Chapter 14.6, Problem 30WE

(a)

To determine

To find: The number of fixed points of rotation ${\Re }_{O,90}$ , reflection $R_y$ , dilation $D_{O,3}$ and translation $T:\left(x,y\right)\to \left(x-3,y+2\right)$ .

Answer to Problem 30WE

The number of fixed points of rotation ${\Re }_{O,90}$ is only one, reflection $R_y$ , dilation is infinite, $D_{O,3}$ is only one and translation $T:\left(x,y\right)\to \left(x-3,y+2\right)$ has no fixed point.

Explanation of Solution

Given information: It is given that a point P is known as fixed point of transformation T if

  $T:P\to P$ .

Calculation:

Consider the rotation ${\Re }_{O,90}$ . As this is not a full rotation, he image will rotate by 90 degree. So, only the center O of the rotation maps itself.

  ${\Re }_{O,90}:O\to O$

The rotation ${\Re }_{O,90}$ has only one fixe point.

Consider the reflection $R_y$ . The transformation if the line of reflection is y -axis:

  $R_y:\left(x_1,y_1\right)\to \left(-x_1,y_1\right)$

All the x- coordinates of the line along the y -axis map to itself. So,

  $R_y:\left(0,y_1\right)\to \left(0,y_1\right)$

So, reflection $R_y$ had infinite number of fixed points.

The dilation $D_{O,k}$ maps at any point $P$ to a point $P^{\prime }$ can be determined as follows:

(1) If $k$ is greater than 0, $P^{\prime }$ lie on $\overline{OP}$ and $O{P}^{\prime }=k\cdot OP$ .

(2) If $k$ is less than 0, $P\text{'}$ lies on the ray opposite to $\overline{OP}$ and $O{P}^{\prime }=\left|k\right|\cdot OP$ .

(3) The center $O$ is its own image is:

For $\left|k\right|>1$ , dilation is called an expansion and for $\left|k\right|<1$ , the dilation is called an contraction.

For the dilation $D_{O,3}$ , the center maps itself. So,

  $D_{O,3}:O\to O$

So, the given dilation has only one fixed point.

Consider the translation $T:\left(x,y\right)\to \left(x-3,y+2\right)$ .

Each point for this transformation will map to every other point. No point maps to itself. So, there is no fixed point for this translation.

Therefore, the number of fixed points of rotation ${\Re }_{O,90}$ is only one, reflection $R_y$ , dilation is infinite, $D_{O,3}$ is only one and translation $T:\left(x,y\right)\to \left(x-3,y+2\right)$ has no fixed point.

(b)

To determine

To find: The coordinates of the fixed point of the composite $D_{O,2}\circ D_{A,\frac{1}{4}}$ .

Answer to Problem 30WE

The coordinates of the fixed point of the composite $D_{O,2}\circ D_{A,\frac{1}{4}}$ is $\left(0,0\right)$ .

Explanation of Solution

Given information: The coordinates of point A is $\left(1,0\right)$ .

Calculation:

The dilation of origin with scale factor 2 is:

  $D_{O,2}=\left(0,0\right)$

The dilation of point A with scale factor $\frac{1}{4}$ is:

  $D_{A,\frac{1}{4}}=\left(\frac{1}{4},0\right)$

The coordinates of fixed point of composite $D_{O,2}\circ D_{A,\frac{1}{4}}$ is:

  $D_{O,2}\circ D_{A,\frac{1}{4}}=\left(0,0\right)$

So, the fixed point is the origin.

Therefore, the coordinates of the fixed point of the composite $D_{O,2}\circ D_{A,\frac{1}{4}}$ is $\left(0,0\right)$ .