the parametric curve is shown in Figure 2

  

Figure 2