Question

in 5 - 10, use the graph or the given coordinates of a dilation to determine the scale factor that was used. then, give an algebraic representation of the dilation.

Explanation:

Step1: Select corresponding side lengths

Let's consider the horizontal side - lengths of the pre - image and the image. For the first figure (left - hand side), if we assume a horizontal side of the pre - image has length \(l_1\) and the corresponding side of the image has length \(l_2\).

Step2: Calculate the scale factor

The scale factor \(k\) is given by the ratio of the side - length of the image to the side - length of the pre - image, \(k=\frac{l_2}{l_1}\).
For the left - hand side figure:
Let's assume a horizontal side of the pre - image (e.g., the base of the parallelogram \(U\)) has length \(4\) units and the corresponding side of the image (base of parallelogram \(U'\)) has length \(2\) units. Then the scale factor \(k = \frac{2}{4}=\frac{1}{2}\).
The algebraic representation of a dilation centered at the origin is \((x,y)\to(kx,ky)\). So the algebraic representation for this dilation is \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\).

For the right - hand side figure:
Let's assume a horizontal side of the pre - image (e.g., the side of the square \(V\)) has length \(2\) units and the corresponding side of the image (side of the larger square \(V'\)) has length \(4\) units. Then the scale factor \(k=\frac{4}{2} = 2\).
The algebraic representation of this dilation is \((x,y)\to(2x,2y)\).

Answer:

For the left - hand side figure:
Scale Factor: \(\frac{1}{2}\)
Algebraic Representation: \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)
For the right - hand side figure:
Scale Factor: \(2\)
Algebraic Representation: \((x,y)\to(2x,2y)\)