Question

which transformation will preserve the size and orientation of its preimage?
a. ((x, y) \to (x + 3, y + 1))
b. ((x, y) \to (-x, y))
c. ((x, y) \to (2x, 2y))
d. ((x, y) \to (y, x))

Explanation:

Step1: Analyze Option A

This is a translation transformation. In a translation \((x,y)\to(x + 3,y + 1)\), we are shifting the point 3 units to the right and 1 unit up. Translations preserve both the size (since we are not scaling) and the orientation (since we are not reflecting or rotating in a way that flips the orientation) of the pre - image.

Step2: Analyze Option B

The transformation \((x,y)\to(-x,y)\) is a reflection over the \(y\) - axis. A reflection changes the orientation of the pre - image (it flips the figure across the \(y\) - axis), so it does not preserve orientation.

Step3: Analyze Option C

The transformation \((x,y)\to(2x,2y)\) is a dilation with a scale factor of 2. A dilation changes the size of the pre - image (it scales the figure by a factor of 2), so it does not preserve size.

Step4: Analyze Option D

The transformation \((x,y)\to(y,x)\) is a reflection over the line \(y = x\). This reflection changes the orientation of the pre - image, so it does not preserve orientation.

Answer:

A. \((x, y) \to (x + 3, y + 1)\)