Chapter 1.7, Problem 33E

To determine

To find: The reproduced form of the given picture by the help of graphing calculator.

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 1

Calculation:

From the figure the centre of the circle is,

  $\left(h,k\right)=\left(2,2\right)$

Consider the parametric equation for the circle in the general form is,

  $\begin{array}{l}x=h+r\cos \theta \\ y=k+r\sin \theta \end{array}$

Then, the parametric equation for the head of the smiley figure is,

  $\begin{array}{l}x=2+2\cos \theta \\ y=2+2\sin \theta \end{array}$

Consider the case for the left eye of the smiley.

The centre of the eye is at $\left(1,3\right)$ and the radius is $0.3$ . Then the parametric equation is,

  $\begin{array}{l}x=1+0.2\cos \theta \\ y=3+0.2\sin \theta \end{array}$

Consider the case for the right eye of the smiley.

The centre of the eye is at $\left(3,3\right)$ and the radius is $0.3$ . Then the parametric equation is,

  $\begin{array}{l}x=3+0.2\cos \theta \\ y=3+0.2\sin \theta \end{array}$

The range of the parameter for both the eye is,

  $0\le t\le 2\pi$

Consider the case for the graph of mouth of the smiley.

The centre of the eye is at $\left(2,2\right)$ and the radius is $1$ . Then the parametric equation is,

  $\begin{array}{l}x=2+\cos \frac{\theta }{2}\\ y=2-\sin \frac{\theta }{2}\end{array}$

The range of the angle $\theta$ is,

  $0\le \theta \le 2\pi$

In the graphing calculator press the mode button and select $\text{Par}$ as shown in Figure 2

  

Figure 2

Input the parametric equation by pressing the $\text{Y=}$ button as shown in Figure 3

  

Figure 3

The next step is to set the axis $0\le T\le 2\pi ,0\le x\le 5$ and $0\le y\le 5$ by pressing the Window button as shown in Figure 4

  

Figure 4

Press the graph button to graph the face as shown in Figure 5

  

Figure 5