Chapter 9.6, Problem 36PPS

(a)

To determine

To Graph: The triangle $ABC$ with points $A(-2,0),B(2,-4),C(4,2)$ . Then draw the image of the triangle after a dilation centered at the origin with the scale factor $-2.$

Explanation of Solution

Given:Points $A(-2,0),B(2,-4),C(4,2)$

Here,

  $A(-2,0),B(2,-4),C(4,2)$

Multiply the coordinates by the scale factor $-2.$

Thus

  $\begin{array}{l}A(-2,0)=A\text{'}(4,0)\\ B(2,-4)=B\text{'}(-4,8)\\ C(4,2)=C\text{'}(-8,-4)\end{array}$

The image of graph after dilation is shown below.

  

(b)

To determine

To Graph:Repeat the dilation centered at the origin with the scale factor $-\frac{1}{2}and-3$

Explanation of Solution

Given:Points $A(-2,0),B(2,-4),C(4,2)$

Here,

  $A(-2,0),B(2,-4),C(4,2)$

Multiply the coordinates by the scale factor $-\frac{1}{2}$

Thus

  $\begin{array}{l}A(-2,0)=A\text{'}\text{'}(1,0)\\ B(2,-4)=B\text{'}(-1,2)\\ C(4,2)=C\text{'}(-2,-1)\end{array}$

The image of graph after dilation is shown below.

  

For scale factor $-3$

  $\begin{array}{l}A(-2,0)=A\text{'}\text{'}\text{'}(6,0)\\ B(2,-4)=B\text{'}\text{'}\text{'}(-6,12)\\ C(4,2)=C\text{'}\text{'}\text{'}(-12,-6)\end{array}$

The graph is shown below.

  

(c)

To determine

To Record: The coordinates of dilation in table.

Explanation of Solution

Given:Points $A(-2,0),B(2,-4),C(4,2)$

  

(d)

To determine

To Make: The conjecture about the function relationship for the dilation centered at the origin with negative scale factor.

Answer to Problem 36PPS

The coordinates are multiplied by the negative scale factor.

Explanation of Solution

Given:Points $A(-2,0),B(2,-4),C(4,2)$

In the dilation with negative factor the coordinates are multiplied by the negative scale factor.

(e)

To determine

To Write: The function rule for the dilation with the scale factor $-k.$

Answer to Problem 36PPS

  $(x,y)=(-kx,-ky)$

Explanation of Solution

Given:Points $A(-2,0),B(2,-4),C(4,2)$

The dilation rule for the scale factor $-k.$ is $(x,y)=(-kx,-ky)$ .

(f)

To determine

To Describe: The dilation centered at the origin with a negative scale factor as the composition of transformations.

Answer to Problem 36PPS

The dilation centered at the origin with a negative scale factor can be described as a dilation centered at the origin with the positive scale factor and a rotation of ${180}^{°}$ about the origin.

Explanation of Solution

Given:Points $A(-2,0),B(2,-4),C(4,2)$

The dilation centered at the origin with a negative scale factor can be described as a dilation centered at the origin with the positive scale factor and a rotation of ${180}^{°}$ about the origin.