Chapter 9.6, Problem 27PPS

(a)

To determine

To Dilate: The $FGHJ$ by scale factor $\frac{1}{2}$ centered at the origin then reflect the dilated image in the $y-axis.$

Explanation of Solution

Given:

  

Multiply the coordinates of the vertex by the scale factor $\frac{1}{2}$ .

i.e.

  $\begin{array}{l}(x,y)=\left(\frac{1}{2}x,\frac{1}{2}y\right)\\ F\left(-1,6\right)=F\text{'}\left(-\frac{1}{2},3\right)\\ G\left(1,5\right)=G\text{'}\left(\frac{1}{2},\frac{5}{2}\right)\\ H\left(2,1\right)=H\text{'}\left(1,\frac{1}{2}\right)\\ J\left(-3,1\right)=J\text{'}\left(-\frac{3}{2},\frac{1}{2}\right)\end{array}$

Now

Reflect in the $y-axis.$

  $\begin{array}{l}(x,y)=\left(-x,y\right)\\ F\text{'}\left(-\frac{1}{2},3\right)=F\text{'}\text{'}\left(\frac{1}{2},3\right)\\ G\text{'}\left(\frac{1}{2},\frac{5}{2}\right)=G\text{'}\text{'}\left(-\frac{1}{2},\frac{5}{2}\right)\\ H\text{'}\left(1,\frac{1}{2}\right)=H\text{'}\text{'}\left(-1,\frac{1}{2}\right)\\ J\text{'}\left(-\frac{3}{2},\frac{1}{2}\right)=J\text{'}\text{'}\left(\frac{3}{2},\frac{1}{2}\right)\end{array}$

The graph of the image is shown below.

  

(b)

To determine

To Complete: The composition of transformation in reverse order.

Explanation of Solution

Given:

  

Reflect the image in $y-axis.$

  $\begin{array}{l}(x,y)=\left(-x,y\right)\\ F\left(-1,6\right)=F\text{'}\left(1,6\right)\\ G\left(1,5\right)=G\text{'}\left(-1,5\right)\\ H\left(2,1\right)=H\text{'}\left(-2,1\right)\\ J\left(-3,1\right)=J\text{'}\left(3,1\right)\end{array}$

Dilate the image.

  $\begin{array}{l}(x,y)=\left(\frac{1}{2}x,\frac{1}{2}y\right)\\ F\text{'}\text{'}\left(\frac{1}{2},3\right)\\ G\text{'}\text{'}\left(-\frac{1}{2},\frac{5}{2}\right)\\ H\text{'}\text{'}\left(-1,\frac{1}{2}\right)\\ J\text{'}\text{'}\left(\frac{3}{2},\frac{1}{2}\right)\end{array}$

  

(c)

To determine

To Find: Does the order of transformation affect the real image?

Answer to Problem 27PPS

No

Explanation of Solution

Given:

  

No,

The coordinates remains the same.

(d)

To determine

To Find:Will the order of composition of a dilation and reflection sometimes, never, or always affect the dilated image.

Answer to Problem 27PPS

Sometimes

Explanation of Solution

Given:

  

The order of composition of a dilation and reflection sometimes affect the dilated image.