Chapter 9.4, Problem 3CYU

To determine

To plot: The graph of $\overline{JK}$ , and its image after a reflection in the x-axis and a rotation $90°$ about the origin.

Answer to Problem 3CYU

The required figure is shown in figure (3).

Explanation of Solution

Given:

The endpoints of $\overline{JK}$ are $J\left(2,5\right)$ , and $K\left(6,5\right)$ .

Calculation:

Plot the graph of $\overline{JK}$ as follows.

  

Figure (1)

Since, a line $\overline{JK}$ has the end point $J\left(2,5\right)$ , and $K\left(6,5\right)$ .

The reflection is in the x-axis, so the reflected x-coordinate will have opposite sign of the original value as $\left(x,y\right)\to \left(-x,y\right)$ .

The coordinates of new $\overline{J\text{'}K\text{'}}$ is as follows.

  $\begin{array}{l}J\left(2,5\right)\to J\text{'}\left(2,-5\right)\\ K\left(6,5\right)\to K\text{'}\left(6,-5\right)\end{array}$

The graph for the line $\overline{J\text{'}K\text{'}}$ is as follows.

  

Figure (2)

Since, a line $\overline{J\text{'}K\text{'}}$ has the end point $J\text{'}\left(2,-5\right)$ , and $K\text{'}\left(6,-5\right)$ .

The rotation is $90°$ about the origin. So, the rotated coordinates will be interchanged. Then the x-coordinates will have opposite sign as $\left(x,y\right)\to \left(-y,x\right)$ .

The coordinates of new $\overline{J\text{'}\text{'}K\text{'}\text{'}}$ is as follows.

  $\begin{array}{l}J\text{'}\left(2,-5\right)\to J\text{'}\text{'}\left(5,2\right)\\ K\text{'}\left(6,-5\right)\to K\text{'}\text{'}\left(5,6\right)\end{array}$

The graph for the line $\overline{J\text{'}\text{'}K\text{'}\text{'}}$ is as follows.

  

Figure (3)

Therefore, the required figure is shown in figure (3).