Question

describe the transformation which maps the pre-image to the image.
options:
○ $(x,y)\to(x,-y)$
○ $(x,y)\to(-x,-y)$
○ $(x,y)\to(x,y)$
○ $(x,y)\to(y,x)$

Explanation:

Step1: Analyze Coordinates

First, identify corresponding points of pre - image (lower triangle) and image (upper triangle). Let's take a point from pre - image, say \( L(-1,-2) \) and its image \( L'(-1,1) \)? Wait, no, better to check reflection. Let's take a point like \( J \) in pre - image (lower) and \( J' \) in image (upper). Wait, actually, let's consider the reflection over x - axis? No, wait the transformation options are about coordinate transformations. Let's check the transformation rules:

• For \( (x,y)\to(x, - y) \): This is reflection over x - axis.
• For \( (x,y)\to(-x,-y) \): This is reflection over origin.
• For \( (x,y)\to(x,y) \): No transformation.
• For \( (x,y)\to(y,x) \): This is reflection over line \( y = x \).

Wait, looking at the graph, the pre - image (lower triangle) and image (upper triangle) seem to be related by reflection over x - axis? Wait no, let's take a point. Let's assume pre - image point \( K(-3,-4) \) and image point \( K'(-3,4) \). Wait, if we apply \( (x,y)\to(x, - y) \), then for \( (x,y)=(-3,-4) \), \( (x,-y)=(-3,4) \), which matches \( K' \). Let's check another point, say \( J(-4,-2) \), applying \( (x,y)\to(x, - y) \), we get \( (-4,2) \), which should be \( J' \). And \( L(-1,-2) \) applying \( (x,y)\to(x, - y) \) gives \( (-1,2) \)? Wait, maybe I misread the graph. Wait the image is above the x - axis and pre - image below. So the transformation is \( (x,y)\to(x, - y) \), which is reflection over the x - axis.

Step2: Match with Options

Now, check the options:

• Option 1: \( (x,y)\to(x, - y) \): This is reflection over x - axis, which matches our analysis.
• Option 2: \( (x,y)\to(-x,-y) \): This would be reflection over origin, which would move the point to the opposite quadrant, but here the x - coordinates are same (negative x stays negative), so this is not correct.
• Option 3: \( (x,y)\to(x,y) \): No change, which is not the case.
• Option 4: \( (x,y)\to(y,x) \): This is reflection over \( y = x \), which would swap x and y coordinates, not matching.

Answer:

\( (x,y)\to(x, - y) \) (the first option)