Chapter 1.4, Problem 1E

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 1E

The correct match of the graph for given parametric equation is (c), the dimension of the window is $\left[-4,4\right]$ by $\left[-3,3\right]$ , 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 $x=3\sin \left(2t\right)$ and $y=1.5\cos t$.

Calculation:

Use the following steps 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 $\left[-4,4\right]$ and $\left[-3,3\right]$ as shown below.

  

Figure (1)

From the graph, it can be seen that the interval of the graph is $-3\le x\le 3$ and $-1.5\le y\le 1.5$.

First find the starting point at $t=0$ to find the period of the function. Substitute 0 for $t$ in the parametric equations $x=3\sin \left(2t\right)$ and $y=1.5\cos t$.

  $\begin{array}{c}x=3\sin \left(2\cdot 0\right)\\ =0\\ y=1.5\cos \left(0\right)\\ =1.5\end{array}$

Find the next value of $t$ for which the coordinates of the curve are $\left(0,1.5\right)$ . After $t=0$ , the next value of $t$ for which $y=1.5$ is $t=2\pi$.

Substitute $2\pi$ for t in the equation $y=1.5\cos t$.

  $\begin{array}{c}y=1.5\cos \left(2\pi \right)\\ =1.5\left(1\right)\\ =1.5\end{array}$

Substitute $2\pi$ for $t$ in the parametric equation $x=3\sin \left(2t\right)$.

  $\begin{array}{c}x=3\sin \left(2\cdot 2\pi \right)\\ =3\sin \left(4\pi \right)\\ =3\sin \left(0\right)\\ =0\end{array}$

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

Therefore, the correct match of the graph for given parametric equation is (c), the dimension of the window is $\left[-4,4\right]$ by $\left[-3,3\right]$ , and parameter interval that traces the curve exactly once is $0\le t\le 2\pi$.