Question

rectangle abcd is the image of rectangle abcd after it has been translated according to the rule t_{-4}(x, y). which points are vertices of the pre - image, rectangle abcd? select four options
□ (-1, -2)
□ (7, 1)
□ (-1, 7)
□ (-1, 1)
□ (7, -2)

Explanation:

Step1: Understand the translation rule

The translation rule \(T_{-4, 1}(x,y)\) means that the image is obtained by translating the pre - image \(4\) units to the left (\(x=x_{pre - image}-4\)) and \(1\) unit up (\(y = y_{pre - image}+1\)). So, to find the pre - image, we use the reverse rule: \(x_{pre - image}=x_{image}+4\) and \(y_{pre - image}=y_{image}-1\).

Step2: Assume vertices of the image

Let's assume the vertices of the image \(A'B'C'D'\) are:

• If we take a point from the image. For example, if a vertex of the image has \(x = 3\) and \(y = 1\) (assuming a vertex of \(C'\)), then \(x_{pre - image}=3 + 4=7\) and \(y_{pre - image}=1-1 = 0\) (not in the options).
• If a vertex of the image has \(x=-5\) and \(y = 1\) (assuming a vertex of \(A'\)), then \(x_{pre - image}=-5 + 4=-1\) and \(y_{pre - image}=1 - 1=0\) (not in the options).
• If a vertex of the image has \(x = 3\) and \(y=-1\) (assuming a vertex of \(C'\) in a wrong - assumption, better to use the fact that for a rectangle, if we consider the general translation):

Let's check each option:

• For the point \((-1,-2)\): If \(x=-1\) and \(y = - 2\), then for the image \(x_{image}=-1-4=-5\) and \(y_{image}=-2 + 1=-1\) (not a vertex of the image shown in the general - rectangle - image in the coordinate - plane).
• For the point \((7,1)\): If \(x = 7\) and \(y = 1\), then \(x_{image}=7-4 = 3\) and \(y_{image}=1 + 1=2\) (not a vertex of the image shown in the general - rectangle - image in the coordinate - plane).
• For the point \((-1,7)\): This is not relevant as per the translation rule (wrong \(y\) - value transformation).
• For the point \((-1,1)\): If \(x=-1\) and \(y = 1\), then \(x_{image}=-1-4=-5\) and \(y_{image}=1 + 1=2\) (not a vertex of the image shown in the general - rectangle - image in the coordinate - plane).
• Let's use another approach. The translation rule \(T_{-4,1}(x,y)=(x - 4,y + 1)\). To get the pre - image, we use \(T_{4,-1}(x,y)=(x + 4,y - 1)\).

If we assume the vertices of the image:
Let's say in the image (from the coordinate - plane of the rectangle \(A'B'C'D'\)):

• If a vertex of the image has \(x=-5\) and \(y = 2\) (left - upper vertex \(A'\)), then \(x_{pre - image}=-5+4=-1\) and \(y_{pre - image}=2 - 1 = 1\) (the point \((-1,1)\)).
• If a vertex of the image has \(x = 3\) and \(y = 2\) (right - upper vertex \(B'\)), then \(x_{pre - image}=3 + 4=7\) and \(y_{pre - image}=2-1 = 1\) (the point \((7,1)\)).
• If a vertex of the image has \(x=-5\) and \(y=-1\) (left - lower vertex \(D'\)), then \(x_{pre - image}=-5 + 4=-1\) and \(y_{pre - image}=-1-1=-2\) (the point \((-1,-2)\)).
• If a vertex of the image has \(x = 3\) and \(y=-1\) (right - lower vertex \(C'\)), then \(x_{pre - image}=3 + 4=7\) and \(y_{pre - image}=-1-1=-2\) (the point \((7,-2)\)).

Answer:

\((-1,-2)\), \((7,1)\), \((-1,1)\), \((7,-2)\)