Question

which must be true of a scale factor of a dilation if the image is smaller than the pre - image?
the scale factor is negative
the scale factor is between - 1 and 0
the scale factor is between 0 and 1
the scale factor is greater than 1

Explanation:

Step1: Recall the properties of dilation scale - factor

When we perform a dilation, if the scale factor \(k> 1\), the image is larger than the pre - image. If \(k = 1\), the image is congruent to the pre - image. If \(0

Step2: Analyze each option

• Option 1: A negative scale factor (\(k<0\)) causes a reflection along with a size change. But just being negative does not guarantee the image is smaller. For example, if \(k=-2\), \(|k| = 2>1\) and the image (after reflection) is larger.
• Option 2: If \(k\in(-1,0)\), \(|k|\in(0,1)\). But a negative scale factor causes a reflection. The problem only mentions the image is smaller (not about reflection).
• Option 3: If \(k\in(0,1)\), then \(|k|=k\) (since \(k>0\)). When we multiply the coordinates of the pre - image by \(k\) (where \(0 < k<1\)), each coordinate of the image is a fraction of the corresponding coordinate of the pre - image. So the image is smaller than the pre - image.
• Option 4: If \(k > 1\), the image is larger than the pre - image.

Answer:

The scale factor is between \(0\) and \(1\)