Chapter 1.4, Problem 2E

To determine

To match: The parametric equations with its appropriate graph, write the approximate dimensions of the viewing window and a parameter interval that traces the curve exactly once.

Answer to Problem 2E

The correct match of the graph for given parametric equation is (a), the dimension of the window is $\left[-2,2\right]$ by $\left[-2,2\right]$ and parameter interval that traces the curve exactly once is $0\le t\le 2\pi$.

Explanation of Solution

Given information: The parametric equations are $x={\sin }^{3}t$ and $y={\cos }^{3}t$.

Calculation:

Use the following step to graph the parametric equations by graphing calculator.

Step 1: Press the $\boxed{\text{MODE}}$ button, under FUNC and choose PAR option.

Step 2: Press $\boxed{\text{Y=}}$ , and set the parametric equations under ${\text{X}}_{1\text{T}}$ and ${\text{Y}}_{1\text{T}}$.

Step 3: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-2,2\right]$ by $\left[-2,2\right]$ as shown below.

  

Figure (1)

First find the starting point at $t=0$ to find the period of the function as shown below.

Substitute 0 for $t$ in the parametric equations $x={\sin }^{3}t$ and $y={\cos }^{3}t$.

  $\begin{array}{c}x={\sin }^3\left(0\right)\\ =0\\ y={\cos }^3\left(0\right)\\ =1\end{array}$

Find the next value of $t$ for which the coordinates of the curve are $\left(0,1\right)$ . After $t=0$ , the next value of $t$ for which $y=1$ is $t=2\pi$.

Substitute $2\pi$ for t in the equation $y={\cos }^{3}t$.

  $\begin{array}{c}y={\cos }^3\left(2\pi \right)\\ ={\left(1\right)}^3\\ =1\end{array}$

Substitute $2\pi$ for $t$ in the parametric equation $x={\sin }^{3}t$.

  $\begin{array}{c}x={\sin }^3\left(2\pi \right)\\ =0\end{array}$

The interval is $0\le t\le 2\pi$.

Therefore, the correct match of the graph for given parametric equation is (a), the dimension of the window is $\left[-2,2\right]$ by $\left[-2,2\right]$ and parameter interval that traces the curve exactly once is $0\le t\le 2\pi$.